MOS/DE

Call an MOS/DE scale "obviously generated" if the generator is a scale step.

It's kind of neat that the only 2, 4, and 6 note MOS/DE scales are obviously generated -- even in continuous space, where you can divide the octave as finely as you like. These are the only cardinalities for which it's the case.

DT
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Dmitri Tymoczko
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On 7/19/06, Dmitri Tymoczko <dmitri@princeton.edu> wrote:
> Call an MOS/DE scale "obviously generated" if the generator is a scale step.
>
> It's kind of neat that the only 2, 4, and 6 note MOS/DE scales are
> obviously generated -- even in continuous space, where you can divide
> the octave as finely as you like. These are the only cardinalities
> for which it's the case.

Actually, you forgot 3. These are all the positive integers whose
totients are at most 2: 1, 2, 3, 4, and 6. Any other positive integer
n has another integer g between 1 and n-1 which is coprime to n, so it
can be a generator.

Keenan

>Actually, you forgot 3. These are all the positive integers whose
>totients are at most 2: 1, 2, 3, 4, and 6. Any other positive integer
>n has another integer g between 1 and n-1 which is coprime to n, so it
>can be a generator.

Quite right. I had realized I'd left off 3 (and 1), but wondered whether anyone would notice.

I find the phrase "a positive integer ... that can serve as generator" a little bit confusing, since I'm imagining a continuous space where the scale's generator need not be an integer.

So, for five notes, the scale's generator can be anything in the range from [4, 6] (these numbers refer to distance in a continuous pitch class space where the octave is 12). And the two step sizes can have an irrational ratio, so that they can't be embedded in an equal-tempered scale.

DT
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WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

>Actually, you forgot 3. These are all the positive integers whose
>totients are at most 2: 1, 2, 3, 4, and 6. Any other positive integer
>n has another integer g between 1 and n-1 which is coprime to n, so it
>can be a generator.

By the way, I think this explanation is essentially correct, it's just not connected to the size of the scale's generator in quite such an obvious way -- since the generator can vary continuously. It's about how many octaves the scale cycles through ...

DT
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WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

On 7/20/06, Dmitri Tymoczko <dmitri@princeton.edu> wrote:
> By the way, I think this explanation is essentially correct, it's
> just not connected to the size of the scale's generator in quite such
> an obvious way -- since the generator can vary continuously. It's
> about how many octaves the scale cycles through ...

I was thinking of the integer representing the number of scale steps.

Keenan

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:

> So, for five notes, the scale's generator can be anything in the
> range from [4, 6] (these numbers refer to distance in a continuous
> pitch class space where the octave is 12).

Why do you confine it to that range?

> > So, for five notes, the scale's generator can be anything in the
> > range from [4, 6] (these numbers refer to distance in a continuous
> > pitch class space where the octave is 12). >
>Why do you confine it to that range?

Here are the desired criteria:

1) S is a scale with five notes
2) S is MOS/DE
3) S's generator is not a scale step

I believe that these three together imply that the generator of S lies in the range [4, 6], where the numbers refer to distance in a continuous pitch class space, with octave size 12. So it's the great god of Mathematics, rather than me, confining things to that range.

I might be wrong, in which case a counterexample would be nice.

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:

> Here are the desired criteria:
>
> 1) S is a scale with five notes
> 2) S is MOS/DE
> 3) S's generator is not a scale step

Ah. Didn't know 3 was a criterion.

> I believe that these three together imply that the generator of S
> lies in the range [4, 6], where the numbers refer to distance in a
> continuous pitch class space, with octave size 12. So it's the great
> god of Mathematics, rather than me, confining things to that range.
>
> I might be wrong, in which case a counterexample would be nice.

No, you are right. If we take 2/5, then on the 5 level of Farey
sequences we get 1/3 < 2/5 < 1/2, so semiconvergents need to be in
this range. In cents this is 400 < g < 600, and in your 100 cent
units, 4 < g < 6.

Dif you deduce this some other way, not using semiconvergents?