Yahoo Tuning Groups Ultimate Backup tuning-math Steiner Systems and other Topics

This is kind of interesting. I am looking to construct a single
Steiner system, per some email exchanges with Noam Elkies.

Apparently, there are at least two ways. One is using Curtis' Kitten
(The MINIMOG) for S(5,6,12), which corresponds to PSL(2,11).

Anyway, I am not getting the kind of system Dr. Elkies challenged me
to find, one based on just 11 hexachords, and their inverse or
complement or inverse complements = 22 hexachords at 6 even
transpositions (in the musical sense).

With the Kitten, I analyzed the 132 hexachords months ago.

Now I found another way, which may be the same as the Kitten, except
that with the Kitten I didn't find (0,1,3,4,5,9) (Quadratic residues
in the projective line over GF11) which is the absolute seed for the
PSL(2,11) method.

It's cool, I am going to find these sets once I write a program.

This one is based on a(y) = y + 1 and b(y) = -1/y, where k^2*y
isn't needed, unless this is the quadratic residue piece?

This is of course mod 11, with 11 = infinity, and 1/11=0 and 1/0=11

But, alas, you don't get 6 or 12 neat transpositions of any set of 22

Also with the Kitten, like I said, I didn't find (0,1,3,4,5,9) which
doesn't make sense.

So analyzing the "seed" hexad/hexachord, just for fun,

In the 5-limit: 120, 125, 144, 150, 160, 200 Just pitches.

I don't feel like calculating the 7-limit.

Musically, C-C#-Eb-E-F-A, This gives F7, FMaj7, Am, AM as subsets.

This is B1 in my hexachord system.

2^3 * 3 * 5
5^3
2^4 * 3^2
2 * 3 * 5^2
2^5 * 5
2^3 * 5^2

For what it's worth

I'd like to learn how this ties into lattices. SPLAG is more about
M24 (S(5,8,24)) and the Golay Code and the Leech Lattice...

I am also seeing that the Exterior Algebra is closely related
to tensors and ideals, but I need to read the Geometric Algebra
paper before I even ask any questions about that. So it's fun to
see that it's all related. Also that linear is an important part
of group theory anyway, with all the PSL stuff, projective spaces,
finite geometry etc. Wow I am boring even for a "student":) (What
did you learn in school today, Jonny?)

I am going to see if ternary Golay Codes, based on S(5,6,12) tie
into 3-color necklaces problems, (0,1,2 states) but they probably
don't, do they?

2 color complementability is boring, but then again, S(5,6,12)
at least always involves exactly 66 hexads and their complements.

Grasping at straws....

* * * *

I've decided to drop my names such as D4-neg X S3 etc. and merely
call them what they are, the M5 and Inv(M5) symmetries. I wish
someone would comment on this, I thought the discovery of simple
symmetry of the square and triangle to find tranpositions, inverses,
M5 and Inv(M5) was pretty good. No pats on the head though. I can
find exact transpositions at any cardinality a la Polya and
group direct products etc...

Pat Pat

There I patted myself on the back

You might be interested to know that Jon Wild is delivering a paper
this weekend in Berlin about Steiner Systems and 19-tET (Total
coincidence, we never discussed this before...) So at least HE is
going beyond 12-tET. Why ain't I?

Well, I guess it's because I am a music theorist, and yes, most
music out there is in 12-tET!!!! Of course other tuning systems
are important, such as Pajara, 7-limit and higher and so forth.

But if the man on the street just thinks it sounds like an out
of tune steam calliope, he/she isn't going to listen to it. Doesn't
matter how beautiful the math is. Of course the ear can be retrained
to some extent, to appreciate say 31-tET meantone, but what is
the right vehicle for bringing this about in society?

Comments?

Thanks

PGH

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> This is kind of interesting. I am looking to construct a single
> Steiner system, per some email exchanges with Noam Elkies.
>
> Apparently, there are at least two ways. One is using Curtis' Kitten
> (The MINIMOG) for S(5,6,12), which corresponds to PSL(2,11).
>
> Anyway, I am not getting the kind of system Dr. Elkies challenged
me
> to find, one based on just 11 hexachords, and their inverse or
> complement or inverse complements = 22 hexachords at 6 even
> transpositions (in the musical sense).
>
> With the Kitten, I analyzed the 132 hexachords months ago.
>
> Now I found another way, which may be the same as the Kitten, except
> that with the Kitten I didn't find (0,1,3,4,5,9) (Quadratic residues
> in the projective line over GF11) which is the absolute seed for
the
> PSL(2,11) method.
>
> It's cool, I am going to find these sets once I write a program.
>
> This one is based on a(y) = y + 1 and b(y) = -1/y, where k^2*y
> isn't needed, unless this is the quadratic residue piece?
>
> This is of course mod 11, with 11 = infinity, and 1/11=0 and 1/0=11
>
> But, alas, you don't get 6 or 12 neat transpositions of any set of
22
>
> Also with the Kitten, like I said, I didn't find (0,1,3,4,5,9) which
> doesn't make sense.
>
> So analyzing the "seed" hexad/hexachord, just for fun,
>
> In the 5-limit: 120, 125, 144, 150, 160, 200 Just pitches.
>
> I don't feel like calculating the 7-limit.
>
> Musically, C-C#-Eb-E-F-A, This gives F7, FMaj7, Am, AM as subsets.
>
> This is B1 in my hexachord system.
>
> 2^3 * 3 * 5
> 5^3
> 2^4 * 3^2
> 2 * 3 * 5^2
> 2^5 * 5
> 2^3 * 5^2
>
> For what it's worth
>
>
>
> I'd like to learn how this ties into lattices. SPLAG is more about
> M24 (S(5,8,24)) and the Golay Code and the Leech Lattice...
>
> I am also seeing that the Exterior Algebra is closely related
> to tensors and ideals, but I need to read the Geometric Algebra
> paper before I even ask any questions about that. So it's fun to
> see that it's all related. Also that linear is an important part
> of group theory anyway, with all the PSL stuff, projective spaces,
> finite geometry etc. Wow I am boring even for a "student":) (What
> did you learn in school today, Jonny?)
>
> I am going to see if ternary Golay Codes, based on S(5,6,12) tie
> into 3-color necklaces problems, (0,1,2 states) but they probably
> don't, do they?
>
> 2 color complementability is boring, but then again, S(5,6,12)
> at least always involves exactly 66 hexads and their complements.
>
> Grasping at straws....
>
> * * * *
>
> I've decided to drop my names such as D4-neg X S3 etc. and merely
> call them what they are, the M5 and Inv(M5) symmetries. I wish
> someone would comment on this, I thought the discovery of simple
> symmetry of the square and triangle to find tranpositions, inverses,
> M5 and Inv(M5) was pretty good. No pats on the head though. I can
> find exact transpositions at any cardinality a la Polya and
> group direct products etc...
>
> Pat Pat
>
> There I patted myself on the back
>
> You might be interested to know that Jon Wild is delivering a paper
> this weekend in Berlin about Steiner Systems and 19-tET (Total
> coincidence, we never discussed this before...) So at least HE is
> going beyond 12-tET. Why ain't I?
>
> Well, I guess it's because I am a music theorist, and yes, most
> music out there is in 12-tET!!!! Of course other tuning systems
> are important, such as Pajara, 7-limit and higher and so forth.
>
> But if the man on the street just thinks it sounds like an out
> of tune steam calliope, he/she isn't going to listen to it. Doesn't
> matter how beautiful the math is. Of course the ear can be
retrained
> to some extent, to appreciate say 31-tET meantone, but what is
> the right vehicle for bringing this about in society?
>
> Comments?
>
> Thanks
>
> PGH

Please excuse this post I was upset when I wrote it. Anyway,
For what it's worth, the two different constructions seem to
be based on modulo-11 labelling (which corresponds to PSL(2,11) and
shuffle labelling, which each give different hexads.

* * *

Who cares about M5 anyway

* * *

31-tET actually sounds better than 12-tET in some ways. Please forgive
the steam-calliope comment

PGH

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗Paul G Hjelmstad <phjelmstad@msn.com>

🔗Carl Lumma <ekin@lumma.org>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> >Yes I guess you are right. Since I have absolute pitch (no big
deal) it
> >would be hard for me to perform music in microtemperaments, but it
> >would be fun to try! I'm pretty engrained in 12. The AP is helpful
> >in atonal music singing for example
>
> You should be able to turn that into microtonal AP with a little
> practice.
>
> -Carl

I agree. I've found a method by which it happens almost
automatically (assuming that you already have AP), but it also
involves a little time, patience, and persistence.

In the late 1960's, when I was experimenting with a retuned
electronic organ (to subsets of 31-ET), I taped a few sessions and,
after listening to them frequently over a period of several weeks,
found that I had memorized the pitches. More recently, I found the
same thing happening after repeated listenings to microtonal pieces
from the MMM list, although achieving MAP this way is less successful
if I don't have a score to follow (inasmuch as memorization requires
linking *pitches* to *note names*; if I'm not sure of the names, then
I can't make the mental linkages, although I can still "play back"
some of these pieces in my mind with the actual pitches -- I guess I
would call that "microtonal memory").

One caveat is the conflict that results when the relationships
between note names change from one tuning to another, e.g., F# is
lower than Gb in 19 and 31-ET, whereas the opposite holds in 17 and
22-ET. Another problem is differing standards of pitch, particularly
with the wide-fifth tunings, e.g., should I tune A to 440, or should
I tune A\! to 440 (to keep C from becoming too low in pitch)? Thus,
developing MAP requires that you select your tuning(s), notation, and
pitch standard(s) ahead of time, and memorize one tuning at a time.
These pitfalls have kept me from making a deliberate effort to
develop MAP.

But if you're interested in doing so, to minimize the foregoing
problems I would advise starting with 19-ET, 31-ET, or 72-ET, with
A=440.

(Just thought I'd throw my 2c in here for the benefit of those few
that might be interested, or for some music psychologist who might
happen to read this sometime in the future.. :-)

--George

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >Yes I guess you are right. Since I have absolute pitch (no big
deal) it
> >would be hard for me to perform music in microtemperaments, but it
> >would be fun to try! I'm pretty engrained in 12. The AP is helpful
> >in atonal music singing for example
>
> You should be able to turn that into microtonal AP with a little
> practice.
>
> -Carl
>
Maybe. What's cool about 31-tET, as you all know, is that you can
use the standard accidentals. 7 naturals, 7 sharps, 7 flats, and 5
double sharps and 5 double flats. I don't know, my F# is pretty much
an F#:) Leopold Mozart espoused 55-tET meantone on the violin. Wonder
how that worked!

Now there is the Microzone and the Clavette, both I believe by
Harold Fortuin. I wonder if these instruments have evolved so that
they would be easier to play, they use hexagonal keys. I suppose
given enough ear-hand coordination, AP on these instruments could
be learned, but I bet it would mess up my normal AP. Someone was
talking about improving these instruments so that they have
depressible and weight-action keyboards, does anyone know about that.
Of course, you could be tone deaf and still play an instrument:)
Having good pitch, though, absolute or relative, helps one grab
the right notes for some reason (this has been verified) by hearing
the note a little ahead of time, and going to it. (Especially
playing from memory, it's both ear-memory and muscular memory)

Relative pitch in 31 (and other microtemperaments) would be
difficult for anyone. There was an organist in the 1300s or so who
trained his choir to sing in 31 (not tempered exactly, they didn't
have the logarithm yet!) along with a 31-tone Pipe Organ, but I can't
imagine they ever sang "a capella"

Then again, if relative pitch is based on just intervals, maybe
it too could be learned. Because 31 is so close to just, at least
for 5/4. (There should be a table to look these things up quickly,
in the Files section, along with linear temperament listings etc)

It would be fun to determine what intervals an a capella choir
actually sings, for different kinds of music. How would this be
done? With an oscilloscope or something? Has an psychoacoustician
done these tests? I have a feeling, that due to the "choir effect"
on pitches, and being in our Western culture, you would get
essential 12-tET, but there might be some comma pumps in certain
situations like the "third" relation, etc. It would be fun to
study this

PGH

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗Paul G Hjelmstad <phjelmstad@msn.com>

🔗Graham Breed <gbreed@gmail.com>

🔗Carl Lumma <ekin@lumma.org>

>Now there is the Microzone and the Clavette, both I believe by
>Harold Fortuin.

The Microzone is by Erv Wilson + Harvey Starr. The Clavette
is not commercially available (that I know of).

>I wonder if these instruments have evolved so that
>they would be easier to play, they use hexagonal keys. I suppose
>given enough ear-hand coordination, AP on these instruments could
>be learned, but I bet it would mess up my normal AP.

I don't think AP can be messed up this way. What's your
experience now when listening to a quartertone?

>Someone was
>talking about improving these instruments so that they have
>depressible and weight-action keyboards, does anyone know about that.

I know everything there is to know about it. Which isn't much.

>Relative pitch in 31 (and other microtemperaments) would be
>difficult for anyone.

The point is to learn to distinguish more hues. Using the
the A440+12ET points that you already know as anchors.

>There was an organist in the 1300s or so who
>trained his choir to sing in 31 (not tempered exactly, they didn't
>have the logarithm yet!) along with a 31-tone Pipe Organ, but I can't
>imagine they ever sang "a capella"

Being able to sing a pitch on demand is one kind of AP skill.
Others include: if I play 6 notes in an octave on a piano, can
you tell me which ones they are? If I play 11 notes in an
octave, can you tell me which one is missing? If I play an
unknown microtone, can you tell me a rough cents offset from
the nearest 12ET point (say, to the nearest 6th tone)?

>Then again, if relative pitch is based on just intervals, maybe
>it too could be learned.

I don't know that RP is based on JI.

>Because 31 is so close to just, at least
>for 5/4. (There should be a table to look these things up quickly,
>in the Files section, along with linear temperament listings etc)

31 has tones that are close to just, but fewer than half of them
will be just in any one key. Maybe a lot fewer depending on your
definition of "just".

>It would be fun to determine what intervals an a capella choir
>actually sings, for different kinds of music. How would this be
>done? With an oscilloscope or something? Has an psychoacoustician
>done these tests?

This question has been addressed, though never completely
rigorously to my knowledge. However I think we have a very
good an idea of choral intonation considering that different
choirs do different things, and same choirs do different
things on different days, etc.

>I have a feeling, that due to the "choir effect"
>on pitches, and being in our Western culture, you would get
>essential 12-tET, but there might be some comma pumps in certain
>situations like the "third" relation, etc. It would be fun to
>study this

There's a sustain-dependant shift from 12-ET to JI for
consonant music in unaccompanied choirs. The pitch standard
is maintained by hook or by crook, if it is.

-Carl

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >Now there is the Microzone and the Clavette, both I believe by
> >Harold Fortuin.
>
> The Microzone is by Erv Wilson + Harvey Starr. The Clavette
> is not commercially available (that I know of).
>
> >I wonder if these instruments have evolved so that
> >they would be easier to play, they use hexagonal keys. I suppose
> >given enough ear-hand coordination, AP on these instruments could
> >be learned, but I bet it would mess up my normal AP.
>
> I don't think AP can be messed up this way. What's your
> experience now when listening to a quartertone?

Perfect pitch is an odd thing. I don't remember not having it.
My friend in college, a composition major, who is now a programmer,
did an experiment in the physics department to study pitch ability
and I actually did worse on higher/lower measurements than some
other people because the AP threw me off for some reason in
some situations. Quartertones, of course, are as out of tune
as you can get from 12-tET. 36/35 is about 50 cents FWIW

> >Someone was
> >talking about improving these instruments so that they have
> >depressible and weight-action keyboards, does anyone know about
that.
>
> I know everything there is to know about it. Which isn't much.

So where are we at? I want to play the damn thing

> >Relative pitch in 31 (and other microtemperaments) would be
> >difficult for anyone.
>
> The point is to learn to distinguish more hues. Using the
> the A440+12ET points that you already know as anchors.

Interesting. BTW, my study of M12 and M24 and other Groups
has lead me to believe that both 12-tET and 24-tET are very very
important. 24 even ties into the RZF per John Baez, of course
I would defer to Gene before making any claims about RZH/RZF

> >There was an organist in the 1300s or so who
> >trained his choir to sing in 31 (not tempered exactly, they didn't
> >have the logarithm yet!) along with a 31-tone Pipe Organ, but I
can't
> >imagine they ever sang "a capella"
>
> Being able to sing a pitch on demand is one kind of AP skill.
> Others include: if I play 6 notes in an octave on a piano, can
> you tell me which ones they are? If I play 11 notes in an
> octave, can you tell me which one is missing? If I play an
> unknown microtone, can you tell me a rough cents offset from
> the nearest 12ET point (say, to the nearest 6th tone)?

I can name any note(s) on the piano, anytime, with my back turned,
and I could even as a small child. Probably up to about 6. 11 notes
out of 12? C'mon how much music, is like that? Except for George
Crumb. I'm not as good as a blind pianist! I'll have one of my
piano students test it today. Maybe I can.

> >Then again, if relative pitch is based on just intervals, maybe
> >it too could be learned.
>
> I don't know that RP is based on JI.

I think it is. Even the best non-AP singers sometimes strugglewith
intervals. (My Bonnie, Here Comes the Bride, Star Trek, they actually
use those clues. There are only 6 absolute intervals of course. I
think perhaps the semitone is learned, and then applied against Just
Fifths and Thirds. My little theory, like that of the brontasaurus,
which is thin at one end, then fat, oh never mind)

> >Because 31 is so close to just, at least
> >for 5/4. (There should be a table to look these things up quickly,
> >in the Files section, along with linear temperament listings etc)
>
> 31 has tones that are close to just, but fewer than half of them
> will be just in any one key. Maybe a lot fewer depending on your
> definition of "just".

Yeah after 5 years on tuning-math I should know better:)

>
> >It would be fun to determine what intervals an a capella choir
> >actually sings, for different kinds of music. How would this be
> >done? With an oscilloscope or something? Has an psychoacoustician
> >done these tests?
>
> This question has been addressed, though never completely
> rigorously to my knowledge. However I think we have a very
> good an idea of choral intonation considering that different
> choirs do different things, and same choirs do different
> things on different days, etc.
>
> >I have a feeling, that due to the "choir effect"
> >on pitches, and being in our Western culture, you would get
> >essential 12-tET, but there might be some comma pumps in certain
> >situations like the "third" relation, etc. It would be fun to
> >study this
>
> There's a sustain-dependant shift from 12-ET to JI for
> consonant music in unaccompanied choirs. The pitch standard
> is maintained by hook or by crook, if it is.

I know. It never ceases to amaze me why people sing wrong notes,
even in the best choirs, not just half step off, but 33 cents off
etc. Even the best choirs have trouble with Messaien's "O Sacrum
Convivium", an a capella choral work with very complex harmony
(but not atonal)

- PGH
>
>
>

🔗Paul G Hjelmstad <phjelmstad@msn.com>

🔗Carl Lumma <ekin@lumma.org>

Hi Paul,

>>> I wonder if these instruments have evolved so that
>>> they would be easier to play, they use hexagonal keys. I suppose
>>> given enough ear-hand coordination, AP on these instruments could
>>> be learned, but I bet it would mess up my normal AP.
>>
>> I don't think AP can be messed up this way. What's your
>> experience now when listening to a quartertone?
>
> Perfect pitch is an odd thing. I don't remember not having it.
> My friend in college, a composition major, who is now a programmer,
> did an experiment in the physics department to study pitch ability
> and I actually did worse on higher/lower measurements than some
> other people because the AP threw me off for some reason in
> some situations.

Sounds like you could use some brush-up on RP.

>Quartertones, of course, are as out of tune
>as you can get from 12-tET. 36/35 is about 50 cents FWIW

You seem to be implying that AP has something to do with 12-tET.

>>> Someone was
>>> talking about improving these instruments so that they have
>>> depressible and weight-action keyboards, does anyone know about
>>> that.
>>
>> I know everything there is to know about it. Which isn't much.
>
> So where are we at? I want to play the damn thing

There's no generalized keyboard with a weighted action in
existence, or even on the drawing board. However, one might
ask what importance "weight" has. . .
I had always taken the belief that it is the *linking* of
action events with sound events that makes the piano so
expressive. But a pianist friend of mine thinks the human
musculature needs some resistance to work against to get
full dynamic range. That may be true, but then again, it
might not. . .
Anyway, the Microzone and the Terpstra keyboard should both
have action->sound linking far superior to halberstadt MIDI
controllers in wide use today. The Microzone does it with
aftertouch pressure, the Terpstra with velocity sensing.

>>> Relative pitch in 31 (and other microtemperaments) would be
>>> difficult for anyone.
>>
>> The point is to learn to distinguish more hues. Using the
>> the A440+12ET points that you already know as anchors.
>
>Interesting. BTW, my study of M12 and M24 and other Groups
>has lead me to believe that both 12-tET and 24-tET are very very
>important. 24 even ties into the RZF per John Baez,

What you consider important may be quite different than what
I consider important. Can you explain it in English?

>> Being able to sing a pitch on demand is one kind of AP skill.
>> Others include: if I play 6 notes in an octave on a piano, can
>> you tell me which ones they are? If I play 11 notes in an
>> octave, can you tell me which one is missing? If I play an
>> unknown microtone, can you tell me a rough cents offset from
>> the nearest 12ET point (say, to the nearest 6th tone)?
>
>I can name any note(s) on the piano, anytime, with my back turned,
>and I could even as a small child. Probably up to about 6. 11 notes
>out of 12? C'mon how much music, is like that?

I know a guy who can do it, is all. It's an AP skill. There
are degrees of AP skill. It's not just something you "have"
or not.
Can you remember not knowing any language? Yet, you're surely
aware there are degrees of language skill, even among native
speakers.

>>> Then again, if relative pitch is based on just intervals, maybe
>>> it too could be learned.
>>
>> I don't know that RP is based on JI.
>
>I think it is.

I think it's based on whatever you practice RP with. If you
use 12-tET, all the intervals (and especially the most commonly
used ones) are close to something just.

-Carl

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> Hi Paul,
>
> >>> I wonder if these instruments have evolved so that
> >>> they would be easier to play, they use hexagonal keys. I suppose
> >>> given enough ear-hand coordination, AP on these instruments
could
> >>> be learned, but I bet it would mess up my normal AP.
> >>
> >> I don't think AP can be messed up this way. What's your
> >> experience now when listening to a quartertone?
> >
> > Perfect pitch is an odd thing. I don't remember not having it.
> > My friend in college, a composition major, who is now a
programmer,
> > did an experiment in the physics department to study pitch ability
> > and I actually did worse on higher/lower measurements than some
> > other people because the AP threw me off for some reason in
> > some situations.
>
> Sounds like you could use some brush-up on RP.

* You know it. When music is transposed I'm in the same boat as
everyone else:)
>
> >Quartertones, of course, are as out of tune
> >as you can get from 12-tET. 36/35 is about 50 cents FWIW
>
> You seem to be implying that AP has something to do with 12-tET.

* It does for me!!! It's stranger than you think
>
> >>> Someone was
> >>> talking about improving these instruments so that they have
> >>> depressible and weight-action keyboards, does anyone know about
> >>> that.
> >>
> >> I know everything there is to know about it. Which isn't much.
> >
> > So where are we at? I want to play the damn thing
>
> There's no generalized keyboard with a weighted action in
> existence, or even on the drawing board. However, one might
> ask what importance "weight" has. . .
> I had always taken the belief that it is the *linking* of
> action events with sound events that makes the piano so
> expressive. But a pianist friend of mine thinks the human
> musculature needs some resistance to work against to get
> full dynamic range. That may be true, but then again, it
> might not. . .
> Anyway, the Microzone and the Terpstra keyboard should both
> have action->sound linking far superior to halberstadt MIDI
> controllers in wide use today. The Microzone does it with
> aftertouch pressure, the Terpstra with velocity sensing.

* Cool. I'll have to buy something, once I save up $$

>
> >>> Relative pitch in 31 (and other microtemperaments) would be
> >>> difficult for anyone.
> >>
> >> The point is to learn to distinguish more hues. Using the
> >> the A440+12ET points that you already know as anchors.
> >
> >Interesting. BTW, my study of M12 and M24 and other Groups
> >has lead me to believe that both 12-tET and 24-tET are very very
> >important. 24 even ties into the RZF per John Baez,
>
> What you consider important may be quite different than what
> I consider important. Can you explain it in English?

Yes, well, I'm out on a limb, this newsgroup is out on a limb
anyway, so I guess it's a limb on a limb, M12 and M24 have to
do with Steiner systems, which for example, tie hexachords to
pentachords, I am hoping to tie musical set theory into tuning
theory. I am also equating M12 with 12-tET and M24 with 24-tET
12 and 24 are important, look at John Baez's This Week in
Mathematical Physics Week 234 for example. Hard to explain in
English since I am still learning it myself! But John is good.

>
> >> Being able to sing a pitch on demand is one kind of AP skill.
> >> Others include: if I play 6 notes in an octave on a piano, can
> >> you tell me which ones they are? If I play 11 notes in an
> >> octave, can you tell me which one is missing? If I play an
> >> unknown microtone, can you tell me a rough cents offset from
> >> the nearest 12ET point (say, to the nearest 6th tone)?
> >
> >I can name any note(s) on the piano, anytime, with my back turned,
> >and I could even as a small child. Probably up to about 6. 11
notes
> >out of 12? C'mon how much music, is like that?
>
> I know a guy who can do it, is all. It's an AP skill. There
> are degrees of AP skill. It's not just something you "have"
> or not.
> Can you remember not knowing any language? Yet, you're surely
> aware there are degrees of language skill, even among native
> speakers.

I'm just saying at 5 I could do it. Not boasting. I just could.
If you're off a half-step, then it's not AP. My brother can, my
sister can't. My dad isn't a musician. My mom tends to be off.

>
> >>> Then again, if relative pitch is based on just intervals, maybe
> >>> it too could be learned.
> >>
> >> I don't know that RP is based on JI.
> >
> >I think it is.
>
> I think it's based on whatever you practice RP with. If you
> use 12-tET, all the intervals (and especially the most commonly
> used ones) are close to something just.

I'll take your word for it. I haven't tried it too much.

PGH
>
>

🔗Carl Lumma <ekin@lumma.org>

>> You seem to be implying that AP has something to do with 12-tET.
>
>* It does for me!!! It's stranger than you think

But this is because you learned AP in a 12-tET world.

>> There's no generalized keyboard with a weighted action in
>> existence, or even on the drawing board. However, one might
>> ask what importance "weight" has. . .
>> I had always taken the belief that it is the *linking* of
>> action events with sound events that makes the piano so
>> expressive. But a pianist friend of mine thinks the human
>> musculature needs some resistance to work against to get
>> full dynamic range. That may be true, but then again, it
>> might not. . .
>> Anyway, the Microzone and the Terpstra keyboard should both
>> have action->sound linking far superior to halberstadt MIDI
>> controllers in wide use today. The Microzone does it with
>> aftertouch pressure, the Terpstra with velocity sensing.
>
>* Cool. I'll have to buy something, once I save up $$

I'm recommending Terpstra. Last I heard it was $8K in
batches of two. Wanna save together?

By the time we amass that kind of cash, Vandervoort's
"Daskin" keyboard might be out, and hopefully at a lower
cost.

>> What you consider important may be quite different than what
>> I consider important. Can you explain it in English?
>
>Yes, well, I'm out on a limb, this newsgroup is out on a limb
>anyway, so I guess it's a limb on a limb, M12 and M24 have to
>do with Steiner systems, which for example, tie hexachords to
>pentachords, I am hoping to tie musical set theory into tuning
>theory. I am also equating M12 with 12-tET and M24 with 24-tET
>12 and 24 are important, look at John Baez's This Week in
>Mathematical Physics Week 234 for example. Hard to explain in
>English since I am still learning it myself! But John is good.

Oh, musical set theory. Yeah, I don't think it's important.
But if you can get those goons to do something microtonal by
publishing about the link, more power to you.

>> I know a guy who can do it, is all. It's an AP skill. There
>> are degrees of AP skill. It's not just something you "have"
>> or not.
>> Can you remember not knowing any language? Yet, you're surely
>> aware there are degrees of language skill, even among native
>> speakers.
>
>I'm just saying at 5 I could do it. Not boasting. I just could.

You have this in common with 99.9999% of APers. There's a
sensitive period for it, that goes along with the stuff involved
in learning language.

>If you're off a half-step, then it's not AP. My brother can, my
>sister can't. My dad isn't a musician. My mom tends to be off.

It's not genetic. There may be genetic factors, and there's a
study my friend at UCSD is loosely involved in to determine them.
But the authors wrongly fall (in my opinion) for the "born with
it" chestnut. Their own study should disavow them of that, if
they do it right.

>> I think it's based on whatever you practice RP with. If you
>> use 12-tET, all the intervals (and especially the most commonly
>> used ones) are close to something just.
>
>I'll take your word for it. I haven't tried it too much.

Some APers wind up using AP as a crutch, and therefore don't
develop RP as strongly as they might have. If you develop
both skills fully, you can very much kick ass at music.

-Carl

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> You seem to be implying that AP has something to do with 12-tET.
> >
> >* It does for me!!! It's stranger than you think
>
> But this is because you learned AP in a 12-tET world.
>
> >> There's no generalized keyboard with a weighted action in
> >> existence, or even on the drawing board. However, one might
> >> ask what importance "weight" has. . .
> >> I had always taken the belief that it is the *linking* of
> >> action events with sound events that makes the piano so
> >> expressive. But a pianist friend of mine thinks the human
> >> musculature needs some resistance to work against to get
> >> full dynamic range. That may be true, but then again, it
> >> might not. . .
> >> Anyway, the Microzone and the Terpstra keyboard should both
> >> have action->sound linking far superior to halberstadt MIDI
> >> controllers in wide use today. The Microzone does it with
> >> aftertouch pressure, the Terpstra with velocity sensing.
> >
> >* Cool. I'll have to buy something, once I save up $$
>
> I'm recommending Terpstra. Last I heard it was $8K in
> batches of two. Wanna save together?
>
> By the time we amass that kind of cash, Vandervoort's
> "Daskin" keyboard might be out, and hopefully at a lower
> cost.

** Well I was going to buy a bike first. I just started a savings
account...

>
> >> What you consider important may be quite different than what
> >> I consider important. Can you explain it in English?
> >
> >Yes, well, I'm out on a limb, this newsgroup is out on a limb
> >anyway, so I guess it's a limb on a limb, M12 and M24 have to
> >do with Steiner systems, which for example, tie hexachords to
> >pentachords, I am hoping to tie musical set theory into tuning
> >theory. I am also equating M12 with 12-tET and M24 with 24-tET
> >12 and 24 are important, look at John Baez's This Week in
> >Mathematical Physics Week 234 for example. Hard to explain in
> >English since I am still learning it myself! But John is good.
>
> Oh, musical set theory. Yeah, I don't think it's important.
> But if you can get those goons to do something microtonal by
> publishing about the link, more power to you.

* Wow! But musical composition is nothing more than combinations
of different notes! At least classical/jazz is.

BTW Dr. Baez is Joan Baez's cousin, another math-music connection.
Hopefully you're not too young to know Joan is a famous folk singer,
who worked with Bob Dylan back in the 60's!)

(Who are the "goons?" Watch out making fun of us big guys:))

> >> I know a guy who can do it, is all. It's an AP skill. There
> >> are degrees of AP skill. It's not just something you "have"
> >> or not.
> >> Can you remember not knowing any language? Yet, you're surely
> >> aware there are degrees of language skill, even among native
> >> speakers.
> >
> >I'm just saying at 5 I could do it. Not boasting. I just could.
>
> You have this in common with 99.9999% of APers. There's a
> sensitive period for it, that goes along with the stuff involved
> in learning language.

That would mean there is 1 person in the world who learned it later?
(Okay, not that funny, I'll keep my day-job)

>
> >If you're off a half-step, then it's not AP. My brother can, my
> >sister can't. My dad isn't a musician. My mom tends to be off.
>
> It's not genetic. There may be genetic factors, and there's a
> study my friend at UCSD is loosely involved in to determine them.
> But the authors wrongly fall (in my opinion) for the "born with
> it" chestnut. Their own study should disavow them of that, if
> they do it right.
>
> >> I think it's based on whatever you practice RP with. If you
> >> use 12-tET, all the intervals (and especially the most commonly
> >> used ones) are close to something just.
> >
> >I'll take your word for it. I haven't tried it too much.
>
> Some APers wind up using AP as a crutch, and therefore don't
> develop RP as strongly as they might have. If you develop
> both skills fully, you can very much kick ass at music.

Actually my 12-tET RP is very strong. It was testing microtonal
RP that was a little off:) Of course AP overrides RP unless one
transposes. And yes, I do "kick ass" in music. Just wish I did in
math as well!

"I can't make you a great dancer. I don't even think I can even make
you a good dancer. But I can make you a better (ital) dancer"

- Bob Fosse, "And All that Jazz"

PGH

🔗Carl Lumma <ekin@lumma.org>

At 01:19 PM 5/26/2007, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>>
>> >> You seem to be implying that AP has something to do with 12-tET.
>> >
>> >* It does for me!!! It's stranger than you think
>>
>> But this is because you learned AP in a 12-tET world.
>>
>> >> There's no generalized keyboard with a weighted action in
>> >> existence, or even on the drawing board. However, one might
>> >> ask what importance "weight" has. . .
>> >> I had always taken the belief that it is the *linking* of
>> >> action events with sound events that makes the piano so
>> >> expressive. But a pianist friend of mine thinks the human
>> >> musculature needs some resistance to work against to get
>> >> full dynamic range. That may be true, but then again, it
>> >> might not. . .
>> >> Anyway, the Microzone and the Terpstra keyboard should both
>> >> have action->sound linking far superior to halberstadt MIDI
>> >> controllers in wide use today. The Microzone does it with
>> >> aftertouch pressure, the Terpstra with velocity sensing.
>> >
>> >* Cool. I'll have to buy something, once I save up $$
>>
>> I'm recommending Terpstra. Last I heard it was $8K in
>> batches of two. Wanna save together?
>>
>> By the time we amass that kind of cash, Vandervoort's
>> "Daskin" keyboard might be out, and hopefully at a lower
>> cost.
>
>** Well I was going to buy a bike first. I just started a savings
>account...

I really want a bike, too. And a piano.

>> >Yes, well, I'm out on a limb, this newsgroup is out on a limb
>> >anyway, so I guess it's a limb on a limb, M12 and M24 have to
>> >do with Steiner systems, which for example, tie hexachords to
>> >pentachords, I am hoping to tie musical set theory into tuning
>> >theory. I am also equating M12 with 12-tET and M24 with 24-tET
>> >12 and 24 are important, look at John Baez's This Week in
>> >Mathematical Physics Week 234 for example. Hard to explain in
>> >English since I am still learning it myself! But John is good.
>>
>> Oh, musical set theory. Yeah, I don't think it's important.
>> But if you can get those goons to do something microtonal by
>> publishing about the link, more power to you.
>
>* Wow! But musical composition is nothing more than combinations
>of different notes! At least classical/jazz is.

Notes are an abstraction. A classical composition is a
temporal combination of notes, phrasing, dynamics, and tempo
indications, but the actual musical sounds are much more
than that.
Electronic music is no different, except the notes abstraction
may or may not have been used at some stage. Actually in
jazz the notes abstraction typically isn't used, though it can
be easier to reverse engineer one from a recording than with
the most extreme electronic music. By easier, we're still
talking about something that computers can't yet do.

>(Who are the "goons?" Watch out making fun of us big guys:))

Musical set theorists. Unfortunately I place the field with
lit. crit. in terms of utility, and the unique way in which
they survive entirely within academia, with little to no economic
interaction with the real world.

>That would mean there is 1 person in the world who learned it later?
>(Okay, not that funny, I'll keep my day-job)

Studies have shown that some AP skills can be learned by
college-age people.

-Carl

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> At 01:19 PM 5/26/2007, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@> wrote:
> >>
> >> >> You seem to be implying that AP has something to do with 12-
tET.
> >> >
> >> >* It does for me!!! It's stranger than you think
> >>
> >> But this is because you learned AP in a 12-tET world.
> >>
> >> >> There's no generalized keyboard with a weighted action in
> >> >> existence, or even on the drawing board. However, one might
> >> >> ask what importance "weight" has. . .
> >> >> I had always taken the belief that it is the *linking* of
> >> >> action events with sound events that makes the piano so
> >> >> expressive. But a pianist friend of mine thinks the human
> >> >> musculature needs some resistance to work against to get
> >> >> full dynamic range. That may be true, but then again, it
> >> >> might not. . .
> >> >> Anyway, the Microzone and the Terpstra keyboard should both
> >> >> have action->sound linking far superior to halberstadt MIDI
> >> >> controllers in wide use today. The Microzone does it with
> >> >> aftertouch pressure, the Terpstra with velocity sensing.
> >> >
> >> >* Cool. I'll have to buy something, once I save up $$
> >>
> >> I'm recommending Terpstra. Last I heard it was $8K in
> >> batches of two. Wanna save together?
> >>
> >> By the time we amass that kind of cash, Vandervoort's
> >> "Daskin" keyboard might be out, and hopefully at a lower
> >> cost.
> >
> >** Well I was going to buy a bike first. I just started a savings
> >account...
>
> I really want a bike, too. And a piano.

I have a Kawai Grand (Medium) KG-52. My revenge is that I have one,
and my professional pianist friend doesn't. Took awhile to pay for!
(We are straying a little from tuning discussions...) Of course he
plays concertos with orchestras, and I don't. Bikes are expensive
these days, but nice. Maybe I will just fix up the 10-speed I have
so I can get a microtonal keyboard. I see. Batches of two. I didn't
think we would share it. Cuz I live in St. Paul. My Roland is in
someone's basement right now. Okay now it's definitely become a chat-
room exchange...
>
> >> >Yes, well, I'm out on a limb, this newsgroup is out on a limb
> >> >anyway, so I guess it's a limb on a limb, M12 and M24 have to
> >> >do with Steiner systems, which for example, tie hexachords to
> >> >pentachords, I am hoping to tie musical set theory into tuning
> >> >theory. I am also equating M12 with 12-tET and M24 with 24-tET
> >> >12 and 24 are important, look at John Baez's This Week in
> >> >Mathematical Physics Week 234 for example. Hard to explain in
> >> >English since I am still learning it myself! But John is good.
> >>
> >> Oh, musical set theory. Yeah, I don't think it's important.
> >> But if you can get those goons to do something microtonal by
> >> publishing about the link, more power to you.
> >
> >* Wow! But musical composition is nothing more than combinations
> >of different notes! At least classical/jazz is.
>
> Notes are an abstraction. A classical composition is a
> temporal combination of notes, phrasing, dynamics, and tempo
> indications, but the actual musical sounds are much more
> than that.
> Electronic music is no different, except the notes abstraction
> may or may not have been used at some stage. Actually in
> jazz the notes abstraction typically isn't used, though it can
> be easier to reverse engineer one from a recording than with
> the most extreme electronic music. By easier, we're still
> talking about something that computers can't yet do.
>
> >(Who are the "goons?" Watch out making fun of us big guys:))
>
> Musical set theorists. Unfortunately I place the field with
> lit. crit. in terms of utility, and the unique way in which
> they survive entirely within academia, with little to no economic
> interaction with the real world.

Wrong wrong wrong, even though I understand your perspective (I went
to the SMT conference, some of these guys are corpses above ground:))
Oh, but I should add that some were brilliant. Okay, a little too
frank. The trouble is some profs don't apply the right math, or
don't apply it correctly. Music theory and math together can be
absolute TNT. The creativity of music and poetry spring from the
same source, and they both can bring about great leaps in knowledge
(in math, physics, whatever! Creativity is creativity). I talk big,
but I am just crawling as a mathematician:)
>
> >That would mean there is 1 person in the world who learned it
later?
> >(Okay, not that funny, I'll keep my day-job)
>
> Studies have shown that some AP skills can be learned by
> college-age people.

I don't believe it. Music-ed majors have rocks in their head. Some
can't even play "Happy Birthday" without the sheet-music. Of course,
neither could my full-tenured full-prof piano teacher. Sorry Dr. G.
Regarding Music-ed majors: Many fail at ear-training/sight-singing.
The college students you talk about probably are not music majors. I
can't imagine learning it that late though. I've had a bit of wine,
as you can tell, so I probably should shut up now:)

Any song any key I can do it by ear always could

Cheers,

PGH
PGH
>

🔗Carl Lumma <ekin@lumma.org>

>I have a Kawai Grand (Medium) KG-52. My revenge is that I have one,
>and my professional pianist friend doesn't. Took awhile to pay for!

I have an antique baby grand, but no space to put it. So it's at
my parents' house in another State. I've borrowed and rented
uprights in its absence. I'm thinking about buying a Yamaha upright
at the moment.

>(We are straying a little from tuning discussions...) Of course he
>plays concertos with orchestras, and I don't. Bikes are expensive
>these days, but nice.

I finally figured out what I want, and it came to $7K. I had a
$4K road bike circa 2000, which I got for a song because I worked
for a bike shop. I sold it for a modest profit when I moved to
California, but now I'm spoiled and affordable bikes don't do it
for me.

>Batches of two. I didn't
>think we would share it. Cuz I live in St. Paul.

Just to be clear, that's $8K each if you buy them 2 at a time.

>> Musical set theorists. Unfortunately I place the field with
>> lit. crit. in terms of utility, and the unique way in which
>> they survive entirely within academia, with little to no economic
>> interaction with the real world.
>
>Wrong wrong wrong, even though I understand your perspective (I went
>to the SMT conference, some of these guys are corpses above ground:))
>Oh, but I should add that some were brilliant. Okay, a little too
>frank. The trouble is some profs don't apply the right math, or
>don't apply it correctly. Music theory and math together can be
>absolute TNT. The creativity of music and poetry spring from the
>same source, and they both can bring about great leaps in knowledge
>(in math, physics, whatever! Creativity is creativity). I talk big,
>but I am just crawling as a mathematician:)

It's not that clever and hard math isn't being used. It's
that it's being used in waste.

>> Studies have shown that some AP skills can be learned by
>> college-age people.
>
>I don't believe it. Music-ed majors have rocks in their head. Some
>can't even play "Happy Birthday" without the sheet-music. Of course,
>neither could my full-tenured full-prof piano teacher. Sorry Dr. G.
>Regarding Music-ed majors: Many fail at ear-training/sight-singing.
>The college students you talk about probably are not music majors. I
>can't imagine learning it that late though. I've had a bit of wine,
>as you can tell, so I probably should shut up now:)

I have two citations if you ever become open to the possibility.

Mind you, fundamental AP skills are widespread in the population.
It's conscious/linguistic access to the skill that most people
don't have.

-Carl

🔗Paul G Hjelmstad <phjelmstad@msn.com>

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

> I have an antique baby grand, but no space to put it. So it's at
> my parents' house in another State. I've borrowed and rented
> uprights in its absence. I'm thinking about buying a Yamaha
upright
> at the moment.

> I finally figured out what I want, and it came to $7K. I had a
> $4K road bike circa 2000, which I got for a song because I worked
> for a bike shop. I sold it for a modest profit when I moved to
> California, but now I'm spoiled and affordable bikes don't do it
> for me.
>
> >Batches of two. I didn't
> >think we would share it. Cuz I live in St. Paul.
>
> Just to be clear, that's $8K each if you buy them 2 at a time.

Ouch. I need a car soon too.

> It's not that clever and hard math isn't being used. It's
> that it's being used in waste.

Exactly. Like the systems (math or otherwise) in so much work in
theology, philosophy, etc. BTW, have you seen "Conservapedia" OMG,
Scary! Perhaps "It's all nonsense" as a mathematician said on
sci.math, but I think it is important to find meaning that is
understandable by a wide audience. I think it is exciting, now, that
the walls between classical and pop are coming down, and that rock
(maybe unfortunately!) is respectable, and rock musicians are
studying in the nation's best schools.

> >> Studies have shown that some AP skills can be learned by
> >> college-age people.

> I have two citations if you ever become open to the possibility.
>
> Mind you, fundamental AP skills are widespread in the population.
> It's conscious/linguistic access to the skill that most people
> don't have.

First, I want to retract my comment about music-ed majors. True,
some aren't very talented, or they chose that major because they
didn't think they could do anything else. (I don't mean to rag
on music-ed, I went overboard, it depends on the class, the school,
etc. So my bad on that.) Of course teaching music in secondary
schools in our society at this time is real tough.

It's funny, even with non-musician people, they will often hum
a pop tune they heard, in the same exact key they listened to it in.
So you are right in that respect. They just don't "know" it's G
major for example. So maybe instead of saying "Sing a G" say,
sing the first note of "Stairway to Heaven" etc. I guess that counts.

If you have a chance, look up Steiner Systems, and M12 for starters.
Of course, this is math, I am just applying it to music theory.

I wish I could win you over to musical set-theory. I showed some of
this stuff to Paul E. and he thought {perhaps) I am on to something.
When I finish my paper "The Little Book of Hexachord Theory" I'll
put it in the Files section. Let me know what you think. I use
hexagrams (I-Ching) and various methods (D4 X S3) to classify
hexachords. I can take criticism, or even a joke.

PGH

🔗Carl Lumma <ekin@lumma.org>

>> Mind you, fundamental AP skills are widespread in the population.
>> It's conscious/linguistic access to the skill that most people
>> don't have.
//
>It's funny, even with non-musician people, they will often hum
>a pop tune they heard, in the same exact key they listened to it in.

Yup - it's been studied.

>So you are right in that respect. They just don't "know" it's G
>major for example. So maybe instead of saying "Sing a G" say,
>sing the first note of "Stairway to Heaven" etc. I guess that counts.

Exactly.

>I wish I could win you over to musical set-theory. I showed some of
>this stuff to Paul E. and he thought {perhaps) I am on to something.
>When I finish my paper "The Little Book of Hexachord Theory" I'll
>put it in the Files section. Let me know what you think. I use
>hexagrams (I-Ching) and various methods (D4 X S3) to classify
>hexachords. I can take criticism, or even a joke.

Sure, I'll check it out. In the meantime I'd lio to hear some
music. I've tried Babbitt and all that, and it's neat but really,
I don't think the fact that you're modulating from one symmetry
group to another or whatever makes any difference in the music.

I'd love to be convinced otherwise. Can you cite a piece of music
where you think the effects of Steiner systems are audible?

-Carl

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> Mind you, fundamental AP skills are widespread in the
population.
> >> It's conscious/linguistic access to the skill that most people
> >> don't have.
> //
> >It's funny, even with non-musician people, they will often hum
> >a pop tune they heard, in the same exact key they listened to it
in.
>
> Yup - it's been studied.
>
> >So you are right in that respect. They just don't "know" it's G
> >major for example. So maybe instead of saying "Sing a G" say,
> >sing the first note of "Stairway to Heaven" etc. I guess that
counts.
>
> Exactly.
>
> >I wish I could win you over to musical set-theory. I showed some
of
> >this stuff to Paul E. and he thought {perhaps) I am on to
something.
> >When I finish my paper "The Little Book of Hexachord Theory" I'll
> >put it in the Files section. Let me know what you think. I use
> >hexagrams (I-Ching) and various methods (D4 X S3) to classify
> >hexachords. I can take criticism, or even a joke.
>
> Sure, I'll check it out. In the meantime I'd lio to hear some
> music. I've tried Babbitt and all that, and it's neat but really,
> I don't think the fact that you're modulating from one symmetry
> group to another or whatever makes any difference in the music.

Simple answer: Going from major to minor. That's mirror inversion.
C4 X C3, is used exensively in Russian Romantic music (they gave us
real chromaticism, finally breaking from the old German diatonicism)
>
> I'd love to be convinced otherwise. Can you cite a piece of music
> where you think the effects of Steiner systems are audible?

Of course Steiner Systems are something in themselves, the symmetry
groups I have been working with relate to the M5 symmetry, among
other things, with M5, as I said, the tritone relation (used in jazz
chord subs) comes to mind. Messiaen and Boulez do all sorts of
things with math, but in their own way of course.

Still trying to find a single Steiner system that is like Dr. Elkies
double one. Finding all 66 x 12 pentachords from a mere 11
hexachords is facinating to me. Since septachords are just
complements of pentachords, you also find all 7 note scales this way
too. Definitely a WIP

>
> -Carl
>

🔗Carl Lumma <ekin@lumma.org>

>Simple answer: Going from major to minor. That's mirror inversion.

It's one case of mirror inversion. Do all mirror inversions have
the same musical effect as the major-minor transition?

>C4 X C3, is used extensively in Russian Romantic music (they gave us
>real chromaticism, finally breaking from the old German diatonicism)

Can you name a Russian Romantic piece you feel uses the chromatic
scale as a gestalt scale? And then perhaps show a dullard like
me wher ethe C4 X C3 is in it?

>> I'd love to be convinced otherwise. Can you cite a piece of music
>> where you think the effects of Steiner systems are audible?
>
>Of course Steiner Systems are something in themselves, the symmetry
>groups I have been working with relate to the M5 symmetry, among
>other things, with M5, as I said, the tritone relation (used in jazz
>chord subs) comes to mind.

Is M5 also the tritone relation in 22-ET?

>Messiaen and Boulez do all sorts of
>things with math, but in their own way of course.

I don't think we should mention Boulez in the same sentence with
Messiaen, but that's another story. :)

>Still trying to find a single Steiner system that is like Dr. Elkies
>double one. Finding all 66 x 12 pentachords from a mere 11
>hexachords is facinating to me. Since septachords are just
>complements of pentachords, you also find all 7 note scales this way
>too. Definitely a WIP

I understand the usefulness of finding all 6-tone scales
(what you call hexachords I think) in a larger scale. And of
course you only want to report one mode of each scale (report
only one cyclic permutation for each scale, I think one
might say). But then IIRC you are eliminating even more
stuff in a way I don't agree is musically warranted.

-Carl

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

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

> I'd love to be convinced otherwise. Can you cite a piece of music
> where you think the effects of Steiner systems are audible?

It's very hard for them to be audible because they are large (compared
to the symmetric group) and multiply transitive. M12 (and M24) are 5-
transitive groups. This means given any five pitch classes in a
prescibed order, there is a group element mapping it to any other five
pitch classes in a prescribed order. The groups will therefore erase
most structure. It is precisely this property which is so unusual--
aside from the symmetric and alternating groups (which erase structure
even more ruthlessly) the only 5-transitive groups are M12 and M24, and
the only 4-transitive the previously named groups and M11 and M23.

To get audible effects, *small* groups are the thing. Cyclic and
dihedral groups are audible and (especially cyclic groups) constantly
used. This suggests trying a slightly larger group which is not too
large. But good luck trying to replace fifths with major sevenths.

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@>
> wrote:
>
> > Interesting. BTW, my study of M12 and M24 and other Groups
> > has lead me to believe that both 12-tET and 24-tET are very very
> > important. 24 even ties into the RZF per John Baez, of course
> > I would defer to Gene before making any claims about RZH/RZF
>
> They are interesting from the point of view of permuation groups,
but
> why does that make them important?

Everything's important:) No, seriously, it's because of how I think of
music. Scales, modulations, mode mixture, etc. Also I think it's
pretty how you can find all the pentachords from just 11 hexachords,
and their complements, inverses, and transpositions. I'm making
headway with SPLAG and some other articles, obviously I'm just getting
started...A lot of modern music, even rock now, uses different
combinations/permutations of notes, plus of course the 4 chord loop
thing that seems to be what everything is based on. Just kidding,
some rock has more than one loop per song:) Also Dr. Elkies DSS
is neatly segmented into 12 transpositions, which is nice for
pentachords anyway cuz they divide out this way (C(12,5)/12)

Let me go to Carl's message for more comments..
>

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >Simple answer: Going from major to minor. That's mirror inversion.
>
> It's one case of mirror inversion. Do all mirror inversions have
> the same musical effect as the major-minor transition?

No, but that's the main one. C7 is the inverse of Chd7, etc.
>
> >C4 X C3, is used extensively in Russian Romantic music (they gave
us
> >real chromaticism, finally breaking from the old German
diatonicism)
>
> Can you name a Russian Romantic piece you feel uses the chromatic
> scale as a gestalt scale? And then perhaps show a dullard like
> me wher ethe C4 X C3 is in it?

It's how they do the "third relation" I'll have to find specific
examples, I'm thinking the "Might Five" and others of that time.
Rachmaninoff pretty much distilled Russian Romanticism and brought
it to it's culmination. Also diminished chord/ augmented chord
manipulations (Arrow-up Arrow-down Arrow-across) etc. Sorry not
much of an answer right now...
>
> >> I'd love to be convinced otherwise. Can you cite a piece of
music
> >> where you think the effects of Steiner systems are audible?
> >
> >Of course Steiner Systems are something in themselves, the
symmetry
> >groups I have been working with relate to the M5 symmetry, among
> >other things, with M5, as I said, the tritone relation (used in
jazz
> >chord subs) comes to mind.
>
> Is M5 also the tritone relation in 22-ET?

You chose a good one! It works in 22. In 12-tET, just multiply each
step by 7 (mod 12), so even ones are fixed and odds swap the
tritone. In 22, multiply by 12 (mod 22) and get 0, 12, 2, 14, 4, 16,
6, 18, 8, 20, 10, 0, 12 ... well almost worked. I think this does
the moebius strip thing or something (it's pretty late)

> >Messiaen and Boulez do all sorts of
> >things with math, but in their own way of course.
>
> I don't think we should mention Boulez in the same sentence with
> Messiaen, but that's another story. :)
>
> >Still trying to find a single Steiner system that is like Dr.
Elkies
> >double one. Finding all 66 x 12 pentachords from a mere 11
> >hexachords is facinating to me. Since septachords are just
> >complements of pentachords, you also find all 7 note scales this
way
> >too. Definitely a WIP
>
> I understand the usefulness of finding all 6-tone scales
> (what you call hexachords I think) in a larger scale. And of
> course you only want to report one mode of each scale (report
> only one cyclic permutation for each scale, I think one
> might say). But then IIRC you are eliminating even more
> stuff in a way I don't agree is musically warranted.

Actually, you have this backwards. I am finding penta/septachords
from hexachords, not vice versa. But it got me thinking, that any 7
note scale will contain 7 hexads. The segmentation is more or less
aesthetic, I like when things subdivide nicely. It creates meaning,
doesn't really eliminate anything, just makes it orderly. I still
need to find the right kind of SSS. (Single Steiner System.)

Septachord/Pentachord complementation is part of Forte's weakly-
related complices and also relate to SS, I've posted on this, and
need to strengthen these ideas somewhat:)

PGH

>
> -Carl
>

🔗Carl Lumma <ekin@lumma.org>

>> >Simple answer: Going from major to minor. That's mirror inversion.
>>
>> It's one case of mirror inversion. Do all mirror inversions have
>> the same musical effect as the major-minor transition?
>
>No, but that's the main one.

Maybe what's important about it isn't the mirror inversion
part?

>C7 is the inverse of Chd7, etc.

In 12-ET it is. Not in JI.

>> Can you name a Russian Romantic piece you feel uses the chromatic
>> scale as a gestalt scale? And then perhaps show a dullard like
>> me wher ethe C4 X C3 is in it?
>
>It's how they do the "third relation" I'll have to find specific
>examples, I'm thinking the "Might Five" and others of that time.
>Rachmaninoff pretty much distilled Russian Romanticism and brought
>it to it's culmination. Also diminished chord/ augmented chord
>manipulations (Arrow-up Arrow-down Arrow-across) etc. Sorry not
>much of an answer right now...

np. Would love to hear an example of this.

>> Is M5 also the tritone relation in 22-ET?
>
>You chose a good one! It works in 22. In 12-tET, just multiply each
>step by 7 (mod 12), so even ones are fixed and odds swap the
>tritone. In 22, multiply by 12 (mod 22) and get 0, 12, 2, 14, 4, 16,
>6, 18, 8, 20, 10, 0, 12 ... well almost worked. I think this does
>the moebius strip thing or something (it's pretty late)

I expected it might work in any even ET, but it looks like it
doesn't (?). Even ETs can temper out 50/49, so I didn't think
it would be fair to ask about 41-ET or something, because
while I would argue this is still tritone substitution, tempering
out 50/49 does give rise to a particular thing.

>> I understand the usefulness of finding all 6-tone scales
>> (what you call hexachords I think) in a larger scale. And of
>> course you only want to report one mode of each scale (report
>> only one cyclic permutation for each scale, I think one
>> might say). But then IIRC you are eliminating even more
>> stuff in a way I don't agree is musically warranted.
>
>Actually, you have this backwards. I am finding penta/septachords
>from hexachords, not vice versa. But it got me thinking, that any 7
>note scale will contain 7 hexads.

A "hexad" is usually 4:5:6:7:9:11 (or its inverse) around
here... you just mean combinations 7 choose 6? Yup, I think
that's 7.

>The segmentation is more or less
>aesthetic, I like when things subdivide nicely. It creates meaning,
>doesn't really eliminate anything, just makes it orderly. I still
>need to find the right kind of SSS. (Single Steiner System.)

SSS was my suggestion for "super-saturated suspension" chord.
I think Graham's ASS (anomalous saturated suspension) stuck,
however.

-Carl

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> >Simple answer: Going from major to minor. That's mirror
inversion.
> >>
> >> It's one case of mirror inversion. Do all mirror inversions have
> >> the same musical effect as the major-minor transition?
> >
> >No, but that's the main one.
>
> Maybe what's important about it isn't the mirror inversion
> part?

?? All forms of symmetry are important:) It leads to 26-dimensional
bosonic string theory. Haha

>
> >C7 is the inverse of Chd7, etc.
>
> In 12-ET it is. Not in JI.

Yes it is, just take reciprocals.

> >> Can you name a Russian Romantic piece you feel uses the chromatic
> >> scale as a gestalt scale? And then perhaps show a dullard like
> >> me wher ethe C4 X C3 is in it?
> >
> >It's how they do the "third relation" I'll have to find specific
> >examples, I'm thinking the "Might Five" and others of that time.
> >Rachmaninoff pretty much distilled Russian Romanticism and brought
> >it to it's culmination. Also diminished chord/ augmented chord
> >manipulations (Arrow-up Arrow-down Arrow-across) etc. Sorry not
> >much of an answer right now...
>
> np. Would love to hear an example of this.

Trouble with the Internet is I surf a lot, and don't keep track
of where I read things. Sometimes you never find things again....
>
> >> Is M5 also the tritone relation in 22-ET?
> >
> >You chose a good one! It works in 22. In 12-tET, just multiply
each
> >step by 7 (mod 12), so even ones are fixed and odds swap the
> >tritone. In 22, multiply by 12 (mod 22) and get 0, 12, 2, 14, 4,
16,
> >6, 18, 8, 20, 10, 0, 12 ... well almost worked. I think this does
> >the moebius strip thing or something (it's pretty late)
>
> I expected it might work in any even ET, but it looks like it
> doesn't (?). Even ETs can temper out 50/49, so I didn't think
> it would be fair to ask about 41-ET or something, because
> while I would argue this is still tritone substitution, tempering
> out 50/49 does give rise to a particular thing.

Yes, exactly. Paul E and I were discussing this, I need to look at
some emails. So 22 does work, you get two cycles 0, 12, 2, 14, 4, 16,
6, 18, 8, 20, 10, 0, 12, 2, 14, 4, 16, 6, 18, 8, 20, 10. But it's
not much use, because I don't believe this ties into anything like
D4 X S3 does for 12-tET, (11 and 2 are the only prime factors,
so D11 X S2? Hmmm!, also with 12 it breaks into four parts, based
on the symmetry of the square and triangle FF, FB, BF, and BB)

> >> I understand the usefulness of finding all 6-tone scales
> >> (what you call hexachords I think) in a larger scale. And of
> >> course you only want to report one mode of each scale (report
> >> only one cyclic permutation for each scale, I think one
> >> might say). But then IIRC you are eliminating even more
> >> stuff in a way I don't agree is musically warranted.
> >
> >Actually, you have this backwards. I am finding penta/septachords
> >from hexachords, not vice versa. But it got me thinking, that any
7
> >note scale will contain 7 hexads.
>
> A "hexad" is usually 4:5:6:7:9:11 (or its inverse) around
> here... you just mean combinations 7 choose 6? Yup, I think
> that's 7.

Well, it's two more things. Hexads in Steiner System S(5,6,12) are
the 132 Steiner blocks with 6 elements such that every pentad
fits into one and only one block. Hexachords as I am using them
are usually 924 (C(12,6)) Here I was indeed talking about C(7,6)
as you mentioned where 7 is C(12,7) so I guess it's C(12,7)* 7
Good Grief.

> >The segmentation is more or less
> >aesthetic, I like when things subdivide nicely. It creates
meaning,
> >doesn't really eliminate anything, just makes it orderly. I still
> >need to find the right kind of SSS. (Single Steiner System.)
>
> SSS was my suggestion for "super-saturated suspension" chord.
> I think Graham's ASS (anomalous saturated suspension) stuck,
> however.

Cute. Our geometry teacher told us Angle-Side-Side isn't a rule,
cuz you don't like the word it spells! (Ninth Grade, some things
stick in your memory)

PGH

🔗Carl Lumma <ekin@lumma.org>

>> >C7 is the inverse of Chd7, etc.
>>
>> In 12-ET it is. Not in JI.
>
>Yes it is, just take reciprocals.

Oh, right, you could call the utonal 7th a half-dim 7th.
But I think 5:6:7:9 is a better story about this chord.

>> >> Can you name a Russian Romantic piece you feel uses the chromatic
>> >> scale as a gestalt scale? And then perhaps show a dullard like
>> >> me wher ethe C4 X C3 is in it?
>> >
>> >It's how they do the "third relation" I'll have to find specific
>> >examples, I'm thinking the "Might Five" and others of that time.
>> >Rachmaninoff pretty much distilled Russian Romanticism and brought
>> >it to it's culmination. Also diminished chord/ augmented chord
>> >manipulations (Arrow-up Arrow-down Arrow-across) etc. Sorry not
>> >much of an answer right now...
>>
>> np. Would love to hear an example of this.
>
>Trouble with the Internet is I surf a lot, and don't keep track
>of where I read things. Sometimes you never find things again....

-Carl

🔗Paul G Hjelmstad <phjelmstad@msn.com>

PGH:

> Still trying to find a single Steiner system that is like Dr. Elkies
> double one. Finding all 66 x 12 pentachords from a mere 11
> hexachords is facinating to me. Since septachords are just
> complements of pentachords, you also find all 7 note scales this way
> too. Definitely a WIP

As soon as I have a little program written, I should be able to find
a modulo-11 labelled Steiner System. Alas, it does not look like it
will be 22 sets at even tranpositions (I already have more than 22
different hexads, so it's not that kind of SSS)

Dr. Elkies told me to start with any set, so why not use the quadratic
residue one, I guess, and then break all associations in the remaining
10. Harder than it sounds! That is, no pentad can be in any two sets.
Also assuming that you get 11 + 11, perhaps as inverses of each other
as in his DSS. Oops, but then some transpose of a symmetrical set would
be in the inverse of a set too. Back to the drawing board.

But for today, I was playing with (0,1,3,4,5,9) and finding different 5-
limit expressions for it. Assuming 5^3=identity, I got this one, which
is curiously half a step from the original:

(11,0,2,3,4,8) based on 15/8, 1/1, 9/8, 6/5, 5/4, 8/5. It's the
prettiest, and forms a nice little circle in a powers grid

X X
X X
X X

So now using just two loops a SSS can be formed. (y=y+1 mod 11,
y=-1/y mod 11, projective of course). It will be fun to see what
patterns the 132 sets form. BTW, shuffle labelling doesn't give me the
right SSS either. Oh well.

PGH

🔗Paul G Hjelmstad <phjelmstad@msn.com>

PGH:

> Still trying to find a single Steiner system that is like Dr. Elkies
> double one. Finding all 66 x 12 pentachords from a mere 11
> hexachords is facinating to me. Since septachords are just
> complements of pentachords, you also find all 7 note scales this way
> too. Definitely a WIP

But for today, I was playing with (0,1,3,4,5,9) and finding different 5-
limit expressions for it. Assuming 5^3=identity, I got this one, which
is curiously half a step from the original:

(11,0,2,3,4,8) based on 15/8, 1/1, 9/8, 6/5, 5/4, 8/5. It's the
prettiest, and forms a nice little circle in a powers grid

X X
X X
X X

PGH

🔗Paul G Hjelmstad <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> PGH:
>
> > Still trying to find a single Steiner system that is like Dr.
Elkies
> > double one. Finding all 66 x 12 pentachords from a mere 11
> > hexachords is facinating to me. Since septachords are just
> > complements of pentachords, you also find all 7 note scales this
way
> > too. Definitely a WIP
>
> As soon as I have a little program written, I should be able to find
> a modulo-11 labelled Steiner System. Alas, it does not look like it
> will be 22 sets at even tranpositions (I already have more than 22
> different hexads, so it's not that kind of SSS)
>
> Dr. Elkies told me to start with any set, so why not use the
quadratic
> residue one, I guess, and then break all associations in the
remaining
> 10. Harder than it sounds! That is, no pentad can be in any two
sets.
> Also assuming that you get 11 + 11, perhaps as inverses of each
other
> as in his DSS. Oops, but then some transpose of a symmetrical set
would
> be in the inverse of a set too. Back to the drawing board.
>
> But for today, I was playing with (0,1,3,4,5,9) and finding
different 5-
> limit expressions for it. Assuming 5^3=identity, I got this one,
which
> is curiously half a step from the original:
>
> (11,0,2,3,4,8) based on 15/8, 1/1, 9/8, 6/5, 5/4, 8/5. It's the
> prettiest, and forms a nice little circle in a powers grid
>
> X X
> X X
> X X
>
> So now using just two loops a SSS can be formed. (y=y+1 mod 11,
> y=-1/y mod 11, projective of course). It will be fun to see what
> patterns the 132 sets form. BTW, shuffle labelling doesn't give me
the
> right SSS either. Oh well.
>
> PGH

Let's insert an O:

XX
XOX
XX