Decompositions of commas

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--- 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.

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
>

--- 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

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

--- 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?

>(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

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

--- 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.