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

This is a reply to msg. #58480 from the main list.

--- 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:
> >> > 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

With this many tones, I would expect that you might be able to go a
bit beyond the 11 limit.

> 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).

In the process of working on the Sagittal notation, Dave Keenan and I
were directed (by Gene Ward Smith) to take a look at 2460-ET, which
divides the 5-comma into 44 parts (allowing both 1/4-comma and 1/11-
comma meantone to be expressed as whole numbers, as well as getting
all of the 27-limit consonances to within 0.154 cents. One step
(which we call a "mina", short for schismina) is ~1/4-schisma, and
we've found it a convenient unit of measure for comparing the ratios
that we're symbolizing with microtonal accidentals. (Also note that
2460 is a multiple of both 12 and 41, and BTW just happens to be
41*60. And, yes, we're also able to notate every single one of those
degrees of 2460-ET on a conventional staff in Sagittal!)

--George

George,

Would 2640-tET be an accurate enough resolution for all practical tunings in the process of bending of the piano strings? I would like to implement a standard which does not depend on the cents system, but perhaps the mina system.

Cordially,
Ozan
----- Original Message -----
From: George D. Secor
To: tuning-math@yahoogroups.com
Sent: 05 Mayıs 2005 Perşembe 22:28
Subject: [tuning-math] The best ETs and 12276 (was Re: Meantone diesis)

This is a reply to msg. #58480 from the main list.

...

In the process of working on the Sagittal notation, Dave Keenan and I
were directed (by Gene Ward Smith) to take a look at 2460-ET, which
divides the 5-comma into 44 parts (allowing both 1/4-comma and 1/11-
comma meantone to be expressed as whole numbers, as well as getting
all of the 27-limit consonances to within 0.154 cents. One step
(which we call a "mina", short for schismina) is ~1/4-schisma, and
we've found it a convenient unit of measure for comparing the ratios
that we're symbolizing with microtonal accidentals. (Also note that
2460 is a multiple of both 12 and 41, and BTW just happens to be
41*60. And, yes, we're also able to notate every single one of those
degrees of 2460-ET on a conventional staff in Sagittal!)

--George

--- In tuning-math@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...>
wrote:
> George,
>
> Would 2640-tET be an accurate enough resolution for all practical
tunings in the process of bending of the piano strings? I would like to
implement a standard which does not depend on the cents system, but
perhaps the mina system.
>
> Cordially,
> Ozan

I would think that it's more than enough. Is the technology you're
using capable of bending the pitch accurately to such small increments?

--George

The technology in question is the one I plan to implement with my Ultratonal Piano (tm) design. It will hopefully be possible (I dare presume) to adjust each string tension by about half a cent at minimum precision. Could you re-affirm this by showing me how much tension ought to be delivered to the A4 strings in order to raise La one degree of 2640tET?

Cordially,
Ozan
----- Original Message -----
From: George D. Secor
To: tuning-math@yahoogroups.com
Sent: 07 Mayıs 2005 Cumartesi 0:07
Subject: [tuning-math] The best ETs and 12276 (was Re: Meantone diesis)

--- In tuning-math@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...>
wrote:
> George,
>
> Would 2640-tET be an accurate enough resolution for all practical
tunings in the process of bending of the piano strings? I would like to
implement a standard which does not depend on the cents system, but
perhaps the mina system.
>
> Cordially,
> Ozan

I would think that it's more than enough. Is the technology you're
using capable of bending the pitch accurately to such small increments?

--George

--- In tuning-math@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...>
wrote:
> The technology in question is the one I plan to implement with my
Ultratonal Piano (tm) design. It will hopefully be possible (I dare
presume) to adjust each string tension by about half a cent at minimum
precision. Could you re-affirm this by showing me how much tension
ought to be delivered to the A4 strings in order to raise La one degree
of 2640tET?
>
> Cordially,
> Ozan

Sorry, I have no idea.

--George