The chords we use in western music theory are usually played in 12-TET, and 12-TET kind of blurs the difference between 3-limit and 5-limit just intonation. So you might wonder, do the chords sound good in 3-limit and 5-limit? Do some chords sound better in one intonation than the other? We're going to find out.
Here's the basic idea. A western chord might look like this:
".7": [P1, m3, P5, M7],
We can treat those as rank-2 intervals, and then the just tunings are 3-limit (a.k.a. Pythagorean). If instead we treat them as rank-3 intervals, then the just tunings are 5-limit.
If we write out the rank-3 intervals that have the same just tunings as the just intonation of the rank-2 intervals, then we get a chord like this:
".7_Pythagoren": [P1, Grm3, P5, AcM7],
I'm going to write down a bunch of chords like that in both systems, and then I'll listen to them and tell you what I think. In principle, we could have chords that mix 5-limit and 3-limit sounds. But we're going to start like this.
Here's a set of 28 fairly basic chords in rank-3 intervals:
chord_quality_to_intervals = {
".dim": [P1, m3, d5],
".dim7": [P1, m3, d5, d7],
".dim9": [P1, m3, d5, d7, M9],
".dim11": [P1, m3, d5, d7, M9, P11],
".m7b5": [P1, m3, d5, m7],
".m9b5": [P1, m3, d5, m7, M9],
".m11b5": [P1, m3, d5, m7, M9, P11],
".dim-maj7": [P1, m3, d5, M7],
".dim-maj9": [P1, m3, d5, M7, M9],
".dim-maj11": [P1, m3, d5, M7, M9, P11],
".m": [P1, m3, P5],
".m7": [P1, m3, P5, m7],
".m9": [P1, m3, P5, m7, M9],
".m11": [P1, m3, P5, m7, M9, P11],
".maj": [P1, M3, P5],
".7": [P1, M3, P5, m7],
".9": [P1, M3, P5, m7, M9],
".11": [P1, M3, P5, m7, M9, P11],
".maj7": [P1, M3, P5, M7],
".maj9": [P1, M3, P5, M7, M9],
".maj11": [P1, M3, P5, M7, M9, P11],
".aug": [P1, M3, A5],
".aug7": [P1, M3, A5, m7],
".aug9": [P1, M3, A5, m7, M9],
".aug11": [P1, M3, A5, m7, M9, P11],
".aug-maj7": [P1, M3, A5, M7],
".aug-maj9": [P1, M3, A5, M7, M9],
".aug-maj11": [P1, M3, A5, M7, M9, P11],
}
For the Pythagorean version, the perfect intervals stay the same, the minor intervals become "grave minor" intervals, the major intervals become "acute major" intervals. Unfortunately, the diminished and augmented intervals have to be treated case by case. But here is the full set of substitutions we'll need:
P1 → P1 # 1/1
m3 → Grm3 # 6/5 → 32/27
M3 → AcM3 # 5/4 → 81/64
d5 → GrGrd5 # 36/25 → 1024/729
P5 → P5 # 3/2
A5 → AcAcA5 # 25/16 → 6561/4096
d7 → GrGrGrd7 # 216/125 → 32768/19683
m7 → Grm7 # 9/5 → 16/9
M7 → AcM7 # 15/8 → 243/128
M9 → AcM9 # 20/9 → 9/4
P11 → P11 # 8/3
With those in hand, here are the Pythagorean chords we're going to examine:
pythagorean_chord_quality_to_intervals = {
".dim_pyth": [P1, Grm3, GrGrd5],
".dim7_pyth": [P1, Grm3, GrGrd5, GrGrGrd7],
".dim9_pyth": [P1, Grm3, GrGrd5, GrGrGrd7, AcM9],
".dim11_pyth": [P1, Grm3, GrGrd5, GrGrGrd7, AcM9, P11],
".m7b5_pyth": [P1, Grm3, GrGrd5, Grm7],
".m9b5_pyth": [P1, Grm3, GrGrd5, Grm7, AcM9],
".m11b5_pyth": [P1, Grm3, GrGrd5, Grm7, AcM9, P11],
".dim-maj7_pyth": [P1, Grm3, GrGrd5, AcM7],
".dim-maj9_pyth": [P1, Grm3, GrGrd5, AcM7, AcM9],
".dim-maj11_pyth": [P1, Grm3, GrGrd5, AcM7, AcM9, P11],
".m_pyth": [P1, Grm3, P5],
".m7_pyth": [P1, Grm3, P5, Grm7],
".m9_pyth": [P1, Grm3, P5, Grm7, AcM9],
".m11_pyth": [P1, Grm3, P5, Grm7, AcM9, P11],
".maj_pyth": [P1, AcM3, P5],
".7_pyth": [P1, AcM3, P5, Grm7],
".9_pyth": [P1, AcM3, P5, Grm7, AcM9],
".11_pyth": [P1, AcM3, P5, Grm7, AcM9, P11],
".maj7_pyth": [P1, AcM3, P5, AcM7],
".maj9_pyth": [P1, AcM3, P5, AcM7, AcM9],
".maj11_pyth": [P1, AcM3, P5, AcM7, AcM9, P11],
".aug_pyth": [P1, AcM3, AcAcA5],
".aug7_pyth": [P1, AcM3, AcAcA5, Grm7],
".aug9_pyth": [P1, AcM3, AcAcA5, Grm7, AcM9],
".aug11_pyth": [P1, AcM3, AcAcA5, Grm7, AcM9, P11],
".aug-maj7_pyth": [P1, AcM3, AcAcA5, AcM7],
".aug-maj9_pyth": [P1, AcM3, AcAcA5, AcM7, AcM9],
".aug-maj11_pyth": [P1, AcM3, AcAcA5, AcM7, AcM9, P11],
}
Now to shake them in the sieve of my aesthetics.
...
I ended up throwing out the chords with 11th degrees. Too busy. Let's focus on the lower notes.
...
Hahahaha, oh no. All of the Pythagorean ones sound better. I got into microtonal music because of the beauty of five-limit just intonation, and I've been writing about microtonal music since February of 2022. It has not all been wasted effort, but I'm still pretty shocked. The 3-limit ones are smooth, jazzy, with a little bit of a sparkly. The 5-limit ones are bold, brassy, a bit like a barbershop quartet, a bit like an air raid siren, and the things that was a "sparkly" before is now a dissonant wub-wub auditory beating that grabs you and shakes you. It's still not actually bad compared to some of the microtonal dissonances I've inflicted on myself, but it's noticeably worse than the 3-limit. The 9th and 11th chords in particular are harder to stomach in 5-limit intonation - the 3 and 4 note chords mostly pass by without notice.
...
I'm not entirely sure what I need to do at this point. Probably rewrite some parts of my microtonal music theory textbook. And I'm going to add the Pythagorean versions of various chords to my 5-limit just intonation counterpoint generating programs. And... see what other people think of the sounds, in case I'm taking crazy pills.
...
Two people said that, if I played the two intonations of a chord side by side, the first one anchored their perception and the second one always sounded worse.
One person said that the Pythagorean chords always sounded better or not worse.
The person whose ear I trust the most, who tunes harpsichords for fun and plays in different meantone temperaments, said that they preferred the 5-limit intonation of major and augmented chords. Presumably that means they preferred the 3-limit intonation of diminished and minor chords.
This is encouraging to me. It means that both intonations might be useful. You can compose music that switches between 5-limit and 3-limit chords. They both have uses. I should listen again.
...
