Pitch Class and Generators

>> 5 0 7 2 9 4 11 reordered 0 2 4 5 7 9 11 (0, has no points at which 3
>> adjacent PCs lie together.
>
> What do you mean by a "PC"?
>

A Pitch-Class, defined as a class that contains all instances of a pitch
related by enhamonicity and octave equivalence. It also represents one of
the equivalence classes to be found in the homomorphic mapping:

N ^(jn+k) -> k (mod n)

Generators I have assumed is as follows:

assume three equivalence classes a b c lie linearly adjacent, such that the
Next predicate follows this sequence

b = Next (a)
c = Next (b)

thus c = Next(Next(a)) or Next^2(a)

If we (after Balzano), abbreviate Next to N,

we can write N^i, representinh powers of N, where i is an integer mod n.

A generator for Cn is an N^i, such that successive applications of
N^i to the starting tone will generate the full Cn.

For any n i=1 is a generator, as is i = -1

For C12, i = 5 and its mod 12 complement 7 are generators.

Where n is prime i = 1..n-1 (i.e any non-zero value of i, mod n) are all
generators

That's my definition too.

Mark

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:

> Generators I have assumed is as follows:
>
> assume three equivalence classes a b c lie linearly adjacent, such that the
> Next predicate follows this sequence
>
> b = Next (a)
> c = Next (b)
>
> thus c = Next(Next(a)) or Next^2(a)
>
> If we (after Balzano), abbreviate Next to N,
>
> we can write N^i, representinh powers of N, where i is an integer mod n.

This is a little strange, though mathematically correct. Normally
you would write the cyclic group of order 12 additively unless it was something coming from a multiplicative structure, such as the multiplicative group of nonzero mod 13 classes. Writing it additively,
we can indentify "N" with 1, and then b=a+1, c=b+1. Then N^i becomes
simply i, and N^(jn+k) is jn+k. This raises the question of whether "j" is a generator for C12--which is the same as asking if j is relatively prime to 12.

> A generator for Cn is an N^i, such that successive applications of
> N^i to the starting tone will generate the full Cn.

Or we could say, a generator for Cn is i, such that (i,n)=1, meaning
i is relatively prime to n. It is an "n-unit"; an invertible element of Cn, taken as a ring.

> That's my definition too.

That's one usage; in general, a group generator for a finitely generated abelian group can mean more kinds of things, some of them
not involving groups with torsion, and actually that's what's been discussed the most around here recently.

In-Reply-To: <B8C4883F.38C4%mark.gould@argonet.co.uk>
Mark Gould wrote:

> A generator for Cn is an N^i, such that successive applications of
> N^i to the starting tone will generate the full Cn.
>
> For any n i=1 is a generator, as is i = -1
>
> For C12, i = 5 and its mod 12 complement 7 are generators.

But I thought the major and minor thirds were also the twin generators for
a diatonic scale. They don't qualify in 12-equal by this definition.

Graham

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:

My disguised followup has shown up yet, let's see if this works.

Has anyone noticed that followups to other postings have been appearing, but things using the "Post" command have not? This is a followup disguised by removing what it follows up to--let's see if it appears.

I've been posting a little about Hermite normal form, but if it doesn't appear, it slows the conversation down. How are other people doing?

From my recollection, I think the C12 group is isomorphic with relation to
two of its sub groups C3 and C4

like this:

(a,b) <--> (4a + 3b) mod 12

Every interval in C12 can thus be measured as from zero to two major thirds
(difference of 4, i.e. the C3 cycle), and zero to three minor thirds
(differece of 3, i.e. the C4 cycle)

This is plotted graphically as

0 4 8 0
9 1 5 9
6 10 2 6
3 7 11 3
0 4 8 0 (extends in all directions, but is in reality a torus)

And so we arrive at fig 5 from Balzano.

Thus the C3xC4 group contains generators of C12.

Mark

> Message: 6
> Date: Mon, 25 Mar 2002 11:34 +0000 (GMT)
> From: graham@microtonal.co.uk
> Subject: Re: Pitch Class and Generators
>
> In-Reply-To: <B8C4883F.38C4%mark.gould@argonet.co.uk>
> Mark Gould wrote:
>
>> A generator for Cn is an N^i, such that successive applications of
>> N^i to the starting tone will generate the full Cn.
>>
>> For any n i=1 is a generator, as is i = -1
>>
>> For C12, i = 5 and its mod 12 complement 7 are generators.
>
> But I thought the major and minor thirds were also the twin generators for
> a diatonic scale. They don't qualify in 12-equal by this definition.
>
>
> Graham

In-Reply-To: <B8C6611F.38DC%mark.gould@argonet.co.uk>
Mark Gould wrote:

> >From my recollection, I think the C12 group is isomorphic with
> relation to
> two of its sub groups C3 and C4

I don't know what that means. One thing though, the main significance of
3 and 4 wrt a diatonic is that the fourth and fifth are 3 and 4 diatonic
steps respectively. It's a coincidence that they happen to add up to the
number of steps to a fifth in 12-equal, and follows from the numbers of
steps to the generators also being the numbers of steps to the MOS
subsets. As this is also true of Balzano's 11/20 diatonic, perhaps that's
what he was doing with the group theory, and you rejected.

> like this:
>
> (a,b) <--> (4a + 3b) mod 12
>
> Every interval in C12 can thus be measured as from zero to two major
> thirds
> (difference of 4, i.e. the C3 cycle), and zero to three minor thirds
> (differece of 3, i.e. the C4 cycle)

To get every interval of a diatonic scale, with correct spelling, you need
a larger range. A semitone is an octave minus two major and one minor
thirds, or (4*-2 + 3*-1) mod 12. A tone is an octave minus two minor and
one major thirds, or (4*-1 + 3*-2) mod 12.

> This is plotted graphically as
>
>
> 0 4 8 0
> 9 1 5 9
> 6 10 2 6
> 3 7 11 3
> 0 4 8 0 (extends in all directions, but is in reality a torus)
>
> And so we arrive at fig 5 from Balzano.
>
> Thus the C3xC4 group contains generators of C12.

That must be a different definition of "generator" from

> >> A generator for Cn is an N^i, such that successive applications of
> >> N^i to the starting tone will generate the full Cn.
> >>
> >> For any n i=1 is a generator, as is i = -1
> >>
> >> For C12, i = 5 and its mod 12 complement 7 are generators.

because neither 3 nor 4 will generate the full 12. It's also different to
the way Gene talks of a pair of generators giving a linear temperament.
The tone is exactly half a major third, and so can't be got by combining
major and minor thirds. You have to add the octave to the full list of
generators, which gives a planar temperament. Or rather a planar scale,
because this is 5-limit JI with no tempering.

The tone and semitone also generate a diatonic scale. Carey and Clampitt
show how to transform [5 7] to [1 2].

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <B8C6611F.38DC%mark.gould@a...>
> Mark Gould wrote:

> > >From my recollection, I think the C12 group is isomorphic with
> > relation to
> > two of its sub groups C3 and C4

> I don't know what that means.

C12 is isomorphic to the direct product C3 x C4. It can be expressed
in terms of a single generator of order 12, but its subgroups can be
expressed in terms of generators of degree 3 and 4 respectively, and
these also generate C12. In fact, since 3 and 4 are relatively prime,
any integer can be expressed as a linear combination of 3s and 4s, so
any 2^(k/12) can be expressed as a product of major and minor 12-et
thirds, without the use of octaves.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., graham@m... wrote:
> > In-Reply-To: <B8C6611F.38DC%mark.gould@a...>
> > Mark Gould wrote:
>
> > > >From my recollection, I think the C12 group is isomorphic with
> > > relation to
> > > two of its sub groups C3 and C4
>
> > I don't know what that means.
>
> C12 is isomorphic to the direct product C3 x C4. It can be
expressed
> in terms of a single generator of order 12, but its subgroups can
be
> expressed in terms of generators of degree 3 and 4 respectively,
and
> these also generate C12. In fact, since 3 and 4 are relatively
prime,
> any integer can be expressed as a linear combination of 3s and 4s,
so
> any 2^(k/12) can be expressed as a product of major and minor 12-et
> thirds, without the use of octaves.

yes, but to claim (as balzano did) that the fundamental importance of
the diatonic scale hinges on this fact is to pull the wool over the
eyes of the numerically inclined reader. the fact is that around the
time of the emergence of tonality in diatonic music, many musicians
advocated a 19- or 31-tone system in which to embed the diatonic
scale, and 12 won out only because of convenience. it is only with
the work of late 19th century russian composers that the cycle-3 and
cycle-4 aspects of C12 became musically important.

in fact, the diatonic scale emerged over and over again around the
world without any 'chromatic universe' whatsoever, let alone an equal-
tempered one. the important properties of the diatonic scale must, i
feel, be found in the scale itself, in whatever tuning it may be
found (with reasonable allowances for the ear's ability to accept
small errors) -- any 'chromatic totality' considerations should wait
until, and be completely dependent upon, the establishment of the
fundamental 'diatonic' entity upon which the music is to be based.
this was my approach in my paper, and more recently, in my adaptation
of fokker's periodicity block theory.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> yes, but to claim (as balzano did) that the fundamental importance
of
> the diatonic scale hinges on this fact is to pull the wool over the
> eyes of the numerically inclined reader.

It's an incorrect assessment, I don't suppose it was intentionally
misleading.

the important properties of the diatonic scale must, i
> feel, be found in the scale itself, in whatever tuning it may be
> found (with reasonable allowances for the ear's ability to accept
> small errors) -- any 'chromatic totality' considerations should
wait
> until, and be completely dependent upon, the establishment of the
> fundamental 'diatonic' entity upon which the music is to be based.

Is it fundamental to the diatonic scale that 81/80~1, in your view?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > yes, but to claim (as balzano did) that the fundamental
importance
> of
> > the diatonic scale hinges on this fact is to pull the wool over
the
> > eyes of the numerically inclined reader.
>
> It's an incorrect assessment, I don't suppose it was intentionally
> misleading.

well, no, i think balzano managed to fool himself, and much of the
academic community in the process.

> > the important properties of the diatonic scale must, i
> > feel, be found in the scale itself, in whatever tuning it may be
> > found (with reasonable allowances for the ear's ability to accept
> > small errors) -- any 'chromatic totality' considerations should
> wait
> > until, and be completely dependent upon, the establishment of the
> > fundamental 'diatonic' entity upon which the music is to be
based.
>
> Is it fundamental to the diatonic scale that 81/80~1, in your view?

well, the diatonic scale is pretty strong already in the 3-limit,
where 80 of course doesn't even come into play. but yes, in the 5-
limit, whether the diatonic scale is in ji or tempered, 81/80 is one
of its defining unison vectors. as joe monzo might put it, the
smallness of 81/80 helps determine the 'finity' of the diatonic
scale -- if 81/80 were large (pardon the arithmetical counterfactual,
and don't take it too seriously), you'd tend to keep adding more
notes until you did hit up against a small unison vector.

--- In tuning-math@yahoogroups.com, "paulerlich" <paul@s...> wrote:

/tuning-math/message/3829

> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., graham@m... wrote:
> > > In-Reply-To: <B8C6611F.38DC%mark.gould@a...>
> > > Mark Gould wrote:
> >
> > > > >From my recollection, I think the C12 group is isomorphic
with
> > > > relation to
> > > > two of its sub groups C3 and C4
> >
> > > I don't know what that means.
> >
> > C12 is isomorphic to the direct product C3 x C4. It can be
> expressed
> > in terms of a single generator of order 12, but its subgroups can
> be
> > expressed in terms of generators of degree 3 and 4 respectively,
> and
> > these also generate C12. In fact, since 3 and 4 are relatively
> prime,
> > any integer can be expressed as a linear combination of 3s and
4s,
> so
> > any 2^(k/12) can be expressed as a product of major and minor 12-
et
> > thirds, without the use of octaves.
>
> yes, but to claim (as balzano did) that the fundamental importance
of
> the diatonic scale hinges on this fact is to pull the wool over the
> eyes of the numerically inclined reader. the fact is that around
the
> time of the emergence of tonality in diatonic music, many musicians
> advocated a 19- or 31-tone system in which to embed the diatonic
> scale, and 12 won out only because of convenience. it is only with
> the work of late 19th century russian composers that the cycle-3
and
> cycle-4 aspects of C12 became musically important.
>
> in fact, the diatonic scale emerged over and over again around the
> world without any 'chromatic universe' whatsoever, let alone an
equal-
> tempered one. the important properties of the diatonic scale must,
i
> feel, be found in the scale itself, in whatever tuning it may be
> found (with reasonable allowances for the ear's ability to accept
> small errors) -- any 'chromatic totality' considerations should
wait
> until, and be completely dependent upon, the establishment of the
> fundamental 'diatonic' entity upon which the music is to be based.
> this was my approach in my paper, and more recently, in my
adaptation
> of fokker's periodicity block theory.

***This is extremely interesting and echos Easley Blackwood's work, I
believe...

J. Pehrson

--- In tuning-math@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> --- In tuning-math@yahoogroups.com, "paulerlich" <paul@s...> wrote:
>
> /tuning-math/message/3829
>
>
> > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:
> > > --- In tuning-math@y..., graham@m... wrote:
> > > > In-Reply-To: <B8C6611F.38DC%mark.gould@a...>
> > > > Mark Gould wrote:
> > >
> > > > > >From my recollection, I think the C12 group is isomorphic
> with
> > > > > relation to
> > > > > two of its sub groups C3 and C4
> > >
> > > > I don't know what that means.
> > >
> > > C12 is isomorphic to the direct product C3 x C4. It can be
> > expressed
> > > in terms of a single generator of order 12, but its subgroups
can
> > be
> > > expressed in terms of generators of degree 3 and 4
respectively,
> > and
> > > these also generate C12. In fact, since 3 and 4 are relatively
> > prime,
> > > any integer can be expressed as a linear combination of 3s and
> 4s,
> > so
> > > any 2^(k/12) can be expressed as a product of major and minor
12-
> et
> > > thirds, without the use of octaves.
> >
> > yes, but to claim (as balzano did) that the fundamental
importance
> of
> > the diatonic scale hinges on this fact is to pull the wool over
the
> > eyes of the numerically inclined reader. the fact is that around
> the
> > time of the emergence of tonality in diatonic music, many
musicians
> > advocated a 19- or 31-tone system in which to embed the diatonic
> > scale, and 12 won out only because of convenience. it is only
with
> > the work of late 19th century russian composers that the cycle-3
> and
> > cycle-4 aspects of C12 became musically important.
> >
> > in fact, the diatonic scale emerged over and over again around
the
> > world without any 'chromatic universe' whatsoever, let alone an
> equal-
> > tempered one. the important properties of the diatonic scale
must,
> i
> > feel, be found in the scale itself, in whatever tuning it may be
> > found (with reasonable allowances for the ear's ability to accept
> > small errors) -- any 'chromatic totality' considerations should
> wait
> > until, and be completely dependent upon, the establishment of the
> > fundamental 'diatonic' entity upon which the music is to be
based.
> > this was my approach in my paper, and more recently, in my
> adaptation
> > of fokker's periodicity block theory.
>
>
> ***This is extremely interesting and echos Easley Blackwood's work,
I
> believe...
>
> J. Pehrson

glad you got to read my comments, even if only a year and a half
late . . .