Chinese Tuning On The Guqin

I want to learn about historic Chinese tunings. We're going to look at tunings of the Chinese zither called the qin, also called the guqin ("ancient" qin). Qin and guqin are pronounced with a "ch" sound for the q. The instrument used to always have silk strings, but now people sometimes use steel and nylon. I'm learning about it from the site SilkQin.com, authored by John Thompson.

The guqin has markers on the neck called hui. I think they're usually flat inlays but they can also protrude as studs - maybe cheap instruments have them painted on. They're placed at the positions of various harmonic nodes. They're usually made of jade, or gold, or sea shell.

Let's talk about harmonics and harmonic nodes. If you touch a string lightly at a hui and then pluck the string, the plucked string can vibrate on both sides of your lightly touching finger, but with a limited set of vibrational modes compared to a full length / open string. This is in contrast to playing stopped notes, where your finger presses the string all the way down to the instrument body. When you play a stopped note, the string's length is effectively shortened and the new fundamental frequency is proportional to the inverse of the string length, just as it was for the open string. Every place that you press a stopped note will give you a different sound on that string. In contrast, there are multiple places you can play the same harmonic sound on a string - if you press a string lightly at 1/5, 2/5, 3/5, or 4/5 of its length, you're going to get the 5th harmonic at all of them, which has a frequency five times the fundamental frequency, i.e. five times the open string frequency. If you press lightly at 1/4, 2/4, 3/4 of the string length, the outer two nodes will give you the fourth harmonic, but the middle node will have the 2nd harmonic as its loudest spectral component (though the fourth harmonic contributes to the sound too). You can just reduce the fraction as look at the denominator to figure out the harmonic.

The guqin has 13 hui markers along its neck. Guqin players don't play harmonics nearly as much as they play stopped notes; the hui are mostly a way to guide you on where to place your fingers for stopped notes - like halfway between the 6th and 7th hui, for example. The player doesn't sit at the center of the strings of the instrument - more to their right, and the hui are numbered from right to left - the most leftward hui for the player is the 13th.

The hui are placed at [1/8, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 7/8] of the string length. The harmonics associated with these are simply the denominators of those fractions: 

    [8, 6, 5, 4, 3, 5,) 2, (5, 3, 4, 5, 6, 8].

These numbers are symmetric around the half-way point, but the pattern doesn't steadily decrease and then increase due to two fifth harmonic nodes straddling the octave.

There are hui at almost all the multiples of 1/5-, and 1/6-, and 1/8-times the open string length, except not at 3/8 and 5/8. I don't know why not. This set happens to also cover multiples of 1/2, 1/3, and 1/4 string lengths.

If you were to play stopped notes at each of the hui, with your left hand fretting and your right hand plucking, you would get frequency ratios as

    [8/7, 6/5, 5/4, 4/3, 3/2, 5/3, 2/1, 5/2, 3/1, 4/1, 5/1, 6/1, 8/1]

over the fundamental, with 8/7 at hui 13 and 8/1 at hui 1. These ratios are all 5-limit except for the 8/7, which is 7-limit. I don't think 8/7 is actually used much in Chinese guqin music as a stopped note, it's just easy to put a hui there since that's where a strong harmonic can be found.

So if the hui aren't identical to the fingerings used for stopped notes, what actual fingerings are used, and what are their frequency ratios?

On one one page of SilkQin, called "Taiyin Daquanji 1: Folio 1C: Miscellaneous Qin Information (I/48a-49a) 2", Thompson shares a historic source that gives poetic names for the tones at various positions along the neck. Thompson then converts these positions to "decimal" notation. In decimal notation, e.g. 7.3 is between hui 7 and hui 8, and it is 0.3 of the way between them. The hui aren't evenly spaced, so it takes a little bit of figuring to turn this into a string length, but we'll have to do the figuring if we want to know the actual string ratios and frequency ratios. Here are the decimals:

    [7.3, 7.6, 7.9, 8.5, 9.0, 9.4, 10.0, 10.8, 12.3, 13.1, 13.5]

I don't know how he come up with these numbers - like the original Chinese will give a poetic name for the 11th hui, but he gives 10.8 for the decimal notation. I assume that he knows where guqin players actually put their fingers in modern times, and he thinks that 11 was a shorthand for 10.8, but I wouldn't mind a little more discussion. Maybe I'll find it on another page.

Let's just do conversions first. Here are approximate frequency ratios for all of those decimal-notation neck positions, as well as ratios for the hui from 7 to 13.

13.5 : 112c ~ m2 # 16/15

13.1 : 207c ~ AcM2 # 9/8

13 : 231c ~ SpM2 8/7

12.3 : 290c ~ Grm3 # 32/27

12 : 316c ~ m3 # 6/5

11 : 386c ~ M3 # 5/4

10.8 : 408c ~ AcM3 81/64

10 : 498c ~ P4 # 4/3

9.4 : 617c ~ SpA4 # 10/7

9 : 702c ~ P5 # 3/2

8.5 : 791c ~ Grm6 # 128/81

8 : 884c ~ M6 # 5/3

7.9 : 913c ~ AcM6 # 27/16

7.6 : 1004c ~ Grm7 # 16/9?

7.3 : 1099c ~ M7 # 15/8?

7 : 1200c ~ P8 # 2/1

Many of these are dead on, and I think all but two (annotated with question marks) are within 4 cents. Maybe decimals aren't so precise in the middle of the instrument, or further from hui 13. The Grm7 is off by 8 cents, and the M7 is off by 11 cents. Closer ratios would be 25/14 and 17/9 respectively, but that's pretty suspicious when all the other intervals besides the tritone (and hui 13 with a stopped note frequency ratio of 8/7) are obviously 5-limit. 

If we only look at the decimal-notation string locations from the Taiyin Daquanji page, and not all of the integer hui, then we knock out hui 7, 8, 12, and 13, which are [P8, M6, m3, SpM2]. That removes a few 5-limit ratios, but we still have 5-limit m2, M3, and M7. So it's not like they're putting markers on 5-limit harmonics but then only playing 3-limit stopped notes. It looks like this source is actually advocating for 5-limit tuning. Which is cool.

In https://silkqin.com/08anal/tunings.htm#decimal, Thompson gives a few more neck positions / string length ratios in decimal-notation. These include 13.9 as a possible alternative to 13.1 and 13.5, as well as many decimal-notations numbers below 7, which the Taiyin Daquanji page didn't have.

The 13.9 decimal is only like 22 cents? That doesn't look right. That could be a Pythagorean or Syntonic comma, but I really doubt that it's going to be a useful melodic interval. Let's do that one by hand to check. The 13th hui is at 1/8 of the string length. We want to move from hui 13 toward the next hui by 1/9 of the remaining distance, but 13 is our last hui, so presumably we just go from hui 13 to the end of the string, giving a string ratio of

    (1/8) - ((1/8) * (9/10)) = 1/80

or a frequency ratio of 80/79 @ 22 cents. Crazy. That's less than a quartertone. I really don't think that's working as a melodic interval. Probably just a comma.

The [6.7 to 4.2] decimals are the same as the [3.4 to 1.2] decimals, just displaced by octaves. Here they are, octave reduced to fit in [0 to 1200 cents]:

6.7 : 107c ~ m2 # 16/15

6.4 : 221c ~ DeAcA2 # 25/22 (or 17 cents sharp of AcM2 # 9/8)

6.2 : 302c ~ Grm3 # 32/27 (but 8 cents too sharp)

5.9 : 415c ~ AcM3 # (but 7 cents too sharp)

5.6 : 506c ~ P4 # 4/3 (but 8 cents too sharp)

5.3 : 601c ~ ExA4 # 17/12

4.8 : 791c ~ Grm6 # 128/81

4.6 : 884c ~ M6 # 5/3

4.4 : 983c ~ Hbm7 # 30/17 (or 13 cents flat of Grm7 # 16/9)

4.2 : 1088c ~ M7 # 15/8

There are some weird cent values in there. They're not all obviously 3- and 5-limit. They're also not far enough away to be obviously something else. Probably we just need more decimal places at certain neck positions in order to zero-in on certain frequency ratios.

Let's finish by discussing string tuning. There are seven strings on a guqin. Thompson describes the standard tuning in terms of how one string relates to another that's fretted at a hui, like the open 7th string should match the 4th string fretted at hui 10, and hui 10 is 4/3 over the open string frequency. If you follow all of these references, you get frequency ratios for the strings, low to high, as

    [1/1, 9/8, 4/3, 3/2, 27/16, 2/1, 9/4]

If we call the low string C and use Pythagorean pitch names, this is

    [C, D, F, G, A, C, D]

with the last two notes being an octave above the first two. Thompson says that this tuning dominates the modern repertoire, but that there were other tunings historically that differed from this by the adjustment of a string or two strings by a half step.

He also describes several tunings that differ from the standard Pythagorean tuning above by adjusting strings by a Pythagorean comma, though I think he means a syntonic comma in several places (which is basically audibly indistinguishable from the Pythagorean one), since that's what needed for the arithmetic to work out. Although he uses frequency ratios that aren't Pythagorean or Syntonic, so... He seems a little confused in multiple directions here, but it has still been an enlightening read.

Music Generating Data Structures

I've made some programs for generating microtonal music in the past - mostly 5-limit just intonation, with some limited forays into higher limits. I'm planning to write a much better program that can go up to 23-limit just intonation, and I'm planning out how it should represent music.

A musical piece will be made of musical sections.

Sections will be made of chord onset events, possibly with measure markers sprinkled in.

Chord onset events are made of notes/rests for multiple voices. I think a rest will be a different kind of object than a note, but maybe it'll just be a note with a special pitch. If a piece has 6 voices, maybe every chord onset event will have rests/notes listed for all six voices? I think so.

Notes have pitch and duration, and a property that tells you whether the note is voiced at the onset time veruss suspended from a previous note with the same pitch. I think notes/rests will have pointers to the previous note/rest in the same voice. Possibly they'll have pointers to the next note/rest as well, so that a voice could be thought of as a double linked list of notes/rests.

Sections are represented in terms of chord onset events, but that's not how they're composed/generated. Sections will mostly be composed from melodies. A secontion retains an edit history of the melodic addition events that were used to compose it.

I think adding melodic content will look like this - perhaps we'll call it a melodic or motivic occurence event:

    Add (invert(Motive-X)), at time T2, in voice W)

"Invert" is a determinstic transform that generates a new melody from a given one. Deterministic transforms include [transpose, invert, reverse, augment rhythm (e.g. double note lengths), diminish rhythm].

There are also going to be transforms like "compose countermelody to a given melody, heavy in contrary motion". It feels like this kind of transform has to be non-deterministic. Maybe these can depend determinstically on a random seed value, so that composition can be variable but also reproducible, but I don't really see how. Maybe it's not important - the edit history can just include "compose counter melody to melody-X // psst: here's the countermelody generated".

I think applying ornaments and diminutions will also be a deterministic affair. You can choose which ones to apply randomly in composition, but there has to be a history of edits that reproducibly generates the song. This will let you edit previously generated content at different levels of abstractions: you can change the ornaments, you can change the amount of transposition of a transform for a motivic onset in one voice, maybe other things. The diminutions of this type will mostly be used for doing things like this: the soprano gets a frilly bit while the middle voices are doing slow notes, maybe then a middle voice responds while the soprano has slow notes.

I haven't thought much about how to generate a response melody from a call melody, but that is as important as generating simultaenous countermelodies, and will have to be addressed. Call-melodies end high, possibly with weaker harmony, and response melodies end low with strong harmony? There's a little more to it than that.

After adding enough motivic content that we have a skeleton of chord onset events for all voices (or at least two successive chord onset events with full voicing), then we'll have the option to try adding suspensions, passing tones, and neighbor notes in multiple voices simultaneously between those chord onsets. I think these will still be applied in the melodic-addition edit history (from which chord-onset events are generated) but it will be convenient to have full voiced chords sequences to decide which extra notes can be used - which combinations make unacceptable dissonances, and which make bold but acceptable passing dissonances while moving between consonant chords.

Obviously I'm trying to design this so that it can make imitative counterpoint, but hopefully it will also be general enough to make many kinds of music - romantic era classical music and smooth jazz and math rock and arabic maqam music and and bossa nova and Georgian folk music and so on.

...

I didn't have to spend much time coding this up before deciding that I had it backwards - the native representation of music should be parallel voices, and the chord-onset view should be more like an annotation.

...

Odd Limit Temperaments

Suppose we have a space of intervals with just tunings that have only odd factors - no factors of 2. We can define 1-dimensional equal temperaments over such a space (equal divisions of a "decade" tuned to 3/1), or even unequal temperaments in higher dimensions that are still lower than the dimension of the interval space in question.

If we represent rank-2 intervals in the odd prime harmonic basis, then the elements are like exponents of 3 and 5 in the factorization of the justly associated frequency ratios. These ratios get large numerators and denominators very quickly, but here are a few easy ones:

[0, 0] # 1/1

[3, -2] # 27/25

[6, -4] # 729/625

[-7, 5] # 3125/2187

[-4, 3] # 125/81

[-1, 1] # 5/3

[2, -1] # 9/5

[5, -3] # 243/125

[8, -5] # 6561/3125

[-5, 4] # 625/243

[-2, 2] # 25/9

[1, 0] # 3/1

If we want to define an equal temperament over the decade, and EDD, we tune the decade purely and temper out some other interval X. 

    t(P12) = 3/1

    t(X) = 1/1

This induces effects on the tuning of other intervals - and we can see this by expressing those intervals in the basis [[1, 0], X] instead of the rank-2 odd prime harmonic basis [[1, 0], [0, 1]]. I'll also call the interval X a comma or tempered comma.

To convert interval bases, we just "left multiply" or "vector-matrix multiply" the target interval we want to convert by a transformation matrix, which is simply the inverse of the matrix [[1, 0], X].

Suppose we've converted a target interval Y to have coordinates in this tempered comma basis, and we'll call the new coordinates [a, b]. Then we have this relation in interval space:

    Y = a * P12  + b * X

and this has a parallel expression in frequency ratio space:

    t(Y) = t(P12)^(a) * t(X)^(b)

where t(Y) is the tuning a interval Y in some tuning system. By substitution of the tunings For P12 and X in our tempered tuning system we have

    t(Y) = (3/1)^(a) * (1/1)^b

which means that any target interval Y will be tuned to a frequency ratio of 3 to some rational number. I haven't proven to you that {a} will be rational, but there are only so many hours in a day.

For example, the frequency ratio 48828125/43046721 is quite small and would make a good comma. It has a prime factorization of 

    3^(-16) * 5 ^(11)

so the associated interval in the rank-2 odd prime harmonic basis is

    [-16, 11]

Our transformation matrix is

    inverse([[1, 0], [-16, 11]])

And you don't have to tune many target intervals with this to guess that it's producing frequency ratios of the form

    3^(i/11)

i.e. this defines the "11 equal division of the decade" temperament, or 11-EDD. We can also verify this by seeing that the absolute determinant of the transformation matrix is 1/11.

There are actually many intervals you could temper out to produce 11-EDD. Another way to define 11-EDD is this: For a rank-N interval space, 11-EDD tunes each interval justly assocaited with an odd prime harmonic to the nearest frequency of the form 3^(i/11). If this had no mistuning, then for a prime P, we'd have

    P = 3^(i/11)

or 

    i = 11 * log_3(P)

but usally there is mistuning, so we round {i} to the nearest integer:

    i = round(11 * log_3(P))

Here are the nearest steps for 11-EDD for primes [3, 5, 7, 11, 13]

    11-EDD: [11, 16, 19, 24, 26]

I'll call a sequence of harmonic EDD-steps like that a canonical definition for the EDD for a given rank, i.e. that's the canonical rank-5 definition of 11-EDD.

At some point down the line, as you might predict from the shape of the log function, multiple harmonics will be tuned to the same step. This has never been a problem for me. Like, 11-EDD has the same tuning for the 16th and 17th odd prime harmonic. But I don't use those? You don't need those harmonics for music making. The 16th and 17th harmonics, you need those. The 16th and 17th *prime* harmonics? No way, Jose.

Samller EDDs reach this point of ambiguity sooner, but you can just ... not use 5-EDD, or you can use it without considering how the steps might map to intervals with just tunings involving factors of 37 and 41. It's not hard. Or you can use just accept that 5-EDD tunes these identically. It's all fine.

It's really easy to use a canonical definition for an EDD to tune intervals in that EDD. Let's use 39-EDD and the interval [-5, 1, 2, 0] in odd-prime-harmonic coordinates as examples.

Our target interval has a just tuning of 

    3^(-5) * 5^(1) * 7^(2) * 11^(0) = 245/243

In 39-EDD, the harmonics aren't exactly 3/1, 5/1, 7/1, they're nearby ratios of the form 3^(i/39) for integers {i}. And the integers are of course found in the canonical definition:

39-EDD: [39, 57, 69, 85] // rank-4

To find the number of steps of 39-EDD for the temepred tuning of the target interval [-5, 1, 2, 0], we just take the dot product.

    [-5, 1, 2, 0] * [39, 57, 69, 85] = 0

This means that the target interval is tuned to zero steps by 39-EDD, i.e. it's tempered out or tuned to a frequency ratio of 1/1.

Some EDDs can't be defined over a rank-2 interval space: no matter what rank-2 interval you try to temper out, you'll get some other EDD. This is very easy to predict: if the first two values of the canonical definition for the EDD have a greatest common divisor of 1, you can give the EDD a definition in terms of a pure decade and a tempered comma. If the first *three* values have GCD = 1, then you can give a rank-3 definition in terms of a pure decade and *two* tempered commas. And so on. Here are a few EDDs, and the shortest canonical definition for each one that gives a GCD of 1:

10-EDD: [10, 15, 18] // rank-3

11-EDD: [11, 16] // rank-2

12-EDD: [12, 18, 21, 26] // rank-4

13-EDD: [13, 19] // rank-2

14-EDD: [14, 21, 25] // rank-3

15-EDD: [15, 22] // rank-2

16-EDD: [16, 23] // rank-2

17-EDD: [17, 25] // rank-2

18-EDD: [18, 26, 32, 39] // rank-4

19-EDD: [19, 28] // rank-2

20-EDD: [20, 29] // rank-2

21-EDD: [21, 31] // rank-2

22-EDD: [22, 32, 39] // rank-3

23-EDD: [23, 34] // rank-2

24-EDD: [24, 35] // rank-2

25-EDD: [25, 37] // rank-2

26-EDD: [26, 38, 46, 57] // rank-4

27-EDD: [27, 40] // rank-2

28-EDD: [28, 41] // rank-2

29-EDD: [29, 42] // rank-2

30-EDD: [30, 44, 53] // rank-3

31-EDD: [31, 45] // rank-2

32-EDD: [32, 47] // rank-2

33-EDD: [33, 48, 58] // rank-3

34-EDD: [34, 50, 60, 74, 79] // rank-5

35-EDD: [35, 51] // rank-2

36-EDD: [36, 53] // rank-2

37-EDD: [37, 54] // rank-2

38-EDD: [38, 56, 67] // rank-3

39-EDD: [39, 57, 69, 85] // rank-4

40-EDD: [40, 59] // rank-2

Using the canonical definitions, it's also really easy to figure out which rank-2 tempered comma you can use to define an EDD that is rank-2 definable. You can see that 11-EDDs has a rank-2 canonical definition of [11, 16]. If you reverse these numbers and flip the sign up one, i.e. temper out [-16, 11] or [16, -11], then you'll get 11-EDD. These two intervals are complementary: they sum to [0, 0], and equivalently, their just tunings will also have inverse frequency ratios. Only one of these ratios will be larger than 1/1, and that's the one I prefer to use in defining EDOs, but they're equivalent definitions, and tempering one out will also temper out the other.

If you give a non-canonical definition of an EDD, e.g. if you don't tune the prime harmonic interval associated with 5/1 to its nearest step, then you get different tempered commas; weird commas that don't have small just tunings near 1/1. If you make a transformation matrix from the decade and a non-canonical comma, you'll still map intervals to frequency ratios like 3^(i/11), but the mapping will be weird - things that should be nearby will be ripped up and sent far from each other, decades away. I once tried using multiple non-canonical definitions like this for an EDO to to generate interesting harmonies. It sounded pretty crazy.

Anyway, the most important EDD is 13-EDD, which is definable with rank-2 intervals. Here's its rank-2 canonical harmonic prefix:

 

       13-EDD: [13, 19] // rank-2

So we could define it by tempering out 

    [-19, 13] # 3^(-19) * 5^(13) = 1220703125/1162261467

But it's more common to define it over rank-3 interval space. It happens to be definable over rank-3 interval space in terms of a pure decade and tempered commas that have just tunigns of 245/243 and 3087/3125.

Finding a rank-2 tempered comma was really easy, but I don't have a short procedure I can explain to you here for finding two commas that generate an equal temperament over rank-3 interval space. I mean, you could do a simialr thing to the rank-2 case by re-ordering and negating some numbers from the canonical definition, but they're going to be really ugly intervals with high complexity just tunings.  I gess you don't really need to have low complexity ratios to define an EDD, but it's nice to give a short simple definition for a temperament that way and I'd feel dirty if I did anything less.

When I first got into microtonality, I would just tune a bunch of intervals in an EDO, e.g. using the canonical definition, then see which ones were tempered out to 0 steps, and look for the ones with the lowest complexity frequency ratios in that set, or search through linear combinations of my found tempered commas to find even lower complexity frequency ratios. Then I'd find the determinant of the transformation matrix to verify that my lowest complexity commas were sufficient to define the EDO. The same thing would work with EDDs, I'm sure.

Later on I starting using the Lenstra–Lenstra–Lovasz lattice basis reduction algorithm to find tempered commas of EDOs. I could probably adapt the code to work for EDDs without much work.

Anyway, if you divide a 13-EDD steps into three parts, you instead get 39-EDD. This does an equally good job of representing ratios with factors of 3, 5, 7, and a much better job of representing ratios with factors of 11. So if we want to write music that sounds like 11-odd-limit just intonation using only a finite set of pitches that supports free modulation, then 39-EDD is a good choice. We could also look at detempering 39-EDD in order to play around with pure 11-limit harmony. 

Let's do it! We'll start by finding 39-EDDs tempered commas. It's rank-4 so we know that we'll need ratios involving factors of 11. I've never thought about this before, but look at the rank-4 canonical definition:

39-EDD: [39, 57, 69, 85] // rank-4

The first and last harmonic steps, 39 and 85, have a GCD of 1. So maybe we could define 39-EDD in the 3.11 just intonation subgroup? That could be cool. But I'll just start with 3.5.7.11.

Here are some intervals that are tempered out by 39-EDD, along with their just tunings:

[-5, 1, 2, 0] # 245/243

[-3, 0, -2, 3] # 1331/1323

[-2, 5, -3, 0] # 3125/3087

[1, 5, -1, -3] # 9375/9317

[-2, 1, 4, -3] # 12005/11979

[-7, 6, -1, 0] # 15625/15309

[3, 4, -5, 0] # 16875/16807

[-10, 2, 4, 0] # 60025/59049

[-8, 1, 0, 3] # 6655/6561

Let's just check if the three lowest complexity commas (in terms of numerator magnitude), plus the decade, have an absolute determinant of 39. This works surprisingly often.

    abs(determinant of [[1, 0, 0, 0], [-5, 1, 2, 0], [-3, 0, -2, 3], [-2, 5, -3, 0]]) = 39

Woo! That is a minimal definition of 39-EDD. For different notions of comma complexity, you could have different minimal definitions, but I think this one is pretty good.

If you wanted to explore some good non-equal odd-limit temperaments, you could temper out any 1 or 2 of these commas. 

There is a little more complexity to defining an unequal temperament than choosing commas though. For example, if you had an interval space that was justly tuned in the 3.5.7 JI subgroup, you could temper out a 7-limit comma and keep the harmonics 3 and 7 pure, and that would give you an interesting thing. But if you tempered out the same 7-limit comma and kept 3 and 5 pure, then your temperament would just look like just intonation in the 3.5 subgroup. Said a little differently: if you temper out a 7-limit comma, you need to keep the 7th harmonic pure, or you're simply defining a tuning system that forgets all the 7-limit ratios. If you temper out some 7-ness, you'll also want to retain some 7-ness. There might be something you could tune the 7th harmonic to besides purity that would keep the interval space from collapsing like this, but I wouldn't. To me, the point of an unequal temperament is to reduce the size of the interval space a little bit for manageability or playability while still maintaining as much harmonic purity as you can.

And we know that 39-EDD already has good accuracy over the 3.5.7.11 subgroup, so these are good commas to temper out.

Let's look at a detempering of 39-EDD. Among all the intervals tuned to each step, I show the just tuning of the one with the simplest just tuning here:

0\39 # 1/1

1\39 # 77/75

2\39 # 35/33

3\39 # 27/25

4\39 # 55/49

5\39 # 63/55

6\39 # 25/21

7\39 # 11/9

8\39 # 125/99

9\39 # 9/7

10\39 # 33/25

11\39 # 15/11

12\39 # 7/5

13\39 # 175/121

14\39 # 49/33

15\39 # 75/49

16\39 # 11/7

17\39 # 121/75

18\39 # 5/3

19\39 # 77/45

20\39 # 135/77

21\39 # 9/5

22\39 # 225/121

23\39 # 21/11

24\39 # 49/25

25\39 # 55/27

26\39 # 343/165

27\39 # 15/7

28\39 # 11/5

29\39 # 25/11

30\39 # 7/3

31\39 # 297/125

32\39 # 27/11

33\39 # 63/25

34\39 # 55/21

35\39 # 121/45

36\39 # 25/9

37\39 # 77/27

38\39 # 225/77

39\39 # 3/1

Let's see if that's symmetric under decade-complementation.

Original scale: 1/1, 77/75, 35/33, 27/25, 55/49, 63/55, 25/21, 11/9, 125/99, 9/7, 33/25, 15/11, 7/5, 175/121, 49/33, 75/49, 11/7, 121/75, 5/3, 77/45, 135/77, 9/5, 225/121, 21/11, 49/25, 55/27, 343/165, 15/7, 11/5, 25/11, 7/3, 297/125, 27/11, 63/25, 55/21, 121/45, 25/9, 77/27, 225/77, 3/1

Decade Complement: 1/1, 77/75, 81/77, 27/25, 135/121, 63/55, 25/21, 11/9, 125/99, 9/7, 33/25, 15/11, 7/5, 495/343, 81/55, 75/49, 11/7, 121/75, 5/3, 77/45, 135/77, 9/5, 225/121, 21/11, 49/25, 99/49, 363/175, 15/7, 11/5, 25/11, 7/3, 297/125, 27/11, 63/25, 55/21, 147/55, 25/9, 99/35, 225/77, 3/1

Not quite. I count 16 ratios that only show up once, so I we're differing on 8 scale degrees.

Let's just use the original scale instead of deciding which ratios we want to use for each of those scale degrees. It'll be fine.

What simple triad chords show up in this scale? Well, we get a lot of 7-limit chords that we saw in Odd Limit Harmony In Bohlen Pierce.

We also get some new 11-limit chords.

[P1, SpM3, AsGr5] # [1/1, 9/7, 55/36]

[P1, DeAcM3, DeA5] # [1/1, 27/22, 50/33]

[P1, m3, DeA5] # [1/1, 6/5, 50/33]

[P1, Sbm3, AsGr5] # [1/1, 7/6, 55/36]

[P1, SpM3, DeAc5] # [1/1, 9/7, 81/55]

[P1, M3, DeAc5] # [1/1, 5/4, 81/55]

[P1, AsGrm3, Gr5] # [1/1, 11/9, 40/27]

[P1, M3, DeA5] # [1/1, 5/4, 50/33]

[P1, Sbm3, DeAc5] # [1/1, 7/6, 81/55]

[P1, M3, AsGr5] # [1/1, 5/4, 55/36]

[P1, DeM3, d5] # [1/1, 40/33, 36/25]

[P1, DeM3, DeA5] # [1/1, 40/33, 50/33]

These don't look so good. We don't have a single perfect fifth interval. I wondered if they had nice otonal representations. A few of them are okay?

[27, 33, 40] : [P1, AsGrm3, Gr5]

[33, 40, 50] : [P1, DeM3, DeA5]

[36, 42, 55] : [P1, Sbm3, AsGr5]

[36, 45, 55] : [P1, M3, AsGr5]

I don't think this is a very good scale. Now we know.

Odd Limit Harmony In Bohlen Pierce

I've heard it said that the fundamental chord of the Bohlen Pierce 7 odd limit just intonation scale is [3:5:7], otonally. In frequency ratios that's

    [1/1, 5/3, 7/3]

If we use regular intervals instead of BP intervals to analyze the harmony, that's the just tuning of

    [P1, M6, Sbm10]

The Bohlen Piece scales don't have octaves, but my ear still has octave equivalence, so I hear that as a spread out voicing of

    [P1, Sbm3, M6] # [1/1, 7/6, 5/3]

Since BP doesn't have octaves, you might also think it's odd to do cyclic inversion of chords at the octave, but my ear has octave equivalence, so this chord has a similar sonority to me as does:

    [P1, SpA4, SpM6] # [1/1, 10/7, 12/7]

And

    [P1, m3, Sbd5] # [1/1, 6/5, 7/5]

This last version has a tertian spelling, and I consider that the canonical form for presentation. So I think the fundamental [3:5:7] chord of BP music is a rotated and spread out version of this diminished chord with a septimal alteration on the fifth. This chord does indeed show up a in the Bohlen Pierce scale. For example, the chord is outlined by the [0th, 6th, and 10th] notes of the scale, and five other places besides that:

^[0, 6, 10]

^[1, 7, 11]

^[3, 9, 13]

^[7, 13, 17]

^[9, 15, 19]

^[11, 17, 21]

What other chords can we make between scale degrees of BP?

Here's the scale for reference (in standard intervals, not BP intervals):

P1 # 1/1

Acm2 # 27/25

SpA2 # 25/21

SpM3 # 9/7

Sbd5 # 7/5

SpSpAA4 # 75/49

M6 # 5/3

m7 # 9/5

SbSbAcd9 # 49/25

SpA8 # 15/7

Sbm10 # 7/3

SbAcd11 # 63/25

A11 # 25/9

P12 # 3/1

This scale doesn't repeat at the octave, it repeats at the decade/tritave/P12, so that all the scale steps in the next decade have just tunings that are 3 times the ones above:

P12 # 3/1

Acm13 # 81/25

SpA13 # 25/7

SpM14 # 27/7

SbAcm16 # 21/5

SpSpAA15 # 225/49

M17 # 5/1

Ac18 # 27/5

SbSbAcd20 # 147/25

SpA19 # 45/7

Sbm21 # 7/1

SbAcd22 # 189/25

A22 # 25/3

AcM23 # 9/1

Here are 91 chords that you can make between BP scale degrees (with octave reduced forms on the right hand side of the colon). These are all tertian spellings - I've inverted the chords at the octave if they were ^[1, 3, 6] or ^[1, 4, 6] chords. This might seem overwhelming at first, but the BP scale has one more note than a chromatic scale, so there should be lots of available chords. And there are other crazier chords besides these that you can make of course, but I thought these looked fairly tame.

^[0, 12, 15] # [1/1, 25/9, 25/7] : [P1, SpM3, d5]

^[0, 14, 20] # [1/1, 81/25, 27/5] : [P1, m3, Gr5]

^[0, 6, 12] # [1/1, 5/3, 25/9] : [P1, m3, d5]

^[0, 6, 19] # [1/1, 5/3, 5/1] : [P1, m3, P5]

^[0, 6, 10] # [1/1, 5/3, 7/3] : [P1, m3, Sbd5]

^[0, 19, 22] # [1/1, 5/1, 45/7] : [P1, M3, SpA5]

^[0, 13, 19] # [1/1, 3/1, 5/1] : [P1, M3, P5]

^[0, 4, 19] # [1/1, 7/5, 5/1] : [P1, M3, Sbd5]

^[0, 4, 10] # [1/1, 7/5, 7/3] : [P1, Sbm3, Sbd5]

^[0, 10, 13] # [1/1, 7/3, 3/1] : [P1, Sbm3, P5]

^[0, 10, 22] # [1/1, 7/3, 45/7] : [P1, Sbm3, SpA5]

^[0, 15, 19] # [1/1, 25/7, 5/1] : [P1, Sbd3, Sbd5]

^[0, 3, 6] # [1/1, 9/7, 5/3] : [P1, m3, Sp5]

^[0, 3, 13] # [1/1, 9/7, 3/1] : [P1, SpM3, P5]

^[0, 3, 4] # [1/1, 9/7, 7/5] : [P1, SpM3, Sbd5]

^[0, 3, 15] # [1/1, 9/7, 25/7] : [P1, Sbd3, d5]

^[0, 3, 22] # [1/1, 9/7, 45/7] : [P1, SpM3, SpA5]

^[1, 14, 20] # [27/25, 81/25, 27/5] : [P1, M3, P5]

^[1, 13, 23] # [27/25, 3/1, 7/1] : [P1, Sbm3, d5]

^[1, 13, 16] # [27/25, 3/1, 27/7] : [P1, SpM3, d5]

^[1, 11, 14] # [27/25, 63/25, 81/25] : [P1, Sbm3, P5]

^[1, 4, 23] # [27/25, 7/5, 7/1] : [P1, M3, Sp5]

^[1, 4, 7] # [27/25, 7/5, 9/5] : [P1, SpM3, Sp5]

^[1, 16, 20] # [27/25, 27/7, 27/5] : [P1, Sbd3, Sbd5]

^[1, 7, 20] # [27/25, 9/5, 27/5] : [P1, m3, P5]

^[1, 7, 13] # [27/25, 9/5, 3/1] : [P1, m3, d5]

^[1, 7, 11] # [27/25, 9/5, 63/25] : [P1, m3, Sbd5]

^[2, 12, 15] # [25/21, 25/9, 25/7] : [P1, Sbm3, P5]

^[2, 6, 12] # [25/21, 5/3, 25/9] : [P1, Sbm3, Sbd5]

^[2, 16, 22] # [25/21, 27/7, 45/7] : [P1, m3, Gr5]

^[2, 5, 6] # [25/21, 75/49, 5/3] : [P1, SpM3, Sbd5]

^[2, 5, 15] # [25/21, 75/49, 25/7] : [P1, SpM3, P5]

^[3, 6, 25] # [9/7, 5/3, 25/3] : [P1, M3, Sp5]

^[3, 6, 9] # [9/7, 5/3, 15/7] : [P1, SpM3, Sp5]

^[3, 13, 16] # [9/7, 3/1, 27/7] : [P1, Sbm3, P5]

^[3, 9, 13] # [9/7, 15/7, 3/1] : [P1, m3, Sbd5]

^[3, 9, 15] # [9/7, 15/7, 25/7] : [P1, m3, d5]

^[3, 9, 22] # [9/7, 15/7, 45/7] : [P1, m3, P5]

^[3, 15, 25] # [9/7, 25/7, 25/3] : [P1, Sbm3, d5]

^[3, 15, 18] # [9/7, 25/7, 225/49] : [P1, SpM3, d5]

^[3, 16, 22] # [9/7, 27/7, 45/7] : [P1, M3, P5]

^[3, 18, 22] # [9/7, 225/49, 45/7] : [P1, Sbd3, Sbd5]

^[3, 7, 13] # [9/7, 9/5, 3/1] : [P1, Sbm3, Sbd5]

^[3, 7, 22] # [9/7, 9/5, 45/7] : [P1, M3, Sbd5]

^[5, 9, 15] # [75/49, 15/7, 25/7] : [P1, Sbm3, Sbd5]

^[5, 15, 18] # [75/49, 25/7, 225/49] : [P1, Sbm3, P5]

^[4, 19, 23] # [7/5, 5/1, 7/1] : [P1, Sbd3, Sbd5]

^[4, 17, 23] # [7/5, 21/5, 7/1] : [P1, M3, P5]

^[4, 8, 23] # [7/5, 49/25, 7/1] : [P1, M3, Sbd5]

^[4, 10, 23] # [7/5, 7/3, 7/1] : [P1, m3, P5]

^[4, 23, 26] # [7/5, 7/1, 9/1] : [P1, M3, SpA5]

^[4, 7, 26] # [7/5, 9/5, 9/1] : [P1, SpM3, SpA5]

^[4, 7, 19] # [7/5, 9/5, 5/1] : [P1, Sbd3, d5]

^[4, 7, 17] # [7/5, 9/5, 21/5] : [P1, SpM3, P5]

^[4, 7, 8] # [7/5, 9/5, 49/25] : [P1, SpM3, Sbd5]

^[4, 7, 10] # [7/5, 9/5, 7/3] : [P1, m3, Sp5]

^[6, 12, 25] # [5/3, 25/9, 25/3] : [P1, m3, P5]

^[6, 20, 26] # [5/3, 27/5, 9/1] : [P1, m3, Gr5]

^[6, 19, 25] # [5/3, 5/1, 25/3] : [P1, M3, P5]

^[6, 10, 25] # [5/3, 7/3, 25/3] : [P1, M3, Sbd5]

^[6, 9, 12] # [5/3, 15/7, 25/9] : [P1, m3, Sp5]

^[6, 9, 19] # [5/3, 15/7, 5/1] : [P1, SpM3, P5]

^[6, 9, 10] # [5/3, 15/7, 7/3] : [P1, SpM3, Sbd5]

^[7, 20, 26] # [9/5, 27/5, 9/1] : [P1, M3, P5]

^[7, 19, 22] # [9/5, 5/1, 45/7] : [P1, SpM3, d5]

^[7, 13, 26] # [9/5, 3/1, 9/1] : [P1, m3, P5]

^[7, 13, 19] # [9/5, 3/1, 5/1] : [P1, m3, d5]

^[7, 13, 17] # [9/5, 3/1, 21/5] : [P1, m3, Sbd5]

^[7, 11, 26] # [9/5, 63/25, 9/1] : [P1, M3, Sbd5]

^[7, 11, 17] # [9/5, 63/25, 21/5] : [P1, Sbm3, Sbd5]

^[7, 17, 20] # [9/5, 21/5, 27/5] : [P1, Sbm3, P5]

^[7, 10, 13] # [9/5, 7/3, 3/1] : [P1, SpM3, Sp5]

^[7, 22, 26] # [9/5, 45/7, 9/1] : [P1, Sbd3, Sbd5]

^[9, 12, 15] # [15/7, 25/9, 25/7] : [P1, SpM3, Sp5]

^[9, 19, 22] # [15/7, 5/1, 45/7] : [P1, Sbm3, P5]

^[9, 13, 19] # [15/7, 3/1, 5/1] : [P1, Sbm3, Sbd5]

^[9, 15, 19] # [15/7, 25/7, 5/1] : [P1, m3, Sbd5]

^[15, 19, 25] # [25/7, 5/1, 25/3] : [P1, Sbm3, Sbd5]

^[15, 18, 19] # [25/7, 225/49, 5/1] : [P1, SpM3, Sbd5]

^[8, 11, 21] # [49/25, 63/25, 147/25] : [P1, SpM3, P5]

^[8, 11, 23] # [49/25, 63/25, 7/1] : [P1, Sbd3, d5]

^[10, 13, 25] # [7/3, 3/1, 25/3] : [P1, Sbd3, d5]

^[10, 13, 23] # [7/3, 3/1, 7/1] : [P1, SpM3, P5]

^[11, 14, 26] # [63/25, 81/25, 9/1] : [P1, Sbd3, d5]

^[11, 14, 24] # [63/25, 81/25, 189/25] : [P1, SpM3, P5]

^[11, 14, 17] # [63/25, 81/25, 21/5] : [P1, m3, Sp5]

^[11, 17, 21] # [63/25, 21/5, 147/25] : [P1, m3, Sbd5]

^[11, 17, 23] # [63/25, 21/5, 7/1] : [P1, m3, d5]

^[11, 21, 24] # [63/25, 147/25, 189/25] : [P1, Sbm3, P5]

^[11, 23, 26] # [63/25, 7/1, 9/1] : [P1, SpM3, d5]

^[12, 15, 25] # [25/9, 25/7, 25/3] : [P1, SpM3, P5]

Across this set of 91 chords, some chord forms are repeated. There are really only 20 distinct sounds here:

[P1, Sbd3, Sbd5] # [1/1, 28/25, 7/5]

[P1, Sbd3, d5] # [1/1, 28/25, 36/25]

[P1, Sbm3, P5] # [1/1, 7/6, 3/2]

[P1, Sbm3, Sbd5] # [1/1, 7/6, 7/5]

[P1, Sbm3, SpA5] # [1/1, 7/6, 45/28]

[P1, Sbm3, d5] # [1/1, 7/6, 36/25]

[P1, m3, Gr5] # [1/1, 6/5, 40/27]

[P1, m3, P5] # [1/1, 6/5, 3/2]

[P1, m3, Sbd5] # [1/1, 6/5, 7/5]

[P1, m3, Sp5] # [1/1, 6/5, 54/35]

[P1, m3, d5] # [1/1, 6/5, 36/25]

[P1, M3, P5] # [1/1, 5/4, 3/2]

[P1, M3, Sbd5] # [1/1, 5/4, 7/5]

[P1, M3, Sp5] # [1/1, 5/4, 54/35]

[P1, M3, SpA5] # [1/1, 5/4, 45/28]

[P1, SpM3, P5] # [1/1, 9/7, 3/2]

[P1, SpM3, Sbd5] # [1/1, 9/7, 7/5]

[P1, SpM3, Sp5] # [1/1, 9/7, 54/35]

[P1, SpM3, SpA5] # [1/1, 9/7, 45/28]

[P1, SpM3, d5] # [1/1, 9/7, 36/25]

I'd say that these are the core triadic harmonies available in Bohlen Pierce music, even though they have factors of 2 in the frequency ratios and the just intonation BP scale doesn't.

Detempering

Tempering means applying a tuning system (to a set of intervals) which maps some intervals to a frequency ratio of 1/1. This hides the effect of those intervals. For example, if Ac1 is tempered out, then you'll hear the same frequency ratio for any tuned intervals expressible as

    n * Ac1

for integers {n}. So in tuning, we've lost information.

Given a piece in a tempered tuning, what can we do to try reconstructing plausible intervallic muisc?

We could off course just associate every tempered frequency ratio with a single interval in our detempering reconstruction, e.g. assume every 1/1 was a P1 before we threw out the information. This is roughly the state of the art in the microtonal community.

That's weaksauce, but it's an okay baseline. We know that we can do at least as well as mapping, e.g. steps of 12 edo to a ch,romatic scale,

    [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] ->

    [P1, m2, M2, m3, M3, P4, d5, P5, m6, M6, m7, M7, P8]

What can we do that's better than that?

Suppose for a toy example that every step of 12-EDO can map to a small finite set of intervals, like the chromatic values above plus or minus an Ac1 and plus or minus a d2. It's up to you if you want to allow both commas to be applied at once, or to apply them multiple times. But we want something finite.

We're free to associate those altered chromatic intervals with the same 12-EDO step because 12-EDO tempers out Ac1 and d2. If you're working with a different temperament, you'll similarly continue to want to used its tempered commas to generate intervallic detempering options for every tuned frequency ratio in your song, but it might not be Ac1 and d2, and they might not be altering a chromatic scale. But let's keep working with 12-TET for simplicity and concreteness and applicability to the canon of modern western music.

For a given song, we'd like to choose among these intervallic detempering options for every note so as to get 

    1) melodic intervals that are fluid

    2) harmonic intervals that are consonant

and if those two criteria leave some decisions un-made, then we'd like to also have

    3) low complexity just tunings of individual notes

Those are currently three vague optimization criteria. Let's make them a little more concrete.

For a first approximation, we'll suppose that melodic fluidity and harmonic consonance are the same - some intervals are better at both functions and some intervals are worse at both functions.

Suppose we bless a set of intervals as perfectly consonant+fluid, perhaps

[P1, P4, P5]

[Grm2, m2, M2, AcM2]

[Grm3, m3, M3, AcM3]

[Grm6, m6, M6, AcM6]

[Grm7, m7, M7, AcM7]

And maybe also the octave displacements of those shall be blessed.

Next we judge all other intervals as being less consonant+fluid based on how many commas we have to traverse to reach a blessed one. We'll need to define a set of traversing commas for this - which steps are we making and counting to get from A to B. Let's work in rank-3 interval space / 5-limit just intonation and use (Ac1, A1, d2) as our traversing commas. This will let us move between any pair of intervals since this basis is unimodular in the rank 3 prime harmonic basis, (P8, P12, M17).

For a given interval B, you

    0) find the octave reduction of B, 

    1) take the difference of reduced-B with each blessed interval, 

    2) express the differences in the (Ac1, A1, d2) basis, 

    3) take the sum of the absolute values of the components of each difference interval

    4) take the minimum value of those sums as the score of dissonance. A larger score means the original interval was separated from the nearest blessed interval by more commas, and therefore was itself more dissonant.

Here's what that looks like for some rank-9 intervals:

1 : A1 # 25/24

1 : A4 # 25/18

1 : Ac1 # 81/80

1 : Acd4 # 162/125

1 : Acm2 # 27/25

1 : As1 # 33/32

1 : As4 # 11/8

1 : AsGrm2 # 88/81

1 : AsGrm3 # 11/9

1 : AsGrm6 # 44/27

1 : AsGrm7 # 11/6

1 : Asm2 # 11/10

1 : De5 # 16/11

1 : DeAcM2 # 12/11

1 : DeAcM3 # 27/22

1 : DeAcM6 # 18/11

1 : DeAcM7 # 81/44

1 : DeM7 # 20/11

1 : Dem7 # 96/55

1 : Gr5 # 40/27

1 : GrA1 # 250/243

1 : Pr1 # 65/64

1 : PrGrm7 # 65/36

1 : Prm2 # 13/12

1 : Prm3 # 39/32

1 : Prm6 # 13/8

1 : Re5 # 96/65

1 : ReAcM2 # 72/65

1 : ReM3 # 16/13

1 : ReM6 # 64/39

1 : ReM7 # 24/13

1 : Rsm2 # 512/475

1 : Sb5 # 35/24

1 : SbAcM2 # 35/32

1 : Sbm2 # 28/27

1 : Sbm3 # 7/6

1 : Sbm7 # 7/4

1 : Sp1 # 36/35

1 : SpM2 # 8/7

1 : SpM3 # 9/7

1 : d4 # 32/25

1 : d5 # 36/25

1 : d5 # 36/25

2 : AcA1 # 135/128

2 : AsGr1 # 55/54

2 : AsGrd7 # 44/25

2 : DeA1 # 100/99

2 : DeAc5 # 81/55

2 : DeSbAcM3 # 105/88

2 : DeSbm7 # 56/33

2 : ExA1 # 17/16

2 : FaA1 # 2375/2304

2 : Grd4 # 512/405

2 : Grd5 # 64/45

2 : PrDe5 # 65/44

2 : PrDem3 # 13/11

2 : PrDem7 # 39/22

2 : PrGrd7 # 26/15

2 : PrSp1 # 117/112

2 : PrSpm2 # 39/35

2 : Prd2 # 26/25

2 : ReA1 # 40/39

2 : ReAs1 # 66/65

2 : ReAsM2 # 44/39

2 : ReAsM3 # 33/26

2 : ReSb5 # 56/39

2 : ReSbAcM2 # 14/13

2 : ReSbM7 # 70/39

2 : SbAcm2 # 21/20

2 : Sbd4 # 56/45

2 : Sbd5 # 7/5

2 : Sbd7 # 42/25

2 : SpA1 # 15/14

2 : SpGr1 # 64/63

2 : Spd4 # 1152/875

3 : AcAcA1 # 2187/2048

3 : AcAcA4 # 729/512

3 : AsGrd5 # 22/15

3 : AsSbGrd7 # 77/45

3 : AsSpGr1 # 22/21

3 : AsSpGr1 # 22/21

3 : DeAcA1 # 45/44

3 : DeDeAcAA1 # 125/121

3 : DeSbAc5 # 63/44

3 : DeSpA1 # 80/77

3 : GrGrd5 # 1024/729

3 : PrDeSp1 # 78/77

3 : PrDed5 # 78/55

3 : PrGrd5 # 13/9

3 : PrPrd4 # 169/128

3 : PrSbGrd7 # 91/54

3 : PrSbd5 # 91/64

3 : PrSpGr1 # 65/63

3 : ReAcA1 # 27/26

3 : ReAsSb5 # 77/52

3 : ReDeAcAA1 # 150/143

3 : SbSbd7 # 49/30

3 : SpAcA1 # 243/224

3 : SpSpGr1 # 256/245

4 : AsAsGrd1 # 121/120

4 : AsSbGrd5 # 77/54

4 : AsSpGrd1 # 176/175

4 : DeSpAcA1 # 81/77

4 : PrAsGrd4 # 143/108

4 : PrDeSbd5 # 91/66

4 : PrDeSbd7 # 91/55

4 : ReAsAsGr1 # 121/117

4 : ReDeAcA1 # 144/143

4 : ReDeAcA4 # 192/143

4 : ReReAsA1 # 176/169

4 : ReReSbAcAA1 # 175/169

4 : ReSbAcA1 # 105/104

5 : DeDeAcAcA1 # 243/242

5 : PrAsSpGrd1 # 143/140

5 : PrPrSpGrd1 # 169/168

This looks really good to me - nothing is miscategorized. And the score is basically the number of adjectives in from of a 3-limit or 5-limit natural interval. Easy. Unfortunately, the categories aren't very granular, e.g. lots of things are 1 step of dissonance away from blessed. So how do we decide among them? Maybe that's where our notion of frequency ratio complexity comes into play.

For our measure of frequency ratio simplicity, I'm a little torn: it's dirt simple to only use numerator magnitude, but this neglects factor structure: a 3-limit Pythagorean Major Third justly tuned to 81/64 is a much more harmonically basic ratio than, e.g. a 13-limit justly tuned Recessed Major Third at 16/13. We could simply ignore the contribution of factors 2 and 3, but then all Pythagorean ratios would be equally consonant, which isn't right.

The first frequency ratio norm I've found that I like somewhat is this:

    norm = sum([abs(coordinate) * (primes[index] ** 3) for index, coordinate in enumerate(harmonic_coordinates)])

Here's how the function works: for a given ratio, find its prime factorization, and represent all the exponents for primes up to some limit as a vector. Here we have a 9 component vector representing exponents of prime factors up to 23, since 23 is the 9th prime: 

    81/80 :: [-4, 4, -1, 0, 0, 0, 0, 0, 0]

The norm for this ratio takes the absolute value of each vector component and multiplies it by the cube of the corresponding prime, then sums all those products:

    (4 * 2^3) + (4 * 3^3) + (1 * 5^3) = 265

Here are a few frequency ratios sorted by increasing norm value to give you an idea of how this norm behaves:

    [1/1, 2/1, 3/2, 4/3, 9/8, 81/64, 16/15, 10/9, 256/243, 81/80, 7/4, 11/4, 11/8, 11/10, 16/13, 13/12, 14/13, 17/16, 57/56]

I said we were working in 5-limit just intonation, so most of these ratios wouldn't show up, but I want to the norm to be well behaved at higher prime limits. If I write a detempering algorithm that works up to 23-limit, I'll use these as my traversing commas:

    [Ac1, A1, d2, Sp1, As1, Pr1, Ex1, Rs1, Nb1][81/80, 25/24, 128/125, 36/35, 33/32, 65/64, 51/50, 96/95, 46/45]

The fractionc complexity norm above does a good job of penalizing higher primes and letting Pythagorean ratios play first, or at least letting them play fairly soon. I like septimal ratios almost as much as Pythagorean ones, and would prefer it if 7/4 wasn't deemed less consonant than 81/80, but this is a good start. I guess I could just put some intervals with septimal just tuning into the blessed set if I wanted them to show up more.

Here are a few intervals that I already had defined in code sorted by the norm of their frequency ratios:

[P1, P8, P5, P4, AcM2, Grm7, AcM6, Grm3, M3, m6, M6, AcM3, m3, Grm6, M7, m7, m2, M2, AcM7, Grm2, Grd5, Gr5, AcAcA4, GrGrd5, AcA1, Ac1, AcAcA1, d4, A1, Grd4, A4, d5, Acm2, Sbm7, SpM2, Sbm3, SpM3, Sbm2, SpGr1, Sbd5, Acd4, SpA1, SbAcM2, SbAcm2, GrA1, SpAcA1, Sb5, Sp1, Sbd4, Sbd7, Spd4, SbSbd7, SpSpGr1, As4, De5, AsGrm7, DeAcM2, AsGrm3, DeAcM6, As1, DeAcM3, AsGrm6, DeAcM7, AsGrm2, Asm2, DeM7, AsGrd5, Dem7, DeAcA1, AsGr1, DeAc5, AsGrd7, DeA1, AsSpGr1, AsSpGr1, DeSbm7, DeSbAc5, AsSbGrd5, DeSpAcA1, DeSpA1, DeSbAcM3, AsSbGrd7, AsSpGrd1, Prm6, ReM3, Prm2, ReM7, PrGrd5, Prm3, ReM6, ReAcA1, PrGrd7, Pr1, ReA1, Re5, PrGrm7, ReAcM2, Prd2, ReSbAcM2, PrSbd5, ReSb5, PrSp1, PrSbGrd7, PrSpm2, ReSbM7, ReSbAcA1, PrSpGr1, DeDeAcAcA1, AsAsGrd1, DeDeAcAA1, PrDem3, PrDem7, ReAsM3, ReAsM2, ReDeAcA4, ReDeAcA1, PrAsGrd4, PrDe5, PrDed5, ReAs1, ReDeAcAA1, ReAsSb5, PrDeSbd5, PrDeSp1, PrDeSbd7, PrAsSpGrd1, PrPrd4, PrPrSpGrd1, ReAsAsGr1, ExA1, ReReSbAcAA1, ReReAsA1, Rsm2, FaA1]

It's kind of weird that an ascendant sub grave diminished unison, AsSbGrd5, is more complex than a prominent minor second, Prm2, but that's on me for definine a bad norm, I guess. But those are going to be in difference dissonance categories, and then the frequency ratio things disambiguates within the category, yeah? I could probably combine them numerically even.... Or continue thinking about detempering.

Wait, no, I've got it! I want the dissonance categories to remain, and the frequency ratios to disambiguate within them. And the dissonacne categories are integers. So I'll map the frequency ratio norm to the range [0, 1) and add that to the dissonance category. That way there's (probably) a total order over intervals and their just tunings, but also nothing gets too far away from its dissonance category. So we need a function that takes [0, inf) to [0, 1), like

f(x) = x / (1 + x)

f(x) = 1 - e^(-x)

f(x) = tanh(x)

f(x) = 2/π * arctan(x)

I tried the first function. It works okay. I notice that since I didn't inclode tritones in the blessed set (intentionally) and since Pythagorean tritones 

AcAcA4 # 729/512

GrGrd5 # 1024/729

have large numerators and lots of factors and they're quite a few commas away from natural intervals like P4 and P5, they get quite high dissonance ratings, whereas the 5-limti A4 and d5 do not. I'm not sure if this is a bug, but I think so. I remember I once listened to tons of different intonations of diminished chords and I thought a pythagorean diminished triad

    [P1, Grm3, GrGrd5]

was particularly beautiful. So the fact that none of my methods like GrGrd5 is a bit of a flaw.

Anyway, regardless of their quality, we now have concrete notions of consonance+fluidity and frequency ratio simplicity.

For a melody in 12-TET, we have oh, 5 or 9 or however many interval options per tempered frequency ratio, and we want to make interval selections for all of our notes to minimize melodic disfluidity/dissonance, and if multiple choices of intervals would leave us with same amount of disfluidity, then we want to minimize the frequency ratio complexity of the just tunings of individual notes.

This is almost solvable, but still a little ill-posed. Having a finite search space helps a lot. How about this: given two interval sequences with equal melodic disfluidity, sum the frequency ratio complexities for the just tunings of all the intervals in each detempering / melodic interval reconstruction. I think this probably won't work well: 2/1 has a normed value of 8, while 16/13 has a norm of 2229. These are really different magnitudes, so the sum of ratio complexities for a detempered melodic line is going to be dominanted by single notes. Maybe it would work, but it seems unlikely. I guess I could take a logarithm of the norm to make things more similarly sized. I'll try it both ways and see what gives better results.

That's a partial solution to detempering, yeah? You could code this up and it would find you an intervallic melody from a 12-TET melody (or 19-TET or 53-TET or meantone or whatever, with a litle tweaking).

That's very exciting to me! It's a big conceptual improvement over mapping 1 ratio to 1 interval. Maybe it won't work that well, but it's got moving parts that can be adjusted until it does.

Let's think about polyphonic music next.

Suppose you have two voices. I care more about harmonic consonance than melodic fluidity, so at every moment, we're going to have pure harmony, and if there are some wonky melodic steps to get there, that's fine.

I've been thinking that my notion of favoring a low frequency ratio complexity in the just tunings of notes over a reference pitch will tend to limit comma drift, but comma drift is beautiful and not something we should limit. We should drift all over frequency space, wherever harmony and fluidity take us, regardless of our starting pitches.

...

I think for a given passage in a song, you're going to have one melody that's most important, and it's going to have the most fluid melodic intervals, and other notes of other voices will adapt around it to make good harmony, even if this makes their melodies less fluid. Maybe everyone can move fluidly and maintain harmony, but I doubt it, and if that's the case, then one star voice at a time will maintain the greatest fluidity.

...

Ornaments Over Major Third

In my post on playing jazz piano from a lead sheet, I shared some melodic substitions rules that I learned from Shan Verma. If you have a melodic passage outlining a major third or minor third, up or down, Shan has some ornaments you can put there to add some movement and variety. His riffs were called triplet major (up/down), triplet minor (up/down), chromatic major (up/down), and standard minor (up/down). Now, I was a little bothered that I didn't know a Verma riff called standard major up or down. If he has one, it's not available in his free materials. Maybe if you pay for his courses. There also isn't a free chromatic minor up or down, but that didn't bother me as much. I want to know the standard stuff first.

So I made up some more riffs for major thirds. Suppose we're in 4/4 time and we go from C for two beats, up a major third to E for some amount of time, let's say also two beats. We'll replace the C with four eighth note and keep E with the same duration, giving us a new rhythm of [e e e e h]. Here are some pitch options and what I call the riffs: 

[C B C D E] # standard major

[C D B C E] # enclosed major

[C D E Eb E] # neighbor major

I like all of these. What's more, if you play them in reverse, going from E down to C, I still like them, and continue to think the names are appropiate.

To put these and the other Verma riffs into code, I wrote some functions. One of them does diatonic offsets, i.e. given a pitch and a scale, it takes you up or down the scale by some number of notes. If the given note isn't in the scale, the first offset gets you back onto the scale on the nearest note in the direction  you want to go (i.e. the nearest note changes depending on if the offset supplied to the function is positive or negative). I also wrote a function that finds chromatic offsets, which mostly just feeds a chromatic scale into the diatonic offset function.

With these functions, we can define the riffs relative to the start note in terms of scalar motion and chromatic motion. This is kind of cool because the riff can adapt depending on where you are in the scale: it doesn't have to have a fixed intervallic structure, though it also could. Like, if you're doing a riff over an ascending minor third in C major, the diatonic middle note could be a major second or a minor second up from the starting note.

[D, E, F] : [M2, m2]

[A, B, C] : [M2, m2]

[E, F, G] : [m2, M2]

[B, C, D] : [m2, M2]

And things get even crazier if your scales are microtonal.

In the major riffs I shared in terms of pitches, the standard major, the enclosed major, and the neighbor major, most of the pitches could be thought of as diatonic in C: neighbor major has a chromatic note, but the rest could come from a C major scale. But some of the B notes below C, maybe I should think of those as a chroamtic offset below the starting pitch. This is the difference between a standard major triad riff up on G in the key of C major looking like:

     [G F G A B] or [G F# G A B]

I intend to try interpreting the B notes in both Standard Major and Enclosed Major as diatonic or chromatic offsets and see which one I like better.

...

I came up with some minor riffs:

[A, G#, A, B, C] # low neighbor minor

[A, B, C, D, C] # high neighbor minor

[A, B, G, A, C] # low enclosure minor

[A, B, D, B, C] # high enclosure minor

[A, B, C, B, C] # trill minor

I think the first three of these continue to sound very good in reverse, and the last two also sound okay. I'm going to code them all up.

I should probably add a few more figures with triplets too.

My plan is that I'll 

    1) generate a diatonic skeleton for a melody or bass line in terms of motions by 2nds and 3rds.

    2) generate whichever of those I didn't in step 1, favoring contrary motion

    3) hopefully find notes for two inner voices that fit with a chord

    4) take turns ornamenting the melody or an inner voice with riffs from Verma and myself, hopefully respecting most species counterpoint rules, but I've made very boring music in the past by respecting all of them, so maybe I'll allow some violations. Generate a few and score them by their number of violations of soft rules or something. I don't know.

I don't know how to get interesting rhythmic variation.

I have an idea of representing a whole piece of music in terms of determinstic transformations, randomly selected. We'll see how far that gets.

...

Maybe I should be calling these embellishments or diminutions or variations instead of ornaments. I'm mostly calling them riffs though. It's fine.

...

Okay! I have four part harmony. That's totally trivial for me to generate. Chord, Chord, Chord, infinitely, voiced in SATB.

Then I can add one step of diatonic motion in half notes within that chord's measure for the upper voices and probably still satisfy counterpoint rules. Maybe all four voices.

Then I can embellish one upper voice at a time, usually the soprano. When I embellish another voice, it will be a response that's related to a previous soprano line by some kind of transformation - perhaps a new theme generated in contrary motion, perhaps an inversion of reversal or transposition or something.

The lower voices will usually be playing half notes at this point. Which is too boring. In the past, I have sometimes satisfied myself with the inner voices playing whole notes while an interesting melody and bassline move in contrary motion, but I want to figure out how to add some rhythm to inner lines, so let's try that now.

Suppose I put a generative distribution over beats and smaller subdivisions of each measure. It might look like "Note onsets are likely on [1, and_of_3, 4]. Somewhat likely on [2, a_of_4]". Every voice has two half note pitches per measure (generated by diatonic motion from the original SATB chord voicing for the measure and respecting species counterpoint rules), and they can associate these two pitches with the set of note onsets times produced by sampling the distribution. And then, uh, for note durations, you just run up to the next note onset, unless that would require a weird note length, and then a rest fills the gap.

So if the Alto's diatonic steps say to play G and then A, then the rhythm might go, 

    [G3,q G3,dq A3,e R,e R,s A3,s]

where I'm using a comma to combine a pitch and its note length (or a rest, R, and it's note length).

Once that's done, maybe we can go back and add passing notes or replace the A3 at the end with a chromatic approach to a note in the next measure of something.

...

I'm suddenly realizing that this post in called "ornaments over major third" but I'm now so far off on a tangent as to be outlining a whole generative music progam. But no one reads my blog, so I'll post what I want.

...