A Set of Twelve Possibilities
May 20, 2021 5:37 PM Subscribe
Holy motherforking shirtballs.
The fact that each one of these doesn't have just one name but several is like, wow.
posted by signal at 7:41 PM on May 20, 2021
The fact that each one of these doesn't have just one name but several is like, wow.
posted by signal at 7:41 PM on May 20, 2021
The discussion reminded me a bit of Gödel numbering, where a symbol, proof, etc. can be represented by a (very long) number. Then pure functions (A xor B etc.) could transform one or more scales into a new scale. A musical Incompleteness Theorem might be in there somewhere.
posted by kurumi at 8:39 PM on May 20, 2021 [1 favorite]
posted by kurumi at 8:39 PM on May 20, 2021 [1 favorite]
The fact that each one of these doesn't have just one name but several is like, wow.
Seriously. I've been a musician on monophonic instruments (bass and winds) for years, but it wasn't until I started learning piano a couple years ago that I began to realize just how ambiguous and fluid a lot of this stuff actually is. Mind continues to be blown.
posted by Greg_Ace at 9:04 PM on May 20, 2021 [1 favorite]
Seriously. I've been a musician on monophonic instruments (bass and winds) for years, but it wasn't until I started learning piano a couple years ago that I began to realize just how ambiguous and fluid a lot of this stuff actually is. Mind continues to be blown.
posted by Greg_Ace at 9:04 PM on May 20, 2021 [1 favorite]
This is absolutely fascinating to me. I've programmed a Launchpad (grid) midi interface which use selectable scales, but I hadn't thought of cataloging them as a 12 digit binary number.
But I have programmed the quite complex PSQ-1684 rack extension in Reason to output 12 bits to a scale quantizer (that rather undocumentedly accepted them,) to make useful generative note changes.
posted by Catblack at 9:05 PM on May 20, 2021 [2 favorites]
But I have programmed the quite complex PSQ-1684 rack extension in Reason to output 12 bits to a scale quantizer (that rather undocumentedly accepted them,) to make useful generative note changes.
posted by Catblack at 9:05 PM on May 20, 2021 [2 favorites]
It seems strange that he mentions ragas without discussing that Indian music sometimes divides the scale into 22 notes rather than 12.
posted by zompist at 9:23 PM on May 20, 2021 [2 favorites]
posted by zompist at 9:23 PM on May 20, 2021 [2 favorites]
You donʻt use any of the scales mentioned to sing the blues.
posted by Droll Lord at 9:46 PM on May 20, 2021 [4 favorites]
posted by Droll Lord at 9:46 PM on May 20, 2021 [4 favorites]
On the subject of dividing the octave (and that there is an octave) into 12 and not more, the repeating subgroups have periods of 2, 2 and 3, so you're in one of 3 quarterings, one of four thirdings, which influences the harmonics or beating you hear 4 or so octaves up. This one is easy and flexible and stuck in white Western culture -- but you could build scales with 2,2,5 (=20) or 2,3,7 (=42) notes and 2^20 or 2^42 possibilities (minus one empty scale of silence we'll call the John Cage scale), respectively.
posted by k3ninho at 10:01 PM on May 20, 2021
posted by k3ninho at 10:01 PM on May 20, 2021
It seems strange that he mentions ragas without discussing that Indian music sometimes divides the scale into 22 notes rather than 12.
He does say upfront that 12-tone equal temperament is one of the assumptions/limitations of the project.
As it happens, I was just teaching one of my classes about where the chromatic scale comes from and why certain divisions of the octave (not only 12) work better than others! We've been studying the theory of rational approximations and I thought it would be a fun application.
The pure third harmonic (which is in some sense the most natural interval after the octave) is log2 3 ≈ 1.585 octaves. That's an irrational number, so you literally can't achieve it in an equal-tempered one-octave scale, but there are fractions that will get you close, and some denominators are better than others. For instance, 3/5 = 0.600, 4/7 ≈ 0.571, and 7/12 ≈ 0.583… and it's no accident that the pentatonic, diatonic, and chromatic scales have 5, 7, and 12 steps. Also note that 3/5 + 4/7 = 7/12 if you "add" fractions the naïve and wrong way! That's no accident, and better approximations can generally be obtained by "adding" existing approximations in this way. Thus 3/5 + 3/5 + 7/12 makes 13/22 ≈ 0.591, explaining why 22 is a relatively good number of tones to have. (But you know what's a really good number of tones? 53, as it turns out.)
There's waaaay more to it, and third harmonics are not the only important interval, but my credentials are in math rather than music theory, so I'll stop there.
posted by aws17576 at 10:04 PM on May 20, 2021 [4 favorites]
He does say upfront that 12-tone equal temperament is one of the assumptions/limitations of the project.
As it happens, I was just teaching one of my classes about where the chromatic scale comes from and why certain divisions of the octave (not only 12) work better than others! We've been studying the theory of rational approximations and I thought it would be a fun application.
The pure third harmonic (which is in some sense the most natural interval after the octave) is log2 3 ≈ 1.585 octaves. That's an irrational number, so you literally can't achieve it in an equal-tempered one-octave scale, but there are fractions that will get you close, and some denominators are better than others. For instance, 3/5 = 0.600, 4/7 ≈ 0.571, and 7/12 ≈ 0.583… and it's no accident that the pentatonic, diatonic, and chromatic scales have 5, 7, and 12 steps. Also note that 3/5 + 4/7 = 7/12 if you "add" fractions the naïve and wrong way! That's no accident, and better approximations can generally be obtained by "adding" existing approximations in this way. Thus 3/5 + 3/5 + 7/12 makes 13/22 ≈ 0.591, explaining why 22 is a relatively good number of tones to have. (But you know what's a really good number of tones? 53, as it turns out.)
There's waaaay more to it, and third harmonics are not the only important interval, but my credentials are in math rather than music theory, so I'll stop there.
posted by aws17576 at 10:04 PM on May 20, 2021 [4 favorites]
Ok hold the phone. Back in music school we would have a grand old time making up ridiculous scales, mostly mining musicology + paleontology (eg the Megalophyrigian scale, the Mesolochrian, the Pan-Lydian Microdorian, etc). Now you're telling me that shit was all real??
posted by range at 10:08 PM on May 20, 2021 [2 favorites]
posted by range at 10:08 PM on May 20, 2021 [2 favorites]
Musicology is an occult art. Nothing will convince me it’s not about spells invoking supernatural powers
posted by Fiasco da Gama at 12:36 AM on May 21, 2021 [1 favorite]
posted by Fiasco da Gama at 12:36 AM on May 21, 2021 [1 favorite]
They all go up to 11! And beyond!
posted by chavenet at 3:54 AM on May 21, 2021 [2 favorites]
posted by chavenet at 3:54 AM on May 21, 2021 [2 favorites]
As a complete newbie without any theoretical foundation, I spent about a week feverishly researching trying to find out why a piano keyboard looked the way it did. Why the whites. Why the blacks. Yes, yes, but why. 99.9% of the info you find online is either purely descriptive, or doesn't pass the sniff test in its speculations or what's essentially numerology.
I had to turn to a now-ancient 1930s book "Music: a science and an art" by John Redfield that finally had a good introduction on how the 12-tone equal temperament scale came to be, what its limitations are, and connected the math to the music.
posted by tigrrrlily at 7:01 AM on May 21, 2021 [4 favorites]
I had to turn to a now-ancient 1930s book "Music: a science and an art" by John Redfield that finally had a good introduction on how the 12-tone equal temperament scale came to be, what its limitations are, and connected the math to the music.
posted by tigrrrlily at 7:01 AM on May 21, 2021 [4 favorites]
That's an irrational number, so you literally can't achieve [the third harmonic] in an equal-tempered one-octave scale, but there are fractions that will get you close, and some denominators are better than others.I have been trying to parse this sentence for approximately twenty minutes. (Take your “eponysterical” and put it in your butt.) The interval between the octave, or second harmonic, and the third harmonic is a “perfect fifth.” From the third harmonic to the fourth harmonic, the next octave, is a “perfect fourth.” It’s in all the scales. How can you mean it’s unachievable? I must be misunderstanding you.
I perhaps think what you mean is that, in an equal-tempered scale, the perfect fifth is always out of tune a bit from the harmonic series produced by the octave, no matter how many subdivisions there are to the scale.
and it's no accident that the pentatonic, diatonic, and chromatic scales have 5, 7, and 12 steps.I think this is what threw me. A (Western) musician who talks about a “pentatonic” or “diatonic” scale is referring to, in the notation of the post, scales 661 or 2741. Those are not equal-temperament scales: some of the jumps are bigger than others. I think you were referring an equal-tempered-pentatonic in which all five of the intervals are the same width, and your point was that the “fifth” in such a scale would be slightly more out of tune from the third harmonic than the “fifth” in the equal-tempered-chromatic scale that I’m familiar with. I have never encountered an equal-tempered-pentatonic scale, and I would be interested to know if it occurs in any non-mathematical contexts.
posted by fantabulous timewaster at 9:15 AM on May 21, 2021
fantabulous timewaster, yes, I'm talking about equal-tempered scales. With just intonation, you can have true perfect fifths, but not in every key. (If you were to traverse the full circle of fifths, it would not "close" unless some of them are slightly flattened.) The existence of an n-tone equal-tempered scale that approximates the perfect fifth well is a pretty good proxy for the existence of an n-tone just intonation where the commas aren't too big -- in both cases, the important thing is that the ratio of a perfect fifth to an octave is close to the integer ratio 7:12.
Like I said, though, I know more about rational approximations than music theory, so if you think I said something wrong, you're probably right!
And you're definitely right about the pentatonic and diatonic scales. I remember that the division of 12 notes into 7 and 5 is ultimately related to 7 and 5 being denominators in the sequence of rational approximations to log2 3, but I don't know enough to justify that.
posted by aws17576 at 9:48 AM on May 21, 2021
Like I said, though, I know more about rational approximations than music theory, so if you think I said something wrong, you're probably right!
And you're definitely right about the pentatonic and diatonic scales. I remember that the division of 12 notes into 7 and 5 is ultimately related to 7 and 5 being denominators in the sequence of rational approximations to log2 3, but I don't know enough to justify that.
posted by aws17576 at 9:48 AM on May 21, 2021
this thread is literally and figuratively music to my ears, but I feel like there's two things I need to add. All moddy mods, please feel free to disagree:
1.
...(Take your “eponysterical” and put it in your butt.) ...
posted by fantabulous timewaster at 9:15 AM on May 21 [+] [!]
? Is there an "eponysterical to the eponysterical power" somewhere? Like, I know i^i is not only a real number, but right around 1/5; this is even more satisfying somehow.
2. Maybe "Metafilter: Take your “eponysterical” and put it in your butt." would have been better?
posted by adekllny at 10:08 AM on May 21, 2021
1.
...(Take your “eponysterical” and put it in your butt.) ...
posted by fantabulous timewaster at 9:15 AM on May 21 [+] [!]
? Is there an "eponysterical to the eponysterical power" somewhere? Like, I know i^i is not only a real number, but right around 1/5; this is even more satisfying somehow.
2. Maybe "Metafilter: Take your “eponysterical” and put it in your butt." would have been better?
posted by adekllny at 10:08 AM on May 21, 2021
That is cool. I have speculated idly about combinatorial possibilities for making scales but of course I never did anything about it.
Specifically, I was thinking: ok a major scale is a series of whole-steps and half-steps, where the half-steps come between the third and fourth scale degree, and the seventh and octave. Suppose you stick to this system of whole steps and two half steps, but arrange them differently. One scale that you could make would be the melodic minor scale, where there is a half-step between the second scale degree and the (now minor) third, and then the same as a major scale from the fourth up.
Ok, so many of the scales that you would make by changing where the two half steps are placed will be modes of the major scale, or modes of the melodic minor scale. But will that be all of them?
Can you have C C# D E F# G# A# and then C again? That cannot be a mode of a scale that has the half-steps distributed more widely. It would seem to fit the constraints - there are two half-steps, and the rest are whole steps. Does it have a name? The bebop whole tone scale? How many more are there?
posted by thelonius at 11:34 AM on May 21, 2021
Specifically, I was thinking: ok a major scale is a series of whole-steps and half-steps, where the half-steps come between the third and fourth scale degree, and the seventh and octave. Suppose you stick to this system of whole steps and two half steps, but arrange them differently. One scale that you could make would be the melodic minor scale, where there is a half-step between the second scale degree and the (now minor) third, and then the same as a major scale from the fourth up.
Ok, so many of the scales that you would make by changing where the two half steps are placed will be modes of the major scale, or modes of the melodic minor scale. But will that be all of them?
Can you have C C# D E F# G# A# and then C again? That cannot be a mode of a scale that has the half-steps distributed more widely. It would seem to fit the constraints - there are two half-steps, and the rest are whole steps. Does it have a name? The bebop whole tone scale? How many more are there?
posted by thelonius at 11:34 AM on May 21, 2021
Amazing Scale Finder calls it "Leading Whole-Tone Inverse". I like mine better.
posted by thelonius at 11:37 AM on May 21, 2021
posted by thelonius at 11:37 AM on May 21, 2021
This is fascinating:
Also, because (3/2)12 ≈ 27, I suddenly understand why my standard-sized piano has seven octaves. Somehow my education about the circle of fifths has never actually extended to sitting at a piano and touching them all in order.
posted by fantabulous timewaster at 12:08 PM on May 21, 2021 [1 favorite]
If you were to traverse the full circle of fifths, it would not "close" unless some of them are slightly flattened.It wasn’t immediately obvious to me at all, since you make both intervals “just” by multiplying the frequency by a rational fraction. But it follows straightaway from the arithmetic fact that no power of three is also a power of two.
Also, because (3/2)12 ≈ 27, I suddenly understand why my standard-sized piano has seven octaves. Somehow my education about the circle of fifths has never actually extended to sitting at a piano and touching them all in order.
posted by fantabulous timewaster at 12:08 PM on May 21, 2021 [1 favorite]
Yes it is a really interesting subject! There are several good recent books about the history of temperaments and tunings, and it came up here recently, in a thread about Middle Eastern culture and EDM; a poster's father there tuned pianos to historical temperaments, I think.
Somehow my education about the circle of fifths has never actually extended to sitting at a piano and touching them all in order.
There is, or used to be, a George Russell website, mostly about his "Lydian Chromatic Concept", which was an important theoretical text for jazz musicians of the 60's. What even is it? They had a great little explainer, saying: you start at C and ascend the circle of fifths, you get, what? C, G, D, A, E, B, F# (hence "Lydian", the #11), which spells a Cmaj9#11 chord, which then becomes the starting point for his theoretical work, which I do not really know anything about more than the above. But that was news to me! I think he may argue that this is the most consonant chord possible , because of being made from all stacked fifths.
posted by thelonius at 12:54 PM on May 21, 2021 [1 favorite]
Somehow my education about the circle of fifths has never actually extended to sitting at a piano and touching them all in order.
There is, or used to be, a George Russell website, mostly about his "Lydian Chromatic Concept", which was an important theoretical text for jazz musicians of the 60's. What even is it? They had a great little explainer, saying: you start at C and ascend the circle of fifths, you get, what? C, G, D, A, E, B, F# (hence "Lydian", the #11), which spells a Cmaj9#11 chord, which then becomes the starting point for his theoretical work, which I do not really know anything about more than the above. But that was news to me! I think he may argue that this is the most consonant chord possible , because of being made from all stacked fifths.
posted by thelonius at 12:54 PM on May 21, 2021 [1 favorite]
It bugs me a bit that the article treats scales almost entirely as mathematical patterns, and rarely touches the acoustics. The scale intervals were not chosen because they relate to sqrt(3). Here's a (non-musician's) attempt at balance.
We like the sound of vibrating things. E.g. 220 vibrations per second (220 Hz) sounds nice; we call that an A. Double that (440 Hz) and the two notes sound good together, so much that we call them "the same note an octave apart."
A 3:2 ratio sounds good too— e.g. 220 Hz (A) plus 330 Hz (E). Almost as good is a 4:3 ratio, e.g. 293 1/3 Hz, or D. You can build a tune out of just these ratios— medieval music often did.
Now, consider the interval between D and E, which is (3/2)/(4/3) or 9/8. This is so useful that we give it a name— a "whole tone". Half a tone, the ratio 16/15, is useful too. You can build sequences of notes with steps of either size. (Nicolas Slonimsky has some books where he explains how Western music, over 500 years, taught itself to tolerate more and more of what used to be considered dissonance. The tritone, for instance, was once called the diabolus in musica.)
At this point you've got the scales described in the article. But you can go further and cut the semitone in half; that gives you the Indian scale.
All this is "just temperament". This is not a problem when (say) you make your music with voices inside a big muddy-sounding cathedral. It becomes a problem when you want to use a piano in a drawing room, because you have to retune the thing for each key, not an easy proposition.
Plus, fifths (the A-E interval) don't quite match octaves. If they did, 12 fifths (12 x 7 semitones) should equal seven octaves (7 x 12 semitones). 7 octaves is just doubling the frequency 7 times, so, 128 times total. But a perfect fifth taken 12 times is (3/2)^12 = 129.75, which is quite noticeably different.
So when the keyboard becomes the predominant instrument, not coincidentally people switch to equal temperament, where a semitone is defined as the twelfth root of 2 (about 1.05946) rather than 16/15 (about 1.06667). My understanding is that equal temperament slightly bothers people who have perfect pitch.
posted by zompist at 2:38 PM on May 21, 2021 [2 favorites]
We like the sound of vibrating things. E.g. 220 vibrations per second (220 Hz) sounds nice; we call that an A. Double that (440 Hz) and the two notes sound good together, so much that we call them "the same note an octave apart."
A 3:2 ratio sounds good too— e.g. 220 Hz (A) plus 330 Hz (E). Almost as good is a 4:3 ratio, e.g. 293 1/3 Hz, or D. You can build a tune out of just these ratios— medieval music often did.
Now, consider the interval between D and E, which is (3/2)/(4/3) or 9/8. This is so useful that we give it a name— a "whole tone". Half a tone, the ratio 16/15, is useful too. You can build sequences of notes with steps of either size. (Nicolas Slonimsky has some books where he explains how Western music, over 500 years, taught itself to tolerate more and more of what used to be considered dissonance. The tritone, for instance, was once called the diabolus in musica.)
At this point you've got the scales described in the article. But you can go further and cut the semitone in half; that gives you the Indian scale.
All this is "just temperament". This is not a problem when (say) you make your music with voices inside a big muddy-sounding cathedral. It becomes a problem when you want to use a piano in a drawing room, because you have to retune the thing for each key, not an easy proposition.
Plus, fifths (the A-E interval) don't quite match octaves. If they did, 12 fifths (12 x 7 semitones) should equal seven octaves (7 x 12 semitones). 7 octaves is just doubling the frequency 7 times, so, 128 times total. But a perfect fifth taken 12 times is (3/2)^12 = 129.75, which is quite noticeably different.
So when the keyboard becomes the predominant instrument, not coincidentally people switch to equal temperament, where a semitone is defined as the twelfth root of 2 (about 1.05946) rather than 16/15 (about 1.06667). My understanding is that equal temperament slightly bothers people who have perfect pitch.
posted by zompist at 2:38 PM on May 21, 2021 [2 favorites]
Well I don't think you need perfect pitch to hear a difference. If you take a guitar, play a chord, and tweak the tuning until that chord seems perfect to you, you have probably tuned the third much closer to where it would be in just intonation than where it is normally. Listen to that chord! So pretty. Now play a different chord, and see what that sounds like......not so great.
posted by thelonius at 4:07 PM on May 21, 2021
posted by thelonius at 4:07 PM on May 21, 2021
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