> From: genewardsmith <genewardsmith@juno.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Friday, February 01, 2002 7:34 PM

> Subject: [tuning-math] Re: 7-limit MT reduced bases for ets

>

>

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

>

> > 140: [2401/2400, 5120/5103, 15625/15552]

>

> I accidentally left off

>

> 171: [2401/2400, 4375/4374, 32805/32768]

>

> Wouldn't want to do that--look at those three high-powered commas!

Right! 171-EDO has been strongly advocated by prolific

German tuning theorist Martin Vogel, who has even had

a keyboard built for it for use in his classes.

The only detailed version of Vogel's work in English is:

Vogel, Martin. _Die Lehre von den Tonbeziehungen_.

Orpheus, Band 16, Verlag fï¿½r systematische Musikwissenschaft,

Bonn-Bad Godesberg, 1975, 480 pages.

English translation _On the Relations of Tone_

by Vincent Jean Kisselbach and edited by Carl A. Poldy,

Verlag fï¿½r systematische Musikwissenschaft, Bonn-Bad Godesberg, 1993.

(note that the title is an homage to Helmholtz)

-monz

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> From: monz <joemonz@yahoo.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Friday, February 01, 2002 10:58 PM

> Subject: [tuning-math] 171-EDO, Vogel (was: 7-limit MT reduced bases for

ets)

>

>

>

> > From: genewardsmith <genewardsmith@juno.com>

> > To: <tuning-math@yahoogroups.com>

> > Sent: Friday, February 01, 2002 7:34 PM

> > Subject: [tuning-math] Re: 7-limit MT reduced bases for ets

> >

> >

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

> >

> > > 140: [2401/2400, 5120/5103, 15625/15552]

> >

> > I accidentally left off

> >

> > 171: [2401/2400, 4375/4374, 32805/32768]

> >

> > Wouldn't want to do that--look at those three high-powered commas!

>

>

> Right! 171-EDO has been strongly advocated by prolific

> German tuning theorist Martin Vogel, who has even had

> a keyboard built for it for use in his classes.

I should have added, in keeping with the original subject line,

that Vogel is an enthusiastic advocate of prime-factor 7 in tuning.

His analyses sometimes include higher primes, but from

what I've read he doesn't seem too interested in 11 or higher.

But early on in his career he wrote this:

Vogel, Martin. _Zahl 7 in der Spekulativen Musiktheorie_.

unpublished PhD thesis, 1955

That titles translates as

"The Number 7 in Speculative Music-Theory".

-monz

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

> > 171: [2401/2400, 4375/4374, 32805/32768]

> >

> > Wouldn't want to do that--look at those three high-powered commas!

> (note that the title is an homage to Helmholtz)

Helmholtz liked the schismic temperament, and Vogel goes him one better by combining schismic with ennealimmal, which the above reduced basis shows is one way of thiking about 171-et. You could temper either 53 tones or 72 tones with it, among other things.

Since I am now writing a piece in 46-et and just finished one in

53-et, I'll also add these:

46: [126/125, 245/243, 1029/1024]

53: [225/224, 1728/1715, 3125/3087]

I'm finding the 43-et set of commas quite useful.

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

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

>

> > > 171: [2401/2400, 4375/4374, 32805/32768]

> > >

> > > Wouldn't want to do that--look at those three high-powered

commas!

>

> > (note that the title is an homage to Helmholtz)

>

> Helmholtz liked the schismic temperament, and Vogel goes him one

better by combining schismic with ennealimmal, which the above

reduced basis shows is one way of thiking about 171-et. You could

temper either 53 tones or 72 tones with it, among other things.

>

> Since I am now writing a piece in 46-et and just finished one in

> 53-et, I'll also add these:

>

> 46: [126/125, 245/243, 1029/1024]

> 53: [225/224, 1728/1715, 3125/3087]

>

> I'm finding the 43-et set of commas quite useful.

Do you really mean 43, or one of the above?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > I'm finding the 43-et set of commas quite useful.

>

> Do you really mean 43, or one of the above?

I meant 46--just my usual terrific job of proof-reading.

43: [81/80, 126/125, 12288/12005]

------------

> From: genewardsmith <genewardsmith@juno.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Saturday, February 02, 2002 12:13 AM

> Subject: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT reduced bases

for ets)

>

>

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

>

> > > I'm finding the 43-et set of commas quite useful.

> >

> > Do you really mean 43, or one of the above?

>

> I meant 46--just my usual terrific job of proof-reading.

>

> 43: [81/80, 126/125, 12288/12005]

thanks for sending t h a t now anyway, gene!

43-edo was advocated as a tuning by Sauveur

file:///C:/interval/dict/meride.htm#43-1/5diff

i've updated that page now thanks to the attention

you caused me to give it, and included a link to your

post above

i'd like to change that link to point to something

that really explains what "7-limit MT reduced bases"

are, but i sure don't know to explain it, so if you

do then i'll let the link point to that post

thanks

from manuel's page:

http://www.xs4all.nl/~huygensf/doc/measures.html

>> mï¿½ride: 1/43 part of an octave

>>

>> This name was chosen by Joseph Sauveur (1653-1716)

>> in 1696. The mï¿½ride and eptamï¿½ride were the first

>> logarithmic interval measures proposed. Sauveur

>> favoured 43-tone equal temperament because the small

>> intervals are well represented in it. He had set the

>> comma to one step, then found a range of 2, 3 or 4

>> steps for the chromatic semitone, corresponding to 31,

>> 43 and 55 tones per octave. He found 43 to be optimal

>> because 4 steps is almost exactly a 16/15 minor second

>> and 7 steps almost exactly the geometric mean of

>> three 9/8 and two 10/9 whole tones. The chromatic scale

>> contained in 43-tET is virtually identical to 1/5-comma

>> meantone tuning.

-monz

------------

-monz

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> From: monz <joemonz@yahoo.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Saturday, February 02, 2002 1:26 AM

> Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT reduced

bases for ets)

>

>

> from manuel's page:

> http://www.xs4all.nl/~huygensf/doc/measures.html

>

> >> ... Sauveur ... found 43 to be optimal

> >> because 4 steps is almost exactly a 16/15 minor second

> >> and 7 steps almost exactly the geometric mean of

> >> three 9/8 and two 10/9 whole tones. The chromatic scale

> >> contained in 43-tET is virtually identical to 1/5-comma

> >> meantone tuning.

[-9 6 0] = 3 * [-3 2 0] (= 9:8 whole tone)

+

[ 2 -4 2] = 2 * [ 1 -2 1] (= 10:9 whole tone)

----------

[-7 2 2] (= 225:128 "augmented 6th")

[7 2 2]^(1/2) = [-7/2 1 1] = ~488.2687147 cents

but what significance does that have? i don't get it

manuel?

-monz

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