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

oops, my bad ... the post about Helmholtz's tuning was supposed
to go to tuning-math. please reply there.

-monz

--- In tuning@y..., "monz" <monz@a...> wrote:
> hi Gene,
>
>
>
>
> > From: "monz" <monz@a...>
> > To: <tuning-math@y...>
> > 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

try the periodicity block with unison vectors *schisma* and *diesis*.
it's torsional, but keep it. temper out the schisma but not the
diesis. then you have helmholtz's scale.