Should the dominant-seventh chord in just intonation use 9:5 for its seventh, or rather 16:9?

Question

It may be just out of habit, what with the 12-edo temperament all around us, but anyway: I find the dominant-seventh rendition 20:25:30:36 sounds extremely harsh. In particular, the tritone from the third to the seventh is so wide that it raises suspicions about being just an extremely detuned perfect fifth. Sure, the chord is meant to be dissonant, but like that?

The alternative with a Pythagorean minor seventh 16:9, on the other hand (though requiring higher ratio numbers 36:45:54:64), comes out less jarring. Its tritone is closer to the √2 : 2 we know from 12-edo, which at least makes it much more suitable for tritone-subtitution techniques. Those were after all used a lot in music – not just in Jazz!

What arguments (historical, musical, practical or otherwise) are there, in favour of either of these possibilities? I'd figure there are a lot of considerations about the leading tones etc..

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While there are no answers, I tried my best (which isn't very much I'm afraid... overall intonation does not exactly deserve the name "just", but I hope it's good enough to make the issue clear. Anyway...) to contrast the different possibilities.

Here's a simple cadence for string quartet, played six times in a row. Each iteration is identical, except for the V₇ chord, which has each time a different ratio of frequencies. It's all the possible ways to do it in random order, with both just and Pythagorean and, for fun, even harmonic seventh (7:4). Each of these was combined both with a just third 5:4, and also with a Pythagorean third 81:64, to make it complete.

Well, for me that has settled the question of what I like best: _{(Hover cursor to see)}

Surprise, surprise! The all-just intonation wins, after all. Within a proper context, it doesn't sound harsh at all but rather "sparkling" and adds the right amount of tension the dominant needs.
The harmonic seventh, as I expected, is right out: it completely undermines the leading melody. The Pythagorean seventh sounds quite ok actually (albeit only together with a just third, otherwise it's much too low in comparison), but it's rather dull compared to the all-just chord 20:25:30:36.

Can you spot which one it is, in the audio file?

1. Pythagorean third, just seventh.
2. Pythagorean third, Pythagorean seventh.
3. Just third, Pythagorean seventh.
4. Pythagorean third, harmonic seventh.
5. Just third, just seventh.
6. Just third, harmonic seventh.

...Of course, that doesn't answer the question though, it's just a personal impression. What I'd like to know is if that's really the generally accepted way to do it.

Answer 1

I also really love the harmonic seventh in dominant chords. There's just something about that 7/4 that gets me going. One technique used by prof S of the CCM for harmonic progressions on violin is to make the most melodically important voice Pythagorean since it is so pleasing horizontally. Then he will adjust the harmonizing voices to be justly intoned relative to the Pythagorean melody. I think this creates a brilliant effect and it sounds very compelling. With this approach we have the best of both worlds. The melody that comes out of the texture most will sound pleasing not awkward and none of the harmonic intervals will be jarring or muddy. If the voice producing the 7 in this V chord has the principal melody then I would suggest tuning from it instead of tuning from the root of the V chord. If we tune the harmonic seventh from the root of the V chord it will be 21/16 in relation to tonic. This out of tune subdominant won't sound good resolving to the mediant. If we keep the subdominant a perfect fourth in relation to tonic and tune the other chord members around it then it's resolution will be fixed. Sure the dominant will be slightly sharp in relation to tonic but this will make the leading tone also sharper and more willing to resolve. Therefore, the intensity of the resolution of the leading tone and the subdominant should let any voicing of the dominant retune to a perfect fifth above tonic at the arrival of I without any aural disturbance.

Answer 2

What I have learned from composition and music history, is that regardless of the intellectual aesthetic, what matters when you get to the double bar is what sounds the best.

Composers do not write squiggles down and figure out sounds to fit those squiggles, patting themselves on the back about how clever they are.

Neither should you either fuss with ratios on paper - notation reflects the sounds you hear. Unless you are performing in a period ensemble with period instruments, different music from around the world, or music in which the performative tradition dictates the temperament, or music written in a specific temperament, the best, simplest advice is to follow your ear. If it sounds appropriate to you, chances are it will sound appropriate to someone else.

Besides, there are perhaps only a small handful of musicians in the world who would be upset with your chosen ratio upon initial hearing.

Answer 3

I agree with your choice, given your context here. But I must say that I prefer the sound of the harmonic seventh in a context with no "leading tones". If you're interested, here's a sample of a double harp playing with the scale tuning 8:9:10:45/32:12:14:15-

cheers from rainy Vienna, Scott