Double diatonic [was: optimal fifth sizes for scales (in fixed-pitch circumstances)]

On 9 Jun 1999 PErlich@Acadian-Asset.com wrote:
> The optimal "fifth" size for the tetradecatonic (double diatonic) scale
> (with 7-limit harmony) is 693.651 cents and can be derived as follows:
>
> The 3/1 is constructed, of course, by one fifth plus an octave.
> The 5/1 is constructed by four fifths.
> The 5/3 is constructed by three fifths minus one octave.
> The 7/1 is constructed by four fifths plus half an octave.
> The 7/3 is constructed by three fifths minus half an octave.
> The 7/5 is constructed by half an octave.

Interesting--Paul, have you mentioned this double diatonic scale on the
list before? The only time I recall is a brief toss-off in your paper
about how the decatonic scale was the only one that met certain
conditions with fewer than fourteen notes. ('Course, I think you've
revised your paper a couple of times since the last draft I read of it.)

I'm curious because when I started my own search (slightly different
than yours, but also similar in many ways) for generalized diatonic
scales, some years ago, the very first one that I tried (derived using
Fokker's method) was tetradecatonic, much as you describe. The main
difference was that even then I was determined to embed it in 31TET, and
since there isn't an exact half-octave in 31TET the two diatonic scales
overlapped by one pitch and I would end up with only thirteen pitches,
which was just too uneven for me. So I never really pursued it very
much.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Hey--do you think I need to lose some weight?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>

Paul Hahn wrote,

> On 9 Jun 1999 PErlich@Acadian-Asset.com wrote:
> > The optimal "fifth" size for the tetradecatonic (double diatonic) scale
> > (with 7-limit harmony) is 693.651 cents and can be derived as follows:
>
> Interesting--Paul, have you mentioned this double diatonic scale on the
> list before?

Yes, and one of the times I attempted to determine which modes are most likely to function tonally.

> The only time I recall is a brief toss-off in your paper
> about how the decatonic scale was the only one that met certain
> conditions with fewer than fourteen notes. ('Course, I think you've
> revised your paper a couple of times since the last draft I read of it.)

That must have been an old version of the paper that only ever existed as an e-mail.

> I'm curious because when I started my own search (slightly different
> than yours, but also similar in many ways) for generalized diatonic
> scales, some years ago, the very first one that I tried (derived using
> Fokker's method) was tetradecatonic, much as you describe.

Describe how you used Fokker's method. Oh yeah, you already did. Well, should we look at it again?

On Tue, 8 Jun 1999, Paul Erlich wrote:
> Paul Hahn wrote,
>> I'm curious because when I started my own search (slightly different
>> than yours, but also similar in many ways) for generalized diatonic
>> scales, some years ago, the very first one that I tried (derived using
>> Fokker's method) was tetradecatonic, much as you describe.
>
> Well, should we look at it again?

Not for my sake. I have several problems with it:

(a) It's still quite uneven, even without the overlap turning 14 pitches
into 13.

(b) Too many pitches anyway (cf. the "7 +/- 2" rule).

(c) It only works if the 50:49 and 81:80 both vanish, but the 225:224
doesn't, which is a little too bassackwards for me.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Hey--do you think I need to lose some weight?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>