TCTMO (was: The Regular Mapping Paradigm)

Hi all,

From: Carl Lumma on Fri May 26, 2006:
>
> Been to
> http://www.lumma.org/music/theory/tctmo/
> lately?

Thanks for the pointer - nice job, Carl!

Comments:
This is definitely the kind of thing needed
as an outline of tuning mathemeatics. I can
cheerfully refer any mathematically-inclined
enquirers here.

The 5-limit linear temperament spreadsheet
is also a good resource; generally, including
examples of the kind of music that has been
made in non-12 is a good way to help readers
make the concepts more concrete.

Question 1:
Is it a done deal to use the denominator of
the comma tempered out as a heuristic
complexity measure? ie is that we all mean
when we say "simple". (I thought that PE, for
example, had other notions.)

Question 2:
Is the development in Sections 8.1 thru 9
appropriate in an "outline"? In my view, they
deviate from the path of "outlining" and
tend towards "advocacy" - tho perhaps your
intent was only to point out a possible
direction? By the same arguments you've
used, I think readers would want to know
why you've jumped over diesic, pelogic and
blackwood - and tho there may be good
reasons to do so, you don't even hint at them.

Practical value:
Armed with this spreadsheet, I'd like to see
what kind of music I can make in the other
"simple" 5-limit temperaments.

Regards,
Yahya

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Yahya Abdal-Aziz wrote:

> Question 1:
> Is it a done deal to use the denominator of > the comma tempered out as a heuristic > complexity measure? ie is that we all mean > when we say "simple". (I thought that PE, for > example, had other notions.)

No, there are different ways of evaluating both the error and complexity of a temperament. One day I'll write a treatise on them. For now, if you can follow my Python code you'll see some implementations. The size of the comma is deficient because you can't say anything about cases where more than one interval is tempered out. In the Middle Path paper, Paul uses a wedgie-based complexity measure. Because the interval tempered out (or its complement, I forget) is the wedgie in the case where there's only one, that means the two complexities are equivalent but might be expressed in different units.

Graham

>> Been to
>> http://www.lumma.org/music/theory/tctmo/
>> lately?
>
>Thanks for the pointer - nice job, Carl!

Thanks for reading, Yahya.

>Question 1:
>Is it a done deal to use the denominator of
>the comma tempered out as a heuristic
>complexity measure? ie is that we all mean
>when we say "simple". (I thought that PE, for
>example, had other notions.)

There are lots of different measures. 'd' isn't
even in favor anymore, I don't think. But it
does work, if only for rank 2 (aka linear) temperaments

>Question 2:
>Is the development in Sections 8.1 thru 9
>appropriate in an "outline"? In my view, they
>deviate from the path of "outlining" and
>tend towards "advocacy"

It's definitely advocacy, but from my perspective
the very goal of tuning-math is to find new tunings
to advocate!

>By the same arguments you've
>used, I think readers would want to know
>why you've jumped over diesic, pelogic and
>blackwood - and tho there may be good
>reasons to do so, you don't even hint at them.

No, porcupine was just an example I had music
for, though I believe it to be one of the best
non-meantone 5-limit systems.

>Practical value:
>Armed with this spreadsheet, I'd like to see
>what kind of music I can make in the other
>"simple" 5-limit temperaments.

Great!

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> No, porcupine was just an example I had music
> for, though I believe it to be one of the best
> non-meantone 5-limit systems.

Where do diaschismic, schismatic, or hanson rate in your scale of things?

>> No, porcupine was just an example I had music
>> for, though I believe it to be one of the best
>> non-meantone 5-limit systems.
>
>Where do diaschismic, schismatic, or hanson rate in your scale of things?

I'm not familiar enough with diaschismic to really know.
I think schismatic's a bit too complex to compete with porcupine,
but it's certainly a good 5-limit system.
Hanson runs neck and neck with porcupine.

How do you rank them? Do you have your lists anywhere on the web?

I see Paul doesn't broach the subject of badness in the Middle
Path paper...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> How do you rank them? Do you have your lists anywhere on the web?

There's always the question of *how* to rank them. Here are a few of
the less complex (Kees complexity <5) tempements, listing name,
comma, Kees complexity, and Kees badness. The Kees badness figure is
strikingly lower for schismatic and meantone than for the rest of
them. The only temperament I can find which beats schismatic is
atomic, an ultra-ultra temperament of great theoretical use but more
precise than practce is ever likely to require. For beating meantone,
aside from schismatic and atomic, I find only pirate.

Schismatic
32805/32768 4.076342 8.821369

Meantone
81/80 1.722706 17.342839

Dicot
25/24 1.261860 30.577661

Hanson
15625/15552 3.785579 31.569803

Father
16/15 1.061606 34.216377

Augmented
128/125 1.892789 39.970873

Diaschismic
2048/2025 2.984566 47.325844

Beep
27/25 1.292030 60.437145

Porcupine
250/243 2.153383 61.949928

Sensi
78732/78125 4.416508 71.017123

Magic
3125/3072 3.154649 80.080175

Pelogic
135/128 1.922959 92.619651

Diminished
648/625 2.523719 108.279619

It might be nice to come up with something that ranks meantone
best of all, and see what that looks like.

-Carl

>> How do you rank them? Do you have your lists anywhere on the web?
>
>There's always the question of *how* to rank them. Here are a few of
>the less complex (Kees complexity <5) tempements, listing name,
>comma, Kees complexity, and Kees badness. The Kees badness figure is
>strikingly lower for schismatic and meantone than for the rest of
>them. The only temperament I can find which beats schismatic is
>atomic, an ultra-ultra temperament of great theoretical use but more
>precise than practce is ever likely to require. For beating meantone,
>aside from schismatic and atomic, I find only pirate.
>
>Schismatic
>32805/32768 4.076342 8.821369
>
>Meantone
>81/80 1.722706 17.342839
>
>Dicot
>25/24 1.261860 30.577661
>
>Hanson
>15625/15552 3.785579 31.569803
>
>Father
>16/15 1.061606 34.216377
>
>Augmented
>128/125 1.892789 39.970873
>
>Diaschismic
>2048/2025 2.984566 47.325844
>
>Beep
>27/25 1.292030 60.437145
>
>Porcupine
>250/243 2.153383 61.949928
>
>Sensi
>78732/78125 4.416508 71.017123
>
>Magic
>3125/3072 3.154649 80.080175
>
>Pelogic
>135/128 1.922959 92.619651
>
>Diminished
>648/625 2.523719 108.279619

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> It might be nice to come up with something that ranks meantone
> best of all, and see what that looks like.

One could try my convex hull business again, though that seems to have
gone over like a lead balloon. The following list has been jiggered so
that the badness of meantone and schismatic are equal. The cutoff is
150, which is just low enough to eliminate atomic; even though the
badness method favors low complexity some of the high complexity
temperaments are good enough that they try to sneak in anyway.

81/80 Meantone
Kees error = 3.392239 Kees complexity = 1.722706 Badness = 26.577422

32805/32768 Schismatic
Kees error = .130234 Kees complexity = 4.076342 Badness = 26.577422

16/15 Father
Kees error = 28.598519 Kees complexity = 1.061606 Badness = 35.860118

25/24 Dicot
Kees error = 15.218479 Kees complexity = 1.261860 Badness = 36.701492

128/125 Augmented
Kees error = 5.894363 Kees complexity = 1.892789 Badness = 65.952315

27/25 Beep
Kees error = 28.021183 Kees complexity = 1.292030 Badness = 73.898774

15625/15552 Hanson
Kees error = .581936 Kees complexity = 3.785579 Badness = 89.747963

2048/2025 Diaschismic
Kees error = 1.780143 Kees complexity = 2.984566 Badness = 111.638361

250/243 Porcupine
Kees error = 6.204076 Kees complexity = 2.153383 Badness = 113.108242