[math-fun] The Saddest Thing about the Integers
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning. http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th... The Saddest Thing I Know about the Integers By Evelyn Lamb | November 30, 2014 The integers are a unique factorization domain, so we canÂt tune pianos. That is the saddest thing I know about the integers. I talked to a Girl Scout troop about math earlier this month, and one of our topics was the intersection of math and music. I chose to focus on the way we perceive ratios of sound wave frequencies as intervals. We interpret frequencies that have the ratio 2:1 as octaves. (Larger frequencies sound higher.) We interpret frequencies that have the ratio 3:2 as perfect fifths. And sadly, I had to break it to the girls that these two facts mean that no piano is in tune. In other words, you can tuna fish, but you canÂt tune a piano. When we tune an instrument, we would like for all our octaves and fifths to be perfect. One way to tune an instrument would be to start with a pitch and start working out the fifths above and below it it. We start with some frequency that we call C. Then 3/2 times that frequency is G, 9/4 times that frequency is D (an octave and a step above our original C), and so on. If you learned about the Âcircle of fifths at some point in your musical life, then you know that if we keep going up by fifths, weÂll eventually land back on something weÂd like to call C. It takes a total of 12 steps, and so if we keep all our fifths perfect, the frequency of the C we get at the end is 312/212, or 531441/4096, times the frequency of the C we had at the beginning. You might notice that 531441/4096 is not an integer, much less a power of 2, so our ears would not perceive the C at the end as being in tune with the C at the beginning. (531441/4096 is about 130, which is 2 more than a power of 2, so we would hear the C at the top as being sharp.) And itÂs not a problem with the assumption that it takes 12 fifths to get from C to shining C. We can never get perfect octaves from a stack of fifths because no power of 3/2 will ever give us a power of 2. Imperfect octaves are pretty unacceptable to any listener, and as a string player, IÂm pretty into perfect fifths. So itÂs disappointing enough that I canÂt have them simultaneously. But the story gets even more complicated when we add thirds. Even if we could resolve the pesky fifths/octave problem, we would be stuck with some pretty strange sounding chords. When we hear frequencies in the ratio 5:4, we hear a perfectly tuned major third (the interval between C and E). But if we go around the circle of 5ths making uncompromising perfect fifths, we get 34/24=81/16. If we divide by 2 a few times to move the E back down to the same octave as the C, we end up with an 81:64 ratio, which is a bit bigger than 5:4 (or 80:64), meaning that the major third from C to E sounds too wide. So fifths are also incompatible with major thirds! Once again, we can never get a perfectly tuned major third from a stack of fifths, or a perfect fifth from a stack of major thirds, because no power of 5/4 equals a power of 3/2. Blame unique factorization. One property of the integers that we take for granted is that we can factor any integer other than -1, 0, or 1 into its prime factors, and that the factorization will be unique. (We call this the fundamental theorem of arithmetic.) Hence, we can call the integers a unique factorization domain. (If youÂre a real stickler, you might be worried about negative integers. The factorization is unique up to signs of numbers, and thatÂs good enough to be a unique factorization domain. If that still bothers you, just ignore the integers smaller than 2.) As a thought experiment, I decided to see if we could fix the problem by expanding from the integers to another set of numbers like the integers in that they can also be multiplied or added together. One such set of numbers is called the Gaussian integers, and it consists of complex numbers of the form a+bi, where a and b are both integers and i2=-1. In the Gaussian integers, 2 is no longer a prime number because it can be factored into (1+i)Ã(1-i), which happen to be primes. Neither is 5, which can be written (1+2i)Ã(1-2i). But 3 is still a prime in the Gaussian integers (this isnÂt obvious, but itÂs true). Thus 2 and 3 still share no prime factors over the Gaussian integers, so we canÂt resolve our octaves/fifths problem there. (Not that I even know what it would mean to divide a frequency by a Gaussian integer. Like I said, this is a thought experiment.) Likewise, 5 and 2 share no Gaussian prime factors, nor do 5 and 3. So even if it made sense to divide a frequency by a complex number, it wouldnÂt help. An even stranger set of numbers is the set of complex numbers of the form a+v5bi, or Z[v-5]. It might seem like this is no different from the Gaussian integers, but it is. ItÂs not a unique factorization domain. For example, the number 6 can be factored into either 2Ã3 or (1+v5i)Ã(1-v5i).* ItÂs not obvious, but 2 and 3 canÂt be factored further; they are irreducible, as are (1+v5i) and (1-v5i). So 6 has two distinct factorizations. Will this help us? Well, we could end up with some powers of 6 in our frequency ratios if we combine fifths and octaves. But letÂs say we could divide that power of 6 by 1+v5i and 1-v5i. Where would we be? WeÂd have reduced our power of 6 by 1, but weÂre no closer to getting a 3 to change into a 2. Bummer! But at least we got to play with some quadratic integers, right? You may have noticed that piano music doesnÂt always sound out of tune, so there must be some resolution to the prime number predicament. Compromise, my friend. Currently, most instruments use equal temperament, which makes all the fifths slightly narrower than perfect so the octaves will be in tune. Each half step has the same frequency ratio as any other half step, and that ratio is 21/12:1. WeÂve lost the pure rational ratios that made Pythagorean intervals sound so sweet, but weÂve gained a lot. The difference between a Pythagorean fifth and an equal temperament fifth is not enough to bother any but the fussiest listeners, but it is detectable to some. Before equal temperament became the law of the land, at least for keyboard instruments and other instruments where the player canÂt make minute adjustments to pitch, there were several other temperament compromises in use. One solution is to tune an instrument so that the octaves, fifths, and/or thirds are perfect or very close for the important chords from some keys (generally Âeasy keys like C, G, and D) but terrible for some other keys. Those systems (often meantone temperaments) ended up with Âwolf fifths that were much narrower than perfect fifths. An instrument with a wolf fifth couldnÂt really play in certain keys. Then along came well temperament, which was not one system but any of many irregular temperaments that made the keys sound different but didnÂt leave any key howling at the moon, so to speak. The Âwell-tempered in BachÂs Well-Tempered Clavier doesnÂt refer to the instrumentÂs beautiful tone (or the instrumentalistÂs equanimity) but to the fact that the set of pieces was composed for a clavier with a temperament that allowed the instrument to play in every key. (The Well-Tempered Clavier is a set of 24 preludes and fugues, one in each major and minor key. Scholars donÂt kno w what exactly the clavierÂs temperament was, but it is unlikely that it was equal temperament, as some musicians assume.) The fact that 3/2, 2, and 5/4 are incommensurable makes me genuinely sad, but later this week I hope to share a fun experiment the Girl Scouts and I did with pitch perception. It doesnÂt rely on having perfectly tuned fifths, thirds, and octaves simultaneously, so it shouldnÂt cause the same existential angst that temperament does. *This sentence and other sentences in this paragraph were edited after publication to correct missing square root signs. Tags: algebra, arithmetic, fundamental theorem of arithmetic, mathematics and music, music theory, prime numbers, pythagoras, quadratic integer rings The views expressed are those of the author and are not necessarily those of Scientific American. Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.
That is some great popular mathematical writing, in my opinion! It made me wonder: Is Z[2^(1/12 <x-apple-data-detectors://0>)] embeddable in a finite-degree extension of Z with unique factorization? (Not that this would really affect anything in music.) Jim On Tuesday, December 2, 2014, Henry Baker <hbaker1@pipeline.com> wrote:
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning.
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
The Saddest Thing I Know about the Integers
By Evelyn Lamb | November 30, 2014
The integers are a unique factorization domain, so we can’t tune pianos. That is the saddest thing I know about the integers.
I talked to a Girl Scout troop about math earlier this month, and one of our topics was the intersection of math and music. I chose to focus on the way we perceive ratios of sound wave frequencies as intervals. We interpret frequencies that have the ratio 2:1 as octaves. (Larger frequencies sound higher.) We interpret frequencies that have the ratio 3:2 as perfect fifths. And sadly, I had to break it to the girls that these two facts mean that no piano is in tune. In other words, you can tuna fish, but you can’t tune a piano.
When we tune an instrument, we would like for all our octaves and fifths to be perfect. One way to tune an instrument would be to start with a pitch and start working out the fifths above and below it it. We start with some frequency that we call C. Then 3/2 times that frequency is G, 9/4 times that frequency is D (an octave and a step above our original C), and so on. If you learned about the “circle of fifths” at some point in your musical life, then you know that if we keep going up by fifths, we’ll eventually land back on something we’d like to call C. It takes a total of 12 steps, and so if we keep all our fifths perfect, the frequency of the C we get at the end is 312/212, or 531441/4096, times the frequency of the C we had at the beginning. You might notice that 531441/4096 is not an integer, much less a power of 2, so our ears would not perceive the C at the end as being in tune with the C at the beginning. (531441/4096 is about 130, which is 2 more than a power of 2, so we would hear the C at the top as being sharp.) And it’s not a problem with the assumption that it takes 12 fifths to get from C to shining C. We can never get perfect octaves from a stack of fifths because no power of 3/2 will ever give us a power of 2.
Imperfect octaves are pretty unacceptable to any listener, and as a string player, I’m pretty into perfect fifths. So it’s disappointing enough that I can’t have them simultaneously. But the story gets even more complicated when we add thirds. Even if we could resolve the pesky fifths/octave problem, we would be stuck with some pretty strange sounding chords. When we hear frequencies in the ratio 5:4, we hear a perfectly tuned major third (the interval between C and E). But if we go around the circle of 5ths making uncompromising perfect fifths, we get 34/24=81/16. If we divide by 2 a few times to move the E back down to the same octave as the C, we end up with an 81:64 ratio, which is a bit bigger than 5:4 (or 80:64), meaning that the major third from C to E sounds too wide. So fifths are also incompatible with major thirds! Once again, we can never get a perfectly tuned major third from a stack of fifths, or a perfect fifth from a stack of major thirds, because no power of 5/4 equals a power of 3/2.
Blame unique factorization. One property of the integers that we take for granted is that we can factor any integer other than -1, 0, or 1 into its prime factors, and that the factorization will be unique. (We call this the fundamental theorem of arithmetic.) Hence, we can call the integers a unique factorization domain. (If you’re a real stickler, you might be worried about negative integers. The factorization is unique up to signs of numbers, and that’s good enough to be a unique factorization domain. If that still bothers you, just ignore the integers smaller than 2.) As a thought experiment, I decided to see if we could fix the problem by expanding from the integers to another set of numbers like the integers in that they can also be multiplied or added together.
One such set of numbers is called the Gaussian integers, and it consists of complex numbers of the form a+bi, where a and b are both integers and i2=-1. In the Gaussian integers, 2 is no longer a prime number because it can be factored into (1+i)×(1-i), which happen to be primes. Neither is 5, which can be written (1+2i)×(1-2i). But 3 is still a prime in the Gaussian integers (this isn’t obvious, but it’s true). Thus 2 and 3 still share no prime factors over the Gaussian integers, so we can’t resolve our octaves/fifths problem there. (Not that I even know what it would mean to divide a frequency by a Gaussian integer. Like I said, this is a thought experiment.) Likewise, 5 and 2 share no Gaussian prime factors, nor do 5 and 3. So even if it made sense to divide a frequency by a complex number, it wouldn’t help.
An even stranger set of numbers is the set of complex numbers of the form a+v5bi, or Z[v-5]. It might seem like this is no different from the Gaussian integers, but it is. It’s not a unique factorization domain. For example, the number 6 can be factored into either 2×3 or (1+v5i)×(1-v5i).* It’s not obvious, but 2 and 3 can’t be factored further; they are irreducible, as are (1+v5i) and (1-v5i). So 6 has two distinct factorizations. Will this help us? Well, we could end up with some powers of 6 in our frequency ratios if we combine fifths and octaves. But let’s say we could divide that power of 6 by 1+v5i and 1-v5i. Where would we be? We’d have reduced our power of 6 by 1, but we’re no closer to getting a 3 to change into a 2. Bummer! But at least we got to play with some quadratic integers, right?
You may have noticed that piano music doesn’t always sound out of tune, so there must be some resolution to the prime number predicament. Compromise, my friend. Currently, most instruments use equal temperament, which makes all the fifths slightly narrower than perfect so the octaves will be in tune. Each half step has the same frequency ratio as any other half step, and that ratio is 21/12:1. We’ve lost the pure rational ratios that made Pythagorean intervals sound so sweet, but we’ve gained a lot. The difference between a Pythagorean fifth and an equal temperament fifth is not enough to bother any but the fussiest listeners, but it is detectable to some. Before equal temperament became the law of the land, at least for keyboard instruments and other instruments where the player can’t make minute adjustments to pitch, there were several other temperament compromises in use.
One solution is to tune an instrument so that the octaves, fifths, and/or thirds are perfect or very close for the important chords from some keys (generally “easy” keys like C, G, and D) but terrible for some other keys. Those systems (often meantone temperaments) ended up with “wolf fifths” that were much narrower than perfect fifths. An instrument with a wolf fifth couldn’t really play in certain keys. Then along came well temperament, which was not one system but any of many irregular temperaments that made the keys sound different but didn’t leave any key howling at the moon, so to speak. The “well-tempered” in Bach’s Well-Tempered Clavier doesn’t refer to the instrument’s beautiful tone (or the instrumentalist’s equanimity) but to the fact that the set of pieces was composed for a clavier with a temperament that allowed the instrument to play in every key. (The Well-Tempered Clavier is a set of 24 preludes and fugues, one in each major and minor key. Scholars don’t kno w what exactly the clavier’s temperament was, but it is unlikely that it was equal temperament, as some musicians assume.)
The fact that 3/2, 2, and 5/4 are incommensurable makes me genuinely sad, but later this week I hope to share a fun experiment the Girl Scouts and I did with pitch perception. It doesn’t rely on having perfectly tuned fifths, thirds, and octaves simultaneously, so it shouldn’t cause the same existential angst that temperament does.
*This sentence and other sentences in this paragraph were edited after publication to correct missing square root signs.
Tags: algebra, arithmetic, fundamental theorem of arithmetic, mathematics and music, music theory, prime numbers, pythagoras, quadratic integer rings
The views expressed are those of the author and are not necessarily those of Scientific American.
Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
According to SAGE the field Q(2^{1/12}) has class number 1, and so does have unique factorization. In addition Z[2^{1/12}] is the maximal order in this field. Victor On Tue, Dec 2, 2014 at 9:11 AM, James Propp <jamespropp@gmail.com> wrote:
That is some great popular mathematical writing, in my opinion!
It made me wonder: Is Z[2^(1/12 <x-apple-data-detectors://0>)] embeddable in a finite-degree extension of Z with unique factorization? (Not that this would really affect anything in music.)
Jim
On Tuesday, December 2, 2014, Henry Baker <hbaker1@pipeline.com> wrote:
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning.
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
The Saddest Thing I Know about the Integers
By Evelyn Lamb | November 30, 2014
The integers are a unique factorization domain, so we can’t tune pianos. That is the saddest thing I know about the integers.
I talked to a Girl Scout troop about math earlier this month, and one of our topics was the intersection of math and music. I chose to focus on
the
way we perceive ratios of sound wave frequencies as intervals. We interpret frequencies that have the ratio 2:1 as octaves. (Larger frequencies sound higher.) We interpret frequencies that have the ratio 3:2 as perfect fifths. And sadly, I had to break it to the girls that these two facts mean that no piano is in tune. In other words, you can tuna fish, but you can’t tune a piano.
When we tune an instrument, we would like for all our octaves and fifths to be perfect. One way to tune an instrument would be to start with a pitch and start working out the fifths above and below it it. We start with some frequency that we call C. Then 3/2 times that frequency is G, 9/4 times that frequency is D (an octave and a step above our original C), and so on. If you learned about the “circle of fifths” at some point in your musical life, then you know that if we keep going up by fifths, we’ll eventually land back on something we’d like to call C. It takes a total of 12 steps, and so if we keep all our fifths perfect, the frequency of the C we get at the end is 312/212, or 531441/4096, times the frequency of the C we had at the beginning. You might notice that 531441/4096 is not an integer, much less a power of 2, so our ears would not perceive the C at the end as being in tune with the C at the beginning. (531441/4096 is about 130, which is 2 more than a power of 2, so we would hear the C at the top as being sharp.) And it’s not a problem with the assumption that it takes 12 fifths to get from C to shining C. We can never get perfect octaves from a stack of fifths because no power of 3/2 will ever give us a power of 2.
Imperfect octaves are pretty unacceptable to any listener, and as a string player, I’m pretty into perfect fifths. So it’s disappointing enough that I can’t have them simultaneously. But the story gets even more complicated when we add thirds. Even if we could resolve the pesky fifths/octave problem, we would be stuck with some pretty strange sounding chords. When we hear frequencies in the ratio 5:4, we hear a perfectly tuned major third (the interval between C and E). But if we go around the circle of 5ths making uncompromising perfect fifths, we get 34/24=81/16. If we divide by 2 a few times to move the E back down to the same octave as the C, we end up with an 81:64 ratio, which is a bit bigger than 5:4 (or 80:64), meaning that the major third from C to E sounds too wide. So fifths are also incompatible with major thirds! Once again, we can never get a perfectly tuned major third from a stack of fifths, or a perfect fifth from a stack of major thirds, because no power of 5/4 equals a power of 3/2.
Blame unique factorization. One property of the integers that we take for granted is that we can factor any integer other than -1, 0, or 1 into its prime factors, and that the factorization will be unique. (We call this the fundamental theorem of arithmetic.) Hence, we can call the integers a unique factorization domain. (If you’re a real stickler, you might be worried about negative integers. The factorization is unique up to signs of numbers, and that’s good enough to be a unique factorization domain. If that still bothers you, just ignore the integers smaller than 2.) As a thought experiment, I decided to see if we could fix the problem by expanding from the integers to another set of numbers like the integers in that they can also be multiplied or added together.
One such set of numbers is called the Gaussian integers, and it consists of complex numbers of the form a+bi, where a and b are both integers and i2=-1. In the Gaussian integers, 2 is no longer a prime number because it can be factored into (1+i)×(1-i), which happen to be primes. Neither is 5, which can be written (1+2i)×(1-2i). But 3 is still a prime in the Gaussian integers (this isn’t obvious, but it’s true). Thus 2 and 3 still share no prime factors over the Gaussian integers, so we can’t resolve our octaves/fifths problem there. (Not that I even know what it would mean to divide a frequency by a Gaussian integer. Like I said, this is a thought experiment.) Likewise, 5 and 2 share no Gaussian prime factors, nor do 5 and 3. So even if it made sense to divide a frequency by a complex number, it wouldn’t help.
An even stranger set of numbers is the set of complex numbers of the form a+v5bi, or Z[v-5]. It might seem like this is no different from the Gaussian integers, but it is. It’s not a unique factorization domain. For example, the number 6 can be factored into either 2×3 or (1+v5i)×(1-v5i).* It’s not obvious, but 2 and 3 can’t be factored further; they are irreducible, as are (1+v5i) and (1-v5i). So 6 has two distinct factorizations. Will this help us? Well, we could end up with some powers of 6 in our frequency ratios if we combine fifths and octaves. But let’s say we could divide that power of 6 by 1+v5i and 1-v5i. Where would we be? We’d have reduced our power of 6 by 1, but we’re no closer to getting a 3 to change into a 2. Bummer! But at least we got to play with some quadratic integers, right?
You may have noticed that piano music doesn’t always sound out of tune, so there must be some resolution to the prime number predicament. Compromise, my friend. Currently, most instruments use equal temperament, which makes all the fifths slightly narrower than perfect so the octaves will be in tune. Each half step has the same frequency ratio as any other half step, and that ratio is 21/12:1. We’ve lost the pure rational ratios that made Pythagorean intervals sound so sweet, but we’ve gained a lot. The difference between a Pythagorean fifth and an equal temperament fifth is not enough to bother any but the fussiest listeners, but it is detectable to some. Before equal temperament became the law of the land, at least for keyboard instruments and other instruments where the player can’t make minute adjustments to pitch, there were several other temperament compromises in use.
One solution is to tune an instrument so that the octaves, fifths, and/or thirds are perfect or very close for the important chords from some keys (generally “easy” keys like C, G, and D) but terrible for some other keys. Those systems (often meantone temperaments) ended up with “wolf fifths” that were much narrower than perfect fifths. An instrument with a wolf fifth couldn’t really play in certain keys. Then along came well temperament, which was not one system but any of many irregular temperaments that made the keys sound different but didn’t leave any key howling at the moon, so to speak. The “well-tempered” in Bach’s Well-Tempered Clavier doesn’t refer to the instrument’s beautiful tone (or the instrumentalist’s equanimity) but to the fact that the set of pieces was composed for a clavier with a temperament that allowed the instrument to play in every key. (The Well-Tempered Clavier is a set of 24 preludes and fugues, one in each major and minor key. Scholars don’t kno w what exactly the clavier’s temperament was, but it is unlikely that it was equal temperament, as some musicians assume.)
The fact that 3/2, 2, and 5/4 are incommensurable makes me genuinely sad, but later this week I hope to share a fun experiment the Girl Scouts and I did with pitch perception. It doesn’t rely on having perfectly tuned fifths, thirds, and octaves simultaneously, so it shouldn’t cause the same existential angst that temperament does.
*This sentence and other sentences in this paragraph were edited after publication to correct missing square root signs.
Tags: algebra, arithmetic, fundamental theorem of arithmetic, mathematics and music, music theory, prime numbers, pythagoras, quadratic integer rings
The views expressed are those of the author and are not necessarily those of Scientific American.
Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 2 Dec 2014 at 9:11, James Propp wrote:
That is some great popular mathematical writing, in my opinion!
And another aspect is that you *can* "perfectly" tune, but only in a limited domain of notes (for example, f# and gflat are different, and so you can't have both0. When we had a harpsichord, there were many medieval works that sounded quite different when we used "simple" tuning [I think it was called "just" tuning or something like that] versus "well tempered" [which is standard tuning]. I remember learing that to properly tune the harpsichord, when you tuned "G" against "C", you first got the perfect fifth, and then you flattened it a bit [I think you listened for something like five beats per second or the like] and did the same as you worked your way up. /bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--
I recommend Easley Blackwood's music, where he divides octaves into n equal intervals for all 12 < n <= 24. Of course, it made my girlfriend in college, a cello player with perfect pitch, run screaming out of the room. Cris On Dec 2, 2014, at 7:11 AM, James Propp <jamespropp@gmail.com> wrote:
That is some great popular mathematical writing, in my opinion!
It made me wonder: Is Z[2^(1/12 <x-apple-data-detectors://0>)] embeddable in a finite-degree extension of Z with unique factorization? (Not that this would really affect anything in music.)
Jim
On Tuesday, December 2, 2014, Henry Baker <hbaker1@pipeline.com> wrote:
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning.
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
The Saddest Thing I Know about the Integers
By Evelyn Lamb | November 30, 2014
The integers are a unique factorization domain, so we can’t tune pianos. That is the saddest thing I know about the integers.
I talked to a Girl Scout troop about math earlier this month, and one of our topics was the intersection of math and music. I chose to focus on the way we perceive ratios of sound wave frequencies as intervals. We interpret frequencies that have the ratio 2:1 as octaves. (Larger frequencies sound higher.) We interpret frequencies that have the ratio 3:2 as perfect fifths. And sadly, I had to break it to the girls that these two facts mean that no piano is in tune. In other words, you can tuna fish, but you can’t tune a piano.
When we tune an instrument, we would like for all our octaves and fifths to be perfect. One way to tune an instrument would be to start with a pitch and start working out the fifths above and below it it. We start with some frequency that we call C. Then 3/2 times that frequency is G, 9/4 times that frequency is D (an octave and a step above our original C), and so on. If you learned about the “circle of fifths” at some point in your musical life, then you know that if we keep going up by fifths, we’ll eventually land back on something we’d like to call C. It takes a total of 12 steps, and so if we keep all our fifths perfect, the frequency of the C we get at the end is 312/212, or 531441/4096, times the frequency of the C we had at the beginning. You might notice that 531441/4096 is not an integer, much less a power of 2, so our ears would not perceive the C at the end as being in tune with the C at the beginning. (531441/4096 is about 130, which is 2 more than a power of 2, so we would hear the C at the top as being sharp.) And it’s not a problem with the assumption that it takes 12 fifths to get from C to shining C. We can never get perfect octaves from a stack of fifths because no power of 3/2 will ever give us a power of 2.
Imperfect octaves are pretty unacceptable to any listener, and as a string player, I’m pretty into perfect fifths. So it’s disappointing enough that I can’t have them simultaneously. But the story gets even more complicated when we add thirds. Even if we could resolve the pesky fifths/octave problem, we would be stuck with some pretty strange sounding chords. When we hear frequencies in the ratio 5:4, we hear a perfectly tuned major third (the interval between C and E). But if we go around the circle of 5ths making uncompromising perfect fifths, we get 34/24=81/16. If we divide by 2 a few times to move the E back down to the same octave as the C, we end up with an 81:64 ratio, which is a bit bigger than 5:4 (or 80:64), meaning that the major third from C to E sounds too wide. So fifths are also incompatible with major thirds! Once again, we can never get a perfectly tuned major third from a stack of fifths, or a perfect fifth from a stack of major thirds, because no power of 5/4 equals a power of 3/2.
Blame unique factorization. One property of the integers that we take for granted is that we can factor any integer other than -1, 0, or 1 into its prime factors, and that the factorization will be unique. (We call this the fundamental theorem of arithmetic.) Hence, we can call the integers a unique factorization domain. (If you’re a real stickler, you might be worried about negative integers. The factorization is unique up to signs of numbers, and that’s good enough to be a unique factorization domain. If that still bothers you, just ignore the integers smaller than 2.) As a thought experiment, I decided to see if we could fix the problem by expanding from the integers to another set of numbers like the integers in that they can also be multiplied or added together.
One such set of numbers is called the Gaussian integers, and it consists of complex numbers of the form a+bi, where a and b are both integers and i2=-1. In the Gaussian integers, 2 is no longer a prime number because it can be factored into (1+i)×(1-i), which happen to be primes. Neither is 5, which can be written (1+2i)×(1-2i). But 3 is still a prime in the Gaussian integers (this isn’t obvious, but it’s true). Thus 2 and 3 still share no prime factors over the Gaussian integers, so we can’t resolve our octaves/fifths problem there. (Not that I even know what it would mean to divide a frequency by a Gaussian integer. Like I said, this is a thought experiment.) Likewise, 5 and 2 share no Gaussian prime factors, nor do 5 and 3. So even if it made sense to divide a frequency by a complex number, it wouldn’t help.
An even stranger set of numbers is the set of complex numbers of the form a+v5bi, or Z[v-5]. It might seem like this is no different from the Gaussian integers, but it is. It’s not a unique factorization domain. For example, the number 6 can be factored into either 2×3 or (1+v5i)×(1-v5i).* It’s not obvious, but 2 and 3 can’t be factored further; they are irreducible, as are (1+v5i) and (1-v5i). So 6 has two distinct factorizations. Will this help us? Well, we could end up with some powers of 6 in our frequency ratios if we combine fifths and octaves. But let’s say we could divide that power of 6 by 1+v5i and 1-v5i. Where would we be? We’d have reduced our power of 6 by 1, but we’re no closer to getting a 3 to change into a 2. Bummer! But at least we got to play with some quadratic integers, right?
You may have noticed that piano music doesn’t always sound out of tune, so there must be some resolution to the prime number predicament. Compromise, my friend. Currently, most instruments use equal temperament, which makes all the fifths slightly narrower than perfect so the octaves will be in tune. Each half step has the same frequency ratio as any other half step, and that ratio is 21/12:1. We’ve lost the pure rational ratios that made Pythagorean intervals sound so sweet, but we’ve gained a lot. The difference between a Pythagorean fifth and an equal temperament fifth is not enough to bother any but the fussiest listeners, but it is detectable to some. Before equal temperament became the law of the land, at least for keyboard instruments and other instruments where the player can’t make minute adjustments to pitch, there were several other temperament compromises in use.
One solution is to tune an instrument so that the octaves, fifths, and/or thirds are perfect or very close for the important chords from some keys (generally “easy” keys like C, G, and D) but terrible for some other keys. Those systems (often meantone temperaments) ended up with “wolf fifths” that were much narrower than perfect fifths. An instrument with a wolf fifth couldn’t really play in certain keys. Then along came well temperament, which was not one system but any of many irregular temperaments that made the keys sound different but didn’t leave any key howling at the moon, so to speak. The “well-tempered” in Bach’s Well-Tempered Clavier doesn’t refer to the instrument’s beautiful tone (or the instrumentalist’s equanimity) but to the fact that the set of pieces was composed for a clavier with a temperament that allowed the instrument to play in every key. (The Well-Tempered Clavier is a set of 24 preludes and fugues, one in each major and minor key. Scholars don’t kno w what exactly the clavier’s temperament was, but it is unlikely that it was equal temperament, as some musicians assume.)
The fact that 3/2, 2, and 5/4 are incommensurable makes me genuinely sad, but later this week I hope to share a fun experiment the Girl Scouts and I did with pitch perception. It doesn’t rely on having perfectly tuned fifths, thirds, and octaves simultaneously, so it shouldn’t cause the same existential angst that temperament does.
*This sentence and other sentences in this paragraph were edited after publication to correct missing square root signs.
Tags: algebra, arithmetic, fundamental theorem of arithmetic, mathematics and music, music theory, prime numbers, pythagoras, quadratic integer rings
The views expressed are those of the author and are not necessarily those of Scientific American.
Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.
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It's an interesting exercise to look for equal divisions of the octave into n intervals that provide bettter approximations to (say) 3rd, 5th, 7th harmonics than n = 12 . WFL On 12/3/14, Cris Moore <moore@santafe.edu> wrote:
I recommend Easley Blackwood's music, where he divides octaves into n equal intervals for all 12 < n <= 24.
Of course, it made my girlfriend in college, a cello player with perfect pitch, run screaming out of the room.
Cris
On Dec 2, 2014, at 7:11 AM, James Propp <jamespropp@gmail.com> wrote:
That is some great popular mathematical writing, in my opinion!
It made me wonder: Is Z[2^(1/12 <x-apple-data-detectors://0>)] embeddable in a finite-degree extension of Z with unique factorization? (Not that this would really affect anything in music.)
Jim
On Tuesday, December 2, 2014, Henry Baker <hbaker1@pipeline.com> wrote:
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning.
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
The Saddest Thing I Know about the Integers
By Evelyn Lamb | November 30, 2014
The integers are a unique factorization domain, so we can’t tune pianos. That is the saddest thing I know about the integers.
I talked to a Girl Scout troop about math earlier this month, and one of our topics was the intersection of math and music. I chose to focus on the way we perceive ratios of sound wave frequencies as intervals. We interpret frequencies that have the ratio 2:1 as octaves. (Larger frequencies sound higher.) We interpret frequencies that have the ratio 3:2 as perfect fifths. And sadly, I had to break it to the girls that these two facts mean that no piano is in tune. In other words, you can tuna fish, but you can’t tune a piano.
When we tune an instrument, we would like for all our octaves and fifths to be perfect. One way to tune an instrument would be to start with a pitch and start working out the fifths above and below it it. We start with some frequency that we call C. Then 3/2 times that frequency is G, 9/4 times that frequency is D (an octave and a step above our original C), and so on. If you learned about the “circle of fifths” at some point in your musical life, then you know that if we keep going up by fifths, we’ll eventually land back on something we’d like to call C. It takes a total of 12 steps, and so if we keep all our fifths perfect, the frequency of the C we get at the end is 312/212, or 531441/4096, times the frequency of the C we had at the beginning. You might notice that 531441/4096 is not an integer, much less a power of 2, so our ears would not perceive the C at the end as being in tune with the C at the beginning. (531441/4096 is about 130, which is 2 more than a power of 2, so we would hear the C at the top as being sharp.) And it’s not a problem with the assumption that it takes 12 fifths to get from C to shining C. We can never get perfect octaves from a stack of fifths because no power of 3/2 will ever give us a power of 2.
Imperfect octaves are pretty unacceptable to any listener, and as a string player, I’m pretty into perfect fifths. So it’s disappointing enough that I can’t have them simultaneously. But the story gets even more complicated when we add thirds. Even if we could resolve the pesky fifths/octave problem, we would be stuck with some pretty strange sounding chords. When we hear frequencies in the ratio 5:4, we hear a perfectly tuned major third (the interval between C and E). But if we go around the circle of 5ths making uncompromising perfect fifths, we get 34/24=81/16. If we divide by 2 a few times to move the E back down to the same octave as the C, we end up with an 81:64 ratio, which is a bit bigger than 5:4 (or 80:64), meaning that the major third from C to E sounds too wide. So fifths are also incompatible with major thirds! Once again, we can never get a perfectly tuned major third from a stack of fifths, or a perfect fifth from a stack of major thirds, because no power of 5/4 equals a power of 3/2.
Blame unique factorization. One property of the integers that we take for granted is that we can factor any integer other than -1, 0, or 1 into its prime factors, and that the factorization will be unique. (We call this the fundamental theorem of arithmetic.) Hence, we can call the integers a unique factorization domain. (If you’re a real stickler, you might be worried about negative integers. The factorization is unique up to signs of numbers, and that’s good enough to be a unique factorization domain. If that still bothers you, just ignore the integers smaller than 2.) As a thought experiment, I decided to see if we could fix the problem by expanding from the integers to another set of numbers like the integers in that they can also be multiplied or added together.
One such set of numbers is called the Gaussian integers, and it consists of complex numbers of the form a+bi, where a and b are both integers and i2=-1. In the Gaussian integers, 2 is no longer a prime number because it can be factored into (1+i)×(1-i), which happen to be primes. Neither is 5, which can be written (1+2i)×(1-2i). But 3 is still a prime in the Gaussian integers (this isn’t obvious, but it’s true). Thus 2 and 3 still share no prime factors over the Gaussian integers, so we can’t resolve our octaves/fifths problem there. (Not that I even know what it would mean to divide a frequency by a Gaussian integer. Like I said, this is a thought experiment.) Likewise, 5 and 2 share no Gaussian prime factors, nor do 5 and 3. So even if it made sense to divide a frequency by a complex number, it wouldn’t help.
An even stranger set of numbers is the set of complex numbers of the form a+v5bi, or Z[v-5]. It might seem like this is no different from the Gaussian integers, but it is. It’s not a unique factorization domain. For example, the number 6 can be factored into either 2×3 or (1+v5i)×(1-v5i).* It’s not obvious, but 2 and 3 can’t be factored further; they are irreducible, as are (1+v5i) and (1-v5i). So 6 has two distinct factorizations. Will this help us? Well, we could end up with some powers of 6 in our frequency ratios if we combine fifths and octaves. But let’s say we could divide that power of 6 by 1+v5i and 1-v5i. Where would we be? We’d have reduced our power of 6 by 1, but we’re no closer to getting a 3 to change into a 2. Bummer! But at least we got to play with some quadratic integers, right?
You may have noticed that piano music doesn’t always sound out of tune, so there must be some resolution to the prime number predicament. Compromise, my friend. Currently, most instruments use equal temperament, which makes all the fifths slightly narrower than perfect so the octaves will be in tune. Each half step has the same frequency ratio as any other half step, and that ratio is 21/12:1. We’ve lost the pure rational ratios that made Pythagorean intervals sound so sweet, but we’ve gained a lot. The difference between a Pythagorean fifth and an equal temperament fifth is not enough to bother any but the fussiest listeners, but it is detectable to some. Before equal temperament became the law of the land, at least for keyboard instruments and other instruments where the player can’t make minute adjustments to pitch, there were several other temperament compromises in use.
One solution is to tune an instrument so that the octaves, fifths, and/or thirds are perfect or very close for the important chords from some keys (generally “easy” keys like C, G, and D) but terrible for some other keys. Those systems (often meantone temperaments) ended up with “wolf fifths” that were much narrower than perfect fifths. An instrument with a wolf fifth couldn’t really play in certain keys. Then along came well temperament, which was not one system but any of many irregular temperaments that made the keys sound different but didn’t leave any key howling at the moon, so to speak. The “well-tempered” in Bach’s Well-Tempered Clavier doesn’t refer to the instrument’s beautiful tone (or the instrumentalist’s equanimity) but to the fact that the set of pieces was composed for a clavier with a temperament that allowed the instrument to play in every key. (The Well-Tempered Clavier is a set of 24 preludes and fugues, one in each major and minor key. Scholars don’t kno w what exactly the clavier’s temperament was, but it is unlikely that it was equal temperament, as some musicians assume.)
The fact that 3/2, 2, and 5/4 are incommensurable makes me genuinely sad, but later this week I hope to share a fun experiment the Girl Scouts and I did with pitch perception. It doesn’t rely on having perfectly tuned fifths, thirds, and octaves simultaneously, so it shouldn’t cause the same existential angst that temperament does.
*This sentence and other sentences in this paragraph were edited after publication to correct missing square root signs.
Tags: algebra, arithmetic, fundamental theorem of arithmetic, mathematics and music, music theory, prime numbers, pythagoras, quadratic integer rings
The views expressed are those of the author and are not necessarily those of Scientific American.
Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.
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I've heard music with n intervals per octave for n in {5, 12, 44}. On Tue, Dec 2, 2014 at 9:04 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
It's an interesting exercise to look for equal divisions of the octave into n intervals that provide bettter approximations to (say) 3rd, 5th, 7th harmonics than n = 12 .
WFL
On 12/3/14, Cris Moore <moore@santafe.edu> wrote:
I recommend Easley Blackwood's music, where he divides octaves into n
equal
intervals for all 12 < n <= 24.
Of course, it made my girlfriend in college, a cello player with perfect pitch, run screaming out of the room.
Cris
On Dec 2, 2014, at 7:11 AM, James Propp <jamespropp@gmail.com> wrote:
That is some great popular mathematical writing, in my opinion!
It made me wonder: Is Z[2^(1/12 <x-apple-data-detectors://0>)] embeddable in a finite-degree extension of Z with unique factorization? (Not that this would really affect anything in music.)
Jim
On Tuesday, December 2, 2014, Henry Baker <hbaker1@pipeline.com> wrote:
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning.
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
The Saddest Thing I Know about the Integers
By Evelyn Lamb | November 30, 2014
The integers are a unique factorization domain, so we can’t tune
pianos.
That is the saddest thing I know about the integers.
I talked to a Girl Scout troop about math earlier this month, and one of our topics was the intersection of math and music. I chose to focus on the way we perceive ratios of sound wave frequencies as intervals. We interpret frequencies that have the ratio 2:1 as octaves. (Larger frequencies sound higher.) We interpret frequencies that have the ratio 3:2 as perfect fifths. And sadly, I had to break it to the girls that these two facts mean that no piano is in tune. In other words, you can tuna fish, but you can’t tune a piano.
When we tune an instrument, we would like for all our octaves and fifths to be perfect. One way to tune an instrument would be to start with a pitch and start working out the fifths above and below it it. We start with some frequency that we call C. Then 3/2 times that frequency is G, 9/4 times that frequency is D (an octave and a step above our original C), and so on. If you learned about the “circle of fifths” at some point in your musical life, then you know that if we keep going up by fifths, we’ll eventually land back on something we’d like to call C. It takes a total of 12 steps, and so if we keep all our fifths perfect, the frequency of the C we get at the end is 312/212, or 531441/4096, times the frequency of the C we had at the beginning. You might notice that 531441/4096 is not an integer, much less a power of 2, so our ears would not perceive the C at the end as being in tune with the C at the beginning. (531441/4096 is about 130, which is 2 more than a power of 2, so we would hear the C at the top as being sharp.) And it’s not a problem with the assumption that it takes 12 fifths to get from C to shining C. We can never get perfect octaves from a stack of fifths because no power of 3/2 will ever give us a power of 2.
Imperfect octaves are pretty unacceptable to any listener, and as a string player, I’m pretty into perfect fifths. So it’s disappointing enough that I can’t have them simultaneously. But the story gets even more complicated when we add thirds. Even if we could resolve the pesky fifths/octave problem, we would be stuck with some pretty strange sounding chords. When we hear frequencies in the ratio 5:4, we hear a perfectly tuned major third (the interval between C and E). But if we go around the circle of 5ths making uncompromising perfect fifths, we get 34/24=81/16. If we divide by 2 a few times to move the E back down to the same octave as the C, we end up with an 81:64 ratio, which is a bit bigger than 5:4 (or 80:64), meaning that the major third from C to E sounds too wide. So fifths are also incompatible with major thirds! Once again, we can never get a perfectly tuned major third from a stack of fifths, or a perfect fifth from a stack of major thirds, because no power of 5/4 equals a power of 3/2.
Blame unique factorization. One property of the integers that we take for granted is that we can factor any integer other than -1, 0, or 1 into its prime factors, and that the factorization will be unique. (We call this the fundamental theorem of arithmetic.) Hence, we can call the integers a unique factorization domain. (If you’re a real stickler, you might be worried about negative integers. The factorization is unique up to signs of numbers, and that’s good enough to be a unique factorization domain. If that still bothers you, just ignore the integers smaller than 2.) As a thought experiment, I decided to see if we could fix the problem by expanding from the integers to another set of numbers like the integers in that they can also be multiplied or added together.
One such set of numbers is called the Gaussian integers, and it consists of complex numbers of the form a+bi, where a and b are both integers and i2=-1. In the Gaussian integers, 2 is no longer a prime number because it can be factored into (1+i)×(1-i), which happen to be primes. Neither is 5, which can be written (1+2i)×(1-2i). But 3 is still a prime in the Gaussian integers (this isn’t obvious, but it’s true). Thus 2 and 3 still share no prime factors over the Gaussian integers, so we can’t resolve our octaves/fifths problem there. (Not that I even know what it would mean to divide a frequency by a Gaussian integer. Like I said, this is a thought experiment.) Likewise, 5 and 2 share no Gaussian prime factors, nor do 5 and 3. So even if it made sense to divide a frequency by a complex number, it wouldn’t help.
An even stranger set of numbers is the set of complex numbers of the form a+v5bi, or Z[v-5]. It might seem like this is no different from the Gaussian integers, but it is. It’s not a unique factorization domain. For example, the number 6 can be factored into either 2×3 or (1+v5i)×(1-v5i).* It’s not obvious, but 2 and 3 can’t be factored further; they are irreducible, as are (1+v5i) and (1-v5i). So 6 has two distinct factorizations. Will this help us? Well, we could end up with some powers of 6 in our frequency ratios if we combine fifths and octaves. But let’s say we could divide that power of 6 by 1+v5i and 1-v5i. Where would we be? We’d have reduced our power of 6 by 1, but we’re no closer to getting a 3 to change into a 2. Bummer! But at least we got to play with some quadratic integers, right?
You may have noticed that piano music doesn’t always sound out of tune, so there must be some resolution to the prime number predicament. Compromise, my friend. Currently, most instruments use equal temperament, which makes all the fifths slightly narrower than perfect so the octaves will be in tune. Each half step has the same frequency ratio as any other half step, and that ratio is 21/12:1. We’ve lost the pure rational ratios that made Pythagorean intervals sound so sweet, but we’ve gained a lot. The difference between a Pythagorean fifth and an equal temperament fifth is not enough to bother any but the fussiest listeners, but it is detectable to some. Before equal temperament became the law of the land, at least for keyboard instruments and other instruments where the player can’t make minute adjustments to pitch, there were several other temperament compromises in use.
One solution is to tune an instrument so that the octaves, fifths, and/or thirds are perfect or very close for the important chords from some keys (generally “easy” keys like C, G, and D) but terrible for some other keys. Those systems (often meantone temperaments) ended up with “wolf fifths” that were much narrower than perfect fifths. An instrument with a wolf fifth couldn’t really play in certain keys. Then along came well temperament, which was not one system but any of many irregular temperaments that made the keys sound different but didn’t leave any key howling at the moon, so to speak. The “well-tempered” in Bach’s Well-Tempered Clavier doesn’t refer to the instrument’s beautiful tone (or the instrumentalist’s equanimity) but to the fact that the set of pieces was composed for a clavier with a temperament that allowed the instrument to play in every key. (The Well-Tempered Clavier is a set of 24 preludes and fugues, one in each major and minor key. Scholars don’t kno w what exactly the clavier’s temperament was, but it is unlikely that it was equal temperament, as some musicians assume.)
The fact that 3/2, 2, and 5/4 are incommensurable makes me genuinely sad, but later this week I hope to share a fun experiment the Girl Scouts and I did with pitch perception. It doesn’t rely on having perfectly tuned fifths, thirds, and octaves simultaneously, so it shouldn’t cause the same existential angst that temperament does.
*This sentence and other sentences in this paragraph were edited after publication to correct missing square root signs.
Tags: algebra, arithmetic, fundamental theorem of arithmetic, mathematics and music, music theory, prime numbers, pythagoras, quadratic integer rings
The views expressed are those of the author and are not necessarily those of Scientific American.
Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.
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where did you hear n=44? - cris On Dec 2, 2014, at 10:29 PM, Scott Huddleston <c.scott.huddleston@gmail.com> wrote:
I've heard music with n intervals per octave for n in {5, 12, 44}.
On Tue, Dec 2, 2014 at 9:04 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
It's an interesting exercise to look for equal divisions of the octave into n intervals that provide bettter approximations to (say) 3rd, 5th, 7th harmonics than n = 12 .
WFL
On 12/3/14, Cris Moore <moore@santafe.edu> wrote:
I recommend Easley Blackwood's music, where he divides octaves into n
equal
intervals for all 12 < n <= 24.
Of course, it made my girlfriend in college, a cello player with perfect pitch, run screaming out of the room.
Cris
On Dec 2, 2014, at 7:11 AM, James Propp <jamespropp@gmail.com> wrote:
That is some great popular mathematical writing, in my opinion!
It made me wonder: Is Z[2^(1/12 <x-apple-data-detectors://0>)] embeddable in a finite-degree extension of Z with unique factorization? (Not that this would really affect anything in music.)
Jim
On Tuesday, December 2, 2014, Henry Baker <hbaker1@pipeline.com> wrote:
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning.
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
The Saddest Thing I Know about the Integers
By Evelyn Lamb | November 30, 2014
The integers are a unique factorization domain, so we can’t tune
pianos.
That is the saddest thing I know about the integers.
I talked to a Girl Scout troop about math earlier this month, and one of our topics was the intersection of math and music. I chose to focus on the way we perceive ratios of sound wave frequencies as intervals. We interpret frequencies that have the ratio 2:1 as octaves. (Larger frequencies sound higher.) We interpret frequencies that have the ratio 3:2 as perfect fifths. And sadly, I had to break it to the girls that these two facts mean that no piano is in tune. In other words, you can tuna fish, but you can’t tune a piano.
When we tune an instrument, we would like for all our octaves and fifths to be perfect. One way to tune an instrument would be to start with a pitch and start working out the fifths above and below it it. We start with some frequency that we call C. Then 3/2 times that frequency is G, 9/4 times that frequency is D (an octave and a step above our original C), and so on. If you learned about the “circle of fifths” at some point in your musical life, then you know that if we keep going up by fifths, we’ll eventually land back on something we’d like to call C. It takes a total of 12 steps, and so if we keep all our fifths perfect, the frequency of the C we get at the end is 312/212, or 531441/4096, times the frequency of the C we had at the beginning. You might notice that 531441/4096 is not an integer, much less a power of 2, so our ears would not perceive the C at the end as being in tune with the C at the beginning. (531441/4096 is about 130, which is 2 more than a power of 2, so we would hear the C at the top as being sharp.) And it’s not a problem with the assumption that it takes 12 fifths to get from C to shining C. We can never get perfect octaves from a stack of fifths because no power of 3/2 will ever give us a power of 2.
Imperfect octaves are pretty unacceptable to any listener, and as a string player, I’m pretty into perfect fifths. So it’s disappointing enough that I can’t have them simultaneously. But the story gets even more complicated when we add thirds. Even if we could resolve the pesky fifths/octave problem, we would be stuck with some pretty strange sounding chords. When we hear frequencies in the ratio 5:4, we hear a perfectly tuned major third (the interval between C and E). But if we go around the circle of 5ths making uncompromising perfect fifths, we get 34/24=81/16. If we divide by 2 a few times to move the E back down to the same octave as the C, we end up with an 81:64 ratio, which is a bit bigger than 5:4 (or 80:64), meaning that the major third from C to E sounds too wide. So fifths are also incompatible with major thirds! Once again, we can never get a perfectly tuned major third from a stack of fifths, or a perfect fifth from a stack of major thirds, because no power of 5/4 equals a power of 3/2.
Blame unique factorization. One property of the integers that we take for granted is that we can factor any integer other than -1, 0, or 1 into its prime factors, and that the factorization will be unique. (We call this the fundamental theorem of arithmetic.) Hence, we can call the integers a unique factorization domain. (If you’re a real stickler, you might be worried about negative integers. The factorization is unique up to signs of numbers, and that’s good enough to be a unique factorization domain. If that still bothers you, just ignore the integers smaller than 2.) As a thought experiment, I decided to see if we could fix the problem by expanding from the integers to another set of numbers like the integers in that they can also be multiplied or added together.
One such set of numbers is called the Gaussian integers, and it consists of complex numbers of the form a+bi, where a and b are both integers and i2=-1. In the Gaussian integers, 2 is no longer a prime number because it can be factored into (1+i)×(1-i), which happen to be primes. Neither is 5, which can be written (1+2i)×(1-2i). But 3 is still a prime in the Gaussian integers (this isn’t obvious, but it’s true). Thus 2 and 3 still share no prime factors over the Gaussian integers, so we can’t resolve our octaves/fifths problem there. (Not that I even know what it would mean to divide a frequency by a Gaussian integer. Like I said, this is a thought experiment.) Likewise, 5 and 2 share no Gaussian prime factors, nor do 5 and 3. So even if it made sense to divide a frequency by a complex number, it wouldn’t help.
An even stranger set of numbers is the set of complex numbers of the form a+v5bi, or Z[v-5]. It might seem like this is no different from the Gaussian integers, but it is. It’s not a unique factorization domain. For example, the number 6 can be factored into either 2×3 or (1+v5i)×(1-v5i).* It’s not obvious, but 2 and 3 can’t be factored further; they are irreducible, as are (1+v5i) and (1-v5i). So 6 has two distinct factorizations. Will this help us? Well, we could end up with some powers of 6 in our frequency ratios if we combine fifths and octaves. But let’s say we could divide that power of 6 by 1+v5i and 1-v5i. Where would we be? We’d have reduced our power of 6 by 1, but we’re no closer to getting a 3 to change into a 2. Bummer! But at least we got to play with some quadratic integers, right?
You may have noticed that piano music doesn’t always sound out of tune, so there must be some resolution to the prime number predicament. Compromise, my friend. Currently, most instruments use equal temperament, which makes all the fifths slightly narrower than perfect so the octaves will be in tune. Each half step has the same frequency ratio as any other half step, and that ratio is 21/12:1. We’ve lost the pure rational ratios that made Pythagorean intervals sound so sweet, but we’ve gained a lot. The difference between a Pythagorean fifth and an equal temperament fifth is not enough to bother any but the fussiest listeners, but it is detectable to some. Before equal temperament became the law of the land, at least for keyboard instruments and other instruments where the player can’t make minute adjustments to pitch, there were several other temperament compromises in use.
One solution is to tune an instrument so that the octaves, fifths, and/or thirds are perfect or very close for the important chords from some keys (generally “easy” keys like C, G, and D) but terrible for some other keys. Those systems (often meantone temperaments) ended up with “wolf fifths” that were much narrower than perfect fifths. An instrument with a wolf fifth couldn’t really play in certain keys. Then along came well temperament, which was not one system but any of many irregular temperaments that made the keys sound different but didn’t leave any key howling at the moon, so to speak. The “well-tempered” in Bach’s Well-Tempered Clavier doesn’t refer to the instrument’s beautiful tone (or the instrumentalist’s equanimity) but to the fact that the set of pieces was composed for a clavier with a temperament that allowed the instrument to play in every key. (The Well-Tempered Clavier is a set of 24 preludes and fugues, one in each major and minor key. Scholars don’t kno w what exactly the clavier’s temperament was, but it is unlikely that it was equal temperament, as some musicians assume.)
The fact that 3/2, 2, and 5/4 are incommensurable makes me genuinely sad, but later this week I hope to share a fun experiment the Girl Scouts and I did with pitch perception. It doesn’t rely on having perfectly tuned fifths, thirds, and octaves simultaneously, so it shouldn’t cause the same existential angst that temperament does.
*This sentence and other sentences in this paragraph were edited after publication to correct missing square root signs.
Tags: algebra, arithmetic, fundamental theorem of arithmetic, mathematics and music, music theory, prime numbers, pythagoras, quadratic integer rings
The views expressed are those of the author and are not necessarily those of Scientific American.
Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.
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http://en.wikipedia.org/wiki/Harry_Partch's_43-tone_scale (43, apparently, not 44). Only heard on Internet recordings, not live. I'd like to know more about Indian music scales (n > 12). On Tue, Dec 2, 2014 at 9:47 PM, Cris Moore <moore@santafe.edu> wrote:
where did you hear n=44?
- cris
On Dec 2, 2014, at 10:29 PM, Scott Huddleston < c.scott.huddleston@gmail.com> wrote:
I've heard music with n intervals per octave for n in {5, 12, 44}.
On Tue, Dec 2, 2014 at 9:04 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
It's an interesting exercise to look for equal divisions of the octave into n intervals that provide bettter approximations to (say) 3rd, 5th, 7th harmonics than n = 12 .
WFL
On 12/3/14, Cris Moore <moore@santafe.edu> wrote:
I recommend Easley Blackwood's music, where he divides octaves into n
equal
intervals for all 12 < n <= 24.
Of course, it made my girlfriend in college, a cello player with perfect pitch, run screaming out of the room.
Cris
On Dec 2, 2014, at 7:11 AM, James Propp <jamespropp@gmail.com> wrote:
That is some great popular mathematical writing, in my opinion!
It made me wonder: Is Z[2^(1/12 <x-apple-data-detectors://0>)] embeddable in a finite-degree extension of Z with unique factorization? (Not that this would really affect anything in music.)
Jim
On Tuesday, December 2, 2014, Henry Baker <hbaker1@pipeline.com> wrote:
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning.
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
The Saddest Thing I Know about the Integers
By Evelyn Lamb | November 30, 2014
The integers are a unique factorization domain, so we can’t tune
pianos.
That is the saddest thing I know about the integers.
I talked to a Girl Scout troop about math earlier this month, and one of our topics was the intersection of math and music. I chose to focus on the way we perceive ratios of sound wave frequencies as intervals. We interpret frequencies that have the ratio 2:1 as octaves. (Larger frequencies sound higher.) We interpret frequencies that have the ratio 3:2 as perfect fifths. And sadly, I had to break it to the girls that these two facts mean that no piano is in tune. In other words, you can tuna fish, but you can’t tune a piano.
When we tune an instrument, we would like for all our octaves and fifths to be perfect. One way to tune an instrument would be to start with a pitch and start working out the fifths above and below it it. We start with some frequency that we call C. Then 3/2 times that frequency is G, 9/4 times that frequency is D (an octave and a step above our original C), and so on. If you learned about the “circle of fifths” at some point in your musical life, then you know that if we keep going up by fifths, we’ll eventually land back on something we’d like to call C. It takes a total of 12 steps, and so if we keep all our fifths perfect, the frequency of the C we get at the end is 312/212, or 531441/4096, times the frequency of the C we had at the beginning. You might notice that 531441/4096 is not an integer, much less a power of 2, so our ears would not perceive the C at the end as being in tune with the C at the beginning. (531441/4096 is about 130, which is 2 more than a power of 2, so we would hear the C at the top as being sharp.) And it’s not a problem with the assumption that it takes 12 fifths to get from C to shining C. We can never get perfect octaves from a stack of fifths because no power of 3/2 will ever give us a power of 2.
Imperfect octaves are pretty unacceptable to any listener, and as a string player, I’m pretty into perfect fifths. So it’s disappointing enough that I can’t have them simultaneously. But the story gets even more complicated when we add thirds. Even if we could resolve the pesky fifths/octave problem, we would be stuck with some pretty strange sounding chords. When we hear frequencies in the ratio 5:4, we hear a perfectly tuned major third (the interval between C and E). But if we go around the circle of 5ths making uncompromising perfect fifths, we get 34/24=81/16. If we divide by 2 a few times to move the E back down to the same octave as the C, we end up with an 81:64 ratio, which is a bit bigger than 5:4 (or 80:64), meaning that the major third from C to E sounds too wide. So fifths are also incompatible with major thirds! Once again, we can never get a perfectly tuned major third from a stack of fifths, or a perfect fifth from a stack of major thirds, because no power of 5/4 equals a power of 3/2.
Blame unique factorization. One property of the integers that we take for granted is that we can factor any integer other than -1, 0, or 1 into its prime factors, and that the factorization will be unique. (We call this the fundamental theorem of arithmetic.) Hence, we can call the integers a unique factorization domain. (If you’re a real stickler, you might be worried about negative integers. The factorization is unique up to signs of numbers, and that’s good enough to be a unique factorization domain. If that still bothers you, just ignore the integers smaller than 2.) As a thought experiment, I decided to see if we could fix the problem by expanding from the integers to another set of numbers like the integers in that they can also be multiplied or added together.
One such set of numbers is called the Gaussian integers, and it consists of complex numbers of the form a+bi, where a and b are both integers and i2=-1. In the Gaussian integers, 2 is no longer a prime number because it can be factored into (1+i)×(1-i), which happen to be primes. Neither is 5, which can be written (1+2i)×(1-2i). But 3 is still a prime in the Gaussian integers (this isn’t obvious, but it’s true). Thus 2 and 3 still share no prime factors over the Gaussian integers, so we can’t resolve our octaves/fifths problem there. (Not that I even know what it would mean to divide a frequency by a Gaussian integer. Like I said, this is a thought experiment.) Likewise, 5 and 2 share no Gaussian prime factors, nor do 5 and 3. So even if it made sense to divide a frequency by a complex number, it wouldn’t help.
An even stranger set of numbers is the set of complex numbers of the form a+v5bi, or Z[v-5]. It might seem like this is no different from the Gaussian integers, but it is. It’s not a unique factorization domain. For example, the number 6 can be factored into either 2×3 or (1+v5i)×(1-v5i).* It’s not obvious, but 2 and 3 can’t be factored further; they are irreducible, as are (1+v5i) and (1-v5i). So 6 has two distinct factorizations. Will this help us? Well, we could end up with some powers of 6 in our frequency ratios if we combine fifths and octaves. But let’s say we could divide that power of 6 by 1+v5i and 1-v5i. Where would we be? We’d have reduced our power of 6 by 1, but we’re no closer to getting a 3 to change into a 2. Bummer! But at least we got to play with some quadratic integers, right?
You may have noticed that piano music doesn’t always sound out of tune, so there must be some resolution to the prime number predicament. Compromise, my friend. Currently, most instruments use equal temperament, which makes all the fifths slightly narrower than perfect so the octaves will be in tune. Each half step has the same frequency ratio as any other half step, and that ratio is 21/12:1. We’ve lost the pure rational ratios that made Pythagorean intervals sound so sweet, but we’ve gained a lot. The difference between a Pythagorean fifth and an equal temperament fifth is not enough to bother any but the fussiest listeners, but it is detectable to some. Before equal temperament became the law of the land, at least for keyboard instruments and other instruments where the player can’t make minute adjustments to pitch, there were several other temperament compromises in use.
One solution is to tune an instrument so that the octaves, fifths, and/or thirds are perfect or very close for the important chords from some keys (generally “easy” keys like C, G, and D) but terrible for some other keys. Those systems (often meantone temperaments) ended up with “wolf fifths” that were much narrower than perfect fifths. An instrument with a wolf fifth couldn’t really play in certain keys. Then along came well temperament, which was not one system but any of many irregular temperaments that made the keys sound different but didn’t leave any key howling at the moon, so to speak. The “well-tempered” in Bach’s Well-Tempered Clavier doesn’t refer to the instrument’s beautiful tone (or the instrumentalist’s equanimity) but to the fact that the set of pieces was composed for a clavier with a temperament that allowed the instrument to play in every key. (The Well-Tempered Clavier is a set of 24 preludes and fugues, one in each major and minor key. Scholars don’t kno w what exactly the clavier’s temperament was, but it is unlikely that it was equal temperament, as some musicians assume.)
The fact that 3/2, 2, and 5/4 are incommensurable makes me genuinely sad, but later this week I hope to share a fun experiment the Girl Scouts and I did with pitch perception. It doesn’t rely on having perfectly tuned fifths, thirds, and octaves simultaneously, so it shouldn’t cause the same existential angst that temperament does.
*This sentence and other sentences in this paragraph were edited after publication to correct missing square root signs.
Tags: algebra, arithmetic, fundamental theorem of arithmetic, mathematics and music, music theory, prime numbers, pythagoras, quadratic integer rings
The views expressed are those of the author and are not necessarily those of Scientific American.
Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.
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care to recommend a particular piece? - cris On Dec 2, 2014, at 10:58 PM, Scott Huddleston <c.scott.huddleston@gmail.com> wrote:
http://en.wikipedia.org/wiki/Harry_Partch's_43-tone_scale (43, apparently, not 44). Only heard on Internet recordings, not live.
I'd like to know more about Indian music scales (n > 12).
On Tue, Dec 2, 2014 at 9:47 PM, Cris Moore <moore@santafe.edu> wrote:
where did you hear n=44?
- cris
On Dec 2, 2014, at 10:29 PM, Scott Huddleston < c.scott.huddleston@gmail.com> wrote:
I've heard music with n intervals per octave for n in {5, 12, 44}.
On Tue, Dec 2, 2014 at 9:04 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
It's an interesting exercise to look for equal divisions of the octave into n intervals that provide bettter approximations to (say) 3rd, 5th, 7th harmonics than n = 12 .
WFL
On 12/3/14, Cris Moore <moore@santafe.edu> wrote:
I recommend Easley Blackwood's music, where he divides octaves into n
equal
intervals for all 12 < n <= 24.
Of course, it made my girlfriend in college, a cello player with perfect pitch, run screaming out of the room.
Cris
On Dec 2, 2014, at 7:11 AM, James Propp <jamespropp@gmail.com> wrote:
That is some great popular mathematical writing, in my opinion!
It made me wonder: Is Z[2^(1/12 <x-apple-data-detectors://0>)] embeddable in a finite-degree extension of Z with unique factorization? (Not that this would really affect anything in music.)
Jim
On Tuesday, December 2, 2014, Henry Baker <hbaker1@pipeline.com> wrote:
> FYI -- Many/most of you may know this stuff, but I found it an > interesting > angle on the problem of musical tuning. > > >
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
> > The Saddest Thing I Know about the Integers > > By Evelyn Lamb | November 30, 2014 > > The integers are a unique factorization domain, so we can’t tune pianos. > That is the saddest thing I know about the integers. > > I talked to a Girl Scout troop about math earlier this month, and one of > our topics was the intersection of math and music. I chose to focus on > the > way we perceive ratios of sound wave frequencies as intervals. We > interpret frequencies that have the ratio 2:1 as octaves. (Larger > frequencies sound higher.) We interpret frequencies that have the ratio > 3:2 as perfect fifths. And sadly, I had to break it to the girls that > these two facts mean that no piano is in tune. In other words, you can > tuna fish, but you can’t tune a piano. > > When we tune an instrument, we would like for all our octaves and fifths > to be perfect. One way to tune an instrument would be to start with a > pitch and start working out the fifths above and below it it. We start > with some frequency that we call C. Then 3/2 times that frequency is G, > 9/4 times that frequency is D (an octave and a step above our original > C), > and so on. If you learned about the “circle of fifths” at some point in > your musical life, then you know that if we keep going up by fifths, > we’ll > eventually land back on something we’d like to call C. It takes a total > of > 12 steps, and so if we keep all our fifths perfect, the frequency of the > C > we get at the end is 312/212, or 531441/4096, times the frequency of the > C > we had at the beginning. You might notice that 531441/4096 is not an > integer, much less a power of 2, so our ears would not perceive the C at > the end as being in tune with the C at the beginning. (531441/4096 is > about 130, which is 2 more than a power of > 2, so we would hear the C at the top as being sharp.) And it’s not a > problem with the assumption that it takes 12 fifths to get from C to > shining C. We can never get perfect octaves from a stack of fifths > because > no power of 3/2 will ever give us a power of 2. > > Imperfect octaves are pretty unacceptable to any listener, and as a > string > player, I’m pretty into perfect fifths. So it’s disappointing enough > that > I can’t have them simultaneously. But the story gets even more > complicated > when we add thirds. Even if we could resolve the pesky fifths/octave > problem, we would be stuck with some pretty strange sounding chords. > When > we hear frequencies in the ratio 5:4, we hear a perfectly tuned major > third > (the interval between C and E). But if we go around the circle of 5ths > making uncompromising perfect fifths, we get 34/24=81/16. If we divide > by > 2 a few times to move the E back down to the same octave as the C, we > end > up with an 81:64 ratio, which is a bit bigger than 5:4 (or 80:64), > meaning > that the major third from C to E sounds too wide. So fifths are also > incompatible with major thirds! Once again, we can never get a > perfectly > tuned major third from a stack of fifths, or a perfect fifth from a > stack > of major thirds, because no power > of 5/4 equals a power of 3/2. > > Blame unique factorization. One property of the integers that we take > for > granted is that we can factor any integer other than -1, 0, or 1 into > its > prime factors, and that the factorization will be unique. (We call this > the fundamental theorem of arithmetic.) Hence, we can call the integers > a > unique factorization domain. (If you’re a real stickler, you might be > worried about negative integers. The factorization is unique up to > signs > of numbers, and that’s good enough to be a unique factorization domain. > If > that still bothers you, just ignore the integers smaller than 2.) As a > thought experiment, I decided to see if we could fix the problem by > expanding from the integers to another set of numbers like the integers > in > that they can also be multiplied or added together. > > One such set of numbers is called the Gaussian integers, and it consists > of complex numbers of the form a+bi, where a and b are both integers and > i2=-1. In the Gaussian integers, 2 is no longer a prime number because > it > can be factored into (1+i)×(1-i), which happen to be primes. Neither is > 5, > which can be written (1+2i)×(1-2i). But 3 is still a prime in the > Gaussian > integers (this isn’t obvious, but it’s true). Thus 2 and 3 still share > no > prime factors over the Gaussian integers, so we can’t resolve our > octaves/fifths problem there. (Not that I even know what it would mean > to > divide a frequency by a Gaussian integer. Like I said, this is a > thought > experiment.) Likewise, 5 and 2 share no Gaussian prime factors, nor do > 5 > and 3. So even if it made sense to divide a frequency by a complex > number, > it wouldn’t help. > > An even stranger set of numbers is the set of complex numbers of the > form > a+v5bi, or Z[v-5]. It might seem like this is no different from the > Gaussian integers, but it is. It’s not a unique factorization domain. > For > example, the number 6 can be factored into either 2×3 or > (1+v5i)×(1-v5i).* > It’s not obvious, but 2 and 3 can’t be factored further; they are > irreducible, as are (1+v5i) and (1-v5i). So 6 has two distinct > factorizations. Will this help us? Well, we could end up with some > powers > of 6 in our frequency ratios if we combine fifths and octaves. But > let’s > say we could divide that power of 6 by 1+v5i and 1-v5i. Where would we > be? We’d have reduced our power of 6 by 1, but we’re no closer to > getting > a 3 to change into a 2. Bummer! But at least we got to play with some > quadratic integers, right? > > You may have noticed that piano music doesn’t always sound out of tune, > so > there must be some resolution to the prime number predicament. > Compromise, > my friend. Currently, most instruments use equal temperament, which > makes > all the fifths slightly narrower than perfect so the octaves will be in > tune. Each half step has the same frequency ratio as any other half > step, > and that ratio is 21/12:1. We’ve lost the pure rational ratios that > made > Pythagorean intervals sound so sweet, but we’ve gained a lot. The > difference between a Pythagorean fifth and an equal temperament fifth is > not enough to bother any but the fussiest listeners, but it is > detectable > to some. Before equal temperament became the law of the land, at least > for > keyboard instruments and other instruments where the player can’t make > minute adjustments to pitch, there were several other temperament > compromises in use. > > One solution is to tune an instrument so that the octaves, fifths, > and/or > thirds are perfect or very close for the important chords from some keys > (generally “easy” keys like C, G, and D) but terrible for some other > keys. > Those systems (often meantone temperaments) ended up with “wolf fifths” > that were much narrower than perfect fifths. An instrument with a wolf > fifth couldn’t really play in certain keys. Then along came well > temperament, which was not one system but any of many irregular > temperaments that made the keys sound different but didn’t leave any key > howling at the moon, so to speak. The “well-tempered” in Bach’s > Well-Tempered Clavier doesn’t refer to the instrument’s beautiful tone > (or > the instrumentalist’s equanimity) but to the fact that the set of pieces > was composed for a clavier with a temperament that allowed the > instrument > to play in every key. (The Well-Tempered Clavier is a set of 24 > preludes > and fugues, one in each major and minor key. Scholars don’t kno > w what exactly the clavier’s temperament was, but it is unlikely that it > was equal temperament, as some musicians assume.) > > The fact that 3/2, 2, and 5/4 are incommensurable makes me genuinely > sad, > but later this week I hope to share a fun experiment the Girl Scouts and > I > did with pitch perception. It doesn’t rely on having perfectly tuned > fifths, thirds, and octaves simultaneously, so it shouldn’t cause the > same > existential angst that temperament does. > > *This sentence and other sentences in this paragraph were edited after > publication to correct missing square root signs. > > Tags: algebra, arithmetic, fundamental theorem of arithmetic, > mathematics > and music, music theory, prime numbers, pythagoras, quadratic integer > rings > > The views expressed are those of the author and are not necessarily > those > of Scientific American. > > Evelyn Lamb is a postdoc at the University of Utah. She writes about > mathematics and other cool stuff. Follow on Twitter @evelynjlamb. > > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com <javascript:;> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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On Tue, 2 Dec 2014, Cris Moore wrote:
care to recommend a particular piece? - cris
On Dec 2, 2014, at 10:58 PM, Scott Huddleston <c.scott.huddleston@gmail.com> wrote:
http://en.wikipedia.org/wiki/Harry_Partch's_43-tone_scale (43, apparently, not 44). Only heard on Internet recordings, not live.
I'm fond of Barstow, a cantata (?) based on eight hitchhiker graffiti collected in 1941 from under a bridge in Barstow. Here's a performance on the original instruments Partch built to play in his peculiar scale: https://www.youtube.com/watch?v=m-Z511Ataqs -- Tom Duff. Ever feel like you're living an episode of "Get Smart"?
I believe that you've heard n=19 Hilarie
Date: Tue, 2 Dec 2014 21:29:27 -0800 From: Scott Huddleston <c.scott.huddleston@gmail.com> Subject: Re: [math-fun] The Saddest Thing about the Integers
I've heard music with n intervals per octave for n in {5, 12, 44}.
On Tue, Dec 2, 2014 at 9:04 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
It's an interesting exercise to look for equal divisions of the octave into n intervals that provide bettter approximations to (say) 3rd, 5th, 7th harmonics than n = 12 .
WFL
On 12/3/14, Cris Moore <moore@santafe.edu> wrote:
I recommend Easley Blackwood's music, where he divides octaves into n
equal
intervals for all 12 < n <= 24.
Of course, it made my girlfriend in college, a cello player with perfect pitch, run screaming out of the room.
Cris
On Dec 2, 2014, at 7:11 AM, James Propp <jamespropp@gmail.com> wrote:
That is some great popular mathematical writing, in my opinion!
It made me wonder: Is Z[2^(1/12 <x-apple-data-detectors://0>)] embeddable in a finite-degree extension of Z with unique factorization? (Not that this would really affect anything in music.)
Jim
On Tuesday, December 2, 2014, Henry Baker <hbaker1@pipeline.com> wrote:
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning.
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
The Saddest Thing I Know about the Integers
By Evelyn Lamb | November 30, 2014
The integers are a unique factorization domain, so we can’t tune
pianos.
That is the saddest thing I know about the integers.
I talked to a Girl Scout troop about math earlier this month, and one of our topics was the intersection of math and music. I chose to focus on the way we perceive ratios of sound wave frequencies as intervals. We interpret frequencies that have the ratio 2:1 as octaves. (Larger frequencies sound higher.) We interpret frequencies that have the ratio 3:2 as perfect fifths. And sadly, I had to break it to the girls that these two facts mean that no piano is in tune. In other words, you can tuna fish, but you can’t tune a piano.
When we tune an instrument, we would like for all our octaves and fifths to be perfect. One way to tune an instrument would be to start with a pitch and start working out the fifths above and below it it. We start with some frequency that we call C. Then 3/2 times that frequency is G, 9/4 times that frequency is D (an octave and a step above our original C), and so on. If you learned about the “circle of fifths” at some point in your musical life, then you know that if we keep going up by fifths, we’ll eventually land back on something we’d like to call C. It takes a total of 12 steps, and so if we keep all our fifths perfect, the frequency of the C we get at the end is 312/212, or 531441/4096, times the frequency of the C we had at the beginning. You might notice that 531441/4096 is not an integer, much less a power of 2, so our ears would not perceive the C at the end as being in tune with the C at the beginning. (531441/4096 is about 130, which is 2 more than a power of 2, so we would hear the C at the top as being sharp.) And it’s not a problem with the assumption that it takes 12 fifths to get from C to shining C. We can never get perfect octaves from a stack of fifths because no power of 3/2 will ever give us a power of 2.
Imperfect octaves are pretty unacceptable to any listener, and as a string player, I’m pretty into perfect fifths. So it’s disappointing enough that I can’t have them simultaneously. But the story gets even more complicated when we add thirds. Even if we could resolve the pesky fifths/octave problem, we would be stuck with some pretty strange sounding chords. When we hear frequencies in the ratio 5:4, we hear a perfectly tuned major third (the interval between C and E). But if we go around the circle of 5ths making uncompromising perfect fifths, we get 34/24=81/16. If we divide by 2 a few times to move the E back down to the same octave as the C, we end up with an 81:64 ratio, which is a bit bigger than 5:4 (or 80:64), meaning that the major third from C to E sounds too wide. So fifths are also incompatible with major thirds! Once again, we can never get a perfectly tuned major third from a stack of fifths, or a perfect fifth from a stack of major thirds, because no power of 5/4 equals a power of 3/2.
Blame unique factorization. One property of the integers that we take for granted is that we can factor any integer other than -1, 0, or 1 into its prime factors, and that the factorization will be unique. (We call this the fundamental theorem of arithmetic.) Hence, we can call the integers a unique factorization domain. (If you’re a real stickler, you might be worried about negative integers. The factorization is unique up to signs of numbers, and that’s good enough to be a unique factorization domain. If that still bothers you, just ignore the integers smaller than 2.) As a thought experiment, I decided to see if we could fix the problem by expanding from the integers to another set of numbers like the integers in that they can also be multiplied or added together.
One such set of numbers is called the Gaussian integers, and it consists of complex numbers of the form a+bi, where a and b are both integers and i2=-1. In the Gaussian integers, 2 is no longer a prime number because it can be factored into (1+i)×(1-i), which happen to be primes. Neither is 5, which can be written (1+2i)×(1-2i). But 3 is still a prime in the Gaussian integers (this isn’t obvious, but it’s true). Thus 2 and 3 still share no prime factors over the Gaussian integers, so we can’t resolve our octaves/fifths problem there. (Not that I even know what it would mean to divide a frequency by a Gaussian integer. Like I said, this is a thought experiment.) Likewise, 5 and 2 share no Gaussian prime factors, nor do 5 and 3. So even if it made sense to divide a frequency by a complex number, it wouldn’t help.
An even stranger set of numbers is the set of complex numbers of the form a+v5bi, or Z[v-5]. It might seem like this is no different from the Gaussian integers, but it is. It’s not a unique factorization domain. For example, the number 6 can be factored into either 2×3 or (1+v5i)×(1-v5i).* It’s not obvious, but 2 and 3 can’t be factored further; they are irreducible, as are (1+v5i) and (1-v5i). So 6 has two distinct factorizations. Will this help us? Well, we could end up with some powers of 6 in our frequency ratios if we combine fifths and octaves. But let’s say we could divide that power of 6 by 1+v5i and 1-v5i. Where would we be? We’d have reduced our power of 6 by 1, but we’re no closer to getting a 3 to change into a 2. Bummer! But at least we got to play with some quadratic integers, right?
You may have noticed that piano music doesn’t always sound out of tune, so there must be some resolution to the prime number predicament. Compromise, my friend. Currently, most instruments use equal temperament, which makes all the fifths slightly narrower than perfect so the octaves will be in tune. Each half step has the same frequency ratio as any other half step, and that ratio is 21/12:1. We’ve lost the pure rational ratios that made Pythagorean intervals sound so sweet, but we’ve gained a lot. The difference between a Pythagorean fifth and an equal temperament fifth is not enough to bother any but the fussiest listeners, but it is detectable to some. Before equal temperament became the law of the land, at least for keyboard instruments and other instruments where the player can’t make minute adjustments to pitch, there were several other temperament compromises in use.
One solution is to tune an instrument so that the octaves, fifths, and/or thirds are perfect or very close for the important chords from some keys (generally “easy” keys like C, G, and D) but terrible for some other keys. Those systems (often meantone temperaments) ended up with “wolf fifths” that were much narrower than perfect fifths. An instrument with a wolf fifth couldn’t really play in certain keys. Then along came well temperament, which was not one system but any of many irregular temperaments that made the keys sound different but didn’t leave any key howling at the moon, so to speak. The “well-tempered” in Bach’s Well-Tempered Clavier doesn’t refer to the instrument’s beautiful tone (or the instrumentalist’s equanimity) but to the fact that the set of pieces was composed for a clavier with a temperament that allowed the instrument to play in every key. (The Well-Tempered Clavier is a set of 24 preludes and fugues, one in each major and minor key. Scholars don’t kno w what exactly the clavier’s temperament was, but it is unlikely that it was equal temperament, as some musicians assume.)
The fact that 3/2, 2, and 5/4 are incommensurable makes me genuinely sad, but later this week I hope to share a fun experiment the Girl Scouts and I did with pitch perception. It doesn’t rely on having perfectly tuned fifths, thirds, and octaves simultaneously, so it shouldn’t cause the same existential angst that temperament does.
*This sentence and other sentences in this paragraph were edited after publication to correct missing square root signs.
Tags: algebra, arithmetic, fundamental theorem of arithmetic, mathematics and music, music theory, prime numbers, pythagoras, quadratic integer rings
The views expressed are those of the author and are not necessarily those of Scientific American.
Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.
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Suddenly I realize that I have no idea why powers of 2^(1/12) provide good approximations to p/q for many low p and q. k 2^(k/12) ------------- 1 1.0594+ 2 1.1224+ 3 1.1892+ ~ 6/5 (minor third) 4 1.2599+ ~ 5/4 (major third) 5 1.3348+ ~ 4/3 (major fourth) 6 1.4142+ 7 1.4983+ ~ 3/2 (major fifth) 8 1.5874+ 9 1.6817+ 10 1.7817+ 11 1.8877+ 12 2 Okay, so maybe the approximations aren't *that* good. But they're almost that good. All four intervals are within less than 1% of just intonation. How come? Is there some basic number-theoretic reason? I suppose the Pythagorean comma = (3/2)^12 / 2^7 ~ 1.01+, the discrepancy between 12 perfect fifths and 7 octaves, is somehow relevant. But I don't see how this works. --Dan
On Dec 2, 2014, at 9:04 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
It's an interesting exercise to look for equal divisions of the octave into n intervals that provide better approximations to (say) 3rd, 5th, 7th harmonics than n = 12 .
WFL
Am Tue, 2 Dec 2014 23:00:05 -0800 schrieb Dan Asimov <dasimov@earthlink.net>: This topic was the main reason I asked for subscription to this mailing list many (10-12?) years ago! There may be something about this in the archives. Considering the continued fraction for log2/log(3/2) suggests 5, 12, 41, 53, ... equal subdivisions of the octave, see http://doctroidalresearch.wordpress.com/pages/recreational-mathematics/conti... But considering 3/2, 4/3, 5/4,... simultaneously is certainly more involved. (fifths or fourths come perfect/augmented/diminished, while 2nds, 3rds, 6ths and 7ths come major/minor) Dirk.
Suddenly I realize that I have no idea why powers of 2^(1/12) provide good approximations to p/q for many low p and q.
k 2^(k/12) ------------- 1 1.0594+ 2 1.1224+ 3 1.1892+ ~ 6/5 (minor third) 4 1.2599+ ~ 5/4 (major third) 5 1.3348+ ~ 4/3 (major fourth) 6 1.4142+ 7 1.4983+ ~ 3/2 (major fifth) 8 1.5874+ 9 1.6817+ 10 1.7817+ 11 1.8877+ 12 2
Okay, so maybe the approximations aren't *that* good.
But they're almost that good. All four intervals are within less than 1% of just intonation.
How come? Is there some basic number-theoretic reason?
I suppose the Pythagorean comma = (3/2)^12 / 2^7 ~ 1.01+, the discrepancy between 12 perfect fifths and 7 octaves, is somehow relevant. But I don't see how this works.
--Dan
On Dec 2, 2014, at 9:04 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
It's an interesting exercise to look for equal divisions of the octave into n intervals that provide better approximations to (say) 3rd, 5th, 7th harmonics than n = 12 .
WFL
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Since all fields are unique factorization domains, this is clearly wrong: 3/2 = 4 mod 5, which is obviously a power of two. On Tue, Dec 2, 2014 at 5:31 AM, Henry Baker <hbaker1@pipeline.com> wrote:
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning.
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
The Saddest Thing I Know about the Integers
By Evelyn Lamb | November 30, 2014
The integers are a unique factorization domain, so we can't tune pianos. That is the saddest thing I know about the integers.
I talked to a Girl Scout troop about math earlier this month, and one of our topics was the intersection of math and music. I chose to focus on the way we perceive ratios of sound wave frequencies as intervals. We interpret frequencies that have the ratio 2:1 as octaves. (Larger frequencies sound higher.) We interpret frequencies that have the ratio 3:2 as perfect fifths. And sadly, I had to break it to the girls that these two facts mean that no piano is in tune. In other words, you can tuna fish, but you can't tune a piano.
When we tune an instrument, we would like for all our octaves and fifths to be perfect. One way to tune an instrument would be to start with a pitch and start working out the fifths above and below it it. We start with some frequency that we call C. Then 3/2 times that frequency is G, 9/4 times that frequency is D (an octave and a step above our original C), and so on. If you learned about the "circle of fifths" at some point in your musical life, then you know that if we keep going up by fifths, we'll eventually land back on something we'd like to call C. It takes a total of 12 steps, and so if we keep all our fifths perfect, the frequency of the C we get at the end is 312/212, or 531441/4096, times the frequency of the C we had at the beginning. You might notice that 531441/4096 is not an integer, much less a power of 2, so our ears would not perceive the C at the end as being in tune with the C at the beginning. (531441/4096 is about 130, which is 2 more than a power of 2, so we would hear the C at the top as being sharp.) And it's not a problem with the assumption that it takes 12 fifths to get from C to shining C. We can never get perfect octaves from a stack of fifths because no power of 3/2 will ever give us a power of 2.
Imperfect octaves are pretty unacceptable to any listener, and as a string player, I'm pretty into perfect fifths. So it's disappointing enough that I can't have them simultaneously. But the story gets even more complicated when we add thirds. Even if we could resolve the pesky fifths/octave problem, we would be stuck with some pretty strange sounding chords. When we hear frequencies in the ratio 5:4, we hear a perfectly tuned major third (the interval between C and E). But if we go around the circle of 5ths making uncompromising perfect fifths, we get 34/24=81/16. If we divide by 2 a few times to move the E back down to the same octave as the C, we end up with an 81:64 ratio, which is a bit bigger than 5:4 (or 80:64), meaning that the major third from C to E sounds too wide. So fifths are also incompatible with major thirds! Once again, we can never get a perfectly tuned major third from a stack of fifths, or a perfect fifth from a stack of major thirds, because no power of 5/4 equals a power of 3/2.
Blame unique factorization. One property of the integers that we take for granted is that we can factor any integer other than -1, 0, or 1 into its prime factors, and that the factorization will be unique. (We call this the fundamental theorem of arithmetic.) Hence, we can call the integers a unique factorization domain. (If you're a real stickler, you might be worried about negative integers. The factorization is unique up to signs of numbers, and that's good enough to be a unique factorization domain. If that still bothers you, just ignore the integers smaller than 2.) As a thought experiment, I decided to see if we could fix the problem by expanding from the integers to another set of numbers like the integers in that they can also be multiplied or added together.
One such set of numbers is called the Gaussian integers, and it consists of complex numbers of the form a+bi, where a and b are both integers and i2=-1. In the Gaussian integers, 2 is no longer a prime number because it can be factored into (1+i)×(1-i), which happen to be primes. Neither is 5, which can be written (1+2i)×(1-2i). But 3 is still a prime in the Gaussian integers (this isn't obvious, but it's true). Thus 2 and 3 still share no prime factors over the Gaussian integers, so we can't resolve our octaves/fifths problem there. (Not that I even know what it would mean to divide a frequency by a Gaussian integer. Like I said, this is a thought experiment.) Likewise, 5 and 2 share no Gaussian prime factors, nor do 5 and 3. So even if it made sense to divide a frequency by a complex number, it wouldn't help.
An even stranger set of numbers is the set of complex numbers of the form a+v5bi, or Z[v-5]. It might seem like this is no different from the Gaussian integers, but it is. It's not a unique factorization domain. For example, the number 6 can be factored into either 2×3 or (1+v5i)×(1-v5i).* It's not obvious, but 2 and 3 can't be factored further; they are irreducible, as are (1+v5i) and (1-v5i). So 6 has two distinct factorizations. Will this help us? Well, we could end up with some powers of 6 in our frequency ratios if we combine fifths and octaves. But let's say we could divide that power of 6 by 1+v5i and 1-v5i. Where would we be? We'd have reduced our power of 6 by 1, but we're no closer to getting a 3 to change into a 2. Bummer! But at least we got to play with some quadratic integers, right?
You may have noticed that piano music doesn't always sound out of tune, so there must be some resolution to the prime number predicament. Compromise, my friend. Currently, most instruments use equal temperament, which makes all the fifths slightly narrower than perfect so the octaves will be in tune. Each half step has the same frequency ratio as any other half step, and that ratio is 21/12:1. We've lost the pure rational ratios that made Pythagorean intervals sound so sweet, but we've gained a lot. The difference between a Pythagorean fifth and an equal temperament fifth is not enough to bother any but the fussiest listeners, but it is detectable to some. Before equal temperament became the law of the land, at least for keyboard instruments and other instruments where the player can't make minute adjustments to pitch, there were several other temperament compromises in use.
One solution is to tune an instrument so that the octaves, fifths, and/or thirds are perfect or very close for the important chords from some keys (generally "easy" keys like C, G, and D) but terrible for some other keys. Those systems (often meantone temperaments) ended up with "wolf fifths" that were much narrower than perfect fifths. An instrument with a wolf fifth couldn't really play in certain keys. Then along came well temperament, which was not one system but any of many irregular temperaments that made the keys sound different but didn't leave any key howling at the moon, so to speak. The "well-tempered" in Bach's Well-Tempered Clavier doesn't refer to the instrument's beautiful tone (or the instrumentalist's equanimity) but to the fact that the set of pieces was composed for a clavier with a temperament that allowed the instrument to play in every key. (The Well-Tempered Clavier is a set of 24 preludes and fugues, one in each major and minor key. Scholars don't kno w what exactly the clavier's temperament was, but it is unlikely that it was equal temperament, as some musicians assume.)
The fact that 3/2, 2, and 5/4 are incommensurable makes me genuinely sad, but later this week I hope to share a fun experiment the Girl Scouts and I did with pitch perception. It doesn't rely on having perfectly tuned fifths, thirds, and octaves simultaneously, so it shouldn't cause the same existential angst that temperament does.
*This sentence and other sentences in this paragraph were edited after publication to correct missing square root signs.
Tags: algebra, arithmetic, fundamental theorem of arithmetic, mathematics and music, music theory, prime numbers, pythagoras, quadratic integer rings
The views expressed are those of the author and are not necessarily those of Scientific American.
Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
On Tue, Dec 2, 2014 at 9:01 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Fields? Doesn't 3 = 2 x 3/2 = 2/7 x 21/2, etc?
What would the primes be?
In a field, every nonzero element is a unit, so there aren't any primes. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
This book, "Geek Sublime" by Vikram Chandra, looks interesting: < http://www.amazon.com/Geek-Sublime-Vikram-Chandra/dp/0571310303/ >. Has anyone read it? --Dan
I don't find this good popular math at all. The article never explains why a non-UFD would enable us to "tune pianos". The problem is that simple fractions like 3/2 are not exact powers of 2^(1/12). How would non-unique factorization fix that? --Dan
On Dec 2, 2014, at 5:31 AM, Henry Baker <hbaker1@pipeline.com> wrote:
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning.
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
The Saddest Thing I Know about the Integers
By Evelyn Lamb | November 30, 2014
The integers are a unique factorization domain, so we cant tune pianos. That is the saddest thing I know about the integers. . . .
If 3 could be written as 2^(p/q) for some positive integers p and q, then 3/2 raised to a suitable power n would be a power of 2, so we could construct an n-note scale in which fifths were mathematically perfect. But unique factorization (combined with the fact that 2 and 3 are prime) tells us that 3 can't be written as 2^(p/q). This reminds me of a woozy old musing of mine, namely, that just as there are extensions of Q in which rational primes split, Q might have "under-things" of some kind in which distinct rational primes merge. As far as I've ever been able to tell, this is utter nonsense. Or rather, it's the wrong kind of nonsense (the kind that doesn't lead to anything interesting) as opposed to the right kind of nonsense (which does). Rethinking my initial high regard for the article, I now think that the author has misidentified the true source of her angst. What she really longs for is Pythagoras' dream-world, where irrational numbers don't exist and in particular log_2 (3/2) is rational. Jim Propp On Tuesday, December 2, 2014, Dan Asimov <dasimov@earthlink.net> wrote:
I don't find this good popular math at all.
The article never explains why a non-UFD would enable us to "tune pianos".
The problem is that simple fractions like 3/2 are not exact powers of 2^( 1/12 <x-apple-data-detectors://5>).
How would non-unique factorization fix that?
--Dan
On Dec 2, 2014, at 5:31 AM, Henry Baker <hbaker1@pipeline.com <javascript:;>> wrote:
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning.
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
The Saddest Thing I Know about the Integers
By Evelyn Lamb | November 30, 2014
The integers are a unique factorization domain, so we can’t tune
pianos. That is the saddest thing I know about the integers. . . . _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
And this musing -- Given that math precludes perfect tuning, are there engineering work-arounds other than equal temperament and n-note scales? Perhaps vibrato, besides being an embellishment, also obscures the note so that it sounds better in conjunction with other notes? Taking that further, perhaps there is a sounding-good function that gives the best frequency perturbations for all of the notes sounding at a given moment. (Psychology experiment opportunity here -- do musicians already make subtle adjustments like this?. Might it depend on previous note and next note? Weighted by the musical "importance" of the current notes?) If so, electronic instruments could re-tune themselves based on what the current chord is, both its own and the notes other instruments or people are sounding. Real-time vocal pitch correction already exists, suggesting this is possible. Would the result sound more pleasant, or always slightly off? -- Mike ----- Original Message ----- From: "James Propp" <jamespropp@gmail.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Wednesday, December 03, 2014 9:42 AM Subject: Re: [math-fun] The Saddest Thing about the Integers
If 3 could be written as 2^(p/q) for some positive integers p and q, then 3/2 raised to a suitable power n would be a power of 2, so we could construct an n-note scale in which fifths were mathematically perfect.
But unique factorization (combined with the fact that 2 and 3 are prime) tells us that 3 can't be written as 2^(p/q).
This reminds me of a woozy old musing of mine, namely, that just as there are extensions of Q in which rational primes split, Q might have "under-things" of some kind in which distinct rational primes merge. As far as I've ever been able to tell, this is utter nonsense. Or rather, it's the wrong kind of nonsense (the kind that doesn't lead to anything interesting) as opposed to the right kind of nonsense (which does).
Rethinking my initial high regard for the article, I now think that the author has misidentified the true source of her angst. What she really longs for is Pythagoras' dream-world, where irrational numbers don't exist and in particular log_2 (3/2) is rational.
Jim Propp
On Tuesday, December 2, 2014, Dan Asimov <dasimov@earthlink.net> wrote:
I don't find this good popular math at all.
The article never explains why a non-UFD would enable us to "tune pianos".
The problem is that simple fractions like 3/2 are not exact powers of 2^( 1/12 <x-apple-data-detectors://5>).
How would non-unique factorization fix that?
--Dan
On Dec 2, 2014, at 5:31 AM, Henry Baker <hbaker1@pipeline.com <javascript:;>> wrote:
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning.
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
The Saddest Thing I Know about the Integers
By Evelyn Lamb | November 30, 2014
The integers are a unique factorization domain, so we can’t tune
pianos. That is the saddest thing I know about the integers. . . . _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Wed, Dec 3, 2014 at 11:37 AM, Michael Beeler <mikebeeler@verizon.net> wrote:
And this musing -- Given that math precludes perfect tuning, are there engineering work-arounds other than equal temperament and n-note scales? Perhaps vibrato, besides being an embellishment, also obscures the note so that it sounds better in conjunction with other notes? Taking that further, perhaps there is a sounding-good function that gives the best frequency perturbations for all of the notes sounding at a given moment. (Psychology experiment opportunity here -- do musicians already make subtle adjustments like this?. Might it depend on previous note and next note? Weighted by the musical "importance" of the current notes?)
If so, electronic instruments could re-tune themselves based on what the current chord is, both its own and the notes other instruments or people are sounding. Real-time vocal pitch correction already exists, suggesting this is possible. Would the result sound more pleasant, or always slightly off?
This piece dynamically adjusts the timbre of the instrument to make each chord sound consonant. http://sethares.engr.wisc.edu/mp3s/three_ears.html He's got lots of other "weird" music, including non-12-tet scales and other things: http://sethares.engr.wisc.edu/otherperson/all_mp3s.html I particularly enjoyed "Truth on a Bus", written in a 19-tet scale. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
I had the same idea -- a piano that's always in tune by flatting or sharping each individual note by whatever optimizes the harmony at a given moment. Of course, this would be a lot easier for an electronic piano than for a mechanical one. But I'd really like to see this for a mechanical one. --Dan On Wed, Dec 3, 2014 at 11:37 AM, Michael Beeler <mikebeeler@verizon.net> wrote:
Taking that further, perhaps there is a sounding-good function that gives the best frequency perturbations for all of the notes sounding at a given moment.
Violinists, trombone players, singers & other instrumentalists with adjustable pitches _do_ make these adjustments all the time to fit the music and especially the key. In particular, when I was taught violin in high school, I was taught to rarely use "open strings" (i.e., without fingering), because these notes could not be adjusted. Except for the lowest ("G") string, one can always use a finger on a lower string to equal the other open strings ("D", "A", "E"), and thus always have dynamically adjustable tuning. Even on supposedly fixed tuned instruments such as the trumpet, there are usually levers on both the 1st and the 3rd valves of concert trumpets which enable the fine tuning of most notes. At 02:30 PM 12/3/2014, Dan Asimov wrote:
I had the same idea -- a piano that's always in tune by flatting or sharping each individual note by whatever optimizes the harmony at a given moment.
Of course, this would be a lot easier for an electronic piano than for a mechanical one. But I'd really like to see this for a mechanical one.
--Dan
On Wed, Dec 3, 2014 at 11:37 AM, Michael Beeler <mikebeeler@verizon.net> wrote:
Taking that further, perhaps there is a sounding-good function that gives the best frequency perturbations for all of the notes sounding at a given moment.
On Dec 3, 2014, at 6:42 AM, James Propp <jamespropp@gmail.com> wrote:
If 3 could be written as 2^(p/q) for some positive integers p and q, then 3/2 raised to a suitable power n would be a power of 2, so we could construct an n-note scale in which fifths were mathematically perfect.
But unique factorization (combined with the fact that 2 and 3 are prime) tells us that 3 can't be written as 2^(p/q).
I don't think unique factorization is the problem here. (As Jim may agree.) Even if Q is embedded in *any* subfield K of C, it will still be the case that 2^L = 3^M in O(K) (the ring of algebraic integers of K) is *false* for all integers L,M >= 1 . So we can never have (3/2)^12 = 2^19, or anything similar, in any subfield of C. --Dan
This reminds me of a woozy old musing of mine, namely, that just as there are extensions of Q in which rational primes split, Q might have "under-things" of some kind in which distinct rational primes merge. As far as I've ever been able to tell, this is utter nonsense. Or rather, it's the wrong kind of nonsense (the kind that doesn't lead to anything interesting) as opposed to the right kind of nonsense (which does).
Rethinking my initial high regard for the article, I now think that the author has misidentified the true source of her angst. What she really longs for is Pythagoras' dream-world, where irrational numbers don't exist and in particular log_2 (3/2) is rational.
Jim Propp
On Tuesday, December 2, 2014, Dan Asimov <dasimov@earthlink.net> wrote:
I don't find this good popular math at all.
The article never explains why a non-UFD would enable us to "tune pianos". . . .
(%i1) (3/2)^12=2^7; 531441 (%o1) ------ = 128 4096 (%i2) %*4096; (%o2) 531441 = 524288 (%i3) lhs(%)-rhs(%); (%o3) 7153 (%i4) factor(%); (%o4) 23 311 (%i5) mod(531441,23); (%o5) 3 (%i6) mod(524288,23); (%o6) 3 (%i7) So what about (mod 23) or (mod 311) ? Can something like this be made to work? At 05:27 PM 12/3/2014, Dan Asimov wrote:
That should be:
So we can never have (3/2)^12 = 2^7, or anything similar, in any subfield of C.
--Dan
On Dec 3, 2014, at 4:58 PM, Dan Asimov <dasimov@earthlink.net> wrote:
So we can never have (3/2)^12 = 2^19, or anything similar, in any subfield of C.
Many of the questions about "what would it sound like if ..." can be partially answered by listening to the material on this page: https://www.prismnet.com/~hmiller/music/warped-canon.html Start with Pythagorean to get your bearings, then wander into the world of microtonal scales. Hilarie
I doubt if most people could tell if two notes 7 octaves apart were quite in tune. Particularly on a piano, where the harmonics aren't quite integer multiples of the fundamental anyway. --ms On 2014-12-02 08:31, Henry Baker wrote:
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning.
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
The Saddest Thing I Know about the Integers
By Evelyn Lamb | November 30, 2014
The integers are a unique factorization domain, so we can’t tune pianos. That is the saddest thing I know about the integers.
I talked to a Girl Scout troop about math earlier this month, and one of our topics was the intersection of math and music. I chose to focus on the way we perceive ratios of sound wave frequencies as intervals. We interpret frequencies that have the ratio 2:1 as octaves. (Larger frequencies sound higher.) We interpret frequencies that have the ratio 3:2 as perfect fifths. And sadly, I had to break it to the girls that these two facts mean that no piano is in tune. In other words, you can tuna fish, but you can’t tune a piano.
When we tune an instrument, we would like for all our octaves and fifths to be perfect. One way to tune an instrument would be to start with a pitch and start working out the fifths above and below it it. We start with some frequency that we call C. Then 3/2 times that frequency is G, 9/4 times that frequency is D (an octave and a step above our original C), and so on. If you learned about the “circle of fifths” at some point in your musical life, then you know that if we keep going up by fifths, we’ll eventually land back on something we’d like to call C. It takes a total of 12 steps, and so if we keep all our fifths perfect, the frequency of the C we get at the end is 312/212, or 531441/4096, times the frequency of the C we had at the beginning. You might notice that 531441/4096 is not an integer, much less a power of 2, so our ears would not perceive the C at the end as being in tune with the C at the beginning. (531441/4096 is about 130, which is 2 more than a power of 2, so we would hear the C at the top as being sharp.) And it’s not a problem with the assumption that it takes 12 fifths to get from C to shining C. We can never get perfect octaves from a stack of fifths because no power of 3/2 will ever give us a power of 2.
Imperfect octaves are pretty unacceptable to any listener, and as a string player, I’m pretty into perfect fifths. So it’s disappointing enough that I can’t have them simultaneously. But the story gets even more complicated when we add thirds. Even if we could resolve the pesky fifths/octave problem, we would be stuck with some pretty strange sounding chords. When we hear frequencies in the ratio 5:4, we hear a perfectly tuned major third (the interval between C and E). But if we go around the circle of 5ths making uncompromising perfect fifths, we get 34/24=81/16. If we divide by 2 a few times to move the E back down to the same octave as the C, we end up with an 81:64 ratio, which is a bit bigger than 5:4 (or 80:64), meaning that the major third from C to E sounds too wide. So fifths are also incompatible with major thirds! Once again, we can never get a perfectly tuned major third from a stack of fifths, or a perfect fifth from a stack of major thirds, because no power of 5/4 equals a power of 3/2.
Blame unique factorization. One property of the integers that we take for granted is that we can factor any integer other than -1, 0, or 1 into its prime factors, and that the factorization will be unique. (We call this the fundamental theorem of arithmetic.) Hence, we can call the integers a unique factorization domain. (If you’re a real stickler, you might be worried about negative integers. The factorization is unique up to signs of numbers, and that’s good enough to be a unique factorization domain. If that still bothers you, just ignore the integers smaller than 2.) As a thought experiment, I decided to see if we could fix the problem by expanding from the integers to another set of numbers like the integers in that they can also be multiplied or added together.
One such set of numbers is called the Gaussian integers, and it consists of complex numbers of the form a+bi, where a and b are both integers and i2=-1. In the Gaussian integers, 2 is no longer a prime number because it can be factored into (1+i)×(1-i), which happen to be primes. Neither is 5, which can be written (1+2i)×(1-2i). But 3 is still a prime in the Gaussian integers (this isn’t obvious, but it’s true). Thus 2 and 3 still share no prime factors over the Gaussian integers, so we can’t resolve our octaves/fifths problem there. (Not that I even know what it would mean to divide a frequency by a Gaussian integer. Like I said, this is a thought experiment.) Likewise, 5 and 2 share no Gaussian prime factors, nor do 5 and 3. So even if it made sense to divide a frequency by a complex number, it wouldn’t help.
An even stranger set of numbers is the set of complex numbers of the form a+v5bi, or Z[v-5]. It might seem like this is no different from the Gaussian integers, but it is. It’s not a unique factorization domain. For example, the number 6 can be factored into either 2×3 or (1+v5i)×(1-v5i).* It’s not obvious, but 2 and 3 can’t be factored further; they are irreducible, as are (1+v5i) and (1-v5i). So 6 has two distinct factorizations. Will this help us? Well, we could end up with some powers of 6 in our frequency ratios if we combine fifths and octaves. But let’s say we could divide that power of 6 by 1+v5i and 1-v5i. Where would we be? We’d have reduced our power of 6 by 1, but we’re no closer to getting a 3 to change into a 2. Bummer! But at least we got to play with some quadratic integers, right?
You may have noticed that piano music doesn’t always sound out of tune, so there must be some resolution to the prime number predicament. Compromise, my friend. Currently, most instruments use equal temperament, which makes all the fifths slightly narrower than perfect so the octaves will be in tune. Each half step has the same frequency ratio as any other half step, and that ratio is 21/12:1. We’ve lost the pure rational ratios that made Pythagorean intervals sound so sweet, but we’ve gained a lot. The difference between a Pythagorean fifth and an equal temperament fifth is not enough to bother any but the fussiest listeners, but it is detectable to some. Before equal temperament became the law of the land, at least for keyboard instruments and other instruments where the player can’t make minute adjustments to pitch, there were several other temperament compromises in use.
One solution is to tune an instrument so that the octaves, fifths, and/or thirds are perfect or very close for the important chords from some keys (generally “easy” keys like C, G, and D) but terrible for some other keys. Those systems (often meantone temperaments) ended up with “wolf fifths” that were much narrower than perfect fifths. An instrument with a wolf fifth couldn’t really play in certain keys. Then along came well temperament, which was not one system but any of many irregular temperaments that made the keys sound different but didn’t leave any key howling at the moon, so to speak. The “well-tempered” in Bach’s Well-Tempered Clavier doesn’t refer to the instrument’s beautiful tone (or the instrumentalist’s equanimity) but to the fact that the set of pieces was composed for a clavier with a temperament that allowed the instrument to play in every key. (The Well-Tempered Clavier is a set of 24 preludes and fugues, one in each major and minor key. Scholars don’t kno w what exactly the clavier’s temperament was, but it is unlikely that it was equal temperament, as some musicians assume.)
The fact that 3/2, 2, and 5/4 are incommensurable makes me genuinely sad, but later this week I hope to share a fun experiment the Girl Scouts and I did with pitch perception. It doesn’t rely on having perfectly tuned fifths, thirds, and octaves simultaneously, so it shouldn’t cause the same existential angst that temperament does.
*This sentence and other sentences in this paragraph were edited after publication to correct missing square root signs.
Tags: algebra, arithmetic, fundamental theorem of arithmetic, mathematics and music, music theory, prime numbers, pythagoras, quadratic integer rings
The views expressed are those of the author and are not necessarily those of Scientific American.
Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.
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I'm not sure what you mean by "most people", but most good musicians can tell the difference. Recall that nonlinearities in your ears produce beat frequencies, which are quite easy to hear. At 02:10 PM 12/2/2014, Mike Speciner wrote:
I doubt if most people could tell if two notes 7 octaves apart were quite in tune. Particularly on a piano, where the harmonics aren't quite integer multiples of the fundamental anyway.
--ms
participants (14)
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Bernie Cosell -
Cris Moore -
Dan Asimov -
Dirk Lattermann -
Fred Lunnon -
Henry Baker -
Hilarie Orman -
James Propp -
Michael Beeler -
Mike Speciner -
Mike Stay -
Scott Huddleston -
Tom Duff -
Victor Miller