1/1 C Tonic 9/8 D Supertonic / Fifth Fifth 6/5 Eb Minor Third 5/4 E Major Third 4/3 F Fourth 54/40 = 27/20 F Super Minor Third 45/32 F# Super Major Third 3/2 G Fifth 24/15 G# Fourth Minor Third 20/12 = 10/6 A Fourth Major Third 27/16 A Super Fifth 7/4 Bb Septimal 16/9 Bb Subtonic 18/10 = 9/5 Bb Fifth Minor Third 15/8 B Fifth Major Third 2/1 C Octave
In this scale you can modulate between I, II, IV and V chords with both minor and major thirds with absolute just accuracy.
Frankly I don't know what's going on with the minor seventh, so I've included all the options. Obviously to facilitate modulation we need the the 18/10 Fifth Minor Third. There is open (and probably unanswerable debate) as to whether the 12-TET minor seventh is approximating a septimal (i.e. derived from the 7th partial) interval or is the complement to the major second. It's probably both, so I've included both. But I don't think anyone needs three B flats, so there's work to be done here. Perhaps with more experience I would be able to make a firmer choice.
This scale will play modal forms such as C major and D dorian adequately well, although better scales could be designed for that purpose.
So that's basically it. I believe this scale is capable of playing most 20th century popular music and most folk more beautifully than 12-TET (it won't play III, VI or VII chords or 7th chords, except the root seventh). Now I need to build an instrument to test it.
Apologia
Some of these intervals are as small as 21 cents. Some are over two semitones. If you're playing a melody it would make no sense to run up and down the notes, just as it makes little sense to play a 12-TET chromatic melody. It might take a while to feel which notes are right, but it may also be possible to suggest harmonic change just by playing a melody. How effective this is remains to be tested.
Some of these intervals appear to be more dissonant than perhaps would be expected. This is because they are intended to suggest a shift in the root note ("key change"). Accepting this more firmly than is reasonable, the largest dissonance that remains is 9/8.
Could we possibly build this around a standard keyboard?
ReplyDeleteThe F would be re-tuned on the fly by the context, i.e. which chord/root you are playing or last played.
If you could formalise a specimen set of rules for me it'd be fairly trivial to program.
OK, let's lose the 16/9 Bb for now and just use the 7/4 and 9/5 Bbs. Once we've tried it we might reverse that decision.
ReplyDeleteIf 9/8 (D), 27/20 (F), 45/32 (F#) or 27/16 (A) is played or is the last note played then F = 27/20, A = 27/16, else F = 4/3
If 4/3 (F), 24/14 (G#), 10/6 (A) or 2/1 (C) is played or is the last note played then A = 10/6
If 3/2 (G), 15/8 (B) or 9/8 (D) is played or is the last note played then Bb = 9/5, else Bb = 7/4
This check has to be instant, for example if the supertonic minor is played consisting of D, F and A, then the F has to be 27/20 and the A has to be 27/16 even though they're played at the same time as the D.
It also has to cope with rests and stacatto playing. If a melody is being played with a rest it needs to know that the last note played must still define the tuning.
You'll notice that the C# key is unassigned. I don't reckon it's needed.
OK, I'm going to rewrite those rules and then you're going to correct my rewrite.
ReplyDeleteRule 1: Hitting the piano keys D,F# or A retunes the 'F' oscillator to 27/20, hitting any other piano keys retunes it to 4/3
Or, in shorthand:
(1) F = 27/20 (D,F#,A) else F = 4/3
(2) A = 27/16 (F,F#,D) else A = 10/6
(3) Bb = 9/5 (G,B,D) else Bb = 7/4
Notice that: (1) I'm distinguishing between physical piano keys and the retunable oscillators that they're hard-wired to, and not allowing keys to affect their 'own' oscillators. (2) Only the keys can retune the oscillators, oscillators cannot retune each other, so the tuning of 'F' cannot affect the tuning of 'A' (let's keep it simple til we decide that we can't...)
What happens if you press F and A together? There needs to be a way to get 27/20 and 27/16 and a different way to get 4/3 and 10/6.
ReplyDeleteOne possible solution would be to have a lower octave with only
1/2 C Tonic
9/16 D Supertonic / Fifth Fifth
6/10 Eb Minor Third
5/8 E Major Third
4/6 F Fourth
3/4 G Fifth
27/32 A Super Fifth
7/8 Bb Septimal
2/2 C Octave
F = 27/20 (D lower octave) else F = 4/3
A = 10/6 (F lower octave) else A = 27/16
Bb = 9/5 (G lower octave) else Bb = 7/4
If you took this route, it would be nice to have a switch to toggle the lower octave sounding on and off.
We can have F and A change together, something like this:
ReplyDelete(1) F=27/20 & A=27/16 (D,F#)
(2) F=4/3 & A=10/6 (C,C#,Eb,E,G,G#,B)
(3) Bb=9/5 (G,B,D)
(4) Bb=7/4 (C,C#,Eb,E,F,F#,G#,A,B)
(These need tidying up a bit, you'll know what to do).
Anyways, I've built the damn thing (well, one very primitve octave of it). Come see how it's put together and then you'll have a better idea of what's possible.