Kleismic & co

2109375/2097152 = 2^-21 3^3 5^7 Orwell

Map:

[ 0 1]
[ 7 0]
[-3 3]

Generators: a = 19.01127197/84; b = 1

badness: 305.93
rms: .8004
g: 7.257
errors: [-.828, -1.082, -.255]

ets: 22,31,53,84

15625/15552 = 2^-6 36-5 5^6 Kleismic

Map:

[ 0 1]
[ 6 0]
[ 5 1]

Generators: a = 14.00435233/53 (~6/5); b = 1

badness: 97
rms: 1.030
g: 4.546
errors: [.523, -.915, -1.438]

ets: 19,34,53,68,72,87,140

1600000/1594323 = 2^9 3^-13 5^-2 Acute Minor Third system

Map:

[ 0 1]
[-5 3]
[-13 6]

Generators: a = 28.00947813/99 (~243/200); b = 1

badness: 305.53
rms: .3831
g: 9.273
error: [-.5009, .0716, -.4293]

All three of the above systems are done quite well by the 53-et, with different minor third generators;

Orwell is 12/53, Kleismic is 14/53, and AMT is 15/53

2^8 3^14 5^-13 Parakleismic

Map:

[ 0 1]
[-13 5]
[-14 6]

Generators: a = 30.99967314/118 (~6/5); b = 1

badness: 373
rms: .2766
g: 11.045
errors: [-.2169, .1735, .3904]

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> 2109375/2097152 = 2^-21 3^3 5^7 Orwell

I like it, but the comma is called the semicomma on Manuel's list.

> 27/25

"Large limmic"? Margo seemed dubious about calling it any kind of a limma.

> 16/15

Semitonic?

> 135/128

Major Limmic? Major Chromic?

Map:

[ 0 1]
[-1 2]
[ 3 1]

Generators: a = 10.0215 / 23; b = 1

badness: 46.1
rms: 18.1
g: 2.94
errors: [-24.8, -17.7, 7.1]

25/24

I called it "Neutral Thirds". "Minor chromic" is another possibility.

> 648/625

Too many things called a diesis, I fear--Major Diesic?

> 250/243

Maximal diesic?

> 128/125

Minor diesic? I think Paul named this; his names sounded good and maybe I should dig them out and keep them handy!

> 3125/3072

Magic--or Small Diesic for anyone who really digs this nomenclature.

78732/78125 = 2^2 3^9 5^-7

You'd think this would be some sort of diesis, but I can't find it.
"The Quite Small Diesis" won't do, I suppose.
> 393216/390625 = 2^17 3 5^-8

This is Wuerschmidt's comma, so obviously the temperament is the Wuerschmidt. Who the heck is Wuerschmidt?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> > 135/128
>
> Major Limmic? Major Chromic?

Paul said it was associated with Pelog. Pelogic?

> > 648/625
>
> Too many things called a diesis, I fear--Major Diesic?

Paul suggested "Octo-diminished", since it can be done very well by the 64-et. Sounds fine to me.

> > 128/125

"It's the augmented system, since the 6-tone MOS is commonly known as
the augmented scale" according to Paul.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
>
> > 2109375/2097152 = 2^-21 3^3 5^7 Orwell
>
> I like it, but the comma is called the semicomma on Manuel's list.
>
> > 27/25
>
> "Large limmic"? Margo seemed dubious about calling it any kind of a
limma.
>
> > 16/15
>
> Semitonic?

No.

>
> > 135/128
>
> Major Limmic? Major Chromic?
>
> Map:
>
> [ 0 1]
> [-1 2]
> [ 3 1]
>
> Generators: a = 10.0215 / 23; b = 1
>
> badness: 46.1
> rms: 18.1
> g: 2.94
> errors: [-24.8, -17.7, 7.1]
>
> 25/24
>
> I called it "Neutral Thirds". "Minor chromic" is another
possibility.
>
>
> > 648/625
>
> Too many things called a diesis, I fear--Major Diesic?
>
> > 250/243
>
> Maximal diesic?
>
>
> > 128/125
>
> Minor diesic? I think Paul named this; his names sounded good and
maybe I should dig them out and keep them handy!
>
> > 3125/3072
>
> Magic--or Small Diesic for anyone who really digs this nomenclature.

I'm going to die sick. I'm going to Diesic.

> > 393216/390625 = 2^17 3 5^-8
>
> This is Wuerschmidt's comma, so obviously the temperament is the
Wuerschmidt. Who the heck is Wuerschmidt?

Do you read German? If so, you should seek out his research. He found
some very interesting stuff, essentially looking into periodicity
blocks before Fokker.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, December 23, 2001 1:20 PM
> Subject: [tuning-math] Re: Temperament names
>
>
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> >
> > > 393216/390625 = 2^17 3 5^-8
> >
> > This is Wuerschmidt's comma, so obviously the temperament is the
> > Wuerschmidt. Who the heck is Wuerschmidt?
>
> Do you read German? If so, you should seek out his research. He found
> some very interesting stuff, essentially looking into periodicity
> blocks before Fokker.

This piqued my interest, so I took a look at the
Tuning and Temperament Bibliography
http://www.xs4all.nl/~huygensf/doc/bib.html#W

and take the liberty of quoting the Joseph W�rschmidt listings:

"Logarithmische und graphische Darstellung der musikalischen Intervalle",
Zeitschrift f�r Physik vol. 3, 1920, p. 89.

"Viertel- und Sechsteltonmusik, eine kritische Studie",
Neue Musikzeitung vol. 42, 1921, p. 183.

"�ber die neunzehnstufige Temperatur",
Neue Musikzeitung vol. 42, 1921, p. 215.

"Buchstabentonschrift und Von Oettingensches Tongewebe",
Zeitschrift f�r Physik vol. 5, 1922, p. 111.

"Die rationellen Tonsysteme in Quinten-Terzengewebe",
Zeitschrift f�r Physik vol. 46, January 1928, p. 527.

"Tonleitern, Tonarten und Tonsysteme. Eine historisch-theoretische
Untersuchung",
Sitzungsberichte der Physikalisch-medizinischen Soziet�t zu Erlangen,
Band 63, 1931, pp. 133-238.

"Die neunzehn-stufige Skala; eine nat�rliche Erweiterung unseres Tonsystems"
(The 19-tone scale; a natural expansion of our tonal system),
Neue Musikzeitung vol. 14 no. 4, 1921, pp. 215-216.

Which of these have you read, Paul? Can you summarize?
I wonder if Tanaka wrote about periodicity-blocks before W�rschmidt?

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> Which of these have you read, Paul?

None, but Mandelbaum touched on his work. A lot of it involves a
conception of "rational tone-systems", which are essentially 5-limit
periodicity blocks, expressed with only three step sizes, but then
equally tempered anyway. W.'s "rational tone systems" included 12-,
19-, 22-, 31-, 34-, 41-, 53-, 65-, and 118-tET, and there weren't
many more given the constrains he imposed.

On Sat Dec 22, 2001, "genewardsmith" <genewardsmith@j...> wrote:

> 1600000/1594323 = 2^9 3^-13 5^-2 Acute Minor Third system

that should be 2^9 3^-13 5^5, right?