Non-octave periodicity blocks

(Sent to both Tuning and Tuning-math; please ignore it in whichever
group it doesn't belong -- I can't figure it out.)

Can anyone point me to any work done on periodicity blocks on
non-octave lattices, particularly the (5/3, 7/3) lattice?

I know about these:

<http://members.aol.com/bpsite/BPlattice.html>

<http://www.kees.cc/tuning/perbl.html>

However, neither seems actually to explore any periodicity blocks
beyond those giving the Bohlen-Pierce 13-note "chromatic" and 9-note
"diatonic" scales.

- Rich Holmes

--- In tuning@yahoogroups.com, Rich Holmes<rsholmes@...> wrote:
>
> (Sent to both Tuning and Tuning-math; please ignore it in whichever
> group it doesn't belong -- I can't figure it out.)
>
> Can anyone point me to any work done on periodicity blocks on
> non-octave lattices, particularly the (5/3, 7/3) lattice?
>
> I know about these:
>
> <http://members.aol.com/bpsite/BPlattice.html>
>
> <http://www.kees.cc/tuning/perbl.html>
>
> However, neither seems actually to explore any periodicity blocks
> beyond those giving the Bohlen-Pierce 13-note "chromatic" and 9-note
> "diatonic" scales.

Hello Rich,

A 7-note scale which can be described as a periodicity block with
625/567 and 49/45 as unison vectors is discussed here:

http://www.kees.cc/music/scale13/scale13.html

This again doesn't satisfy the requirement you discussed earlier that
the scale intervals should be larger than the unison vectors. But
consider the BP pentatonic scale discussed here:

/tuning/topicId_62588.html#62588

It can be described as a periodicity block with unison vectors 245/243
and 25/21. This too has an interval that is smaller than 25/21 but I
don't think this is so bad because 25/21 is supposed to be a
*chromatic* i.e. non-vanishing unison vector. If 245/243 is tempered
out this scale has four consonant triads all produced by the same
scale pattern.

There's also a 7-note scale related to the first one mentioned. Its
unison vectors are (625/567)/(49/45)=3125/3087 and 49/45. This becomes
2 2 2 2 2 2 1 in 13-tone division of 3:1. This too has four consonant
triads although the major and minor versions are in a different
inversional relationship than in the BP pentatonic.

Kalle Aho

"Kalle Aho" <kalleaho@mappi.helsinki.fi> writes:

> A 7-note scale which can be described as a periodicity block with
> 625/567 and 49/45 as unison vectors is discussed here:
>
> http://www.kees.cc/music/scale13/scale13.html

Yes, that's linked from <http://www.kees.cc/tuning/perbl.html>. I
knew about that but forgot to mention of it in my query; sorry.

> But consider the BP pentatonic scale discussed here:
>
> /tuning/topicId_62588.html#62588

Thanks!

> There's also a 7-note scale related to the first one mentioned. Its
> unison vectors are (625/567)/(49/45)=3125/3087 and 49/45. This becomes
> 2 2 2 2 2 2 1 in 13-tone division of 3:1. This too has four consonant
> triads although the major and minor versions are in a different
> inversional relationship than in the BP pentatonic.

Is this discussed further anywhere?

- Rich Holmes

--- In tuning@yahoogroups.com, Rich Holmes<rsholmes@...> wrote:
>
> "Kalle Aho" <kalleaho@...> writes:

> > There's also a 7-note scale related to the first one mentioned. Its
> > unison vectors are (625/567)/(49/45)=3125/3087 and 49/45. This becomes
> > 2 2 2 2 2 2 1 in 13-tone division of 3:1. This too has four consonant
> > triads although the major and minor versions are in a different
> > inversional relationship than in the BP pentatonic.
>
> Is this discussed further anywhere?

Carol Krumhansl mentions this scale in page 45 of

http://www.speech.kth.se/music/publications/kma/papers/kma54-ocr.pdf

Kalle Aho