A question about just vs. tempered scales?

>Dear all,

>I was wondering what makes a sequence of notes a (just
>intonation) scale. Maybe somebody can tell me about properties
>of just intonation scales, like conditions they have to satisfy? In
>the Encyclopaedia of Tuning I did find some properties of scales.
>Under 'scale' is written: "Scales often, but not always, exhibit
>tetrachordal similarity, and other properties such as MOS,
>propriety, distributional evenness, etc." However, all these
>properties are only valid for equal tempered scales (or am I
>wrong?). Are there similar properties for just intonation scales? I
>hope somebody can help me. Thanks in advance.
>
>Best regards,
>
>Aline Honingh

Dear Dr. Aline Honingh,

At the following website:

******************************

http://staff.science.uva.nl/~ahoningh/research

******************************

I was fascinated to find your autobiographical summary:

******************************

"Research

In 2001 I finished my master thesis in theoretical physics at the
University of Amsterdam. During my master study I followed
some extra courses at the department of music where my
scientific interest in music grew and the idea arose to do a PhD
combining physics with music. In February 2002, I received a PhD
grant at the Institute for Logic, Language and Computation (ILLC )
at the University of Amsterdam in the group of Rens Bod. My main
research interests are temperaments and intonation, and
mathematical group theory applied to music. During the first year
of my PhD I spent one term in Cambridge studying music theory.
I play the violin since I was nine years old.

Publications

A. Honingh and R. Bod, 2004. The Notion of Convexity in Music
(PS, PDF). Proceedings UCM 2004, Caserta, Italy.

A. Honingh, 2004. Limitations on Fixed n-Tone Equal Tempered
Divisions (PS, PDF). Proceedings UCM 2004, Caserta, Italy.

A. Honingh, 2003. Group Theoretic Description of Just Intonation
(PS, PDF revised version). Proceedings UCM 2003, Caserta,
Italy.

A. Honingh, 2003. Measures of Consonances in a Goodness-of-fit
Model for Equal-tempered Scales (PS, PDF). Proceedings ICMC
2003, Singapore.

A. Honingh, 2003. Group Theoretic Description of Just Intonation
(PS, PDF revised version). Proceedings UCM 2003, Caserta,
Italy"

******************************

Your article:

******************************

"Group Theoretic Description of Just Intonation"

******************************

includes the following mathematical description of a just intoned
scale:

******************************

"Table 1 shows the ratios of the just major scale with
respect to the fundamental.
Note do re mi fa so la ti do
Ratio 1:1 9:8 5:4 4:3 3:2 5:3 15:8 2:1"

******************************

Here are my observations.

All rational and irrational tuning systems may include a vast number of
tones. Scales include a small number of tones. Therefore, all
rational and irrational tuning systems may include a number of scales
that may approach incomprehensibility.

The number of ways 12 tones may be ordered is
12! = 479,001,600.0

The number of ways 7 tones may be ordered is 7! = 5,040.0.

Of these 12 tones, the number of ways 7 tones may be uniquely
ordered is (12 * 11 * 10 * 9 * 8 * 7 * 6) / 7! = 792.0.

Therefore, I surmise that what constitutes a scale cannot be
universally defined. Even within a given civilization,
one can only speak of human musical tendencies.

The following text is from my manuscript
Musical Mathematics: A Practice in the Mathematics of Tuning
Instruments and Analyzing Scales. Chapter 11, "Indian Music."

******************************

An imaginary east-to-west diagonal from Calcutta, near the Bay of
Bengal, to Bombay, on the coast of the Arabian Sea, roughly
divides the subcontinent of India into two geographical and
cultural areas called North India and South India. The former
region is known for Hindustani music, and the latter, for Carnatic
music. Although both cultures evolved from the same ancient
tradition, by the 17th century each had developed a separate
musical vocabulary and unique styles in the performance of ragas.

Before we discuss these two traditions, let us first address the
question, "What is a raga?" The answer is extremely complex
because ragas are musical expressions that continuously change
in time. Most ragas consist of two basic elements: (1) a set of
technical definitions and instructions, transformed into music
through (2) spontaneously improvised performances. Because
Indian music is inherently melodic in nature, musicians have
created literally thousands of different ragas. In the 16th and 17th
centuries, theorists attempted to organize this incomprehensible
plethora of ragas by creating organizational systems based on
mathematically defined tunings and combinatorics. However,
because such efforts compelled theorists and musicians to closely
examine the quintessential relation between the technical and
musical aspects of ragas, these systems profoundly influenced not
only intellectual definitions of ragas, but artistic interpretations as
well.

In the next five sections, we will examine the evolution of South
Indian ragas in the writings of Ramamatya (1550), Venkatamakhi
(1620), and Govinda (c. 1800). These three writers focused on a
theme common to all organizational systems, namely, the
principle of abstraction. Ramamatya was the first Indian theorist
to formulate a system based on a mathematically determined
tuning. He defined (1) a theoretical 14-tone scale, (2) a practical
12-tone tuning, and (3) a distinction between abstract mela ragas
and musical janya ragas. He then combined these three concepts
to identify 20 mela ragas, under which he classified more than 60
janya ragas. Venkatamakhi extended Ramamatya's approach,
and proposed a system that consists of 72 heptatonic
permutations called melakartas. He then identified 19 melakartas,
which he used to classify dozens of janya ragas. Finally, Govinda
extended Venkatamakhi's basic concept, and defined the present
day system, which consists of 72 modern mela ragas, or 72
theoretically constructed heptatonic scales.

We may think of these developments as representing three
interrelated stages in the history of South Indian music.
Ramamatya, Venkatamakhi, and Govinda created organizational
systems based on the principle of musical abstraction,
combinatorial abstraction, and theoretical abstraction,
respectively. As such, these underlying principles do not tell us
how to understand the resulting systems. Therefore, consider the
following three interpretations. In the writings of Ramamatya and
Venkatamakhi, the terms 'mela ragas' and 'melakartas,'
respectively, refer to prototypes - or collections of seven tones -
that function as abstract raga-categories. In the 16th, 17th, and
18th centuries, theorists and musicians used melas for only one
purpose: to organize and classify musical ragas. Consequently,
in the original sense, 'melas' never represented regular scales, or
musical ragas. Figure 29 shows two melas identified by
Ramamatya enclosed in separate circles to convey the impression
of pools of seven tones that similar musical janya ragas have in
common. In most texts, melas are represented as linear
sequences of notes on a treble staff. Westerners, when
presented with progressions of notes in regular ascending and/or
descending order, unwittingly interpret such sequences as scales.
Because this tendency seriously interferes with a historical
understanding of South Indian music, I have chosen to equate the
original 'melas' with the terms 'raga-categories,' 'prototypes,' and
'tone pools.'

******************************

The concept of 'tone pools' is the best that I can offer for a ubiquitous
definition -- not of scales -- but for the fountainheads or primary sources
of scales.

In other words, a tone pool is to a scale what a zygote is to an
embryo. Alternatively, tone pools reflect musical experiences prior
to numerical divisions or verbal categorizations.

To read the entire article, please logon at:

http://www.Chrysalis-Foundation.org/Ramamatya's_Vina.htm

Sincerely,

Cris Forster, Music Director
www.Chrysalis-Foundation.org

Dear Cris Forster,

Thank you for your e-mail.

Cris Forster wrote:
> > Dear Dr. Aline Honingh,
You can leave the "Dr." out, I haven't finished my PhD yet ;-)

> At the following website:
> > ******************************
> > http://staff.science.uva.nl/~ahoningh/research
> > ******************************
> > I was fascinated to find your autobiographical summary:

I'm happy but also a bit embarrassed to hear that you found my website, since I'm not so proud anymore of my first two publications. The last written paper however (The notion of convexity in music) was the reason that I sent a question to the mailing list.
I will explain. When all 5-limit just intonation intervals are put on a lattice (a lattice like the one used by Longuet-Higgins, Balzano, and others, only I chose the major and minor third to be the unity vectors) musical items like chords and scales can be found in convex sets. I tried to check most of the 5-limit JI scales defined in Scala, and it turned out that a remarkable large part of these scales is convex.
Since I don't really understand why this is the case, I was hoping to be able to relate this property to other properties of JI scales. And since I couldn't find any properties of these scales, I sent this question to the list. Also, I was wondering if there are any necessary conditions for a set/sequence of notes to be a scale. But it seems there aren't..

> Here are my observations.
> > All rational and irrational tuning systems may include a vast number of
> tones. Scales include a small number of tones. Therefore, all
> rational and irrational tuning systems may include a number of scales
> that may approach incomprehensibility.
> > The number of ways 12 tones may be ordered is
> 12! = 479,001,600.0
> > The number of ways 7 tones may be ordered is 7! = 5,040.0.
> > Of these 12 tones, the number of ways 7 tones may be uniquely
> ordered is (12 * 11 * 10 * 9 * 8 * 7 * 6) / 7! = 792.0.
> > Therefore, I surmise that what constitutes a scale cannot be
> universally defined. Even within a given civilization,
> one can only speak of human musical tendencies.

This is an interesting point, but I'm not sure if I agree. Indeed, the number of ways to choose 7 from 12 in an ordered way is already quite a large number. However, there are infinitely many just intonation ratios to choose scales from, but not all combinations are considered to be scales, at least not all are ever defined to be a scale. And since (from what I've learned) there are some properties for scales, not every random chosen combination from the set of all possible intervals would be a good scale.

Kraig Grady made the point that what makes a scale is determined by our ears. That would suggest (I think) that if it's not possible to define scale in a mathematical way, at least there can be made a psychological measure.

> > The following text is from my manuscript
> Musical Mathematics: A Practice in the Mathematics of Tuning
> Instruments and Analyzing Scales. Chapter 11, "Indian Music."
> > > The concept of 'tone pools' is the best that I can offer for a ubiquitous
> definition -- not of scales -- but for the fountainheads or primary sources
> of scales.
> > In other words, a tone pool is to a scale what a zygote is to an
> embryo. Alternatively, tone pools reflect musical experiences prior
> to numerical divisions or verbal categorizations. > > To read the entire article, please logon at:
> > http://www.Chrysalis-Foundation.org/Ramamatya's_Vina.htm

Thank you for this text, it's really interesting. I will also read the entire article.

Best regards,
Aline Honingh

On 5/24/05, Cris Forster <76153.763@compuserve.com> wrote:

<snip>
> "Research
>
> In 2001 I finished my master thesis in theoretical physics at the
> University of Amsterdam. During my master study I followed
> some extra courses at the department of music where my
> scientific interest in music grew and the idea arose to do a PhD
> combining physics with music. In February 2002, I received a PhD
> grant at the Institute for Logic, Language and Computation (ILLC )
> at the University of Amsterdam in the group of Rens Bod. My main
> research interests are temperaments and intonation, and
> mathematical group theory applied to music. During the first year
> of my PhD I spent one term in Cambridge studying music theory.
> I play the violin since I was nine years old.

Oh yes, well spotted! So Aline, if you're still following this, have
you checked tuning-math? You should be capable of understanding it,
at least. Also see Carey & Clampitt's paper from Perspectives of New
Music a few years back, and Rothenberg's original papers.

> Publications
>
> A. Honingh and R. Bod, 2004. The Notion of Convexity in Music
> (PS, PDF). Proceedings UCM 2004, Caserta, Italy.

This looks like properties of JI scales.

> A. Honingh, 2004. Limitations on Fixed n-Tone Equal Tempered
> Divisions (PS, PDF). Proceedings UCM 2004, Caserta, Italy.

Selects 5-limit consistent ETs that support meantone without
contorsion. Also relaxes the meantone criterion.

> A. Honingh, 2003. Group Theoretic Description of Just Intonation
> (PS, PDF revised version). Proceedings UCM 2003, Caserta,
> Italy.

Not really very exciting, just doing the groundwork.

> A. Honingh, 2003. Measures of Consonances in a Goodness-of-fit
> Model for Equal-tempered Scales (PS, PDF). Proceedings ICMC
> 2003, Singapore.

Just a review of old ideas by the looks of it.

Graham

Dear Graham Breed,

Graham Breed wrote:>
> Oh yes, well spotted! So Aline, if you're still following this, have
> you checked tuning-math? You should be capable of understanding it,
> at least. Also see Carey & Clampitt's paper from Perspectives of New
> Music a few years back, and Rothenberg's original papers.

Yes, I'm still following and I'm learning a lot. I have also followed the discussion on the tuning-math.
I assume you mean the paper "self-similar pitch structures, their duals, and rhythmic analogues" by Cary and Clampitt? I am aware of the paper but have never read it thoroughly yet. I will now, thanks for naming it here. The original papers by Rothenberg I don't know. Can you maybe send me (off list) the references of papers that you think I should read?

> > Publications
> >
> > A. Honingh and R. Bod, 2004. The Notion of Convexity in Music
> > (PS, PDF). Proceedings UCM 2004, Caserta, Italy.
> > This looks like properties of JI scales.

Yes, as I explained in the e-mail to Cris Forster I was hoping to relate convexity to other scale properties.

> > > Not really very exciting, just doing the groundwork.
> > Just a review of old ideas by the looks of it.
> I know and already said I'm not too proud anymore of these papers..

But thanks for your reactions on them.

regards,
Aline

On 5/24/05, Aline Honingh <ahoningh@science.uva.nl> wrote:

> I'm happy but also a bit embarrassed to hear that you found my website,
> since I'm not so proud anymore of my first two publications. The last
> written paper however (The notion of convexity in music) was the reason
> that I sent a question to the mailing list.
> I will explain. When all 5-limit just intonation intervals are put on a
> lattice (a lattice like the one used by Longuet-Higgins, Balzano, and
> others, only I chose the major and minor third to be the unity vectors)
> musical items like chords and scales can be found in convex sets. I
> tried to check most of the 5-limit JI scales defined in Scala, and it
> turned out that a remarkable large part of these scales is convex.
> Since I don't really understand why this is the case, I was hoping to be
> able to relate this property to other properties of JI scales. And since
> I couldn't find any properties of these scales, I sent this question to
> the list. Also, I was wondering if there are any necessary conditions
> for a set/sequence of notes to be a scale. But it seems there aren't..

A periodicity block constructed from the unison vectors should always
be convex, in that it's a parallelogram (or higher dimensional
equivalent). Some of the scales you're checking are periodicity
blocks.

I think a Combination Product Set (CPS) will always be convex as well.
At least, you can check that.

An MOS will naturally be convex in that it's a square of generators
and periods. So a convex JI scale will still be convex when
approximated as an MOS, although the reverse isn't the case. You can
also prove this from the relationship between MOS and periodicity
blocks.

Connectedness measures are related to convexity. A scale is connected
in an odd-limit if any note can be reached from any other note using
only intervals within the odd limit. Further, if all notes are
connected with complete chords of the odd-limit then it's chord
connected. In your case, the relevant limit is 5, and the chords the
major and minor triads. All convex scales will be connected, and most
will be chord connected (not all, you could just have a run of fifths)
although not all chord connected scales are convex. Chord connected
scales that aren't star convex sound bizarre, but I'm not saying
nobody would ever propose one. Maximising the number of chords within
a scale will lead you towards convexity.

If you choose ratios to be small relative to the 1/1, that will tend
to give you star convexity, but I don't know of any measures that
ensure this. But it is natural to want to move by the shortest number
of consonances from the tonic to any other note in the scale, don't
you think? Probably a constant structure Tenney reduced will be
convex.

Your lattice is really a triangular one turned on its side, so major
and minor triads are the simplest structures.

Rothenberg's measures -- propriety, stability and efficiency -- are
based on step sizes, and so not at all relevant here.

Graham

--- In tuning@yahoogroups.com, Aline Honingh <ahoningh@s...> wrote:

> I will explain. When all 5-limit just intonation intervals are put on a
> lattice (a lattice like the one used by Longuet-Higgins, Balzano, and
> others, only I chose the major and minor third to be the unity vectors)
> musical items like chords and scales can be found in convex sets.

Convexity is invariant under affine transformations, so broadly
speaking it won't matter what you use for generators. The idea of
using a lattice apparently should be credited to Hugo Riemann, who got
the basis of the idea from Euler, but I am intrigued by your mention
of Longuet-Higgins. I havn't read his work, yet with his background it
would seem it might be very much worth my while to do so, and I'm
wondering if you could supply some pointers to it.

I
> tried to check most of the 5-limit JI scales defined in Scala, and it
> turned out that a remarkable large part of these scales is convex.
> Since I don't really understand why this is the case, I was hoping
to be
> able to relate this property to other properties of JI scales.

I suggested to Manuel a while back that determining convexity would be
a good addition to Scala, and I still think so. It's a pretty basic
property, and one which is helpful if you want the scale to have a lot
of consonant chords and intervals. Since this is likely to be a design
consideration for anyone creating a scale, it isn't surprising to see
it turn up in the final result. Moreover, some methods of JI scale
construction result in a convex scale automatically; an example being
a Fokker block, and another and even more obvious one the Euler genus.

One thing to note about convexity is that it is not invariant under
the process of projecting down to a lattice of smaller dimension via
tempering. However, for tempered scales convexity is also a
consideration; often discussion of rank two (two-generator)
temperaments has centered on what are called "MOS" or "DE" scales,
where under octave equivalence we have convexity just along a single
chain of generators. The reason for that I think is the same reason
for convexity in general: it tends to give you a lot of (tempered)
consonances if you push it out far enough compared to the complexity
of the temperament.

By the way, heavily mathematical discussions are something people
sometimes object to on this list, and if you feel the urge we could
move things to the tuning-math list instead.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> On 5/24/05, Aline Honingh <ahoningh@s...> wrote:

> I think a Combination Product Set (CPS) will always be convex as well.
> At least, you can check that.

A CPS takes sums of points, and therefore has to give results in the
convex hull of the set of points it is taking sums of. However, there
might be other points in there if the points you start from are widely
enough separated. When, exactly the CPS is guaranteed to be convex
sounds like a problem worth considering.

> An MOS will naturally be convex in that it's a square of generators
> and periods. So a convex JI scale will still be convex when
> approximated as an MOS, although the reverse isn't the case.

This doesn't follow. You can have a nice convex epimorphic scale and
when you project it down to a MOS for the same val find you only get a
modmos. For a simple example, take the scale generated by
(5/4)^i (6/5)^j, -1<=i<=1, -1<=j<=2, octave reduced. This is
epimorphic for <12 19 28|, and if you project to rank one via this
val, the result is trivially epimorphic. If you project to rank two in
the form of meantone, the result is not epimorphic, but is a modmos: a
mod 12 alteration of a Fokker block by diesis adjustments. Of course
it is also clear that normally, taking an et and applying it to this
scale will not give you convexity, as it will for 12.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> A CPS takes sums of points, and therefore has to give results in the
> convex hull of the set of points it is taking sums of.

Sorry, I was babbling here; the correct convex hull is determined in a
way a little more complicated than this.

Gene Ward Smith wrote:
> > Convexity is invariant under affine transformations, so broadly
> speaking it won't matter what you use for generators. I know, and therefore it is an interesting property, right?

The idea of
> using a lattice apparently should be credited to Hugo Riemann, who got
> the basis of the idea from Euler, but I am intrigued by your mention
> of Longuet-Higgins. I havn't read his work, yet with his background it
> would seem it might be very much worth my while to do so, and I'm
> wondering if you could supply some pointers to it.

Longuet-Higgins (1962a), Letter to a musical friend, Music review 23, 244-48
Longuet-Higgins (1962b), Second letter to a musical friend, Music review 23, 271-80

And of course you are right about Riemann/Euler, I should have mentioned them too.
> > I suggested to Manuel a while back that determining convexity would be
> a good addition to Scala, and I still think so. It's a pretty basic
> property, and one which is helpful if you want the scale to have a lot
> of consonant chords and intervals. Since this is likely to be a design
> consideration for anyone creating a scale, it isn't surprising to see
> it turn up in the final result. Moreover, some methods of JI scale
> construction result in a convex scale automatically; an example being
> a Fokker block, and another and even more obvious one the Euler genus.

Thanks! this helps,

Aline

--- In tuning@yahoogroups.com, Aline Honingh <ahoningh@s...> wrote:

> > Convexity is invariant under affine transformations, so broadly
> > speaking it won't matter what you use for generators.

> I know, and therefore it is an interesting property, right?

Oh, absolutely. Since you are answering questions so readily, I wonder
if you can tell us what software you use for convexity computations?
I use Qhull and Maple in combination, but it's a bit of a pain. A
program which found the lattice convex closure--that is, the set of
lattice points in the convex hull of a given set of lattice
points--would be nice to have.

> Longuet-Higgins (1962a), Letter to a musical friend, Music review
23, 244-48
> Longuet-Higgins (1962b), Second letter to a musical friend, Music
review
> 23, 271-80

Thanks!

Gene Ward Smith wrote:>
> Oh, absolutely. Since you are answering questions so readily, I wonder
> if you can tell us what software you use for convexity computations?
> I use Qhull and Maple in combination, but it's a bit of a pain. A
> program which found the lattice convex closure--that is, the set of
> lattice points in the convex hull of a given set of lattice
> points--would be nice to have.

I use Matlab for calculating whether a set is convex or not. In Matlab there is a function "convhull" that gives the convex hull of a set of points. If you define a lattice you can check whether or not all the points of the set are in the convex hull so as to determine whether the set is convex or not. I am not sure if this is the same method you used, but for me it works fine. The only thing is that the computation time increases when you want to have a bigger lattice.

Best,
Aline

--- In tuning@yahoogroups.com, Aline Honingh <ahoningh@s...> wrote:

> I use Matlab for calculating whether a set is convex or not. In Matlab
> there is a function "convhull" that gives the convex hull of a set of
> points. If you define a lattice you can check whether or not all the
> points of the set are in the convex hull so as to determine whether the
> set is convex or not. I am not sure if this is the same method you
used,
> but for me it works fine. The only thing is that the computation time
> increases when you want to have a bigger lattice.

It sort of is, since Matlab calls on Qhull to do it. Maybe I should
think about getting Matlab, which *also* calls on Maple to do things.

Thanks for the information.