The best ETs and 12276 (was Re: Meantone diesis)

From: "Danny Wier" <dawiertx@...>

> Yeah, I noticed early on that all the good ETs conform to this formula: n > =
> 5a + 7b, where a and b are whole numbers. (Obviously, this is because the
> first two small Pythagorean intervals are 2^8/3^5 and 3^7/2^11.)
>
> Now when you get into ETs with more notes per octave, you'd want to > abandon
> 5 and 7 and mpve on to the smaller Pythagorean commas, which involve > higher
> powers of three, such as 12, 41, 53, 306 and 665.

I'm making myself look like a total idiot here.

Of course 12, 41, 53, 306 and 665 are NOT powers of 3. What I meant is that the Pythagorean intervals derived from chains of fifths of those lengths have increasingly smaller sizes - those are the commas you're heard of, at least the 12-tone one.

The 665-tone comma is an exceptionally small interval: 3^665/2^1054 measures about 0.0756 cents.

> And I just now discovered a ridiculously high-order equal temperament:
> 12,276-tone. Pretty accurate up to 11-limit (and the Pythagorean comma is
> 240 degrees in this tuning, or 15 seconds in my own measurement system for
> small intervals). 12,276 = 665*18 + 306, by the way.

About those tiny intervals I posted earlier... the vanishing commas (11-limit) in 12276-tone equal temperament are 781258401/781250000, 67110351/67108864 and 199297406/199290375. None of those intervals have names (as far as I know).

And the 3/2 fifth is 7181 steps, the 5/4 third is 3952, the 7/4 seventh is 9911, and the 11/8 fourth is 5640.

~Danny~

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
> From: "Danny Wier" <dawiertx@>
> ...
> > And I just now discovered a ridiculously high-order equal
temperament:
> > 12,276-tone. Pretty accurate up to 11-limit (and the Pythagorean
comma is
> > 240 degrees in this tuning, or 15 seconds in my own measurement
system for
> > small intervals). 12,276 = 665*18 + 306, by the way.

Take a look at 11664-ET: 27-limit consistent, with no odd number within
that limit in error by more than 19% of a degree (of ~0.103c)

From: "George D. Secor" <gdsecor@...>

>> > And I just now discovered a ridiculously high-order equal
> temperament:
>> > 12,276-tone. Pretty accurate up to 11-limit (and the Pythagorean
> comma is
>> > 240 degrees in this tuning, or 15 seconds in my own measurement
> system for
>> > small intervals). 12,276 = 665*18 + 306, by the way.
>
> Take a look at 11664-ET: 27-limit consistent, with no odd number within
> that limit in error by more than 19% of a degree (of ~0.103c)

(This might belong in tuning-math, but I kinda have to keep to one list now, since I am so busy actually writing music nowadays - 72-tone for now - so I have less time for theorizing.)

I'll compare the errors up to 11-limit:

11664:
3/2 ~ +0.002608
5/4 ~ -0.030701
7/4 ~ -0.012189
11/8 ~ -0.189600

The error is low for everything up to 7-limit, but 11-limit is a little high.

12276:
3/2 ~ -0.000341
5/4 ~ -0.010707
7/4 ~ 0.089023
11/8 ~ -0.017450

All errors are less than +/-0.1. But for 13/8, there is no comparison: error in 11664-tone is -0.071128; 12276-tone is -0.402020.

I was impressed with 12276-tone not only because of its 11-limit precision, but the fact that the Pythagorean comma is 240 steps, a number I like so much for its sexagesimal properties. In 11664, it's 228. And I really meant to focus on dividing commas more than extreme ETs; finding 12276-edo was pure serendipity.

I'm still debating with myself whether to use the Pyth-comma or the 81/80-comma as a basis of measurement. Brombaugh's temperament units are 1/720ths of the latter, and Eitz notation uses fractions of both. If I use syntonic, the schisma-sized step in 665-tone becomes the 5-minute interval, since 81/80 is 12 steps (and 64/63 is 15).

~Danny~

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
> From: "George D. Secor" <gdsecor@>
>
> >> > And I just now discovered a ridiculously high-order equal
> > temperament:
> ...
> (This might belong in tuning-math, but I kinda have to keep to one
list now,
>
> ~Danny~

This does belong in tuning-math, so I kinda had to reply here:

/tuning-math/message/12103

--George