2401:2400 (was:: Marc Jones EDOs)

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, February 19, 2002 9:08 PM
> Subject: [tuning] (Erlich) Re: Marc Jones EDOs
>
>
> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> >
> > > the explanation was that you tend to find better
> > > 7-limit ets if 2400:2401 is nearly an integer number
> > > of steps than if it's nearly a half-integer number
> > > of steps. makes sense if you think about it --
> > > lower-limit example abound. but why don't we
> > > see a wave corresponding to 4375:4374??? mystery
> > > of mysteries . . .
> >
> >
> > 2401/2400 has a fourth power in the numerator, which I think
> > explains some of it.
>
> why is that? you can reply on tuning-math, though i'm sure your
> explanation is already there somewhere . . . but my eyes tend to
> glaze over in incomprehension at the way you initially explain
> certain things . . .

paul, i'm at l e a s t as mystified as you with Gene's
initial explanations, but i can at least offer this ...

2401/2400 is prime-factored into [2 3 5 7]^[-5 -1 -2 4] .

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
> > > 2401/2400 has a fourth power in the numerator, which I think
> > > explains some of it.

> paul, i'm at l e a s t as mystified as you with Gene's
> initial explanations, but i can at least offer this ...
>
>
> 2401/2400 is prime-factored into [2 3 5 7]^[-5 -1 -2 4] .

Sorry to be so cryptic; I was referring to the "jumping jack" business I've mentioned before. We have 81/80 = (9/8)/(10/9), a ratio of two "jacks"--one with a square numerator, the other with a triangular numerator in this case. Similarly, 2401/2400 =
(49/48)/(50/49) and 194481/194480 = (441/440)/(442/441), so we might expect these to be comparitively important also. For that matter, we can look at numerators which are squares of triangular numbers, among which we find 225/224 = (15/14)/(16/15), 3025/3024 = (55/54)/(56/55),
and 123201/123200 = (351/350)/(352/351), all of which may lay claim to being commas of interest. The square and triangular part is really of little significance by itself, and 9801/9800 = (99/98)/(100/99) is also a jumping jack, for instance. Since all of this bears on high-octane temperaments, equal or otherwise, it is probably of little practical importance, I'm afraid.

In-Reply-To: <007401c1b9ec$93c27660$af48620c@dsl.att.net>
monz wrote:

> paul, i'm at l e a s t as mystified as you with Gene's
> initial explanations, but i can at least offer this ...
>
>
> 2401/2400 is prime-factored into [2 3 5 7]^[-5 -1 -2 4] .

You're writing what looks like a wedge product, but isn't. You could
write it

(-5 -1 -2 4)H

where the ith element of H is the logarithm of the ith real number. For
octave-equivalent harmony, you can ignore the 2s, so

(-1 -2 4)h

The size of the numerator is 4, because it's the same prime number
multiplied 4 times. The size of the denominator is 3, because it's
3*5*5. So this is a fourth-order 7-limit ratio.

The other example is 4375:4374. That's 7 * 5**4 / 2 / 3**7 or (-1 -7 4
1)H or (-7 4 1)h. As the denominator is a prime number multiplied 7
times, it's a seventh-order 7-limit ratio. That makes it a lot more
complex, and more than makes up for it being a smaller interval.

It looks like small intervals, simply expressed in terms of defined
consonances make for good approximations. That's why superparticular
ratios are favoured to begin with. 4375:4374 should look better in the
9-limit, because it's only a fifth-order ratio. With octave-specific
harmony, you can go by the size of the superparticular.

Graham

> From: <graham@microtonal.co.uk>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, February 20, 2002 4:38 AM
> Subject: [tuning] Re: 2401:2400 (was:: Marc Jones EDOs)
>
>
> In-Reply-To: <007401c1b9ec$93c27660$af48620c@dsl.att.net>
> monz wrote:
>
> > paul, i'm at l e a s t as mystified as you with Gene's
> > initial explanations, but i can at least offer this ...
> >
> >
> > 2401/2400 is prime-factored into [2 3 5 7]^[-5 -1 -2 4] .
>
> You're writing what looks like a wedge product, but isn't. You could
> write it
>
> (-5 -1 -2 4)H
>
> where the ith element of H is the logarithm of the ith real number.

thanks, Graham. i asked quite a while ago if that notation
was correct or misleading etc., and no-one ever answered me.

so how about if i write [2 3 5 7]**[-5 -1 -2 4] ?

i prefer to keep the prime series in the notation if possible.

-monz

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--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > > > 2401/2400 has a fourth power in the numerator, which I think
> > > > explains some of it.
>
> > paul, i'm at l e a s t as mystified as you with Gene's
> > initial explanations, but i can at least offer this ...
> >
> >
> > 2401/2400 is prime-factored into [2 3 5 7]^[-5 -1 -2 4] .
>
> Sorry to be so cryptic; I was referring to the "jumping jack"
>business I've mentioned before. We have 81/80 = (9/8)/(10/9), a
>ratio of two "jacks"--one with a square numerator, the other with a
>triangular numerator in this case. Similarly, 2401/2400 =
> (49/48)/(50/49) and 194481/194480 = (441/440)/(442/441), so we
>might expect these to be comparitively important also.

why? why? why?

>For that matter, we can look at numerators which are squares of
>triangular numbers, among which we find 225/224 = (15/14)/(16/15),
>3025/3024 = (55/54)/(56/55),
> and 123201/123200 = (351/350)/(352/351), all of which may lay claim
>to being commas of interest. The square and triangular part is
>really of little significance by itself,

thus my questions!

i don't see that you've gone any way toward explaining why 2401:2400
is so much more "powerful" in the et fft than 4375:4374 . . . after
all, 4375:4374 = (54:35)/(125:81), among other things.

--- In tuning@y..., graham@m... wrote:

> The other example is 4375:4374. That's 7 * 5**4 / 2 / 3**7 or (-1 -
7 4
> 1)H or (-7 4 1)h. As the denominator is a prime number multiplied
7
> times, it's a seventh-order 7-limit ratio. That makes it a lot
more
> complex, and more than makes up for it being a smaller interval.

ok graham . . . this explanation does make some sense to me.

> With octave-specific
> harmony, you can go by the size of the superparticular.

that's fascinating graham. how would we test this? evaluate lots of
fractional, as well as integer, ets, according to some integer limit,
perhaps?

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > > > > 2401/2400 has a fourth power in the numerator, which I think
> > > > > explains some of it.
> >
> > > paul, i'm at l e a s t as mystified as you with Gene's
> > > initial explanations, but i can at least offer this ...
> > >
> > >
> > > 2401/2400 is prime-factored into [2 3 5 7]^[-5 -1 -2 4] .
> >
> > Sorry to be so cryptic; I was referring to the "jumping jack"
> >business I've mentioned before. We have 81/80 = (9/8)/(10/9), a
> >ratio of two "jacks"--one with a square numerator, the other with a
> >triangular numerator in this case. Similarly, 2401/2400 =
> > (49/48)/(50/49) and 194481/194480 = (441/440)/(442/441), so we
> >might expect these to be comparitively important also.
>
> why? why? why?
>
> >For that matter, we can look at numerators which are squares of
> >triangular numbers, among which we find 225/224 = (15/14)/(16/15),
> >3025/3024 = (55/54)/(56/55),
> > and 123201/123200 = (351/350)/(352/351), all of which may lay claim
> >to being commas of interest. The square and triangular part is
> >really of little significance by itself,
>
> thus my questions!
>
> i don't see that you've gone any way toward explaining why 2401:2400
> is so much more "powerful" in the et fft than 4375:4374 . . . after
> all, 4375:4374 = (54:35)/(125:81), among other things.

Why should we care about that?

We can express 50/49 as a ratio of two 7-limit consonances:
50/49 = (10/7)/(7/5). It is therefore a superparticular ratio of two
7-limit consonances. We can do the same for 49/48 = (7/6)/(8/7). Hence 2401/2400 = (49/48)/(50/49) is a superparticular ratio of two superparticular ratios, each of which is the ratio of two 7-limit consonances.

In the same way, 81/80 is a superparticular ratio of 5-limit superparticular ratios of two 5-limit consonances, 225/224 is the same for the 7-limit, and 3025/3024 and 9801/9800 for the 11-limit.
123201/123200 and 194481/194480 get the superparticular ratio part right, but not the p-limit consonance part--they aren't low-limit jumping jacks, and I shouldn't have listed them. On the other hand I could have listed 4096/4095. I think I'll find a complete listing for the lower prime limits.

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

> > >The square and triangular part is
> > >really of little significance by itself,
> >
> > thus my questions!
> >
> > i don't see that you've gone any way toward explaining why
2401:2400
> > is so much more "powerful" in the et fft than 4375:4374 . . .
after
> > all, 4375:4374 = (54:35)/(125:81), among other things.
>
> Why should we care about that?
>
> We can express 50/49 as a ratio of two 7-limit consonances:
> 50/49 = (10/7)/(7/5). It is therefore a superparticular ratio of two
> 7-limit consonances. We can do the same for 49/48 = (7/6)/(8/7).
>Hence 2401/2400 = (49/48)/(50/49) is a superparticular ratio of two
>superparticular ratios, each of which is the ratio of two 7-limit
>consonances.

i don't see where superparticularity comes in. all that seems to be
relevant is that 2401:2400 requires a chain of only four 7-limit
consonances, as graham explained. does superparticularity add
significance somehow in the present context?

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

> i don't see where superparticularity comes in.

For any given range of Tenney heights, superparticulars have the smallest size, and for any given range of size, the least Tenney height. They are the most efficient commas, so to speak.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > i don't see where superparticularity comes in.
>
> For any given range of Tenney heights, superparticulars have the
>smallest size, and for any given range of size, the least Tenney
>height. They are the most efficient commas, so to speak.

indeed, but what we're talking about here are the most efficient
commas _of_ the most efficient commas. why are those special in the
et fft context, as opposed to simply the most efficient commas of
7-limit intervals in general (as my 4375:4374 'factoring' was
intended to illustrate)?

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

> indeed, but what we're talking about here are the most efficient
> commas _of_ the most efficient commas. why are those special in the
> et fft context, as opposed to simply the most efficient commas of
> 7-limit intervals in general (as my 4375:4374 'factoring' was
> intended to illustrate)?

It seems to me these *are* the most efficient commas of p-limit intervals of smaller height. When 225/224 makes use of 16/15 and 15/14, it is equating two intervals each of which is itself important as a step between consonances; in the same way 81/80 equates two 5-limit intervals which are themselves important. I suppose we could introduce an efficiency measure of a sort if we had a badness measure for temperaments in general, but in the 5-limit 81/80 is quite efficient. Is 2^-90 3^-15 5^49 more efficient?