More on MOS/temperaments

If T is a linear temperament, and if we tune a scale and/or temperament
on n notes to a tuning of T, we may denote this by T[n]. T[n] has
certain commas associated to it, in addition to the ones belonging to
T itself, namely q such that q mapes to +-n generator steps in terms
of the (octave based) period and generator description of T.

These associated commas give us equivalences when going around the
circle of generators in T[n], since they represent the n-step jumps
which occur when we reach the end of the chain of generators. When
these commas are "interesting", meaning such that they allow for nice
equivalences, we are in a particularly nice situation. Perhaps the most
interesting comma to have in this way is 36/35, which as I mentioned
before has a numerator (36) which is both square and triangular and
which allows for a lot of useful equivalences, since for instance
1-5/4-3/2-7/4 is converted to 1-9/7-3/2-9/5 by two 36/35's, and
likewise 1-6/5-3/2-12/7 to 1-7/6-3/2-5/3. Other commas, eg 15/14,
21/20, 25/24, 49/48, 45/44, etc. have their own properties, and this
could become a rather involved study.

To start with, Here are some T[n] with 36/35 as an associated comma:

Orwell[9], Tertiathirds[9]

Supersupermajor[10]

Superkleismic[11]

Meantone[12], Pajara[12], Tripletone[12], Injera[12]

Octafifths[13]

Amity[14]

Catakleismic[15]

Muggles[16]

Beatles[17], Squares[17]

Miracle[21]

Schismic[24], Diaschismic[24]

At which point I'll stop.

How these work out in practice depends on the details of the mapping,
but in general tetrads tend to be sent to alternative versions which
have more or less nice consonance properties. And recall, this is only
one comma!

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> If T is a linear temperament, and if we tune a scale and/or
temperament
> on n notes to a tuning of T, we may denote this by T[n]. T[n] has
> certain commas associated to it, in addition to the ones belonging
to
> T itself,

for example the chromatic unison vector!

> namely q such that q mapes to +-n generator steps in terms
> of the (octave based) period and generator description of T.

that's it! i think you've finally stumbled on the chromatic unison
vector, which means that from now on, you may understand what it
means!

> These associated commas give us equivalences when going around the
> circle of generators in T[n], since they represent the n-step jumps
> which occur when we reach the end of the chain of generators. When
> these commas are "interesting", meaning such that they allow for
nice
> equivalences, we are in a particularly nice situation. Perhaps the
most
> interesting comma to have in this way is 36/35, which as I mentioned
> before has a numerator (36) which is both square and triangular and
> which allows for a lot of useful equivalences, since for instance
> 1-5/4-3/2-7/4 is converted to 1-9/7-3/2-9/5 by two 36/35's, and
> likewise 1-6/5-3/2-12/7 to 1-7/6-3/2-5/3. Other commas, eg 15/14,
> 21/20, 25/24, 49/48, 45/44, etc. have their own properties, and this
> could become a rather involved study.

we already started on this in a series of posts, in particular we
were looking at scales where the major tetrad and the minor tetrad
arise from the same pattern of scale steps.

> And recall, this is only
> one comma!

i'm not sure what you mean to imply by that -- and of course this
chroma is only defined plus or minus any arbitrary number of commatic
unison vectors.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:

> > namely q such that q maps to +-n generator steps in terms
> > of the (octave based) period and generator description of T.
>
> that's it! i think you've finally stumbled on the chromatic unison
> vector, which means that from now on, you may understand what it
> means!

Looks that way. Thanks for pointing this out.

Communicating with me is a pain, I suppose. :)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
>
> > > namely q such that q maps to +-n generator steps in terms
> > > of the (octave based) period and generator description of T.
> >
> > that's it! i think you've finally stumbled on the chromatic
unison
> > vector, which means that from now on, you may understand what it
> > means!
>
> Looks that way. Thanks for pointing this out.
>
> Communicating with me is a pain, I suppose. :)

nah, this was a rare case of something that it seemed you were not
understanding, since you got it wrong a few times. but hopefully the
next time the whole discussion that kalle started and that carl was
interested in reviving comes around, you'll be in a position to
contribute to it, and thus we will all be far better informed than if
you weren't around. (hint, hint, carl, gene, etc.)

Hmm, something's amiss. Anybody else get this list sent to
them by e-mail? I got msg. 6056 but not 6057-6067. Just got
6068 & 9.

>we already started on this in a series of posts, in particular we
>were looking at scales where the major tetrad and the minor tetrad
>arise from the same pattern of scale steps.

! Just when I was thinking, "when would I ever be interested
in this incomplete-chain extra-intervals stuff"... Good looking
out, Paul!

More later.

-C.

>>And recall, this is only one comma!
>
>i'm not sure what you mean to imply by that -- and of course this
>chroma is only defined plus or minus any arbitrary number of
>commatic unison vectors.

Maybe Gene's alluding to the planar and higher cases.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >>And recall, this is only one comma!
> >
> >i'm not sure what you mean to imply by that -- and of course this
> >chroma is only defined plus or minus any arbitrary number of
> >commatic unison vectors.
>
> Maybe Gene's alluding to the planar and higher cases.

No, I've just been looking, so far, at 36/35 chroma cases. If you want
major triads and minor triads to convert, you could look at 25/24; if
you want other fun stuff, including asses, 15/14 or 21/20. 49/48 is
also worth exploring.