Heathwaite-Smith and the fifteen note marvelous dwarf

I mentioned this collaborative piece with Andrew Heathwaite, Marvelous
Penta, on non12comp but I'll repeat myself here.

http://66.98.148.43/~xenharmo/gene.html

I want to expand on the anaolgy to the diatonic scale of the 15 note
marvelous dwarf I mention there. The diatonic scale has the following
attributes:

(1) Three otonal and three utonal 5-limit triads

(2) The otonal triads are tonic, dominant and subdominant

(3) 81/80 is tempered out, leading to

(4) a comma pump, passing through utonality to connect the subdominant
to the dominant

The 15-note marvelous dwarf has the following attributes:

(1) Three otonal and three utonal 9-limit pentads

(2) The otonal pentads are tonic, dominant and subdominant

(3) 225/224 is tempered out, leading to

(4) a comma pump, passing through utonality to connect the subdominant
to the dominant

Something which is *not* analogous to the diatonic scale are the large
quantity of triads--this would be as if the diatonic scale had many
consonant dyads which did not appear in any triads, which it does not
have. The 15-note marvelous dwarf, on the other hand, has five major
and five minor triads in addition to the three of each which extend to
pentads. The total harmonic resources are impressive given the fifteen
notes and the exactness of the tuning, which is better even than that
of miracle.

Below I give this marvelous dwarf in the ever-popular 72-et:

4, 5, 12, 16, 23, 28, 30, 35, 42, 46, 53, 54, 58, 65, 72

Here it is in 228-et, which allows for greater tuning accuracy:

13, 16, 38, 51, 73, 89, 95, 111, 133, 146, 168, 171, 184, 206, 228

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

/tuning/topicId_56017.html#56017

> I mentioned this collaborative piece with Andrew Heathwaite,
Marvelous
> Penta, on non12comp but I'll repeat myself here.
>
> http://66.98.148.43/~xenharmo/gene.html
>

***I thought that stragetic pauses and phrasing added a lot to this
piece...

JP

Gene,

>I want to expand on the anaolgy to the diatonic scale of the 15 note
>marvelous dwarf I mention there.

I notice you haven't given the scale on either of these lists.
Or did I miss it?

-Carl

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> /tuning/topicId_56017.html#56017
>
> > I mentioned this collaborative piece with Andrew Heathwaite,
> Marvelous
> > Penta, on non12comp but I'll repeat myself here.
> >
> > http://66.98.148.43/~xenharmo/gene.html
> >
>
> ***I thought that stragetic pauses and phrasing added a lot to this
> piece...

Thanks, though the pauses are all Andrew's work. Have you compared
this to the 5-et version?

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I notice you haven't given the scale on either of these lists.
> Or did I miss it?

I gave it in 72 and 228 tuning, but for theoretical purposes one also
wants the 5-limit version:

25/24, 135/128, 9/8, 75/64, 5/4, 675/512, 4/3, 45/32, 3/2, 25/16,
5/3, 27/16, 225/128, 15/8, 2

On my web site, I described it in terms of the pentads represented by

[0, -1, -2], [0, -1, -1], [0, 0, -1], [0, 0, 0], [0, 1, 0], [0, 1, 1]]

In more concrete terms, these chords are

[28/27, 7/6, 4/3, 14/9, 28/15], [1, 7/6, 4/3, 3/2, 5/3],
[1, 7/6, 7/5,14/9, 7/4], [1, 9/8, 5/4, 3/2, 7/4],
[21/20, 7/6, 21/16, 3/2, 7/4], [9/8, 21/16, 3/2, 27/16, 15/8]

The first chord, [0, -1, -2], is equivalent under marvel to
[0, -1, -2] + [1, 1, 4] = [1, 0, 2], and [1, 0, 2] represents the
quintad [25/24, 75/64, 75/56, 25/16, 15/8] which is 225/224 higher in
JI terms. This, of course, contains a 15/8, or in other words the
28/15 in the first pentad does double duty as a 15/8, and hence links
with the dominant pentad, creating a comma pump. This comma pump is
there but probably less significant than the diatonic pump since the
subdominant and dominant pentad are linked by a common note anyway
(3/2)in just intonation. However, in my view comma pumps impart
dynamism to the harmony.

If we extract 25/24 and 27/16 from the scale, we now have the same
pump using tetrads. This is an interesting scale--more so, I think,
than the dwarf for this val--permutation epimorphic for <13 20 30| and
supplied with six tetrads, twelve triads, and a comma pump. In
comparison, the twelve note marvelous dwarf (the marvel tempered Ellis
duodene) has only four tetrads, albeit ones that extend to quintads,
as well as twelve triads also, using one less note. Setting this
thirteen note scale up on a keyboard or guitar using the epimorphic
ordering might be an interesting thing for an instrumentalist to try.
I'd be interested to know how well it works to tune a piano with one
extra note to the octave.

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

> If we extract 25/24 and 27/16 from the scale, we now have the same
> pump using tetrads. This is an interesting scale--more so, I think,
> than the dwarf for this val--permutation epimorphic for <13 20 30| and
> supplied with six tetrads, twelve triads, and a comma pump.

This seems worth expanding on, so I give the 1/4-kleismic tuning of it
below, in epimorphic order. It has the exotic character one would
expect from a permutation epimorphic scale on such a val, but it is
well supplied with harmony of various kinds. One kind is simply the
fifths; the circle of nominal fifths goes, in the 5-limit tuning,
3/2 3/2 3/2 512/405 3/2 3/2 3/2 3/2 512/405 3/2 3/2 3/2 25/18. That's
ten actual fifths, two "fifths" of size 512/405, which marvel-temper
to a good version of a 14/11 major third, and a 25/18, which
1/4-kleismic marvel, having both fifths and major thirds flat by about
a schisma, keeps as 25/18, which is 100/99 sharper than 11/8. In
comparison diadie2, also with a comma pump under marvel tempering, has
seven fifths in its circle, also three 1024/675 which, marvel tempered
or not, is simply a wolf, and two 375/256, which 1/4-kleismic tempers
to exactly 35/24, the "septimal semi-diminished fifth" of 653 cents.
The pum13marv is giving "fifths" which are less dissonant.

Both scales have six tetrads and a comma pump, but the diadie2 tetrads
are less diatonic in character. It also has four major and four minor
triads, as opposed to six each for pum13marv, our 13-note scale. If
you are looking for an exotic but logically organized 13-note scale,
this is certainly worth considering; and you can have fun with the
strange things transpositions do for you when the scale is arranged as
below.

! pum13marv.scl
pum13 marvel tempered and in epimorphic order
13
!
200.054240
268.798787
468.853027
384.385833
584.440075
499.972880
700.027121
768.771666
968.825906
884.358713
1084.412954
1284.467194
1200.000000
! tempering of 5-limit scale ordered as [9/8, 75/64, 675/512,
! 5/4, 45/32, 4/3, 3/2, 25/16, 225/128, 5/3, 15/8, 135/64, 2]

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

/tuning/topicId_56017.html#56026

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> >
> > /tuning/topicId_56017.html#56017
> >
> > > I mentioned this collaborative piece with Andrew Heathwaite,
> > Marvelous
> > > Penta, on non12comp but I'll repeat myself here.
> > >
> > > http://66.98.148.43/~xenharmo/gene.html
> > >
> >
> > ***I thought that stragetic pauses and phrasing added a lot to
this
> > piece...
>
> Thanks, though the pauses are all Andrew's work. Have you compared
> this to the 5-et version?

***I thought I'd listened to that... or I listened to *some* 5-tET
pieces anyway... :) I'm getting a bit confused as to what's what...

JP