Re: [tuning] Digest Number 1223

Thanks for the reply Paul.

> Robert Valentine wrote,
>
> >Okay, I was thinking of it in terms of the set of "3 b's and 7 a's" rather
> >than the numeric values.
>
> Then the term you're looking for is "distributionally even" rather than
> "maximally even".
>

Thanks. Referring to a "thing" I called "tuning three"

41141

> >Tuning three certainly doesn't have this property, BUT, all
> >intervals are unique. So "7", despite being a "fourth" between
> >the two flavors of "third" is still uniquely a fourth. (This
> >is the special word I was looking for).
>
> Oh, that would be the case for any non-ET tuning of the scale, right?
>

Well, the way I spelled this, one could think of it as a pentatonic
in 11-et. The point I was wondering about is that strict propriety
seems to say that all intervals are ordered and unique (three steps,
whether there are one or ten flavors in the scale, is larger than
two steps). You said the diatonic is not strictly proper which I take
to mean because it is ordered but not unique ("three steps is always
less than or equal to four steps"). So in this example the intervals
are unique, but not ordered.

>
> >Thanks for the post regarding ways to think of transposability
> >with decatonics. I like the 'disjoint MOS' interpretation the
> >best.
>
> You mean MOS within a half-octave periodicity, or two interlaced MOSs?
>

I meant the interlaced MOSs. It looks like there could be a lot of
interesting potential here, thinking from a sort of
polytonal/counterpoint standpoint. By modulating the MOSs individually,
the melodic logic can be maintained but the harmonic interaction
changed. This is not new and is related to what makes melodic imitation
in a different mode or even polytonality work, it just gives a nice
way of approaching it with a new tuning structure where the 'handles'
may not be so obvious.

> Anyway, I find all five types of modulations musically useful.
>

I had a long reply about the multi-step modulations, which I
can sum up by saying that I'm trying to wrap my head
around more familiar territory first.

A side note that I should include due to the recent curiosity
regarding 72, is about transposable systems of three terms
with the example being a quasi-JI diatonic.

For instance, in 72 : LMsLMLs is

12 11 7 12 11 12 7

If we try to make it exactly one-step transposable we
require an "accidental vector" to move the comma

12 11 7 12 11 12 7
+ 5 (-5) 1 (-1)
-----------------------------------
12 11 12 7 12 11 7

I fiddled with this and haiving the comma adjust "going
along for the ride" only worked for true key changes and
not isolated accidentals or, more importantly, minor
tonality.

What is perhaps more interesting is to ignore the comma (have the
accidental only effect one point) and to 'cycle' through the
system of scales under transposition.

C 12 11 7 12 11 12 7
G 12 11 12 7 11 12 7
D 7 11 12 7 11 12 12
A 7 11 12 12 6 12 12
E 12 6 12 12 6 12 12
B 12 6 12 12 11 7 12
F# 12 11 7 12 11 7 12
C# 12 11 7 12 11 12 7

Of course, all we've done here is to figure out a way to sneak the
wolves back into 12tet!

Bob Valentine

Robert --

>You said the diatonic is not strictly proper which I take
>to mean because it is ordered but not unique ("three steps is always
>less than or equal to four steps").

It's not about the uniqueness -- it's that the ordering is "less than or
equal to" as opposed to "strictly less than". The former implies propriety;
the latter, strict propriety. Of course, strict propriety implies uniqueness
but uniqueness does not imply strict propriety. To clarify for others,
"uniqueness" here means that each specific interval size is mapped to one
and only one generic (i.e., counting number of steps in the scale) interval
size.

>I meant the interlaced MOSs. It looks like there could be a lot of
>interesting potential here, thinking from a sort of
>polytonal/counterpoint standpoint. By modulating the MOSs individually,
>the melodic logic can be maintained but the harmonic interaction
>changed. This is not new and is related to what makes melodic imitation
>in a different mode or even polytonality work, it just gives a nice
>way of approaching it with a new tuning structure where the 'handles'
>may not be so obvious.

I've thought of using such a polytonal (bi-pentatonic) approach before for
decatonic composition, but never tried it. One problem is that each of the
tetrads (analogous to the diatonic's triads) end up 3 notes in one
pentatonic scale and 1 note in the other pentatonic scale -- but which one
has 3 and which one has 1 depends on the chord. Hence it would seem quite
difficult to project the full harmonic variety in a bi-pentatonic approach.
Anyway, the pentachordal decatonic scale is quite effective as an integrated
whole melodically, and doesn't beg for a division into two pentatonics,
particularly when tetradic harmony is used.