JMM access (fwd)

Dear tuning-math folks,

I'm passing this along because I figure someone might be interested in the paper by Marek Zabka that is linked below, called "Well-formedness in two dimensions". It's related to the question of MOS generalisations.

Jon Wild

---------- Forwarded message ----------
Date: Sun, 16 Oct 2011 19:40:16 -0500
From: Robert W Peck <rpeck@lsu.edu>
Reply-To: SMCM-discussion <smcm-discussion@music.mcgill.ca>
To: smcm-discussion@music.mcgill.ca
Subject: [SMCM-discussion] JMM access

Dear List Members,

For a limited time, the following selection of research articles, book reviews, and editorials published in Journal of Mathematics and Music are available for free download, using the links provided.

Best wishes,

Robert Peck, Associate Professor and Coordinator of Music Theory
Louisiana State University, School of Music
Co-Editor, Journal of Mathematics and Music
E-mail: rpeck@lsu.edu

------------------------------------------------------------

Welcome

Thomas Noll & Robert Peck

http://www.tandfonline.com/doi/pdf/10.1080/17459730701267843

Continued fractions, best measurements, and musical scales and intervals

J. Douthett & R. Krantz

http://www.tandfonline.com/doi/pdf/10.1080/17459730601137799

The legacy of John Clough in mathematical music theory

David Clampitt

http://www.tandfonline.com/doi/pdf/10.1080/17459730701494710

Coherence and sameness in wellformed and pairwise well-formed scales

Norman Carey

http://www.tandfonline.com/doi/pdf/10.1080/17459730701376743

Enharmonic systems: A theory of key signatures, enharmonic equivalence and diatonicism

Julian Hook

http://www.tandfonline.com/doi/pdf/10.1080/17459730701374805

David Lewin and maximally even sets

Emmanuel Amiot

http://www.tandfonline.com/doi/pdf/10.1080/17459730701654990

Guest Editors’ Foreword

Elaine Chew, Alfred Cramer & Christopher Raphael

http://www.tandfonline.com/doi/pdf/10.1080/17459730802312225

Motivic analysis according to Rudolph Réti: formalization by a topological model

Chantal Buteau & Guerino Mazzola

http://www.tandfonline.com/doi/pdf/10.1080/17459730802518292

Special Issue: Tiling Problems in Music; Guest Editors’ Foreword

Moreno Andreatta & Carlos Agon

http://www.tandfonline.com/doi/pdf/10.1080/17459730903086140

Minimum description length modelling of musical structure

Panayotis Mavromatis

http://www.tandfonline.com/doi/pdf/10.1080/17459730903313122

Ionian theorem

Thomas Noll

http://www.tandfonline.com/doi/pdf/10.1080/17459730903309815

Well-formedness in two dimensions: a generalization of Carey and Clampitt's theorem

Marek Žabka

http://www.tandfonline.com/doi/pdf/10.1080/17459737.2010.491975

On Hellegouarch's definition of musical scales

Adrien Kassel & Christian Kassel

http://www.tandfonline.com/doi/pdf/10.1080/17459737.2010.496582

Two musical paths to the Farey series and devil's staircase

Julyan H.E. Cartwright, Jack Douthett, Diego L. González, Richard Krantz & Oreste Piro

http://www.tandfonline.com/doi/pdf/10.1080/17459737.2010.485001

Can computational music analysis be both musical and computational?

Christina Anagnostopoulou & Chantal Buteau

http://www.tandfonline.com/doi/pdf/10.1080/17459737.2010.520455

Distinctive patterns in the first movement of Brahms’ String Quartet in C minor

Darrell Conklin

http://www.tandfonline.com/doi/pdf/10.1080/17459737.2010.515421

Tonal trends and α-motif in the first movement of Brahms’ String Quartet op. 51 nr. 1

Atte Tenkanen

http://www.tandfonline.com/doi/pdf/10.1080/17459737.2010.512745

Harmonic vectors and stylistic analysis: a computer-aided analysis of the first movement of Brahms’ String Quartet Op. 51-1

Philippe Cathé

http://www.tandfonline.com/doi/pdf/10.1080/17459737.2010.520446

Lerdahl's tonal pitch space model and associated metric spaces

Richard R. Randall & Bilal Khan

http://www.tandfonline.com/doi/pdf/10.1080/17459737.2010.529654

Modelling of sonic textures by homogenization

Isabelle Gruais

http://www.tandfonline.com/doi/pdf/10.1080/17459737.2010.529676

Imaginary transformations

Robert Peck

http://www.tandfonline.com/doi/pdf/10.1080/17459737.2010.539666

Music, Experiment and Mathematics in England, 1653–1705

H. Floris Cohen

http://www.tandfonline.com/doi/pdf/10.1080/17459737.2010.513277

Around set theory

Catherine Losada

http://www.tandfonline.com/doi/pdf/10.1080/17459737.2010.531214

--- In tuning-math@yahoogroups.com, Jon Wild <wild@...> wrote:
>
> Dear tuning-math folks,
>
> I'm passing this along because I figure someone might be interested in the
> paper by Marek Zabka that is linked below, called "Well-formedness in two
> dimensions". It's related to the question of MOS generalisations.

Dammit, I need to get some non-xenharmonic work done, but now I need to know whether this Slovakian guy just solved all the problems I've been working on!

Keenan

Note that a "comma-demarcated GTS" in this paper (definition 2.9) is exactly what we've been calling a "Lumma block". It does not include arbitrary free Fokker blocks.

Keenan

Okay, so this paper just proves one very specific thing about rank-3 Lumma blocks, which I would have guessed before. It's nice to have a formal proof of it (and I may borrow some of these mathematical definitions), but this paper says nothing about free Fokker blocks, generalizations of Myhill's property / DE, or non-Fokker-block scales.

Also, this guy thinks he's the first to explain the existence of 22 sruti in an octave. Someone should tell him about Paul Erlich's work.

Lastly, I absolutely love this paragraph:

Problem 5.4 (Fokker's periodicity blocks) Fokker's `periodicity blocks' [12,13] are selections from the free abelian group (harmonic lattice) related to what is called here specifying function. Fokker's selections are also given by a parallelogram (or a parallelepiped) delimited by a set of commas. In this sense, periodicity blocks correspond to GTSs. However, not all periodicity blocks are `demarcated' as defined in the present paper. Loosely speaking, Fokker does not require the vertices of the delimiting parallelogram to belong to the free abelian group. Therefore, the concept of periodicity block is more general than the concept of comma demarcated GTS. The Hungarian scale given above as an example of non-demarcated GTS (Example 2.10) can be modelled as a periodicity block.

Hah! In your face, Carl Lumma!

Keenan

On Mon, Oct 17, 2011 at 2:35 AM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> Okay, so this paper just proves one very specific thing about rank-3 Lumma blocks, which I would have guessed before. It's nice to have a formal proof of it (and I may borrow some of these mathematical definitions), but this paper says nothing about free Fokker blocks, generalizations of Myhill's property / DE, or non-Fokker-block scales.

What is it?

> Also, this guy thinks he's the first to explain the existence of 22 sruti in an octave. Someone should tell him about Paul Erlich's work.

These papers always end up getting so painfully existential.

-Mike

"Keenan Pepper" <keenanpepper@gmail.com> wrote:

> Also, this guy thinks he's the first to explain the
> existence of 22 sruti in an octave. Someone should tell
> him about Paul Erlich's work.

Or Bosanquet. We're getting to assume that peer reviewed
articles miss key citations. In this case, there really
should be something on Paul's work with Fokker blocks. But
at least the review process did bring up Hellegouarch,
whose work seems to be similar. The French papers are at
smf.emath.fr and you can find them by searching Google for
Hellegouarch, Gammes naturelles and telling it you really
did mean Hellegouarch, Gammes naturelles. (Google
obfuscates the links so I didn't copy them.) I haven't
studied them in detail, and they are in French, but I see
it's the "suite" paper with the matrices. (And no citation
for Fokker, or Karp, or me.)

Graham

And the same author has a more recent paper on word theory and generalised well-formed scales, here: http://www.springerlink.com/content/c23748337406x463/

Jon Wild <wild@music.mcgill.ca> wrote:
> And the same author has a more recent paper on word
> theory and generalised well-formed scales, here:
> http://www.springerlink.com/content/c23748337406x463/

It's a good abstract, anyway. The rule "that the number of
specific varieties for any non-zero generic interval in
n-dimensional comma-demarcated generated tone systems is
between 2 and 2^n" agrees with what I found provided n is
counted octave equivalently.

Comma-demarcated GTS are the stricter kind of Fokker
blocks, right? The same should generalize to all Fokker
blocks.

Graham

--- In tuning-math@yahoogroups.com, Jon Wild <wild@...> wrote:
>
> And the same author has a more recent paper on word theory and generalised
> well-formed scales, here:
> http://www.springerlink.com/content/c23748337406x463/

This is a good paper. Note that in this one, "comma-demarcated GTS" means a *free* Fokker block and "strictly comma-demarcated GTS" means a Lumma block.

Basically, this paper rigorously proves some things we already knew.

Theorem 1 says that rank-(n+1) free Fokker blocks have interval varieties between 2 and 2^n inclusive. (Actually we know that all interesting Fokker blocks, i.e. the ones that are fully rank-(n+1) rather than degenerate, have interval varieties between n+1 and 2^n inclusive.)

Theorems 2 and 3 establish that, in rank 3, free Fokker blocks are exactly those scales which temper down to MOSes in at least two different ways. This property is called "quasi pairwise well-formedness".

"Pairwise well-formed" scales are defined as those which temper down to MOSes in three different ways, so these are the extra-special max-variety-3 scales I've talked about recently ("triple wakalixes", although I guess I've only used that term talking to Mike).

"Product words" are a particularly nice and simple mathematical formalization of something we (or at least I) already knew about. It's a particular kind of "detempering". For example, the JI major scale, abcabac, is a Fokker block with commas 81/80 and 25/24. That means it tempers down to a meantone MOS, aacaaac, as well as a dicot MOS, abbabab. Going the other way, the product word of aacaaac and abbabab is (a,a),(a,b),(c,b),(a,a),(a,b),(a,a),(c,b), which is equivalent to abcabac.

(In this particular case, since the JI major scale is "pairwise well-formed" rather than merely "quasi pairwise well-formed", there are also two other ways to do it. It's the product of a meantone MOS and a mavila MOS, or a dicot MOS and a mavila MOS.)

Should we send this Žabka guy an email and ask him to join some discussions here?

Keenan

On Mon, Oct 17, 2011 at 2:48 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> "Product words" are a particularly nice and simple mathematical formalization of something we (or at least I) already knew about. It's a particular kind of "detempering". For example, the JI major scale, abcabac, is a Fokker block with commas 81/80 and 25/24. That means it tempers down to a meantone MOS, aacaaac, as well as a dicot MOS, abbabab. Going the other way, the product word of aacaaac and abbabab is (a,a),(a,b),(c,b),(a,a),(a,b),(a,a),(c,b), which is equivalent to abcabac.

That's great, makes total sense. I wrote you a message on XA to
consider some transformations between the meantone and porcupine
lattices, so as to predict what the most trippy and novel sounding
chord progressions in porcupine will be. One way I considered doing it
was to look at the meantone-porcupine wakalixes, because both of those
scales aren't just "meantone[7] with comma issues," but "porcupine[7]
with comma issues" as well, and the more time we spend in porcupine[7]
the more we'll start to hear it that way too.

In general, this approach might be useful to turn one Rothenberg
categorical map into another.

> (In this particular case, since the JI major scale is "pairwise well-formed" rather than merely "quasi pairwise well-formed", there are also two other ways to do it. It's the product of a meantone MOS and a mavila MOS, or a dicot MOS and a mavila MOS.)

Weren't there also more than one dicot-meantone wakalixes? So does
that mean that there's no unique product word? Or is it that it's
dependent on which cyclic translation of the original words you
multiply?

> Should we send this Žabka guy an email and ask him to join some discussions here?

I say go for it.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> That's great, makes total sense. I wrote you a message on XA to
> consider some transformations between the meantone and porcupine
> lattices, so as to predict what the most trippy and novel sounding
> chord progressions in porcupine will be. One way I considered doing it
> was to look at the meantone-porcupine wakalixes, because both of those
> scales aren't just "meantone[7] with comma issues," but "porcupine[7]
> with comma issues" as well, and the more time we spend in porcupine[7]
> the more we'll start to hear it that way too.
>
> In general, this approach might be useful to turn one Rothenberg
> categorical map into another.

Yeah, totally.

There are 7 different meantone-porcupine Fokker blocks, of which 3 are triple wakalixes (one with dicot and one mirror pair with tetracot):

9/8 10/9 27/25 10/9 10/9 27/25 10/9 (meantone-porcupine-dicot)
10/9 16/15 10/9 10/9 10/9 27/25 10/9 (meantone-porcupine-tetracot)
10/9 27/25 10/9 10/9 10/9 16/15 10/9 (meantone-porcupine-tetracot)
9/8 10/9 27/25 10/9 10/9 10/9 27/25
9/8 10/9 10/9 27/25 10/9 10/9 27/25
10/9 9/8 27/25 10/9 10/9 10/9 27/25
10/9 9/8 27/25 10/9 10/9 27/25 10/9

> Weren't there also more than one dicot-meantone wakalixes? So does
> that mean that there's no unique product word? Or is it that it's
> dependent on which cyclic translation of the original words you
> multiply?

The latter, exactly. For n steps in an octave, there can be up to n non-"conjugate" product words because you can rotate one of the factor words n different ways with respect to the other. There can be fewer than n if one of the words is not "primitive". In fact, if I'm not mistaken, the number of non-conjugate product words is exactly the GCD of the lengths of the primitive roots of the two words.

Keenan

Keenan wrote:

>Lastly, I absolutely love this paragraph:
>
>Problem 5.4 (Fokker's periodicity blocks) Fokker's `periodicity
>blocks' [12,13] are selections from the free abelian group (harmonic
>lattice) related to what is called here specifying function. Fokker's
>selections are also given by a parallelogram (or a parallelepiped)
>delimited by a set of commas. In this sense, periodicity blocks
>correspond to GTSs. However, not all periodicity blocks are
>`demarcated' as defined in the present paper. Loosely speaking, Fokker
>does not require the vertices of the delimiting parallelogram to
>belong to the free abelian group. Therefore, the concept of
>periodicity block is more general than the concept of comma demarcated
>GTS. The Hungarian scale given above as an example of non-demarcated
>GTS (Example 2.10) can be modelled as a periodicity block.
>
>Hah! In your face, Carl Lumma!

Does he cite Fokker on that?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
> Does he cite Fokker on that?

He sure does!

Keenan

At 04:19 PM 10/17/2011, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>> Does he cite Fokker on that?
>
>He sure does!

What's the cite?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> At 04:19 PM 10/17/2011, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <carl@> wrote:
> >> Does he cite Fokker on that?
> >
> >He sure does!
>
> What's the cite?

Well, you can look at the footnotes just as well as I can, but there are two: "Selections from the Harmonic Lattice of Perfect Fifths and Major Thirds..." and "Unison Vectors and Periodicity Blocks in the Three-Dimensional (3-5-7-)Harmonic Lattice..."

Keenan

>> >> Does he cite Fokker on that?
>> >He sure does!
>> What's the cite?
>Well, you can look at the footnotes just as well as I can,

Why do you assume I have the paper?

>but there are two: "Selections from the Harmonic Lattice of
>Perfect Fifths and Major Thirds..." and "Unison Vectors and
>Periodicity Blocks in the Three-Dimensional (3-5-7-)
>Harmonic Lattice..."

It's been years since I've read them, and my copies are
in Montana. It's funny I don't remember seeing FFB-type
constructions.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
> Why do you assume I have the paper?

The original post of this thread was to tell everyone that they have access to the paper online.

Keenan

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
> Why do you assume I have the paper?

The original post of this thread was about how everyone has access to this paper on the internet.

> It's been years since I've read them, and my copies are
> in Montana. It's funny I don't remember seeing FFB-type
> constructions.

I've never seen them, so for all I know this guy could be misinterpreting them.

Keenan