Re: kleismic (say, is it good for klezmer?)

> Now, can you figure out how "kleismic" is defined? Hint: the kleisma
> = 15625:15552
>

It is the ratio of octave_normalized( 5^6/3^5 ), that I would probably
think of more as... difference between "B" found by cycling fifths and
Ax found by cycling major thirds.

Now, is there a special term for when

best( 5/4 ) = 9 * best( 3/2 )

(in other words, D# ~= 5/4, which I believe is the case in 22 but is not
a general rule for diaschismic).

> You can always look up commas here:
>
> http://www.xs4all.nl/~huygensf/doc/intervals.html
>

thanks, Bob

--- In tuning-math@y..., Robert C Valentine <BVAL@I...> wrote:
>
> > Now, can you figure out how "kleismic" is defined? Hint: the
kleisma
> > = 15625:15552
> >
>
> It is the ratio of octave_normalized( 5^6/3^5 ), that I would
probably
> think of more as... difference between "B" found by cycling fifths
and
> Ax found by cycling major thirds.
>
> Now, is there a special term for when
>
> best( 5/4 ) = 9 * best( 3/2 )
>
> (in other words, D# ~= 5/4, which I believe is the case in 22 but
is not
> a general rule for diaschismic).

I don't think there's a special term for that -- but there are plenty
of simpler ways of characterizing 22 -- for example, 22 is the only
tuning where all three of the commas

50:49
64:63
245:243

vanish.

This is the 7-limit Minkowski-reduced basis for 22 -- meaning, the
simplest set of three 7-limit commas that define 22. Perhaps Gene can
provide the 5-limit Minkowski reduced basis, which would have only
two commas. I bet the diaschisma is one of them . . .

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

Perhaps Gene can
> provide the 5-limit Minkowski reduced basis, which would have only
> two commas. I bet the diaschisma is one of them . . .

Here you go. Enjoy!

7: [25/24, 81/80]
12: [81/80, 128/125]
19: [81/80, 3125/3072]
22: [250/243, 2048/2025]
31: [81/80, 393216/390625]
34: [2048/2025, 15625/15552]
41: [3125/3072, 20000/19683]
46: [2028/2025, 78732/78125]
53: [15625/15552, 32805/32768]
65: [32805/32768, 78732/78125]
72: [15625/15552, 531441/524288]
84: [78732/78125, 531441/524288]
87: [15625/15552, 67108864/66430125]
99: [393216/390625, 1600000/1594323]
118: [32805/32768, 1224440064/1220703125]

20000/19683, 531441/524288 and 67108864/66430125 weren't on my temperament list, so I'd better check if they should have been.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> Perhaps Gene can
> > provide the 5-limit Minkowski reduced basis, which would have
only
> > two commas. I bet the diaschisma is one of them . . .
>
> Here you go. Enjoy!
>
> 7: [25/24, 81/80]
> 12: [81/80, 128/125]
> 19: [81/80, 3125/3072]
> 22: [250/243, 2048/2025]

How about 15? Herman Miller wrote some wonderful music in 15-tET
exploiting the 250:243.

>20000/19683, 531441/524288 and 67108864/66430125 weren't on my
>temperament list,

The first two are called the "minimal diesis" and "Pythagorean comma".

>so I'd better check if they should have been.

I had hoped your 5-limit search was "airtight" already . . . oh well.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> 20000/19683, 531441/524288 and 67108864/66430125 weren't on my temperament list, so I'd better check if they should have been.

20000/19683

Map: [[0, 4, 9], [1, 1, 1]]

Ets: 7, 27, 34, 41, 75

Generators: a = 6.023/41 = 176.2823 cents; b = 1

badness 649
rms 2.504
g 6.38

531441/524288

Map: [[0, 0, 1], [12, 19, 24]]

Ets: 12, 24, 36, 48, 60, 72, 84 ...

Generators: a = 385.3362 cents, b = 100 cents

badness 1300
rms 1.382
g 9.80

67108864/66430125

Map: [[0, -1, 4], [3, 6, 2]]

Ets: 12, 75, 87, 99, 111

Generators: a = 496.7879 cents; b = 400 cents

badness 1280
rms .9052
g 11.225

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> How about 15? Herman Miller wrote some wonderful music in 15-tET
> exploiting the 250:243.

5: [16/15, 27/25]
15: [128/125, 250/243]
171: [32805/32768, 2 5^18 3^(-27)]

> I had hoped your 5-limit search was "airtight" already . . . oh well.

It should be, but it never hurts to check. None of the new temperaments were under my 500 limit.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > How about 15? Herman Miller wrote some wonderful music in 15-tET
> > exploiting the 250:243.

> 15: [128/125, 250/243]

There it is!

> None of the new >temperaments were under my 500 limit.

I noticed that. BTW, have you had a chance to think about
my "question for Gene"?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> I noticed that. BTW, have you had a chance to think about
> my "question for Gene"?

I was going to do some calculations when I had a chance--I'm going to see A Beautiful Mind and prepping for tomorrow's classs tonight.