set theory for ETS 19

Hello,

I'm new to this list and am wondering if anyone can point me to software
(or any other resource) that applies set theory labels to systems of equal
temperament other than 12, especially ETS 19.

Thanks in advance,

Ralph Lorenz
Kent State University
rlorenz@kent.edu

--- In tuning@egroups.com, Ralph Lorenz <rlorenz@m...> wrote:
> Hello,
>
> I'm new to this list and am wondering if anyone can point me to
software
> (or any other resource) that applies set theory labels to systems of
equal
> temperament other than 12, especially ETS 19.
>
> Thanks in advance,

Hi Ralph,

Can you explain what you mean by "applies set theory labels to"? Maybe
an example of how it is done with 12 equal.

Regards,
-- Dave Keenan

--- In tuning@egroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:

http://www.egroups.com/message/tuning/16236

>
> Hi Ralph,
>
> Can you explain what you mean by "applies set theory labels to"?
Maybe
> an example of how it is done with 12 equal.
>
> Regards,
> -- Dave Keenan

Oh... I think he means the 'ol "P.C.-I.C. set complexes" that they
teach, or used to, in the music schools... with [0,1,2,3...11] as
"labels" from the first pitch in the set....

__________ ___ __
Joseph Pehrson

--- In tuning@egroups.com, Ralph Lorenz <rlorenz@m...> wrote:

> I'm new to this list and am wondering if anyone can point me to
> software (or any other resource) that applies set theory labels
> to systems of equal temperament other than 12, especially ETS 19.

I sent Ralph a quick reply when he posted the question on another
list recently (Daniel Wolf suggested looking at Scala, and pointed
Ralph here--nice to see he hasn't forgotten about the tuning list
since unsubscribing), but there's more information in this post:

http://www.egroups.com/message/tuning/14154

By the way, in case anyone decides they'd like to enumerate the set-
classes of 31-tET, I'd like to scare you with a big number: the
number of equivalence classes, under transposition and inversion, of
scales/chords of cardinality 15, has an easily calculated lower bound
of 4,847,422. Kind of a long way from 12-tET's 50 hexachord classes...

The calculation for the above number is

31
C
15
-------
62

where C is the "choose" operation, so the numerator here means the
number of ways of choosing 15 things from 31, i.e. 31! / (16! *
15!). The actual number of set-classes will be a little larger than
the figure I gave, since some classes (with inversional symmetry)
have only 31 members instead of 62. The true number of set-classes
for this cardinality is probably over 5 million.

Such a lower-bound approximation is only close for prime ETs. The
more abundant the ET, the poorer the approximation, since there will
be more degrees of transpositional symmetry available, hence
equivalence classes with fewer members.

If I'm not mistaken, Scala can give you the number of triadic set-
classes available for a scale, but not chords of other cardinalities.
Maybe if Manuel is reading, he could think about adding such a
feature in a future release?

best --jon

>> Hi Ralph,
>>
>> Can you explain what you mean by "applies set theory labels to"?
>Maybe
>> an example of how it is done with 12 equal.
>>
>> Regards,
>> -- Dave Keenan
>
>Oh... I think he means the 'ol "P.C.-I.C. set complexes" that they
>teach, or used to, in the music schools... with [0,1,2,3...11] as
>"labels" from the first pitch in the set....
>
>__________ ___ __
>Joseph Pehrson

Yes, this is the process of labeling set classes according to
transpositional and inversional equivalence. For a nice web site that will
compute set classes in ETS 12, see:

http://www.arts.ilstu.edu/~staylor/setfinder/index.html

Best,

Ralph Lorenz