Bill Sethares is a researcher who gave us some code for calculating dissonance curves between two sounds. Here it is in python from github user Endolith. Beating between harmonics produces dissonance, and we can use this fact to find {frequency ratios between the fundamentals of two harmonic sounds} which produce little beating.
As you add more harmonics, you get valleys in the plot at new consonant points. I did this up to 13 harmonics and figured out all of the valleys produced within an octave. Here's the full set:
[1/1, 13/12, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 13/11, 6/5, 11/9, 5/4, 9/7, 13/10, 4/3, 11/8, 7/5, 10/7, 13/9, 3/2, 11/7, 8/5, 13/8, 5/3, 12/7, 7/4, 9/5, 11/6, 13/7, 2/1]
I think how the valleys arise with the addition of partial is interesting. Let's look at that.
With harmonics from 1 to 6, we get dips in the dissonance curve at these fractions:
[6/5, 5/4, 4/3, 3/2, 5/3, 2/1]
The seventh harmonic adds dips at
[7/6, 7/5, 7/4]
If the fractions had any smaller denominators, then the fractions would be more than (2/1), which I'm not currently looking at, though I hope to do so soon.
The eighth harmonic adds dips at:
[8/7, 8/5]
and we already had 8/6 in the form of 4/3.
The 9th harmonic adds dips at:
[9/8, 9/7 9/5]
and we already had 9/6 in the form of 3/2.
Obviously the Nth harmonic just adds (N/i), for i in range 1 to N as dips in the dissonance curve. And if N and i aren't corprime, then we'll have seen the fraction before.
We don't get octave complements for free this way. Super interesting, right? I should try more composing in systems without octave complements.
Also, not all of the valleys are as deep as others. The small super-particular ratios at the start of the series, like 10/9, hardly make a dent in the plot.
If we continue looking at harmonic divisions with smaller denominators (that have frequency ratios greater than 2/1), then we get to append these guys on the end of the previous list:
[13/6, 11/5, 9/4, 7/3, 12/5, 5/2, 13/5, 8/3, 11/4, 3/1, 13/4, 10/3, 7/2, 11/3]
Those go up to the second tuned octave, (4/1), now. We can reduce these by an octave to get some elements already seen:
[13/12, 11/10, 9/8, 7/6, 6/5, 5/4, 13/10, 4/3, 11/8, 3/2, 13/8, 5/3, 7/4, 11/6]
I think it would be interesting to look at tertian chords made from the full two octave set. By "tertian" here I mean that the steps between successive frequency ratios are smaller than a justly tuned P4, (4/3), and larger than a justly tuned M2, (10/9).
I think those will sound interesting. And if they sound interesting enough, they might even merit a naming system, particularly if the intervals which are justly tuned to those frequency ratios are actually spelled by thirds, (with some kind of a seventh interval following some kind of fifth interval, following a third interval, and so on).
Here are some four-note Sethares chords which are approximately tertian in their tuning, if not necessarily tertian intervallically:
[1/1, 11/9, 10/7, 13/7]
[1/1, 11/9, 10/7, 5/3]
[1/1, 11/9, 10/7, 8/5]
[1/1, 11/9, 10/7, 9/5]
[1/1, 11/9, 11/7, 2/1]
[1/1, 11/9, 11/7, 9/5]
[1/1, 11/9, 11/8, 8/5]
[1/1, 11/9, 13/8, 11/6]
[1/1, 11/9, 13/8, 13/7]
[1/1, 11/9, 13/8, 2/1]
[1/1, 11/9, 13/9, 5/3]
[1/1, 11/9, 13/9, 7/4]
[1/1, 11/9, 3/2, 11/6]
[1/1, 11/9, 3/2, 9/5]
[1/1, 11/9, 7/5, 11/6]
[1/1, 11/9, 7/5, 11/7]
[1/1, 11/9, 7/5, 12/7]
[1/1, 11/9, 7/5, 5/3]
[1/1, 11/9, 7/5, 7/4]
[1/1, 11/9, 7/5, 8/5]
[1/1, 11/9, 8/5, 2/1]
[1/1, 11/9, 8/5, 9/5]
[1/1, 13/10, 11/7, 11/6]
[1/1, 13/10, 11/7, 13/7]
[1/1, 13/10, 11/7, 2/1]
[1/1, 13/10, 12/7, 11/5]
[1/1, 13/10, 12/7, 13/6]
[1/1, 13/10, 12/7, 2/1]
[1/1, 13/10, 13/8, 13/7]
[1/1, 13/10, 13/8, 2/1]
[1/1, 13/10, 3/2, 11/6]
[1/1, 13/10, 3/2, 12/7]
[1/1, 13/10, 3/2, 7/4]
[1/1, 13/10, 5/3, 13/7]
[1/1, 13/10, 5/3, 2/1]
[1/1, 13/10, 8/5, 11/6]
[1/1, 13/10, 8/5, 2/1]
[1/1, 13/11, 10/7, 11/6]
[1/1, 13/11, 10/7, 5/3]
[1/1, 13/11, 10/7, 8/5]
[1/1, 13/11, 11/7, 11/6]
[1/1, 13/11, 11/7, 13/7]
[1/1, 13/11, 11/7, 2/1]
[1/1, 13/11, 11/7, 7/4]
[1/1, 13/11, 11/8, 12/7]
[1/1, 13/11, 11/8, 13/8]
[1/1, 13/11, 11/8, 7/4]
[1/1, 13/11, 11/8, 8/5]
[1/1, 13/11, 13/9, 11/6]
[1/1, 13/11, 13/9, 12/7]
[1/1, 13/11, 13/9, 13/7]
[1/1, 13/11, 13/9, 5/3]
[1/1, 13/11, 13/9, 9/5]
[1/1, 13/11, 3/2, 13/7]
[1/1, 13/11, 3/2, 9/5]
[1/1, 13/11, 4/3, 11/7]
[1/1, 13/11, 4/3, 3/2]
[1/1, 13/11, 4/3, 7/4]
[1/1, 13/11, 7/5, 11/7]
[1/1, 13/11, 7/5, 12/7]
[1/1, 13/11, 7/5, 13/7]
[1/1, 13/11, 7/5, 5/3]
[1/1, 13/11, 7/5, 7/4]
[1/1, 13/11, 7/5, 9/5]
[1/1, 5/4, 10/7, 11/6]
[1/1, 5/4, 10/7, 13/7]
[1/1, 5/4, 10/7, 13/8]
[1/1, 5/4, 10/7, 5/3]
[1/1, 5/4, 10/7, 7/4]
[1/1, 5/4, 10/7, 8/5]
[1/1, 5/4, 11/7, 7/4]
[1/1, 5/4, 11/7, 9/5]
[1/1, 5/4, 13/8, 11/6]
[1/1, 5/4, 13/9, 13/8]
[1/1, 5/4, 13/9, 7/4]
[1/1, 5/4, 3/2, 12/7]
[1/1, 5/4, 3/2, 13/7]
[1/1, 5/4, 3/2, 9/5]
[1/1, 5/4, 7/5, 13/7]
[1/1, 5/4, 7/5, 13/8]
[1/1, 5/4, 7/5, 7/4]
[1/1, 5/4, 7/5, 8/5]
[1/1, 5/4, 7/5, 9/5]
[1/1, 5/4, 8/5, 11/6]
[1/1, 5/4, 8/5, 13/7]
[1/1, 5/4, 8/5, 2/1]
[1/1, 5/4, 8/5, 9/5]
[1/1, 6/5, 10/7, 13/7]
[1/1, 6/5, 10/7, 5/3]
[1/1, 6/5, 10/7, 7/4]
[1/1, 6/5, 10/7, 8/5]
[1/1, 6/5, 11/7, 11/6]
[1/1, 6/5, 11/7, 2/1]
[1/1, 6/5, 11/7, 7/4]
[1/1, 6/5, 11/7, 9/5]
[1/1, 6/5, 11/8, 11/7]
[1/1, 6/5, 11/8, 12/7]
[1/1, 6/5, 11/8, 7/4]
[1/1, 6/5, 11/8, 8/5]
[1/1, 6/5, 13/9, 11/6]
[1/1, 6/5, 13/9, 12/7]
[1/1, 6/5, 13/9, 13/7]
[1/1, 6/5, 3/2, 11/6]
[1/1, 6/5, 3/2, 7/4]
[1/1, 6/5, 3/2, 9/5]
[1/1, 6/5, 7/5, 11/6]
[1/1, 6/5, 7/5, 12/7]
[1/1, 6/5, 7/5, 13/7]
[1/1, 6/5, 7/5, 5/3]
[1/1, 6/5, 7/5, 8/5]
[1/1, 7/6, 10/7, 11/6]
[1/1, 7/6, 10/7, 12/7]
[1/1, 7/6, 10/7, 13/7]
[1/1, 7/6, 10/7, 13/8]
[1/1, 7/6, 10/7, 7/4]
[1/1, 7/6, 10/7, 8/5]
[1/1, 7/6, 10/7, 9/5]
[1/1, 7/6, 11/8, 8/5]
[1/1, 7/6, 11/8, 9/5]
[1/1, 7/6, 13/10, 13/8]
[1/1, 7/6, 13/10, 5/3]
[1/1, 7/6, 13/9, 11/6]
[1/1, 7/6, 13/9, 12/7]
[1/1, 7/6, 13/9, 5/3]
[1/1, 7/6, 3/2, 11/6]
[1/1, 7/6, 3/2, 7/4]
[1/1, 7/6, 3/2, 9/5]
[1/1, 7/6, 4/3, 12/7]
[1/1, 7/6, 4/3, 13/8]
[1/1, 7/6, 4/3, 3/2]
[1/1, 7/6, 4/3, 5/3]
[1/1, 7/6, 4/3, 7/4]
[1/1, 7/6, 7/5, 13/8]
[1/1, 7/6, 7/5, 5/3]
[1/1, 7/6, 7/5, 9/5]
[1/1, 8/7, 10/7, 11/6]
[1/1, 8/7, 10/7, 12/7]
[1/1, 8/7, 10/7, 13/8]
[1/1, 8/7, 10/7, 5/3]
[1/1, 8/7, 10/7, 8/5]
[1/1, 8/7, 11/8, 13/8]
[1/1, 8/7, 11/8, 5/3]
[1/1, 8/7, 11/8, 7/4]
[1/1, 8/7, 11/8, 8/5]
[1/1, 8/7, 11/8, 9/5]
[1/1, 8/7, 13/10, 11/7]
[1/1, 8/7, 13/10, 12/7]
[1/1, 8/7, 13/9, 11/6]
[1/1, 8/7, 13/9, 12/7]
[1/1, 8/7, 13/9, 13/7]
[1/1, 8/7, 3/2, 11/6]
[1/1, 8/7, 3/2, 12/7]
[1/1, 8/7, 3/2, 7/4]
[1/1, 8/7, 4/3, 12/7]
[1/1, 8/7, 4/3, 3/2]
[1/1, 8/7, 4/3, 5/3]
[1/1, 8/7, 4/3, 7/4]
[1/1, 8/7, 4/3, 8/5]
[1/1, 8/7, 7/5, 11/6]
[1/1, 8/7, 7/5, 11/7]
[1/1, 8/7, 7/5, 12/7]
[1/1, 8/7, 7/5, 13/7]
[1/1, 8/7, 7/5, 5/3]
[1/1, 8/7, 7/5, 8/5]
[1/1, 8/7, 7/5, 9/5]
[1/1, 8/7, 9/7, 11/7]
[1/1, 8/7, 9/7, 13/9]
[1/1, 8/7, 9/7, 3/2]
[1/1, 8/7, 9/7, 5/3]
[1/1, 9/7, 11/7, 2/1]
[1/1, 9/7, 11/7, 9/5]
[1/1, 9/7, 13/8, 11/6]
[1/1, 9/7, 13/8, 2/1]
[1/1, 9/7, 13/9, 12/7]
[1/1, 9/7, 13/9, 13/7]
[1/1, 9/7, 13/9, 5/3]
[1/1, 9/7, 3/2, 12/7]
[1/1, 9/7, 3/2, 13/7]
[1/1, 9/7, 3/2, 7/4]
[1/1, 9/7, 8/5, 2/1]
[1/1, 9/7, 8/5, 9/5]
[1/1, 9/8, 10/7, 11/6]
[1/1, 9/8, 10/7, 13/7]
[1/1, 9/8, 10/7, 5/3]
[1/1, 9/8, 10/7, 7/4]
[1/1, 9/8, 10/7, 8/5]
[1/1, 9/8, 10/7, 9/5]
[1/1, 9/8, 11/8, 11/7]
[1/1, 9/8, 11/8, 13/8]
[1/1, 9/8, 11/8, 7/4]
[1/1, 9/8, 11/8, 8/5]
[1/1, 9/8, 11/8, 9/5]
[1/1, 9/8, 13/10, 12/7]
[1/1, 9/8, 13/10, 13/8]
[1/1, 9/8, 13/10, 3/2]
[1/1, 9/8, 13/10, 8/5]
[1/1, 9/8, 13/9, 12/7]
[1/1, 9/8, 13/9, 13/8]
[1/1, 9/8, 4/3, 11/7]
[1/1, 9/8, 4/3, 12/7]
[1/1, 9/8, 4/3, 3/2]
[1/1, 9/8, 4/3, 5/3]
[1/1, 9/8, 4/3, 7/4]
[1/1, 9/8, 7/5, 11/7]
[1/1, 9/8, 7/5, 13/7]
[1/1, 9/8, 7/5, 13/8]
[1/1, 9/8, 7/5, 8/5]
[1/1, 9/8, 9/7, 11/7]
[1/1, 9/8, 9/7, 5/3]
[1/1, 9/8, 9/7, 8/5]
.
That's not an exhaustive list, just a quick sampling of the valid chord space. The members which are 5-limit, rather than 7-limit or higher are:
[1/1, 5/4, 3/2, 9/5]
[1/1, 5/4, 8/5, 2/1]
[1/1, 5/4, 8/5, 9/5]
[1/1, 6/5, 3/2, 9/5]
[1/1, 9/8, 4/3, 3/2]
[1/1, 9/8, 4/3, 5/3]
You can see that the third column, the "some kind of fifth" column, has a justly tuned minor sixth (8/5) and a justly tuned perfect fourth (4/3). So the intervals aren't actually tertian intervallically and we could clean this up a bit.
If we want to spell chords intervallically by thirds, then the only Sethares fractions which are [1st, 3rds, 5ths, 7th, 9th, 11ths, 13ths] in the rank-6 Lilley-Johnston system are:
P1 # 1/1 = 1.0
PrDem3 # 13/11 = 1.1818181818181819
AsGrm3 # 11/9 = 1.2222222222222223
Sbm3 # 7/6 = 1.1666666666666667
m3 # 6/5 = 1.2
M3 # 5/4 = 1.25
SpM3 # 9/7 = 1.2857142857142858
Sbd5 # 7/5 = 1.4
PrGrd5 # 13/9 = 1.4444444444444444
P5 # 3/2 = 1.5
AsSpGr5 # 11/7 = 1.5714285714285714
Sbm7 # 7/4 = 1.75
m7 # 9/5 = 1.8
AsGrm7 # 11/6 = 1.8333333333333333
PrSpGrm7 # 13/7 = 1.8571428571428572
Prm9 # 13/6 = 2.1666666666666665
Asm9 # 11/5 = 2.2
AcM9 # 9/4 = 2.25
Prd11 # 13/5 = 2.6
P11 # 8/3 = 2.6666666666666665
As11 # 11/4 = 2.75
Prm13 # 13/4 = 3.25
M13 # 10/3 = 3.3333333333333335
Here are some chords made of 7-limit frequency ratios that are tertian sounding in their tuning (harmonic intervals between 10/9 and 4/3) and also tertian intervallically, in the sense of being enumerated correctly in Lilley-Johnston bases (and consequently spelled correctly in pitch classes):
[1/1, 5/4, 3/2, 7/4]
[1/1, 5/4, 3/2, 9/5]
[1/1, 5/4, 7/5, 7/4]
[1/1, 5/4, 7/5, 9/5]
[1/1, 6/5, 3/2, 7/4]
[1/1, 6/5, 3/2, 9/5]
[1/1, 6/5, 7/5, 7/4]
[1/1, 6/5, 7/5, 9/5]
[1/1, 7/6, 3/2, 7/4]
[1/1, 7/6, 3/2, 9/5]
[1/1, 7/6, 7/5, 7/4]
[1/1, 7/6, 7/5, 9/5]
[1/1, 9/7, 3/2, 7/4]
[1/1, 9/7, 3/2, 9/5]
A few of these can be represented with common denominators that are fairly small:
[4/4, 5/4, 6/4, 7/4] : [P1, M3, P5, Sbm7]
[5/5, 6/5, 7/5, 9/5] : [P1, m3, Sbd5, m7]
[12/12, 14/12, 18/12, 21/12] : [P1, Sbm3, P5, Sbm7]
[30/30, 35/30, 42/30, 54/30] : [P1, Sbm3, Sbd5, m7]
And a few more can be represented with common numerators that are fairly small:
[7/7, 7/6, 7/5, 7/4] : [P1, Sbm3, Sbd5, Sbm7]
[9/9, 9/7, 9/6, 9/5] : [P1, SpM3, P5, m7]
[21/21, 21/18, 21/14, 21/12] : [P1, Sbm3, P5, Sbm7]
There's actually a repeat here: [P1, Sbm3, P5, Sbm7] had {12} in the denominator and {21} in the numerator.
Between the Sethares harmonic model, the tertian tuning and intervallic structure, this otonality/utonality structure, and the low prime-limit, I expect these 6 guys to be good chords on many different grounds. Perhaps we should listen to them and figure out names for them? I think so.
I think these names, while not exactly good, are in keeping with standard modern chordal naming conventions:
[P1, M3, P5, Sbm7] : harmonic-seventh
[P1, Sbm3, P5, Sbm7] : sub-minor harmonic-seventh
[P1, Sbm3, Sbd5, Sbm7] : sub-minor sub-diminished harmonic-seventh
[P1, Sbm3, Sbd5, m7] : sub-diminished sub-minor seventh
[P1, SpM3, P5, m7] : super-major dominant-seventh
[P1, m3, Sbd5, m7] : sub-diminished minor-seventh
.
...
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