I wanted to create a system for systematically giving interval names to arbitrary frequency ratios in 7-limit just intonation, inspired by Lilley's system for naming intervals in 5-limit just intonation.
Specifically, I was hoping I could find a system in which all the perfect intervals only have factors of (2, 3) and all the major and minor intervals just only have factors of (2, 3, 5), and then some other intervals, either the (augmented and diminished) or the (acute and grave) would all have factors of (2, 3, 5, 7). I wanted the qualities of the intervals to tell you something about their factors.
Such a system might exist, but I wasn't able to find one. Or rather, I found several systems, but the ratios with factors of 7 get interval names with new qualities, instead of using old interval qualities like (augmented and diminished). The new qualities are "sub", as in a "septimal sub-minor 7th", Sbm7, and "super", like a "septimal super unison", Sp1. And this is fine. To talk about a 4 dimensional space, we were going to need new interval qualities one way or the other. This way all the familiar frequency ratios from 5-limit JI keep their old interval names, and the new septimal frequency ratios get new intervals names.
I found three functional systems for naming septimal intervals, and I'll introduce them to you in the order that I found them.
:: The Palatine System
I started by looking at rank-2 intervals tuned to 31-EDO, which I'd heard offered good approximations to most of the simple 7-limit just intonation intervals. Whenever I could find a 7-limit frequency ratio that was very close to a step of 31-EDO and also simpler than the corresponding 5-limit frequency ratio for the interval, I thought my new system should tune that interval to the septimal ratio instead of the 5-limit one. These were the first substitutions I found:
d3: 144/125 -> 8/7
A2: 125/108 -> 7/6
d4: 32/25 -> 9/7
A4: 25/18 -> 7/5
d5: 36/25 -> 10/7
A5: 25/16 -> 14/9
d7: 216/125 -> 12/7
A6: 125/72 -> 7/4
.
I eventually found substitutions for all of the once modified (i.e. once augmented or diminished) intervals and then for the twice modified intervals. But when I looked at the interval differences, the system wasn't very regular. It was a mess of weird skips and jumps. I spent a really long time trying to figure out an algebra of interval differences where you could consistently say, "adding an augmented unison to a perfect fourth produces an augmented fourth", but it wasn't coming together. Eventually I gave up on the project of making augmented and diminished interval septimal.
What finally worked was taking the Lilley system of 5-limit just intonation and adding a septimal comma.
I wrote a little bit about Lilley's system on this blog last month and also you can see the original presentation in Haskell code at FiveLimit.hs. His system has an interval basis (Ac1, A1, d2) tuned to a frequency ratio basis (81/80, 25/24, 128/125).
The augmented unison, A1, tuned to a value of 25/24, is the difference between the 5-limit major and minor seconds,
A1 = M2 - m2
(25 / 24) = (10/9) / (16/15)
.
More generally, it' the difference between any major Nth and minor Nth.
The diminished second interval, d2, tuned to a value of 128/125, is the difference between 5-limit minor second and the augmented first that we just defined, t(m2) = 16/15 and t(A1) = 25/24.
d2 = m2 - A1
(128 / 125) = (16/15) / (25/24)
.
But to represent a rank-3 system like 5-limit just intonation, we need one more comma, and Lilley uses the acute unison, Ac1, tuned to a frequency ratio of 81/80. It's the difference between the 5-limit major second and the Pythagorean major second, 9/8. In Lilley's system, the Pythagorean major second now gets the name "acute major second", AcM2.
Lilley's basis is nice for lots of reasons, and one of them is that the absolute value of the determinant of the matrix of basis vectors is unitary, and that's the right way to define just intonation tuning systems and also a good way to design enharmonic keyboard layouts (details
here).
To extend Lilley's 5-limit system to a 7-limit system, we need a fourth interval as a basis vector, which will be the septimal super unison, Sp1, and we need to tune Sp1 to a septimal frequency ratio. I tried several small septimal frequency ratios, and found that adding in 21/20 produced a full rank basis of frequency ratios with an absolute determinant of one (when expressed in any other such basis, like the (2/1, 3/1, 5/1, 7/1) prime basis), and also this system reproduced a lot of the features that I was trying to get by substituting in 7-limit frequency ratios in place of 5-limit frequency ratios as approximations to steps in 31-EDO.
For example, I thought the frequency ratio of 7/4, the octave-reduced 7th harmonic, should be called some kind of 6th interval, since 7/4 was very close to the 31-EDO value for the augmented 6th, and also the frequency ratio 7/4 was simpler than the 5-limit version of the augmented sixth. The rank-4 extension of Lilley's 5-limit system with Sp1 tuned to 21/20 - it calls the frequency ratio (7/4) a Super major 6th, SpM6. And I thought it was a sixth previously! Great success!
I liked this system so much that I gave it a name: It is the "Palatine system" and (21/20) is the "Palatine comma". The letters of the name "Palatine" appear in order in the phrase "septimal chromatic semitone".
I'll talk more about how we actually assign (frequency ratios to interval names) and (interval names to frequency ratios) in a bit.
:: The Leipzig system
But my Palatine comma was not the only septimal comma on the block. Whoever wrote the Wikipedia articles on septimal intervals is very partial to a septimal comma of 64/63, which is called the Leipzig comma in some sources. If we tune the septimal super unison, Sp1, to the Leipzig comma in the rank-4 extension of Lilley's system, then we get a different set of interval names for septimal frequency ratios. This new set matches the ordinals for the interval names, (1st, 2nd, 3rd, ....), on Wikipedia, which is good. This system gives the intervals different qualities than the ones on Wikipedia, but I'd argue that the qualities are more regular here, just as Lilley's interval qualities are so, *so* much nicer than the mess of crap in Wikipedia's "List of intervals in 5-limit just intonation". The octave-reduced 7th harmonic, 7/4, is now instead called a sub-grave-minor 7th, SbGrm7, in the Leipzig system.
The Leipzig comma, (64/63), can be obtained as (16/15) / (21/20), i.e. it's the 5-limit minor second minus the Palatine comma. The Leipzig comma is also the frequency ratio associated with the septimal accidental in Helmholtz-Ellis staff notation.
For both the Palatine and Leipzig extensions of Lilley's system, written in this order (Sp1, Ac1, A1, d2), the old 5-limit frequency ratios have all the same interval coordinates, except with a 0 tacked on the front. Here are some simple ones:
1/1 :: (0, 0, 0, 0) # P1
16/15 :: (0, 0, 1, 1) # m2
10/9 :: (0, 0, 2, 1) # M2
6/5 :: (0, 1, 3, 2) # m3
5/4 :: (0, 1, 4, 2) # M3
4/3 :: (0, 1, 5, 3) # P4
3/2 :: (0, 2, 7, 4) # P5
8/5 :: (0, 2, 8, 5) # m6
5/3 :: (0, 2, 9, 5) # M6
9/5 :: (0, 3, 10, 6) # m7
15/8 :: (0, 3, 11, 6) # M7
. A lot of zeroes up front, as promised.
For septimal ratios, if there's a 1 in the front, then it's a Super-(whatever the interval would be called if there were a zero in front). And if there's a -1 in the first column, then it's a Sub-(whatever the interval would be called if there were a zero in front). If the first component were a -2, then it would be a Sub-Sub-(whatever).
Let's do an example. In the Palatine extension of Lilley's 5-limit tuning system, we have these two related intervals:
5/3 :: (0, 2, 9, 5) # M6 in either system
7/4 :: (1, 2, 9, 5) # Octave-reduced 7th harmonic in Palatine system
These two have the same coordinates at the end, but 7/4 is raised by one unit in the septimal component relative to the major sixth, so it's a super major sixth, SpM6.
In the Leipzig extension of the 5-limit tuning system, we have
9/5 :: (0, 3, 10, 6) # m7 in either system
7/4 :: (-1, 2, 10, 6) # Octave-reduced 7th harmonic in Leipzig system
.
The components of the interval for 7/4 are lowered, relative to m7, in both the acute unison component and the septimal component. That makes (7/4) a sub grave minor seventh, SbGrm7.
By the same logic we can say that, e.g. (15/14) is a sub grave major second, SbGrM2, in the Palatine system, and it's a super acute augmented first, SpAcA1, in the Leipzig system. If you were wondering.
:: The Johnston system
That's as far as this post got originally, except for a lot of failed attempts to alter one or more of Lilley's original basis vectors to make things even nicer. But I promised you three tuning systems, and the best is yet to come.
The great microtonal composer Ben Johnston used the frequency ratio (36/35) as the frequency ratio for the septimal accidental in his staff notation. Johnston's comma is really close to a quarter tone, i.e. 2^(1/24), and I discovered this while writing a later post about
quartertone harmony and maqamat in middle eastern music.
In that same post, I make some suggestions for interval qualities for 11-limit and 13-limit just intonation: we could associate a undecimal comma with the qualities (ascendant | descendant), notated As and De, and associate a tridecimal comma with the qualities (prominent | recessed), notated as Pr and Re. Maybe. One day.
I'm not going to reproduce everything in that post here, but I like the septimal Johnston-Lilley system even more than my Palatine system at this point. It agrees with Wikipedia and HEJI that 7/4 is some kind of seventh interval, but the Johnston-Lilley system gives (7/4) the nicer name of a sub-minor seventh, Sbm7. Also it's fairly good for analyzing quarter tone music. Also I just love Ben Johnston's music and like that I can use something of his productively. Also, honestly, the Palatine comma was kind of wide to be called a comma, it's more than 50 cents.
All three septimal commas are super-particular, if that's something you care about, and all three systems have their merits, but I'm sticking with Johnston from here on out. I will leave you with two gifts. First, a table of 150 intervals with short names in the Johnston-Lilley system:
P1 : (0, 0, 0, 0) :: 1/1
SpA0 : (1, 0, 0, -1) :: 225/224
Grd2 : (0, -1, 0, 1) :: 2048/2025
Ac1 : (0, 1, 0, 0) :: 81/80
SbA1 : (-1, 0, 1, 0) :: 875/864
SpGr1 : (1, -1, 0, 0) :: 64/63
d2 : (0, 0, 0, 1) :: 128/125
AcAc1 : (0, 2, 0, 0) :: 6561/6400
Sp1 : (1, 0, 0, 0) :: 36/35
GrA1 : (0, -1, 1, 0) :: 250/243
Acd2 : (0, 1, 0, 1) :: 648/625
Sbm2 : (-1, 0, 1, 1) :: 28/27
SpAc1 : (1, 1, 0, 0) :: 729/700
A1 : (0, 0, 1, 0) :: 25/24
Spd2 : (1, 0, 0, 1) :: 4608/4375
Grm2 : (0, -1, 1, 1) :: 256/243
AcA1 : (0, 1, 1, 0) :: 135/128
SpSp1 : (2, 0, 0, 0) :: 1296/1225
m2 : (0, 0, 1, 1) :: 16/15
SpA1 : (1, 0, 1, 0) :: 15/14
Acm2 : (0, 1, 1, 1) :: 27/25
SbM2 : (-1, 0, 2, 1) :: 175/162
Spm2 : (1, 0, 1, 1) :: 192/175
GrM2 : (0, -1, 2, 1) :: 800/729
M2 : (0, 0, 2, 1) :: 10/9
Sbd3 : (-1, 1, 2, 2) :: 28/25
AcM2 : (0, 1, 2, 1) :: 9/8
SbA2 : (-1, 0, 3, 1) :: 4375/3888
Grd3 : (0, 0, 2, 2) :: 256/225
SpM2 : (1, 0, 2, 1) :: 8/7
GrA2 : (0, -1, 3, 1) :: 2500/2187
d3 : (0, 1, 2, 2) :: 144/125
A2 : (0, 0, 3, 1) :: 125/108
Acd3 : (0, 2, 2, 2) :: 729/625
Sbm3 : (-1, 1, 3, 2) :: 7/6
AcA2 : (0, 1, 3, 1) :: 75/64
Spd3 : (1, 1, 2, 2) :: 5184/4375
Grm3 : (0, 0, 3, 2) :: 32/27
SpA2 : (1, 0, 3, 1) :: 25/21
m3 : (0, 1, 3, 2) :: 6/5
Acm3 : (0, 2, 3, 2) :: 243/200
SbM3 : (-1, 1, 4, 2) :: 175/144
Spm3 : (1, 1, 3, 2) :: 216/175
GrM3 : (0, 0, 4, 2) :: 100/81
Sbd4 : (-1, 1, 4, 3) :: 56/45
M3 : (0, 1, 4, 2) :: 5/4
SbSb4 : (-2, 1, 5, 3) :: 1225/972
Grd4 : (0, 0, 4, 3) :: 512/405
AcM3 : (0, 2, 4, 2) :: 81/64
SbA3 : (-1, 1, 5, 2) :: 4375/3456
d4 : (0, 1, 4, 3) :: 32/25
SbGr4 : (-1, 0, 5, 3) :: 2800/2187
SpM3 : (1, 1, 4, 2) :: 9/7
GrA3 : (0, 0, 5, 2) :: 625/486
Acd4 : (0, 2, 4, 3) :: 162/125
Sb4 : (-1, 1, 5, 3) :: 35/27
GrGr4 : (0, -1, 5, 3) :: 25600/19683
A3 : (0, 1, 5, 2) :: 125/96
SbAc4 : (-1, 2, 5, 3) :: 21/16
Spd4 : (1, 1, 4, 3) :: 1152/875
Gr4 : (0, 0, 5, 3) :: 320/243
AcA3 : (0, 2, 5, 2) :: 675/512
P4 : (0, 1, 5, 3) :: 4/3
SpA3 : (1, 1, 5, 2) :: 75/56
Ac4 : (0, 2, 5, 3) :: 27/20
SbA4 : (-1, 1, 6, 3) :: 875/648
SpGr4 : (1, 0, 5, 3) :: 256/189
AcAc4 : (0, 3, 5, 3) :: 2187/1600
Sp4 : (1, 1, 5, 3) :: 48/35
GrA4 : (0, 0, 6, 3) :: 1000/729
SpAc4 : (1, 2, 5, 3) :: 243/175
A4 : (0, 1, 6, 3) :: 25/18
Sbd5 : (-1, 2, 6, 4) :: 7/5
AcA4 : (0, 2, 6, 3) :: 45/32
SpSp4 : (2, 1, 5, 3) :: 1728/1225
SbSb5 : (-2, 2, 7, 4) :: 1225/864
Grd5 : (0, 1, 6, 4) :: 64/45
SpA4 : (1, 1, 6, 3) :: 10/7
d5 : (0, 2, 6, 4) :: 36/25
SbGr5 : (-1, 1, 7, 4) :: 350/243
Acd5 : (0, 3, 6, 4) :: 729/500
Sb5 : (-1, 2, 7, 4) :: 35/24
GrGr5 : (0, 0, 7, 4) :: 3200/2187
SbAc5 : (-1, 3, 7, 4) :: 189/128
Spd5 : (1, 2, 6, 4) :: 1296/875
Gr5 : (0, 1, 7, 4) :: 40/27
Sbd6 : (-1, 2, 7, 5) :: 112/75
P5 : (0, 2, 7, 4) :: 3/2
Grd6 : (0, 1, 7, 5) :: 1024/675
Ac5 : (0, 3, 7, 4) :: 243/160
SbA5 : (-1, 2, 8, 4) :: 875/576
SpGr5 : (1, 1, 7, 4) :: 32/21
d6 : (0, 2, 7, 5) :: 192/125
AcAc5 : (0, 4, 7, 4) :: 19683/12800
Sp5 : (1, 2, 7, 4) :: 54/35
GrA5 : (0, 1, 8, 4) :: 125/81
Acd6 : (0, 3, 7, 5) :: 972/625
Sbm6 : (-1, 2, 8, 5) :: 14/9
SpAc5 : (1, 3, 7, 4) :: 2187/1400
A5 : (0, 2, 8, 4) :: 25/16
Spd6 : (1, 2, 7, 5) :: 6912/4375
Grm6 : (0, 1, 8, 5) :: 128/81
AcA5 : (0, 3, 8, 4) :: 405/256
SpSp5 : (2, 2, 7, 4) :: 1944/1225
m6 : (0, 2, 8, 5) :: 8/5
SpA5 : (1, 2, 8, 4) :: 45/28
Acm6 : (0, 3, 8, 5) :: 81/50
SbM6 : (-1, 2, 9, 5) :: 175/108
Spm6 : (1, 2, 8, 5) :: 288/175
GrM6 : (0, 1, 9, 5) :: 400/243
M6 : (0, 2, 9, 5) :: 5/3
Sbd7 : (-1, 3, 9, 6) :: 42/25
AcM6 : (0, 3, 9, 5) :: 27/16
SbA6 : (-1, 2, 10, 5) :: 4375/2592
Grd7 : (0, 2, 9, 6) :: 128/75
SpM6 : (1, 2, 9, 5) :: 12/7
GrA6 : (0, 1, 10, 5) :: 1250/729
d7 : (0, 3, 9, 6) :: 216/125
A6 : (0, 2, 10, 5) :: 125/72
Acd7 : (0, 4, 9, 6) :: 2187/1250
Sbm7 : (-1, 3, 10, 6) :: 7/4
AcA6 : (0, 3, 10, 5) :: 225/128
Spd7 : (1, 3, 9, 6) :: 7776/4375
Grm7 : (0, 2, 10, 6) :: 16/9
SpA6 : (1, 2, 10, 5) :: 25/14
m7 : (0, 3, 10, 6) :: 9/5
Acm7 : (0, 4, 10, 6) :: 729/400
SbM7 : (-1, 3, 11, 6) :: 175/96
Spm7 : (1, 3, 10, 6) :: 324/175
GrM7 : (0, 2, 11, 6) :: 50/27
Sbd8 : (-1, 3, 11, 7) :: 28/15
M7 : (0, 3, 11, 6) :: 15/8
SbSb8 : (-2, 3, 12, 7) :: 1225/648
Grd8 : (0, 2, 11, 7) :: 256/135
AcM7 : (0, 4, 11, 6) :: 243/128
SbA7 : (-1, 3, 12, 6) :: 4375/2304
d8 : (0, 3, 11, 7) :: 48/25
SbGr8 : (-1, 2, 12, 7) :: 1400/729
SpM7 : (1, 3, 11, 6) :: 27/14
GrA7 : (0, 2, 12, 6) :: 625/324
Acd8 : (0, 4, 11, 7) :: 243/125
Sb8 : (-1, 3, 12, 7) :: 35/18
GrGr8 : (0, 1, 12, 7) :: 12800/6561
A7 : (0, 3, 12, 6) :: 125/64
SbAc8 : (-1, 4, 12, 7) :: 63/32
Spd8 : (1, 3, 11, 7) :: 1728/875
Gr8 : (0, 2, 12, 7) :: 160/81
AcA7 : (0, 4, 12, 6) :: 2025/1024
Sbd9 : (-1, 3, 12, 8) :: 448/225
P8 : (0, 3, 12, 7) :: 2/1
, and second, a python program for working with 7-limit intervals. Enjoy.
Oh, man! I found a cool application. There was an ancient Greek dude named Archytas who liked breaking super-particular frequency ratios into new super-particular frequency ratios using harmonic means. He came up with some 7-limit scales and tetrachords. A tetrachord is a sequence of intervals that sum to P4.
Music theorists usually just state the tetrachords in terms of frequency ratios, but let's give them 7-limit interval names!
Archytas's enharmonic tetrachord:
(28/27) * (36/35) * (5/4) = (4/3)
Sbm2 + Sp1 + M3 = P4
Archytas's diatonic tetrachord:
(28/27) * (8/7) * (9/8) = (4/3)
Sbm2 + SpM2 + AcM2 = P4
Archytas's chromatic tetrachord:
(28/27) * (243/224) * (32/27) = (4/3)
Sbm2 + SpAcA1 + Grm3 = P4
I'm not here to say whether his tetrachords were good, interesting, or useful, but we've successfully named the components.
Ooh, update:
If you split (4/3) into (8/7) (which is the harmonic mean of 4/3 with 1/1) and (7/6) (which is the arithmetic mean of 4/3 with 1/1), and stack them up alternately, reducing by octaves when necessary, then you get a lot of septimal frequency ratios separated by (64/63) like
(256/243) / (28/27) = (64/63)
(32/27) / (7/6) = (64/63)
(128/81) / (14/9) = (64/63)
(256/189) / (4/3) = (64/63)
(1024/567) / (16/9) = (64/63)
I know that's not a great reason to use 64/63 as a septimal comma, but I found it repeatedly as a septimal comma while doing that stacking thing and thought it might go well as an endnote in a post partially about one's choice of septimal comma.
You also get lots of frequency ratios separated by (28/27), like:
(256/243) / (64/63) = 28/27
(32/27) / (8/7) = 28/27
(128/81) / (32/21) = 28/27
.
So whatever.
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