Inspired by Henry Baker's musing about modem tones: Each Touch Tone button on a telephone sends two simultaneous audio frequencies through the phone line, one associated with the row it's on, and one associated with the column it's on. There are four rows and four columns (though only the first three columns exist on most telephones) for a total of eight frequencies. The frequencies were chosen such that the sums or differences of any two of the eight frequencies are as far as possible from any of the eight frequencies. Similarly with whole-number multiples ("harmonics") of the eight frequencies or of their sums or differences. This is to prevent the phone company equipment from getting confused, since non-linearities in the circuits could result in such sum and difference frequencies appearing. This is probably why Touch Tones are so unmusical. Music relies on frequencies with small whole-number ratios, almost the exact opposite of the Touch Tone criteria. (Or did they only concern themselves with sums or differences of row and column frequencies? Two different row frequencies, or two different column frequencies, should never appear together.) Also, the frequencies should be between 400 and 3400 Hz, so they'll fit in a phone line's bandwidth. See https://en.wikipedia.org/wiki/Dual-tone_multi-frequency_signaling for more information, including the eight frequencies. Can you do better on selecting eight frequencies with those constraints? What about other numbers of frequencies? Is there a good algorithm for selecting them?
Note(!) that the ratios of DTMF tones form a "persymmetric" matrix (symmetric about the minor/anti diagonal). The matrix below shows the number of semitones (times 10) between the row tone and the column DTMF tone: [ 95 78 61 43 ] [ ] [ 113 95 78 61 ] [ ] [ 130 113 95 78 ] [ ] [ 147 130 113 95 ] The only real musical chords are the entries "130" (touch tones "3" and "B") in this table above. This chord is an octave + minor second. Curiously, the entries "61" (touch tones "7" and "0") are quite close to the "tritone" dreaded by classical composers. Leonard Bernstein famously (in musical circles, at least) included a musical joke in "West Side Story" with the first two notes of the song "Maria", which form a *tritone* (the two "Ma ri" notes of the "Ma ri a"). Re the "musicality" of DTMF: Back in the "Captain Crunch" days of hacking ATT lond distance, there was a (probably apocryphal) story meant to illustrate the difference between MIT and Harvard students. MIT students hacking LD built "blue boxes" which generated all 16 DTMF tones (in addition to other tones). Blue boxes could be used to dial free long distance calls. Harvard students supposed gathered together a group of their musical friends, who then *played the DTMF tones on their instruments*. As you can see from the matrix above, this requires some skill from the musicians, as these notes are all seriously out of tune. BTW, the Bell 202 frequencies are almost exactly halfway between a "minor seventh" and a "major seventh". So Keith is absolutely right about DTMF chords being unmusical. At 04:26 PM 1/16/2018, Keith F. Lynch wrote:
Inspired by Henry Baker's musing about modem tones:
Each Touch Tone button on a telephone sends two simultaneous audio frequencies through the phone line, one associated with the row it's on, and one associated with the column it's on. There are four rows and four columns (though only the first three columns exist on most telephones) for a total of eight frequencies.
The frequencies were chosen such that the sums or differences of any two of the eight frequencies are as far as possible from any of the eight frequencies. Similarly with whole-number multiples ("harmonics") of the eight frequencies or of their sums or differences. This is to prevent the phone company equipment from getting confused, since non-linearities in the circuits could result in such sum and difference frequencies appearing.
This is probably why Touch Tones are so unmusical. Music relies on frequencies with small whole-number ratios, almost the exact opposite of the Touch Tone criteria.
That is a famous tritone (in the equal-tempered scale, the square root of 2) but I don’t think it’s a joke. Another classic use is in the Tristan chord. I was surprised to learn from a friend that in the middle ages, the major third was considered very dissonant. We consider it quite consonant, and associate it with the ratio 5:4. In the 12-tone scale this is about 1% off from the cube root of 2. Cris
On Jan 17, 2018, at 9:36 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Leonard Bernstein famously (in musical circles, at least) included a musical joke in "West Side Story" with the first two notes of the song "Maria", which form a *tritone* (the two "Ma ri" notes of the "Ma ri a").
Was the major third dissonant for them because they tuned their instruments differently? Or was it just a matter of perception? On Wed, Jan 17, 2018 at 9:03 AM, Cris Moore <moore@santafe.edu> wrote:
That is a famous tritone (in the equal-tempered scale, the square root of 2) but I don’t think it’s a joke. Another classic use is in the Tristan chord.
I was surprised to learn from a friend that in the middle ages, the major third was considered very dissonant. We consider it quite consonant, and associate it with the ratio 5:4. In the 12-tone scale this is about 1% off from the cube root of 2.
Cris
On Jan 17, 2018, at 9:36 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Leonard Bernstein famously (in musical circles, at least) included a musical joke in "West Side Story" with the first two notes of the song "Maria", which form a *tritone* (the two "Ma ri" notes of the "Ma ri a").
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I don’t know the answer to this question, but I note that the 5th and 4th are more like 0.1% off, while the major 3rd is 1% off. Also, quoth Wikipedia: "The older concept of a ditone (two 9:8 major seconds) made a dissonantly wide major third with the ratio 81:64." - Cris
On Jan 17, 2018, at 10:14 AM, Tomas Rokicki <rokicki@gmail.com> wrote:
Was the major third dissonant for them because they tuned their instruments differently? Or was it just a matter of perception?
On Wed, Jan 17, 2018 at 9:03 AM, Cris Moore <moore@santafe.edu> wrote:
That is a famous tritone (in the equal-tempered scale, the square root of 2) but I don’t think it’s a joke. Another classic use is in the Tristan chord.
I was surprised to learn from a friend that in the middle ages, the major third was considered very dissonant. We consider it quite consonant, and associate it with the ratio 5:4. In the 12-tone scale this is about 1% off from the cube root of 2.
Cris
On Jan 17, 2018, at 9:36 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Leonard Bernstein famously (in musical circles, at least) included a musical joke in "West Side Story" with the first two notes of the song "Maria", which form a *tritone* (the two "Ma ri" notes of the "Ma ri a").
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Cris Moore's comment that the major third was considered dissonant in medieval music was a surprise to me. I know, however, that well into the Baroque, the perfect fourth was considered a dissonance, and its avoidance was part of the standard rules of counterpoint of the time. Mathematically, the fourth is even more consonant than the major third, since it corresponds to a 4:3 ratio. On Wed, Jan 17, 2018 at 9:55 PM, Cris Moore <moore@santafe.edu> wrote:
I don’t know the answer to this question, but I note that the 5th and 4th are more like 0.1% off, while the major 3rd is 1% off. Also, quoth Wikipedia: "The older concept of a ditone (two 9:8 major seconds) made a dissonantly wide major third with the ratio 81:64."
- Cris
On Jan 17, 2018, at 10:14 AM, Tomas Rokicki <rokicki@gmail.com> wrote:
Was the major third dissonant for them because they tuned their instruments differently? Or was it just a matter of perception?
On Wed, Jan 17, 2018 at 9:03 AM, Cris Moore <moore@santafe.edu> wrote:
That is a famous tritone (in the equal-tempered scale, the square root
of
2) but I don’t think it’s a joke. Another classic use is in the Tristan chord.
I was surprised to learn from a friend that in the middle ages, the major third was considered very dissonant. We consider it quite consonant, and associate it with the ratio 5:4. In the 12-tone scale this is about 1% off from the cube root of 2.
Cris
On Jan 17, 2018, at 9:36 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Leonard Bernstein famously (in musical circles, at least) included a musical joke in "West Side Story" with the first two notes of the song "Maria", which form a *tritone* (the two "Ma ri" notes of the "Ma ri a").
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wow! that surprises me. after all, the fourth is the interval between the fifth and the octave above it. doesn’t the “amen” at the end of a hymn often go from the fourth to the tonic? I thought that was quite old in Western music. - Cris
On Jan 17, 2018, at 9:36 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Cris Moore's comment that the major third was considered dissonant in medieval music was a surprise to me. I know, however, that well into the Baroque, the perfect fourth was considered a dissonance, and its avoidance was part of the standard rules of counterpoint of the time. Mathematically, the fourth is even more consonant than the major third, since it corresponds to a 4:3 ratio.
On Wed, Jan 17, 2018 at 9:55 PM, Cris Moore <moore@santafe.edu> wrote:
I don’t know the answer to this question, but I note that the 5th and 4th are more like 0.1% off, while the major 3rd is 1% off. Also, quoth Wikipedia: "The older concept of a ditone (two 9:8 major seconds) made a dissonantly wide major third with the ratio 81:64."
- Cris
On Jan 17, 2018, at 10:14 AM, Tomas Rokicki <rokicki@gmail.com> wrote:
Was the major third dissonant for them because they tuned their instruments differently? Or was it just a matter of perception?
On Wed, Jan 17, 2018 at 9:03 AM, Cris Moore <moore@santafe.edu> wrote:
That is a famous tritone (in the equal-tempered scale, the square root
of
2) but I don’t think it’s a joke. Another classic use is in the Tristan chord.
I was surprised to learn from a friend that in the middle ages, the major third was considered very dissonant. We consider it quite consonant, and associate it with the ratio 5:4. In the 12-tone scale this is about 1% off from the cube root of 2.
Cris
On Jan 17, 2018, at 9:36 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Leonard Bernstein famously (in musical circles, at least) included a musical joke in "West Side Story" with the first two notes of the song "Maria", which form a *tritone* (the two "Ma ri" notes of the "Ma ri a").
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Cristopher Moore Professor, Santa Fe Institute Why should we be deported? This is very, very hard for a family. What will our fellow citizens think if honest subjects are faced with such a decree — not to mention the great material losses it would incur. I would like to become a Bavarian citizen again. Your most humble and obedient, Friedrich Trump (1905)
It was commonly used, then it was considered dissonant, then it wasn't. It is still dissonant today "in two-tone textures or if the interval occurs above the bass in a chord with three or more notes." The involvement this statement with Common Lisp is left as an exercise to the reader. Hilarie
wow! that surprises me. after all, the fourth is the interval between the fifth and the octave above it. doesn’t the “amen” at the end of a hymn often go from the fourth to the tonic? I thought that was quite old in Western music.
- Cris
On Jan 17, 2018, at 9:36 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Cris Moore's comment that the major third was considered dissonant in medieval music was a surprise to me. I know, however, that well into the Baroque, the perfect fourth was considered a dissonance, and its avoidance was part of the standard rules of counterpoint of the time. Mathematically, the fourth is even more consonant than the major third, since it corresponds to a 4:3 ratio.
On Wed, Jan 17, 2018 at 9:55 PM, Cris Moore <moore@santafe.edu> wrote:
I don’t know the answer to this question, but I note that the 5th and 4th are more like 0.1% off, while the major 3rd is 1% off. Also, quoth Wikipedia: "The older concept of a ditone (two 9:8 major seconds) made a dissonantly wide major third with the ratio 81:64."
- Cris
On Jan 17, 2018, at 10:14 AM, Tomas Rokicki <rokicki@gmail.com> wrote:
Was the major third dissonant for them because they tuned their instruments differently? Or was it just a matter of perception?
On Wed, Jan 17, 2018 at 9:03 AM, Cris Moore <moore@santafe.edu> wrote:
That is a famous tritone (in the equal-tempered scale, the square root
of
2) but I don’t think it’s a joke. Another classic use is in the Tristan chord.
I was surprised to learn from a friend that in the middle ages, the major third was considered very dissonant. We consider it quite consonant, and associate it with the ratio 5:4. In the 12-tone scale this is about 1% off from the cube root of 2.
Cris
On Jan 17, 2018, at 9:36 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Leonard Bernstein famously (in musical circles, at least) included a musical joke in "West Side Story" with the first two notes of the song "Maria", which form a *tritone* (the two "Ma ri" notes of the "Ma ri a").
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Cristopher Moore Professor, Santa Fe Institute
Why should we be deported? This is very, very hard for a family. What will our fellow citizens think if honest subjects are faced with such a decree — not to mention the great material losses it would incur. I would like to become a Bavarian citizen again. Your most humble and obedient, Friedrich Trump (1905)
How timbre relates to dissonance: http://sethares.engr.wisc.edu/consemi.html On Wed, Jan 17, 2018 at 11:53 PM, Hilarie Orman <ho@alum.mit.edu> wrote:
It was commonly used, then it was considered dissonant, then it wasn't. It is still dissonant today "in two-tone textures or if the interval occurs above the bass in a chord with three or more notes." The involvement this statement with Common Lisp is left as an exercise to the reader.
Hilarie
wow! that surprises me. after all, the fourth is the interval between the fifth and the octave above it. doesn’t the “amen” at the end of a hymn often go from the fourth to the tonic? I thought that was quite old in Western music.
- Cris
On Jan 17, 2018, at 9:36 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Cris Moore's comment that the major third was considered dissonant in medieval music was a surprise to me. I know, however, that well into the Baroque, the perfect fourth was considered a dissonance, and its avoidance was part of the standard rules of counterpoint of the time. Mathematically, the fourth is even more consonant than the major third, since it corresponds to a 4:3 ratio.
On Wed, Jan 17, 2018 at 9:55 PM, Cris Moore <moore@santafe.edu> wrote:
I don’t know the answer to this question, but I note that the 5th and 4th are more like 0.1% off, while the major 3rd is 1% off. Also, quoth Wikipedia: "The older concept of a ditone (two 9:8 major seconds) made a dissonantly wide major third with the ratio 81:64."
- Cris
On Jan 17, 2018, at 10:14 AM, Tomas Rokicki <rokicki@gmail.com> wrote:
Was the major third dissonant for them because they tuned their instruments differently? Or was it just a matter of perception?
On Wed, Jan 17, 2018 at 9:03 AM, Cris Moore <moore@santafe.edu> wrote:
That is a famous tritone (in the equal-tempered scale, the square root
of
2) but I don’t think it’s a joke. Another classic use is in the Tristan chord.
I was surprised to learn from a friend that in the middle ages, the major third was considered very dissonant. We consider it quite consonant, and associate it with the ratio 5:4. In the 12-tone scale this is about 1% off from the cube root of 2.
Cris
> On Jan 17, 2018, at 9:36 AM, Henry Baker <hbaker1@pipeline.com> wrote: > > Leonard Bernstein famously (in musical circles, at least) > included a musical joke in "West Side Story" with the first > two notes of the song "Maria", which form a *tritone* > (the two "Ma ri" notes of the "Ma ri a").
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Cristopher Moore Professor, Santa Fe Institute
Why should we be deported? This is very, very hard for a family. What will our fellow citizens think if honest subjects are faced with such a decree — not to mention the great material losses it would incur. I would like to become a Bavarian citizen again. Your most humble and obedient, Friedrich Trump (1905)
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Perhaps "joke" should have been "inside joke". Bernstein, like any other musical innovator, took a classical rule of musical composition, and not only violated it -- he obliterated it -- he made this violation THE key element of the Maria song, as the tritone sequence gets repeated over and over again. (Kind of like quantum physics: you not only violate classical algebraic commutativity, you base your entire physics on this novelty.) Once our generation grew up hearing Bernstein's music, the tritone sequence no longer hurts our ears, the way such a tritone would have hurt the ears of someone who grew up hundreds of years ago. At 09:03 AM 1/17/2018, Cris Moore wrote:
That is a famous tritone (in the equal-tempered scale, the square root of 2) but I don't think it's a joke.
On Jan 17, 2018, at 9:36 AM, Henry Baker <hbaker1@pipeline.com> wrote: Leonard Bernstein famously (in musical circles, at least) included a musical joke in "West Side Story" with the first two notes of the song "Maria", which form a *tritone* (the two "Ma ri" notes of the "Ma ri a").
On Wed, Jan 17, 2018 at 1:11 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps "joke" should have been "inside joke".
Saying this is an "inside joke" is sort of like saying that Beethoven repeats the same sequence of repeating a note three times, followed by a descending third as a musical joke in the first movement of his fifth symphony. The tritone is a unifying theme through the entire score of West Side story, from the opening notes of the overture to just before the end of the final musical underscoring where it finally resolves. It's the opening notes not only of Maria, but also of Cool and Something's Coming, and occurs in a bunch of other places in the score.
Bernstein, like any other musical innovator, took a classical rule of musical composition, and not only violated it -- he obliterated it -- he made this violation THE key element of the Maria song, as the tritone sequence gets repeated over and over again.
West Side Story is a brilliant piece of music, perhaps Bernstein's best. But his use of the tritone is hardly innovative, or something that obliterates a rule that existed at the time. I don't see how anyone who has ever listened to the music of Stravinsky and Schoenberg from a half century earlier can consider the use of the tritone, or the use of dissonance in general in West Side Story, to be an innovation. Andy
1. You may have missed the "inside" joke. At least it wasn't a direct attack on violas (which I play). 2. Not much of Schoenberg had shown up in Broadway musicals when West Side Story was written. The ear hears what the ear is accustomed to, and the mind expects familiarity.
From: Andy Latto <andy.latto@pobox.com>
On Wed, Jan 17, 2018 at 1:11 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps "joke" should have been "inside joke".
Saying this is an "inside joke" is sort of like saying that Beethoven repeats the same sequence of repeating a note three times, followed by a descending third as a musical joke in the first movement of his fifth symphony.
The tritone is a unifying theme through the entire score of West Side story, from the opening notes of the overture to just before the end of the final musical underscoring where it finally resolves. It's the opening notes not only of Maria, but also of Cool and Something's Coming, and occurs in a bunch of other places in the score.
Bernstein, like any other musical innovator, took a classical rule of musical composition, and not only violated it -- he obliterated it -- he made this violation THE key element of the Maria song, as the tritone sequence gets repeated over and over again.
West Side Story is a brilliant piece of music, perhaps Bernstein's best. But his use of the tritone is hardly innovative, or something that obliterates a rule that existed at the time. I don't see how anyone who has ever listened to the music of Stravinsky and Schoenberg from a half century earlier can consider the use of the tritone, or the use of dissonance in general in West Side Story, to be an innovation.
Andy
participants (8)
-
Allan Wechsler -
Andy Latto -
Cris Moore -
Henry Baker -
Hilarie Orman -
Keith F. Lynch -
Mike Stay -
Tomas Rokicki