Prime Harmonic Bases and Comma Bases

: Two Interval Coordinates Systems Of Great Utility

There are two very convenient ways to represent intervals as coordinates. The first combines intervals that correspond to frequency ratios of prime harmonics. For example, the octave, P8, is justly tuned to a frequency ratio of 2/1 and the perfect-twelfth is justly tuned to a frequency ratio of 3/1. The major seventeenth, in 5-limit just intonation and and higher prime-limit tuning systems, is justly tuned to a frequency ratio of 5/1. Primes, all. Not all micro-tonal music theorists will agree on the names for intervals, but they can all agree what prime factors make up rational frequency ratios and this is the start of a bridge for communication.

The second system that is very convenient way to represent intervals as integer coordinates is a comma basis. Commas are intervals that are usually tuned to small values. In the history of music theory, we've often named new intervals based on how their tuned values differ by a small amount from the frequency ratios of known intervals. For example, a diminished second is a factor of 2187/2048 less than a minor second in Pythagorean tuning (or a factor of 25/24 less in 5-limit just intonation and higher prime-limit systems). Comma bases are useful for figuring out the names of intervals. In this text, we've already use two comma bases that were employed by the music theorist theorist E. J. Lilley  for a rank-2 and rank-3 interval spaces, and in this post we'll extend Lilley comma bases to rank-4 and rank-5 and rank-6 interval spaces.

(A third fairly common system is to use intervals that are justly tuned to octave-reduced prime harmonics, i.e. a prime divided by the largest integer power of two smaller than that prime, so that the frequency ratio lies in the range (1, 2). For example, the perfect fifth , P5, is justly tuned to 3/2, an octave-reduced prime. Likewise, the major third, M3, in five-limit just intonation and above, is tuned to 5/4, another octave-reduced prime. And so (P8, P5) is a common basis for rank-2 interval analysis and (P8, P5,  M3) is a common basis for 5-limit just intonation. We'll talk about this one less, but it's intimately related to the prime harmonic basis. Easy conversion.

The Lilley basis for rank-2 intervals consists of the augmented unison and the diminished second, (A1, d2). For rank-3 interval space, the Lilley basis consists of the acute unison, the augmented unison, and the diminished second, (Ac1, A1, d2). In this post we'll formulate a comma basis in the style of Lilley for rank-4 and rank-5 and rank-6 interval spaces. These can respectively be used to name intervals that are justly tuned to 7-limit and 11-limit and 13-limit frequency ratios.

Ultimately, music is made of frequencies and all this interval talk is an abstraction that's derived from them. But once we have discovered the abstraction, we find that it's a useful abstraction, and we can use it without reference to frequencies in order to do cool things with music, both analytical and generative. You can analyze and compose music in a way that doesn't depend on the tuning system.

Thus it may be more true to history to start by learning about prime harmonic bases, the bases that are based on harmonic frequency ratios. Nevertheless, we shall ignore the course of history and learn first about (Lilley) comma bases, so that we can learn how to name intervals. Then we'll return to prime bases and see how convenient they are for finding intervals that are tempered-out in EDO tuning systems. But we must learn that intervals are different from frequency ratios before we talk about tempering them out, or else we'll make the mistake of talking about tempering out frequency ratios, which is simply mathematical contradiction, though one that we frequently observe microtonal music theorists expressing.

:: Lilley-Johnston Comma Bases

We've already seen the comma bases of (A1, d2) and (Ac1, A1, d2) being fruitfully applied for rank-2 and rank-3 intervals, respectively. Rank-4 intervals are great for expressing septimal just intonation (with prime factors up to 7) , and rank-5 is good for undecimal just intonation (with prime factors up to 11). I've never found much user for higher primes, but other people have, and more power to you if you have principles for using them. I hope you'll write them down and share.

: The Septimal Comma and Rank-4 Intervals

We have some freedom in choosing commas to extend the rank-3 Lilley set. Like, it's traditional to call septimal intervals thing like "sub minor seventh" and "super major third", so we  know that the septimal qualities should be "Sub" and "Super", much as we added the adjectives "Grave" and "Acute" as interval qualities when we went up to 5-limit just intonation. But what frequency ratio should a justly tuned septimal super unison be? Difference choices of frequency ratio will mean alter which interval names get assigned to which frequency ratios.

Initially, I tried (21/20), and I still liked it a bit. Since a 5-limit M6 is tuned to (5/3), and 

(5/3) * (21/20) = (7/4),

this system gives the reduced seventh harmonic the name "super major sixth", SpM6. The frequency ratio 21/20 is 84 cents, which is kind of large to call a comma, but I still like it.

Then I tried (64/63) as the tuned value for a septimal super unison. I got that idea from Wikipedia. It's also used as the frequency ratio for the septimal accidental in Helmholtz-Ellis staff notation. Since a 5-limit Grm7 is tuned to (16/9) and 

(16/9) / (64/63) = (7/4)

this system calls the reduced seventh harmonic a sub grave minor seventh, SbGrm7.

Later, I found out that (36/35) is really nice for analyzing 24-EDO / quartertones, which I'd been struggling with. The great composer Ben Johnston used this as the tuned value for the septimal accidental in his staff notation. Since a 5-limit justly tuned m7 has a frequency ratio of (9/5) and

(9/5) / (36/35) = (7/4)

this system calls a reduced seventh harmonic a sub-minor seventh, Sbm7. This is a nice short name for a simple interval, so between that and (36/35)'s utility for analyzing 24-EDO and my love of Ben Johnston, I've never looked back. Here are the comma and its inverse as they are justly tuned:

t(Sp1) = 36/35

t(Sb1) = 35/36

The 7th harmonic is two octaves above the reduced harmonic, so the harmonic is sub-minor twenty first, Sbm21st, if my arithmetic is right. You'd think you could do (7 + 8 + 8), but intervals have this off-by-one error in their names (like the unison not being called a 0th), so you do (6 + 7 + 7) and then add 1 back on at the end.

: The Undecimal Comma and Rank-5 Intervals

I haven't done much with 11-limit just intonation, but every just intonation staff notation system that I know of uses (33/32) as the frequency ratio for the undecimal accidental, including Ben Johnston's notation and Helmholtz Ellis notation. Let's use it as well. Easy. The only thing left for us to do is decide on adjectives for the interval qualities. I suggest we associate the undecimal comma with the adjectives (ascendant | descendant). Here's the comma and its inverse, as they are justly tuned:

t(As1) = (33/32) 

t(De1) = (32/33)

Since a perfect fourth is tuned to (4/3) in any of our systems and

(4/3) * (33/32) = (11/8)

we can say that the reduced 11th harmonic is an ascendant fourth.

The 11th harmonic, (11/1), is three octaves above the ascendant fourth, so if my arithmetic is right, it's a ...(3 + 7 + 7 + 7 + 1) , an ascendant 25th, As25. Nice.

: The Tridecimal Comma and Rank-6 intervals

I don't know what tridecimal frequency ratios are good for, but I wanted to associate them with rank-6 intervals all the same. Maybe having names rank-6 intervals will help me work with them more so that I can get a feel for the uses of 13-limit frequency ratios. I suggest (prominent | recessed), notated as Pr and Re, as the qualities for the the comma and its inverse. But what frequency ratio for the tridecimal comma? Ben Johnston used (65/64). Helmholtz-Ellis uses (27/26). I don't have strong reasons for choosing either of them or anything else. I've heard that Ben Johnston's accidental is good at turning 5-limit frequency ratios into nearby 13-limit ones, and the Helmholtz-Ellis accidental is good at turning 3-limit Pythagorean frequency ratios into near-by 13-limit ones. And 5-limit just intonation is the one true way, so sure, let's go with Johnston again. I also slightly prefer his whole system on aesthetic grounds, like that he uses super-particular ratios for all of his prime commas, and also some stuff about the factors included in the frequency ratios. So, here's are the tridecimal comma and its inverse as I justly tune them:

t(Pr1) = 65/64

t(Re1) = 64/65

Since a justly tuned minor sixth, m6, has a frequency ratio of (8/5) and 

(8/5) * (65/64) = (13/8)

we can say that the reduced thirteenth harmonic is a justly tuned prominent minor sixth. Prm6. Nice.

The thirteenth harmonic is three octaves above Prm6, so it' s a... (5 + 7 + 7 + 7 + 1) ...prominent-minor twenty-seventh, Prm27. Well done, us.

: All Together Now

Here's the full set of commas that we'll use in this text, for 5-limit just intonation and higher:

t(Ac1) = 81/80

t(A1) = 25/24

t(d2) = 128/125

t(Sp1) = 36/35

t(As1) = 33/32

t(Pr1) = 65/64

As we successively add Sp1, As1, and Pr1 onto the rank-3 Lilley basis, (Ac1, A1, d2), I think it's right and fair the call the produced bases "Lilley-Johnson comma bases" of different ranks. They're pretty great. I also think it's kind of funny that Johnston's tridecimal comma, (65/64), is so close to the Helmholtz-Ellis septimal comma, (64/63).

Here is a program for naming intervals expressed in the rank-6 Lilley-Johnston comma basis.

:: Prime Bases

Okay, we're ready! Prime harmonic bases are just made of the intervals that we tune to the prime harmonics. If we go up to rank 13, then in the order (2, 3, 5, 7, 11, 13), they are:

(P8, P12, M17, Sbm21, As25, Prm27) 

They're highly useful, let me tell you.

First, to find the coordinates of an interval that is justly tuned to a given frequency ratio, you basically just have to factorize the fraction. If a frequency ratio can be expressed as (2^a * 3^b * 5^c * 7^d), with integers in the exponents, then there's an interval in the rank-4 prime harmonic basis with coordinates (a, b, c, d) that is justly tuned to that frequency ratio. It's so easy and convenient to work with this basis that people can do interesting intervallic math in it without realizing that intervals are different from frequency ratios.

Once you have the coordinates for an interval in the prime harmonic basis, how do you find the name of the interval? We perform a change of basis to the Lilley-Johnston comma basis of the same rank and use our naming functions from there.

Here's how change of basis generally works between tuning systems: you find the basis intervals of the old system expressed in terms of the basis vectors of the new system. Let's go all the way up to rank-6 right away and express the old prime harmonic basis vectors (P8, P12, M17, Sbm21, As25, Prm27) in the rank-6 Lilley-Johnston basis, (Ac1, A1, d2, Sp1, As1, Pr1) , which is element-wise justly tuned to ((81/80), (25/24), (128/125), (36/35), (33/32), (65, 64)). Here they are:

P8 : (3, 12, 7, 0, 0, 0) # 2/1

P12 : (5, 19, 11, 0, 0, 0) # 3/1

M17 : (7, 28, 16, 0, 0, 0) # 5/1

Sbm21 : (9, 34, 20, -1, 0, 0) # 7/1

As25 : (10, 41, 24, 0, 1, 0) # 11/1

Prm27 : (11, 44, 26, 0, 0, 1) # 13/1

A nice thing I just notices about this system: a normal major major triad has the intervals [P1, M3, P5] which are tuned in five limit just intonation to (4/4, 5/4, 6/4). If we continue the harmonic series upward while dividing through by four, then we get [P1, M3, P5, Sbm7, P8, AcM9, M10, As11, P12, Pr13]  tuned to (4:5:6:7:8:9:10:11:12:13).  All the other twice reduced harmonics are associated with intervals that have the same ordinal! The M10 is tuned to 10/4. The As11 is tuned to 11/4. So fourth! What beautiful commas Ben Johnston selected. I wonder if the trend continued with higher primes. Probably not? I think that would mean that 17/16 would have to be some kind of a tuned third. Anyway.

Now that we have the old basis vectors expressed in the new tuning system, we can write a change of basis function.

def convert_rank_6_prime_harmonic_basis_to_comma_basis(interval1):

old_basis_intervals = [

[3, 12, 7, 0, 0, 0], # t(P8) = 2/1

[5, 19, 11, 0, 0, 0], # t(P12) = 3/1

[7, 28, 16, 0, 0, 0], # t(M17) = 5/1

[9, 34, 20, -1, 0, 0], # t(Sbm21) = 7/1

[10, 41, 24, 0, 1, 0], # t(As25) = 11/1

[11, 44, 26, 0, 0, 1], # t(Prm27) = 13/1

]

interval2 = [sum(interval1[i] * old_basis_intervals[i][j] for i in range(len(interval1))) for j in range(len(old_basis_intervals))]

return interval2

To instead change in the other direction, from coordinates in the comma basis to coordinates in the prime harmonic basis, we only need to invert this matrix.  

def convert_rank_6_comma_basis_to_prime_harmonic_basis(interval1):

old_basis_intervals = [

[-4, 4, -1, 0, 0, 0], # t(Ac1) = 81/80

[-3, -1, 2, 0, 0, 0], # t(A1) = 25/24

[7, 0, -3, 0, 0, 0], # t(d2) = 128/125

[2, 2, -1, -1, 0, 0], # t(Sp1) = 36/35

[-5, 1, 0, 0, 1, 0], # t(As1) = 33/32

[-6, 0, 1, 0, 0, 1], # t(Pr1) = 65/64

]

interval2 = [sum(interval1[i] * old_basis_intervals[i][j] for i in range(len(interval1))) for j in range(len(old_basis_intervals))]

return interval2

We could have also found rows of this matrix individually by figuring out the prime factorizations the justly tuned frequency ratio associated with each comma. For example, I've written a comment after the first row of the matrix

[-4, 4, -1, 0, 0, 0]

reminding us of the Acute Unison comma, which is appropriate because 

(2^-4) * (3^4) * (5^-1) * (7 ^0) * (11^0) * (13^0) = 81/80.  

In addition to prime harmonic bases being good for finding coordinates of intervals that are justly tuned to desired fractions, they're really good for...

: High-Rank EDO Analysis

You can tune high-rank intervals to a 1-dimensional line of EDO values like 12-EDO just the same as you can tune rank-2 intervals. There's probably a way to do it that's analogous to the rank-2 case where you tune the octaves purely and then you temper out enough intervals to cover all the remaining dimensions of the intervals space. I suppose you'd need 2 tempered commas for rank-3 tuning systems, and 3 tempered commas for rank-4 interval space and so on. And there's probably one simplest 2-limit comma and one simplest 3-limit comma and so on. So each time you increase the dimension, you get a new tempered comma to add to the set. But instead of using commas, you can use a prime harmonic basis, and it's so easy that it's stupid. I felt like a dirty cheater when I figured out how to do this. It's sickening. You're going to love it.

Let's say we want to find 13-limit intervals that are tempered out by, say, 19-EDO.

First, list the rank-6 intervals that are justly tuned to each prime harmonic (2, 3, 5, 7, 11, 13):

(P8, P12, M17, Sbm21, As25, Prm27)

Now instead of tuning these justly, we tune them to the nearest step of 19-EDO that approximates the just frequency ratio. We'll do one together. The perfect 12th, P12, is justly tuned to (3/1). If this is expressible as a step of 19-EDO, the step will be an integer {i} such that

2^(i/19) = (3/1)

We take logarithms of both sides 

i/19 = log_2(3)

and multiply both sides by 19 to solve for {i},

i = 19 * log_2(3) ~ 30.114

and alas it's not an integer, but pretty close. So we'll invent a tuning system in which P12 is tuned to 30 steps of 19 edo,

t(P12) = 2^(30/19)

It's slightly mistuned relative to the just-version, but every EDO mistunes intervals in service of perceptually equals spacing. That's what they do.

The formula takes the same form for all the harmonic basis intervals. Let's call the numerator in the exponent of the Nth harmonic {i_n}. Then

i_2 = round(19 * log_2(2)) = 19

i_3 = round(19 * log_2(3)) = 30

i_5 = round(19 * log_2(5)) = 44

i_7 = round(19 * log_2(7)) = 53

i_11 = round(19 * log_2(11)) = 66

i_13 = round(19 * log_2(13)) = 70

If we'd been working in 31-EDO, then all of those 19s would be 31s.

To find the number of steps of 19-EDO for an arbitrary rank-5 interval with coordinates (a, b, c, d, e, f) expressed in the rank-6 prime harmonic basis, (P8, P12, M17, Sbm21, As25, Prm27), we just multiply the interval component (like {b}) corresponding to a prime (like 3/1) by the tuned value for interval associated with the prime (like 30 steps of 19-EDO) and sum all of these up:

steps = 19 * a + 30 * b + 44 * c + 53 * d + 66 * e + 70 * f

We'll call this the harmonic EDO tuning formula. Any rank-6 interval (a, b, c, d, e, f) which makes {steps = 0} above will be tempered out by 19-EDO. Or perhaps I should say, "in the rank-6 19-EDO tuning system that we defined by detuning the intervals justly associated with prime harmonics", but that's a mouthful and it's not worth saying until I learn of a comparably good way to define a rank-5 19-EDO that isn't equivalent to this. Of course, with this math you can also find intervals that are tuned to 1 step of 19-EDO or 33 steps or anything else. The point is, harmonic bases let you analyze intervallically and justly the harmonic and melodic functions of weird EDO tuning systems.  You can find out what justly tuned harmonies and melodies they're approximating. You can find out what the song would be like in a different tuning. You can talk about classes of EDO tuning systems that all temper out the septimal super unison or whatever. It's weirdly powerful given that the math is just rounding logarithms to find nearby integer exponents.

Anyway, 

• Find interval coordinates (a, b, c, d, e, f) that make the Harmonic EDO Tuning Formula equal zero steps, just by doing a brute force search of possible component values like {a} from -8 to 8  and so on. These are coordinates in the prime harmonic basis for intervals that are tempered out by your EDO. You can also tune these justly right away if you want to see some nifty frequency ratios.

• Convert the interval coordinates to the rank-6 Lilley-Johnson basis using a change of basis function. This will make it easy to name the intervals.

• Name the interval using the program I linked above on my github gists page.

I did all of that for you! Here are some tempered out intervals of 19-EDO, sorted by increasing complexity of the just frequency ratios, rather than by the shortness of their names:

DeAcA1 = [1, 1, 0, 0, -1, 0] # 45/44

SbSbAcm2 = [1, 1, 1, -2, 0, 0] # 49/48

DeSbAcm2 = [1, 1, 1, -1, -1, 0] # 56/55

PrDeSp1 = [0, 0, 0, 1, -1, 1] # 78/77

Ac1 = [1, 0, 0, 0, 0, 0] # 81/80

DeA1 = [0, 1, 0, 0, -1, 0] # 100/99

DeDeAcAA1 = [1, 2, 0, 0, -2, 0] # 125/121

DeDeSbAcAcM2 = [2, 2, 1, -1, -2, 0] # 126/121

SbAcd2 = [1, 0, 1, -1, 0, 0] # 126/125

SpA0 = [0, 0, -1, 1, 0, 0] # 225/224

SbSbm2 = [0, 1, 1, -2, 0, 0] # 245/243

AsSb1 = [0, 0, 0, -1, 1, 0] # 385/384

DeSpSpA0 = [0, 0, -1, 2, -1, 0] # 540/539

SbSbSbdd3 = [1, 1, 2, -3, 0, 0] # 686/675

DeAcAcA1 = [2, 1, 0, 0, -1, 0] # 729/704

DeSbSbAcAcM2 = [2, 2, 1, -2, -1, 0] # 735/704

DeDeSpAcA1 = [1, 1, 0, 1, -2, 0] # 864/847

SbA1 = [0, 1, 0, -1, 0, 0] # 875/864

DeSbm2 = [0, 1, 1, -1, -1, 0] # 896/891

In addition to this big list, I also looked for the simplest tempered interval that was 3-limit, 5-limit, 7-limit, and so forth. The 5-limit ones and higher all showed up in the list above. But the Pythagorean 3-limit one was ridiculously long.  

AcAcAcAcAcAA0 = [5, 1, -1, 0, 0, 0] # 1162261467/1073741824 // 3-limit

Ac1 = [1, 0, 0, 0, 0, 0] # 81/80 // 5-limit

SbSbAcm2 = [1, 1, 1, -2, 0, 0] # 49/48 // 7-limit

DeAcA1 = [1, 1, 0, 0, -1, 0] # 45/44 // 11-limit

PrDeSp1 = [0, 0, 0, 1, -1, 1] # 78/77 // 13-limit

The first one has a rank-3 (or higher) interval name, because 5-limit just intonation is the One True Way. But the just frequency ratio is still 3-limit: it's factorization is (3^19)/(2^30). You can think of it as 11 octaves below a stack of 19 perfect fifths, i.e. its coordinates are (19, -11) in the (P5, P8) basis. Its rank-2 interval name is actually the rank-3 name without all the "acute" prefixes: it's just "AA0", an "augmented-augmented zeroth". This has coordinates (1, -1) in the (A1, d2) basis. But it's tuned to a stupidly complex ratio and it deserves a stupid complex name. That's a feature of 5-limit just intonation, not a bug.

Anyway, 19-EDO tempers out lots of intervals, like the Pythagorean AA0 and the Acute unison (a.k.a. the syntonic comma). Cool, right? That's the power of the prime harmonic basis!

Not impressed yet? Let's look at 12-EDO. Here are some rank-6 intervals that are tempered out by 12-EDO, sorted by increasing complexity of their justly tuned frequency ratios:

Prd2 = [0, 0, 1, 0, 0, 1] # 26/25

Sp1 = [0, 0, 0, 1, 0, 0] # 36/35

DeAcA1 = [1, 1, 0, 0, -1, 0] # 45/44

SpSpGrA0 = [-1, 0, -1, 2, 0, 0] # 50/49

PrSpSpGr1 = [-1, 0, 0, 2, 0, 1] # 52/49

DeSbAcm2 = [1, 1, 1, -1, -1, 0] # 56/55

SpGr1 = [-1, 0, 0, 1, 0, 0] # 64/63

PrSpGr1 = [-1, 0, 0, 1, 0, 1] # 65/63

Pr1 = [0, 0, 0, 0, 0, 1] # 65/64

DeSpA1 = [0, 1, 0, 1, -1, 0] # 80/77

DeSpAcA1 = [1, 1, 0, 1, -1, 0] # 81/77

Ac1 = [1, 0, 0, 0, 0, 0] # 81/80

PrDeSbAcm2 = [1, 1, 1, -1, -1, 1] # 91/88

PrSbd2 = [0, 0, 1, -1, 0, 1] # 91/90

AsSpSpGrM0 = [-1, -1, -1, 2, 1, 0] # 99/98

DeA1 = [0, 1, 0, 0, -1, 0] # 100/99

PrDem2 = [0, 1, 1, 0, -1, 1] # 104/99

PrDeAcm2 = [1, 1, 1, 0, -1, 1] # 117/110

PrSp1 = [0, 0, 0, 1, 0, 1] # 117/112

DeDeAcAA1 = [1, 2, 0, 0, -2, 0] # 125/121

DeDeSbAcAcM2 = [2, 2, 1, -1, -2, 0] # 126/121

SbAcd2 = [1, 0, 1, -1, 0, 0] # 126/125

DeDeAcM2 = [1, 2, 1, 0, -2, 0] # 128/121

d2 = [0, 0, 1, 0, 0, 0] # 128/125

PrDeDeAcM2 = [1, 2, 1, 0, -2, 1] # 130/121

PrAsSpGrd1 = [-1, -1, 0, 1, 1, 1] # 143/140

ReDeAcA1 = [1, 1, 0, 0, -1, -1] # 144/143

PrPrDeSpm2 = [0, 1, 1, 1, -1, 2] # 169/154

PrPrd2 = [0, 0, 1, 0, 0, 2] # 169/160

PrPrGrd2 = [-1, 0, 1, 0, 0, 2] # 169/162

AsSpGrd1 = [-1, -1, 0, 1, 1, 0] # 176/175

SpA0 = [0, 0, -1, 1, 0, 0] # 225/224

DeDeSbSbAcAcM2 = [2, 2, 1, -2, -2, 0] # 245/242

SpSpGr1 = [-1, 0, 0, 2, 0, 0] # 256/245

DeAcm2 = [1, 1, 1, 0, -1, 0] # 288/275

The intervals with 5-limit frequency ratios will look familiar to you:

Ac1 = [1, 0, 0, 0, 0, 0] # 81/80

d2 = [0, 0, 1, 0, 0, 0] # 128/125

since they're two of the commas in the rank-3 Lilley basis. Stacks and combinations of these will also be tempered out.

Here are the intervals with 7-limit frequency ratios from the big set above:

Sp1 = [0, 0, 0, 1, 0, 0] # 36/35

SpSpGrA0 = [-1, 0, -1, 2, 0, 0] # 50/49

SpGr1 = [-1, 0, 0, 1, 0, 0] # 64/63

SbAcd2 = [1, 0, 1, -1, 0, 0] # 126/125

SpA0 = [0, 0, -1, 1, 0, 0] # 225/224

SpSpGr1 = [-1, 0, 0, 2, 0, 0] # 256/245

We see that the simplest interval, in terms of the complexity of its just frequency ratio, that is tempered out by 12-EDO is Johnston's septimal comma.

Here are the simplest 11-limit ones:

DeAcA1 = [1, 1, 0, 0, -1, 0] # 45/44

DeSbAcm2 = [1, 1, 1, -1, -1, 0] # 56/55

PrSpGr1 = [-1, 0, 0, 1, 0, 1] # 65/63

DeSpA1 = [0, 1, 0, 1, -1, 0] # 80/77

Johnston's undecimal comma with a frequency ratio of 33/32 doesn't make an appearance: 12-EDO tunes it to 1 step, not 0 steps.

Here are the simplest intervals with 13-limit ratios:

Prd2 = [0, 0, 1, 0, 0, 1] # 26/25

PrSpSpGr1 = [-1, 0, 0, 2, 0, 1] # 52/49

Pr1 = [0, 0, 0, 0, 0, 1] # 65/64

The power of the prime harmonic basis!

Not impressed yet? Check this out. I wanted a 7-limit interpretation of 19-EDO. I started by finding simple 7-limit frequency for each step. Most of the steps matched their octave complements, but a few steps came in pairs where either the original was simpler or the complement was simpler. When this happened, I defaulted to the simpler interval name, or the one that increased numerically (e.g. 3rds coming after 2nds coming after 1sts). After doing all of that, I got this beauty:

0 - P1 # 1/1

1 - Sbm2 # 28/27

2 - m2 # 16/15

3 - M2 # 10/9

4 - SpM2 # 8/7 or Sbm3 # 7/6

5 - m3 # 6/5

6 - M3 # 5/4

7 - SpM3 # 9/7

8 - P4 # 4/3

9 - Sbd5 # 7/5

10 - SpA4 # 10/7

11 - P5 # 3/2

12 - Sbm6 # 14/9

13 - m6 # 8/5

14 - M6 # 5/3

15 - SpM6 # 12/7 or Sbm7 # 7/4

16 - m7 # 9/5

17 - M7 # 15/8

18 - SpM7 # 27/14

19 - P8 # 2/1

And, yeah, okay, it's still not a fully defined scale on the 4th and 15th steps, but it's a damn fine start. You could make an arbitrary choice to use (SpM2 and Sbm7) or (Sbm3 and SpM6), then tune a keyboard with 19-keys per octave to this scale, and jam out. I think I prefer the former option, since Sbm7 is justly tuned to a reduced 7th harmonic. It's an umportant thing to have if you're doing septimal justly intoned music.

The power of the prime harmonic basis!

Let's do one more trick. Suppose you're enamored with the idea of having the number 22 in the denominator of your frequency ratios for some reason. "Otonal harmony" it's called, regardless of the value of the denominator. We can make an intervallic scale for you that tempers out to 12-TET.

Here are 13-limit justly tuned frequency ratios with factors of 11 or 12 in the denominator for some rank-6 intervals, along with the 12-EDO steps to which they're tuned. I've got the 12-EDO step at the start of the line, then the just frequency ratio, and finally the interval name at the end:

0: 1/1 (i.e. 11/11) (P1)

1: 12/11 (DeAcM2)

2: 13/11 (PrDem3) or 25/22 (DeAcA2)

3: 27/22 (DeAcM3)

4: 14/11 (DeSbAc4)

5: 15/11 (DeAcA4)

6: 16/11 (De5)

7: 3/2 (i.e. 33/22) (P5)

8: 18/11 (DeAcM6) or 35/22 (DeSbAcM6)

9: 39/22 (PrDem7)

10: 20/11 (DeM7)

11: 21/11 (DeSbAc8)

12: 2/1 (i.e. 22/11) (P8)

There are a few conspicuously missing high-prime numerators because we're only using 13-limit frequency ratios:

[23/22, 29/22, 31/22, 17/11, 37/22, 19/11, 41/22, 43/22]

If you didn't work in interval space, you might be tempted to make an 11-limit version of 12-EDO by multiplying the irrational frequency ratios of 12-EDO by 22 and rounding to the nearest integer. That will give you numerators for fractions that are tuned very close to the 12-EDO steps. In particular, you'd get

0: 1/1 (i.e. 22/22)

1: 23/22

2: 25/22

3: 13/11

4: 14/11

5: 15/11

6: 31/22

7: 3/2 (i.e. 33/22)

8: 35/22

9: 37/22

10: 39/22

11: 21/11

12: 2/1 i.e. (44/22)

This is not actually 11-limit: it has higher primes in the numerators. But it's still cool, honestly. And it sounds pretty. This scale has a lot of overlap with the previous set of tuned intervals and also some interesting differences. For one thing, (13/11) appears to be closest to 3 steps of 12-EDO, even though the intervallic interpretation says that the related interval, PrDem3, tunes to 2 steps. On the other end of the scale, apparently (39/22) is closest in tuning to 10 steps of 12-EDO, although the intervallic interpretation says that the related interval, PrDem7, tunes to 9 steps. I don't think this casts any shadow on the intervallic interpretation: sometimes EDOs send intervals a little bit away from where you'd expect based on the just frequency ratios - in fact, you can make EDOs behave arbitrarily badly with a little forethought. Consider this: by stacking intervals that are tempered out in an EDO, like d2 for 12-EDO, you can find intervals that are justly tuned to arbitrarily large values (positive or negative), but which 12-EDO still maps to 0 steps = 1/1. EDOs aren't perfect. They just collapse part of the interval space so that we have finitely many frequency ratios per octave, separated by consistent amounts so that we can modulate thoughtlessly and have something that sounds the same afterwards. So I'm not going to fret that my awesome self-consistent intervallic vector space machinery occasionally says that 13/11 is a large major second when it sounds to the layman like a small minor third.

If you want the use the second scale with 22 in the denomintors, that's great. You'll be in good company among microtonalists. This scale is what George Secor called a quasi-equal rational tuning. It's what Zheanna Erose calls a NEJI (near-equal just intonation). It's a subset of 22-AFDO (arithmetic frequency division of the octave) or the "Over-22" overtone scale. So many names. We'll talk about this stuff in a future post on non-intervallic microtonal music. It's not something I would have come up with using intervals, and I'm glad it exists. Since it's a little bit off from 12-EDO, different modulations have a little bit different sonic character from each other, while still perhaps sounding like they have 11 or 22 in the denominators, so much as otonality can be perceived in association with specific denominators.

I'm also glad that my 22-denominator scale exists. There is an underlying structure in music which "Quasi-Equal Over-22-AFDO 12\22-NEJI" ignores (and which it could not *not* ignore since it's based on 12-EDO, which already ignores a lot of musical structure). Sometimes it's nice to keep that structure intact. Other times it's nice to play with relatively unstructured sounds that don't have that underlying algebraic structure of complements and inverses and commas and alphabetical chromatic scales. I hope you can appreciate both methods a little more through reading my works.

P. S.

I've been thinking  a little bit about adjectives for higher-prime commas. How about:

17-limit comma: (Exalted | Humbled) :: (Ex | Hb) # 51/50

19-limit comma: (Lofty | Lowly) :: (Lf | Lw) # 96/95

23-limit comma: (Noble | Disgraced) :: (Nb | Ds) # 46/45

Those are nice, yeah? Here's the full set of commas related to harmonics 7 through 23, presented in Lilley-Johnston comma coordinates:

Sp = (0, 0, 0, 1) # 36/35

Sb = (0, 0, 0, -1) # 35/36

As = (0, 0, 0, 0, 1) # 33/32

De = (0, 0, 0, 0, 1) # 32/33

Pr = (0, 0, 0, 0, 0, 1) # 65/64

Re = (0, 0, 0, 0, 0, -1) # 64/65

Ex = (0, 0, 0, 0, 0, 0, 1) # 51/50

Hb = (0, 0, 0, 0, 0, 0, -1) # 50 / 51

Lf = (0, 0, 0, 0, 0, 0, 0, 1) # 96/95

Lw = (0, 0, 0, 0, 0, 0, 0, -1) # 95/96

Nb = (0, 0, 0, 0, 0, 0, 0, 0, 1) # 46/45

Ds =  (0, 0, 0, 0, 0, 0, 0, 0, -1) # 45/46

Now for naming the harmonics themselves, and the reduced harmonics. We find that the reduced 17th harmonic is a 17-limit comma away from a 5- limit A1:

(17/16) = (25/24) * (51/50)

So we can say that the reduced 17th harmonic is an exalted augmented unison, ExA1. The 17th harmonic is four octaves up from its reduced counterpart, so it's a justly tuned "exalted augmented 29th" or ExA29.

17/16 = t(ExA1)

17/8 = t(ExA8)

17/4 = t(ExA15)

17/2 = t(ExA22)

17/1 = t(ExA29)

How about the 19th and 23rd harmonics?

The reduced 19th harmonic is a lofty unison smaller than a 5-limit minor third,

(19/16) = (6/5) / (96/95)

so the reduced harmonic is a lowly minor third, Frm3. The prime harmonic is four octaves up, so it's a thirty-first interval with the same quality: a lowly minor thirty-first, Frm31.

The reduced 23rd harmonic is a noble unison larger than an acute augmented fourth, 

(45/32) *  (46/45) = (23/16)

AcA4 + Nb1 = reduced 23rd harmonic

so the reduced 23rd harmonic is an noble acute augmented fourth, NbAcA4. The prime harmonic is four octaves up, so it's a 32nd interval, in particular NbAcA32.

:: Higher prime limit textual accidentals

We've been talking about what abbreviations to use for indicating qualities of intervals that are justly tuned to high prime limit frequency ratios. Since pitches and intervals exist in a 1-to-1 correspondence, we should also talk a little bit about how to write pitch accidentals associated with large primes.

Ideally, we'd just use the accidentals that Ben Johnston used in his scores, but for the most part these aren't available as unicode characters. The eleven-limit/undecimal accidentals are fairly easy, since he just uses up and down arrows, ↑ ↓, and these are rendered in most unicode fonts. But for septimal, tridecimal, septendecimal accidentals, et cetera, Ben Johnston uses the associated number as the glyph, both upright and rotated 180 degrees. (Weirdly he uses the upright numeral glyph for all the sharpening accidentals, those tuned larger than 1/1, except for the 19-limit accidental, where the upright one is justly tuned to 95/96 and its rotated version is 96/95.) Unfortunately, Unicode doesn't have a symbol that is a little "17" rotated 180 degrees, so how are we going to write out pitches associated with high dimensional interval spaces?

After considering lots of options, I think that using superscripts for sharpening accidentals and subscripts for flattening accidentals is the option that will be most visually consistent and easily understood. 

Here are the sharpening accidentals

[A⁷, Bb¹¹, C+¹³, D#¹⁷, E-¹⁹]

and here are the flattening accidentals:

[F#₇, G₁₁, Ab₁₃, B-₁₇, C+₁₉]

I'm unlikely to use higher primes than 19 in this textbook much, but it's easy enough to look up superscript and subscript characters if you want to use more accidentals than these. . It is done and it is done well.

As an aside, I don't actually known a standard adjective for 19-limit ratios, but either "decennoval" or "undevigintal" seems good to me, and when I asked around I also heard recommendations for "novemdecimal" and "novendecimal" and "undevicesimal".