For Paul: Blackwood's 15-Equal Decatonic Scale

Hi Paul (and everyone interested)!

In your "famous" paper you said that Blackwood's ten-tone symmetrical
scale in 15-equal just misses property (4) which is:

(4) Key coherence: A chord progression of no more than three
consonant chords is required to cover the entire scale.

This is true in 5-limit but 15-equal is consistent in 7-limit and
this particular scale has a complete 7-limit tetrad on every scale
degree. These chords satisfy (4).

Kalle

--- In tuning@y..., "Kalle Aho" <kalleaho@m...> wrote:
> Hi Paul (and everyone interested)!
>
> In your "famous" paper you said that Blackwood's ten-tone
symmetrical
> scale in 15-equal just misses property (4) which is:
>
> (4) Key coherence: A chord progression of no more than three
> consonant chords is required to cover the entire scale.
>
> This is true in 5-limit but 15-equal is consistent in 7-limit and
> this particular scale has a complete 7-limit tetrad on every scale
> degree. These chords satisfy (4).
>
> Kalle

kalle, according to the standards of my paper, 15-equal isn't quite
good enough in the 7-limit. and blackwood's own presentation centers
around triads, not tetrads.

however, in practice, i've enjoyed 7-limit harmony in 15-equal --
it's a bit better than in 12-equal, in some respects. and yes, i've
used this ten-tone symmetrical scale to acheive it. the effect is
similar to what you get from the diminished (octatonic) scale in 12-
equal -- lots of consonant harmony, but quite an atonal feel due to
the symmetry (aka limited transposability) at a fraction of the
octave. it would be nice if this scale could be made omnitetrachordal
as easily as the 22-equal symmetrical decatonic can be made
omnitetrachordal, but i don't think it can be done.

on tuning-math, you'll find that "blackwood" is the name of one of
the linear temperaments in our "best" list. the linear temperament is
defined so that the period is 1/5 octave, and the generator is
somewhere in the vicinity of 1/15 octave, so the two-notes-per-period
MOS of this temperament is essentially blackwood's scale.

cheers,
paul

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
>
> kalle, according to the standards of my paper, 15-equal isn't quite
> good enough in the 7-limit. and blackwood's own presentation
centers
> around triads, not tetrads.

Understood.

> however, in practice, i've enjoyed 7-limit harmony in 15-equal --
> it's a bit better than in 12-equal, in some respects. and yes, i've
> used this ten-tone symmetrical scale to acheive it. the effect is
> similar to what you get from the diminished (octatonic) scale in 12-
> equal -- lots of consonant harmony, but quite an atonal feel due to
> the symmetry (aka limited transposability) at a fraction of the
> octave. it would be nice if this scale could be made
omnitetrachordal
> as easily as the 22-equal symmetrical decatonic can be made
> omnitetrachordal, but i don't think it can be done.

Do you mean octatonic can't be made omnitetrachordal? 15-equal
decatonic _is_ omnitetrachordal, isn't it?

> on tuning-math, you'll find that "blackwood" is the name of one of
> the linear temperaments in our "best" list. the linear temperament
is
> defined so that the period is 1/5 octave, and the generator is
> somewhere in the vicinity of 1/15 octave, so the two-notes-per-
period
> MOS of this temperament is essentially blackwood's scale.

Great!

Kalle

--- In tuning@y..., "Kalle Aho" <kalleaho@m...> wrote:
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
> wrote:
> >
> > kalle, according to the standards of my paper, 15-equal isn't
quite
> > good enough in the 7-limit. and blackwood's own presentation
> centers
> > around triads, not tetrads.
>
> Understood.
>
> > however, in practice, i've enjoyed 7-limit harmony in 15-equal --
> > it's a bit better than in 12-equal, in some respects. and yes,
i've
> > used this ten-tone symmetrical scale to acheive it. the effect is
> > similar to what you get from the diminished (octatonic) scale in
12-
> > equal -- lots of consonant harmony, but quite an atonal feel due
to
> > the symmetry (aka limited transposability) at a fraction of the
> > octave. it would be nice if this scale could be made
> omnitetrachordal
> > as easily as the 22-equal symmetrical decatonic can be made
> > omnitetrachordal, but i don't think it can be done.
>
> Do you mean octatonic can't be made omnitetrachordal? 15-equal
> decatonic _is_ omnitetrachordal, isn't it?

umm . . . yes . . . you're right of course . . . i meant
octatonic . . . :) actually, i was thinking not just
omnitetrachordal, but also fully transposable (no interval of
repetition smaller than the octave), so that a fully tonal, and not
atonal/polytonal, vibe can be projected . . .

> > on tuning-math, you'll find that "blackwood" is the name of one
of
> > the linear temperaments in our "best" list. the linear
temperament
> is
> > defined so that the period is 1/5 octave, and the generator is
> > somewhere in the vicinity of 1/15 octave, so the two-notes-per-
> period
> > MOS of this temperament is essentially blackwood's scale.
>
> Great!
>
> Kalle