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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