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Decompositions of commas

🔗Keenan Pepper <keenanpepper@gmail.com>

8/17/2006 4:37:08 PM
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🔗Gene Ward Smith <genewardsmith@coolgoose.com>

8/17/2006 5:40:04 PM

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...>
wrote:

> Does this happen to coincide with anything that's already been defined?

The idea is sort of familiar, but I don't think a precise definition
has been given before.

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

8/22/2006 1:21:40 AM

Hi Keenan,

this is an interesting idea. How do you define this simplest
decomposition? Taxicab distance or what?

Would decomposition trees of commas have any uses?

Kalle

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...>
wrote:
>
> I was pondering how 36/35 = 64/63 * 81/80, and how that's the simplest
> pair of ratios that combine to 36/35, so any temperament that tempers
> out 36/35 might as well temper out 64/63 and 81/80 too, and I figured
> out a way to generalize it to any ratio and prime limit. The attached
> Python program finds the simplest decomposition of a given comma
> within a given prime limit, which generates the "best" temperament
> that tempers out that comma one rank lower. Here are some examples:
>
> 5-limit:
> 25/24 = 81/80 * 250/243 -> 7-equal
> 81/80 = 2048/2025 * 32805/32768 -> 12-equal
> 128/125 = 2048/2025 * 81/80 -> 12-equal
> 135/128 = 648/625 * 3125/3072 -> 16-equal
> 250/243 = 2048/2025 * 3125/3072 -> 22-equal
>
> 7-limit:
> 28/27 = 49/48 * 64/63 -> Blacksmith
> 36/35 = 64/63 * 81/80 -> Dominant
> 49/48 = 81/80 * 245/243 -> Semaphore
> 50/49 = 64/63 * 225/224 -> Pajara
> 64/63 = 245/243 * 1728/1715 -> Superpyth
> 81/80 = 126/125 * 225/224 -> Septimal meantone
> 225/224 = 2401/2400 * 16875/16807 -> Miracle
>
> Does this happen to coincide with anything that's already been defined?
>
> Keenan
>

🔗Kalle Aho <kalleaho@mappi.helsinki.fi>

8/22/2006 3:12:01 AM

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...>
wrote:
>
> I was pondering how 36/35 = 64/63 * 81/80, and how that's the simplest
> pair of ratios that combine to 36/35, so any temperament that tempers
> out 36/35 might as well temper out 64/63 and 81/80 too, and I figured
> out a way to generalize it to any ratio and prime limit.

But why not 36/35 = (6/5)/(7/6) or 81/80 = (648/625)/(128/125)?

Kalle

🔗Keenan Pepper <keenanpepper@gmail.com>

8/22/2006 8:57:09 PM

On 8/22/06, Kalle Aho <kalleaho@mappi.helsinki.fi> wrote:
> But why not 36/35 = (6/5)/(7/6) or 81/80 = (648/625)/(128/125)?

My thinking was, "If 36/35 is tempered out, what other commas might as
well be tempered out also?". 64/63 and 81/80 make sense, because they
are both smaller than 36/35 and they add up to it. If 36/35 is
tempered out, but 64/63 and 81/80 are not, that means one of 64/63 and
81/80 is represented by a negative interval in the temperament, i.e.,
it goes the wrong way. I think (proof?) every temperament has some
inverted intervals like these (for example, in 12-equal 1728/1715 is
-1 semitones), but you want them to be as complex as possible.

It should be easy to see why such reasoning doesn't apply to things
like 36/35 = (6/5)/(7/6). If 36/35 is tempered out, that means 6/5 and
7/6 are the same, but that's fine, that's how it is in 12-equal.

> this is an interesting idea. How do you define this simplest
> decomposition? Taxicab distance or what?

Simply the smallest numbers in the fractions. Specifically, the
maximum of the numerators is minimized.

> Would decomposition trees of commas have any uses?

Well, it's one way of answering questions like "What is the best equal
temperament that tempers out 25/24?" (7-equal) and "What is the best
7-limit linear temperament that tempers out 49/48?" (semaphore).

Keenan

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

8/23/2006 12:48:41 AM

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...>
wrote:

> I think (proof?) every temperament has some
> inverted intervals like these (for example, in 12-equal 1728/1715 is
> -1 semitones), but you want them to be as complex as possible.

Every temperament has a nontrivial kernel. Picking elements > 1 in the
kernel, you can take any interval q < 1 and get q' > 1 mapped to the
same place by multiplying by kernel elements.

Now the question is, what are the inverted p-limit intervals of lowest
height?

🔗Carl Lumma <ekin@lumma.org>

8/22/2006 9:02:20 PM

>(for example, in 12-equal 1728/1715 is
>-1 semitones), but you want them to be as complex as possible.

Why? Seems like these could be compositionally fun.

>> Would decomposition trees of commas have any uses?
>
>Well, it's one way of answering questions like "What is the best equal
>temperament that tempers out 25/24?" (7-equal) and "What is the best
>7-limit linear temperament that tempers out 49/48?" (semaphore).

Whoa, how would that work?

-Carl

🔗Keenan Pepper <keenanpepper@gmail.com>

8/23/2006 4:14:32 PM

On 8/23/06, Carl Lumma <ekin@lumma.org> wrote:
> Why? Seems like these could be compositionally fun.

Haha, I dare you to write a piece based on inverted intervals. I can't
even imagine how it would sound. You'd expect the sharper pitch to
play one role and the flatter pitch to play a different one, but they
wouldn't work if you used them that way and they'd cry out to switch
places with one another. Crazy upside-down music.

> >Well, it's one way of answering questions like "What is the best equal
> >temperament that tempers out 25/24?" (7-equal) and "What is the best
> >7-limit linear temperament that tempers out 49/48?" (semaphore).
>
> Whoa, how would that work?

25/24 decomposes into 81/80 * 250/243. If a temperament tempers out
25/24 but not 81/80 and 250/243, then one of 81/80 and 250/243 must be
inverted. The only temperament in which neither is inverted is the one
that tempers out all three: 7-equal.

Put another way, going from 7-equal to dicot adds another degree of
freedom. Each of the 7 pitches splits into a series of pitches
separated by an interval that corresponds to 81/80, but it also
corresponds to 250/243 backwards. Of course, that will always happen
with some interval, but 81/80 and 250/243 are the simplest pair, so
assuming we want to avoid inverting simple intervals, 7-equal is the
best equal temperament that tempers out 25/24.

Keenan

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

10/31/2008 1:24:52 PM

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...>
wrote:
>... find the simplest decomposition of a given comma
> within a given prime limit,...
> 81/80 = 2048/2025 * 32805/32768

Hi Keenan & all others,

I found at least that 4 epimoric decompositions of the Schisma:
32805/32768 = 5*3^8/2^15 = (886 + 23/37)/(885 + 23/37)
~1.95372079...Cents

1; 41-limit
(1025/1024)*(6561/6560) := (41*5^2/2^11)*(3^8/41/5/2^5)
~1.68983327...Cents + ~0.263887517...Cents

2; 59,7-limit
(945/944)*(14337/14336) := (7*5*3^3/59/2^4)*(59*3^5/7/2^11)
~1.8329637...Cents + ~0.120757092...Cents

3; 299( = 23*13 ),7-limit
(897/896)*(76545/76544) := (299*3/7/2^7)*(7*5*3^7/299/2^8)
~1.93110343...Cents + ~0.0226173539...Cents

4; 89,11-limit
(891/890)*(180225/180224) := (11*3^4/89/5/2)*(89*5^2*3^4/11/2^14)
~1.9441148...Cents + ~0.00960598614...Cents

the last one amounts even tiny less than 1/100st Cent.

Two challenging quests:
1. Are there any more pairs with that property?
2. How many pairs do exist totally altogether?

bye
A.S.