Deux Chalumeaux in D minor tritonic hexatonic scale

Hi,

[I wrote the following before adding a third part to produce the piece called "Trois Chalumeaux", which I posted today. But the tuning part, and the invitation, is still relevant to the outcome. - Yahya.]
_________________

Here's a weird scale:

D E F G# A# B ~ D E F Ab Bb Cb

In 12-EDO, of course, either spelling is the same thing.

Steps are TSATSA, where T=tone, S=semitone, A=augmented tone

Using the sharp spelling, I make out that progressions in "thirds" actually have minor third, major third, augmented third (tritone), a pattern that repeats at the half-octave. Also this scale has at most two perfect fifths E-B and (A#=)Bb-F, yet has three tritones built in.

I've written a small piece in this "D minor tritonic hexatonic" scale, emphasising this progression in "thirds", and scored it for two (MIDI) flutes, not worrying overmuch about playability on analogue instruments. As a fan of Bartok, I've called it, naturally enough, "Deux Chalumeaux". I will post it to the files shortly. [I have posted "Trois Chalumeaux" instead.]

Question is: what is the "best" JI tuning for this scale, to maximise the consonance of the "thirds", and keep all the tritones as equal as possible (ie I want two low-limit rational approximations to the square root of 2)? I want minor and major thirds to be exactly 6/5 and 5/4, and all tritones to be one of two ratios. Suppose I make D-G# = E-A# = F-B = m/n. Then:
G#-D = A#-E = B-F = (2/1) / (m/n) = 2n/m, and we want m/n ~=2n/m, ie m^2 ~= 2n^2.
D = 1/1
E = 4x/1
F = 6/5
G# = m/n = (4x/1) (5/4) = 5x/1 -> x= m/5n (1)
A# = (4x/1) (m/n) = 4mx/n
B = (6/5) (m/n) = 6m/5n
(1) -> E = 4m/5n, A# = 4m^2/5n^2.

So the scale is:
D = 1/1, E = 4m/5n, F = 6/5, G# = m/n, A# = 4m^2/5n^2, B = 6m/5n, with m^2 ~= 2n^2.

Interestingly, E-B is a perfect fifth, whatever m and n we choose.

Check:
G#-D' ~= m/n and also = (2/1) / (m/n) = 2n/m, consistent with m^2 ~= 2n^2.
A#-D' ~= 5/4 and also = (2/1) / (4m^2/5n^2) = 5n^2/2m^2, consistent with m^2 ~= 2n^2.
B-D' = 6/5 and also = (2/1) / (6m/5n) = 5n/3m, giving m/n = 25/18, and (25/18)^2 ~= 1.93.

So a tuning for this scale is, when m/n = 25/18:
D = 1/1, E = 10/9, F = 6/5, G# = 25/18, A# = 125/81, B = 5/3.

The interval E-B is 5/3 x 9/10 = 3/2, a perfect fifth.
The interval A#-F' is 12/5 x 81/125 = 972/625, or 648/625 times a perfect fifth. That is, it is a "quintal comma" greater than a perfect fifth.

(Note: For A#-F' to be a perfect fifth, we would have to tune A# as 2 x (6/5) / (3/2) = 8/5. But A# = 4m^2/5n^2, -> m^2 = 2n^2. This would produce the tuning: G# ~= 1.4142 = 7071/5000 ~= 1.414 = 707/500 ~=1.41 = 3x47/100 ~= 1.4 = 7/5, none of which are 5-limit tunings of the tritone.)

The "D minor tritonic hexatonic" scale, in 12-EDO, uses the step numbers:
D = 0, E = 2, F = 3, G# = 6, A# = 8, B = 9, (D' = 12)

The "D minor tritonic hexatonic" scale, in 5-limit JI, uses the ratios:
D = 1/1, E = 10/9, F = 6/5, G# = 25/18, A# = 125/81, B = 5/3, (D' = 2/1)

I note that Jon Smith has extended this 5-limit tuning to a 12-note scale, which he might like to tell us about - Jon?
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Now, I would like to invite you all to use the same "minor tritonic hexatonic" scale to write some music, simple or complex. Unlike Jacob, I can't promise a performance on specially-tuned instruments! Still, it would be instructive to learn what others can make from the same, small tonal resource.

Regards,
Yahya

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Hi Yahya.

Yahya Abdal-Aziz escreveu:
> Hi,
> > [I wrote the following before adding a third part to produce the
> piece called "Trois Chalumeaux", which I posted today. But the
> tuning part, and the invitation, is still relevant to the outcome. -
> Yahya.] _________________
> > Here's a weird scale:
> > D E F G# A# B ~ D E F Ab Bb Cb
> > In 12-EDO, of course, either spelling is the same thing.
> > Steps are TSATSA, where T=tone, S=semitone, A=augmented tone

In 12-EDO, it is a mode with limited transpositions (Messiaen), or a
subset of the ``diminished'' scale (as jazzists could call it). For Messian, I found this:

http://mediatheque.ircam.fr/HOTES/SNM/ITPR25RIVTXT.html

<<<
5.7. 2ème échelle à transpositions limitées tronquée n°2

[...]

L'ordre des intervalles est

(1 1/2 + 1 + 1/2 + 1 1/2 + 1 + 1/2) = 6 tons

ou

(1/2 + 1 1/2 + 1 + 1/2 + 1 1/2 + 1) = 6 tons

ou

(1 + 1/2 + 1 1/2 + 1 + 1/2 + 1 1/2) = 6 tons
>>>

[Yahya's JI tuning for that scale:]

> D = 1/1, E = 10/9, F = 6/5, G# = 25/18, A# = 125/81, B = 5/3

The same intervals in cents:
0.000000000000000
182.403712134060072
315.641287000552552
568.717425998894782
751.121138132954684
884.358712999447448
1200.000000000000000

(I note that there is a single tuned perfect triad: {E G# B}.)

> Now, I would like to invite you all to use the same "minor tritonic
> hexatonic" scale to write some music, simple or complex. Unlike
> Jacob, I can't promise a performance on specially-tuned instruments!
> Still, it would be instructive to learn what others can make from the
> same, small tonal resource.

Maybe I will have a try.

Thanks,
Hudson

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