Re: Helmholtz's schismic temperament (was: NMOS)

hi Gene,

> From: "monz" <monz@attglobal.net>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, October 24, 2002 11:15 PM
> Subject: Re: [tuning-math] Re: NMOS
>

> ...
>
> Helmholtz's tuning can be viewed as the Pythagorean
> chain 3^(-16...+7). but Helmholtz himself viewed it
> as a skhismic temperament described by the Euler genus
> 3^(-8...+7) * 5^(0...+1).
>
> with C as n^0 (= 1/1), this gives a 12-tone Pythagorean chain
> from Ab 3^-4 to C# 3^7 which has a counterpart one syntonic comma
> lower at (using {3,5}-prime-vector notation) Ab [-8 1] to C# [3 1].
> this is a 24-tone torsional periodicity-block defined by
> the Pythagorean and syntonic commas, [12 0] and [4 -1].

re-reading this, i realized that including prime-factor 2
in the vectors would be a good idea, since you've pointed
out that it's necessary in order to see the torsion. so ...

Helmholtz's tuning, as {2,3,5}-prime-vectors with C=n^0, viewed as:

- a Pythagorean chain, Fbb [26 -16 0] ... C# [-11 7 0]

- a 5-limit Euler genus,
Ab [7 -4 0] ... C# [-11 7 0] + Ab [11 -8 1] ... C# [-7 3 1]

generating unision-vectors:

[-19 12 0] Pythagorean comma
[-4 4 -1] syntonic comma

from that data, can you explain how the torsion
works in this tuning?

-monz

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

> [-19 12 0] Pythagorean comma
> [-4 4 -1] syntonic comma

> from that data, can you explain how the torsion
> works in this tuning?

These aren't unison vectors. As a 5-limit tuning, which is how Helmhotz thought of it, it is defined by the schisma, [-15 8 1]; this is why the temperament is "schismic". For Pauline's purposes, we want to add the comma 225/224, giving a septimal version of schismic. This septimal schismic; it is covered by 41 and 53 equal, and from an rms error point of view done to perfection by the 94-et. If the 7 is relatively unimportant we might prefer the 53-et, and if 41 tones is not too many to handle we could even try building a 41-et organ.

A page on Helmholtz's Harmonium would be a find addition.

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "monz" <monz@a...> wrote:
>
> > [-19 12 0] Pythagorean comma
> > [-4 4 -1] syntonic comma
>
> > from that data, can you explain how the torsion
> > works in this tuning?
>
> These aren't unison vectors.

if monz changed this to schisma and diesis (not a major change), why
not?

>As a 5-limit tuning, which is how Helmhotz thought of it, it is
>defined by the schisma, [-15 8 1]; this is why the temperament
>is "schismic".

that's right -- the schisma is the unison vector that is tempered
out. you can use the diesis as the unison vector that *isn't*
tempered out. then you clearly have a 24-tone periodicity block, a
torsional one.