Yahoo Tuning Groups Ultimate Backup tuning-math Orbifolds and lattices

We seem to be speaking different languages here. I'm going to try my best to translate.

> > It's remarkable that the configuration space of a series of points in
> > a one-dimensional pitch space is going to get you surprisingly close
> > to familiar notions of harmonic distance. It's not clear to me that
> > we need traditional tuning lattices for this purpose.
>
> Configuration space? You can use a continuous dissonance function.
> Harmonic lattices are a simple way of specifying the basic consonances.

I can't really reply to this, since I don't know what you're saying here. What I was trying to say was that the orbifolds I consider are configuration spaces. I'm not sure if you're denying this, or are confused about this, or are wondering about its relevance, or what ...

> > For example: if you look at the 12 diatonic scales in the
> > 7-dimensional space, they are naturally ordered by voice-leading
> > proximity into a circular chain, where each scale shares 6 notes with
> > its neighbors. This is just the circle of fifths.
>
> That isn't something that falls out naturally from the geometry.

I don't know why you say that. The relations I'm talking about do fall out of the geometry: just take the 12 familiar diatonic scales and plot them in 7-dimensional scale space. You'll see that they do indeed, as I claimed, form a circular chain. They are connected by short line segments representing voice leading in which only one note moves, and it moves by one semitone. Check out "Scale Networks and Debussy" which depicts a portion of 7-note scale space, as a cubic lattice, and shows some of the nearby scales.

> Diatonic scales are very unusual musical objects in that they're related
> by fifths in this way. In general terms, the optimal interval for
> transposing an MOS scale is the generator.

Sort of. Actually the 12 familiar diatonic scales would be linked by very efficient voice leading even if they were not strictly speaking generated. Here's a thought experiment: take the 12 familiar diatonic scales and retune each one as you like. For instance, tune C major using pythagorean tuning; tune G major using just intonation; tune D major using meantone temperament; and so forth. The scales are still going to be linked by very smooth voice leading. And they're no longer all generated -- nor are they, strictly speaking transpositionally related!

The business about their being "generated" is a bit of a distractor. What's important is that diatonic scales are nearly even, and hence cluster around the center of the orbifold, and hence are linked by efficient voice leading.

>Incidentally, relating scales by the notes they have in common is
> similar to Rothenberg's efficiency. I think his work is more closely
> related to yours than tuning lattices are, although it's not what this
> thread's about. He's generally in favor among tuning theorists but
> ignored by academics. The diatonic set theorists are slowly reinventing
> his ideas. But don't let that put you off.

I'm certainly curious about this -- who is he, exactly? What are the important papers?

> What's this "diatonic metric"? It looks suspiciously like a distance in
> a lattice defined on scale steps.

Any scale can be used to define a metric on (continuous) circular pitch space such that it's adjacencies are one "unit" apart. (Cf. the supplementary materials of my Science paper, near the end of the second-to-last section on maximal evenness.) This is what I'm talking about.

> > BTW, one thing I think is cool is that you needn't take your space of
> > notes to be a torsor of your group of intervals. It's perfectly
> > possible, for instance, to let the group of real numbers act as
> > intervals on circular pitch space. This allows you to group together
> > octave-related pitches while still distinguishing motion by an
> > ascending octave from motion by a descending octave.
>
> Does it? How does that work?

You can just take an interval to be a line segment (unidirectional path) in pitch class space. So write

C-(+12)->C

to refer to the path that moves a complete clockwise circumference (representing an ascending octave) around pitch class space. Or

C-(-12)->C

For the one-octave descending path.

The point is that the intervals act on the space, but not freely or faithfully. This can be useful analytically.

If you want to see how this works check out the "circular pitch class space" window in Chord Geometries -- watch how pitch intervals are represented by paths, which may wind one or more times around the circle.

DT
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Dmitri Tymoczko
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🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:

> Sort of. Actually the 12 familiar diatonic scales would be linked by
> very efficient voice leading even if they were not strictly speaking
> generated.

Which is because they are Rothenberg proper, would be one way of
saying it.

> The business about their being "generated" is a bit of a distractor.
> What's important is that diatonic scales are nearly even, and hence
> cluster around the center of the orbifold, and hence are linked by
> efficient voice leading.

You really should read Rothenberg's stuff. Here's a Wikipedia article
I wrote on one relevant aspect:

http://en.wikipedia.org/wiki/Rothenberg_propriety

> I'm certainly curious about this -- who is he, exactly? What are the
> important papers?

His papers:

http://www.lumma.org/tuning/rothenberg/Rothenberg1.pdf
http://www.lumma.org/tuning/rothenberg/Rothenberg2.pdf
http://www.lumma.org/tuning/rothenberg/Rothenberg3.pdf

> Any scale can be used to define a metric on (continuous) circular
> pitch space such that it's adjacencies are one "unit" apart.

What advantage does this hold over doing things in the usual way, via
the induced metric from pitch space to pitch class space? The more
even the scale, the less it will matter, of course.

This can be described as working in a non-principal homogeneous space;
I don't think Lewin made any assumption that a musical space was
principal, by the way. Tuning-math old-timers may or may not be happy
to learn that wedgies define homogeneous spaces.

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

> > Sort of. Actually the 12 familiar diatonic scales would be linked by
>> very efficient voice leading even if they were not strictly speaking
>> generated.
>
>Which is because they are Rothenberg proper, would be one way of
>saying it.

Yes, sort of, but you can be much more precise here. The real issue is how closely scalar transposition approximates chromatic transposition. See "Scale Theory, Serial Theory, and Voice Leading" and "Voice Leadings as Generalized Key Signatures" on my website.

>You really should read Rothenberg's stuff. Here's a Wikipedia article
>I wrote on one relevant aspect:
>
>http://en.wikipedia.org/wiki/Rothenberg_propriety
>
>> I'm certainly curious about this -- who is he, exactly? What are the
>> important papers?
>
>His papers:
>
>http://www.lumma.org/tuning/rothenberg/Rothenberg1.pdf
>http://www.lumma.org/tuning/rothenberg/Rothenberg2.pdf
>http://www.lumma.org/tuning/rothenberg/Rothenberg3.pdf

Thanks! It is odd and unfortunate that academic music theory hasn't absorbed some of this stuff. Well, academic music theory has a lot of other problems as well.

> > Any scale can be used to define a metric on (continuous) circular
>> pitch space such that it's adjacencies are one "unit" apart. >
>What advantage does this hold over doing things in the usual way, via
>the induced metric from pitch space to pitch class space? The more
>even the scale, the less it will matter, of course.

1. Composers use scalar transposition all the time: "do, a dear" becomes "re, a drop." This process is absolutely ubiquitous in Western music, and it is considered a kind of transposition (translation). For it to be so, you need to use the scale-based metric. This "scale step" metric is used everywhere, in every tonal piece.

2. This perspective allows you to find interesting parallelism of musical structures. For instance, suppose you made a voice leading map of diatonic trichords (measuring voice leading in scale steps). You'd find it was basically isomorphic to the voice leading map of familiar near-diatonic scales (measuring voice leading in chromatic scale steps). See Appendix I of "Scale Networks and Debussy."

>This can be described as working in a non-principal homogeneous space;

Precisely.

>I don't think Lewin made any assumption that a musical space was
>principal, by the way.

He unquestionably did. Look at the definition of a Generalized Interval System in "Generalized Musical Intervals and Transformations."

DT
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WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
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Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗Graham Breed <gbreed@gmail.com>

Dmitri Tymoczko wrote:

>> > It's remarkable that the configuration space of a series of points in
>> > a one-dimensional pitch space is going to get you surprisingly close
>> > to familiar notions of harmonic distance. It's not clear to me that
>> > we need traditional tuning lattices for this purpose.
>>
>> Configuration space? You can use a continuous dissonance function.
>> Harmonic lattices are a simple way of specifying the basic consonances.
> > I can't really reply to this, since I don't know what you're saying > here. What I was trying to say was that the orbifolds I consider are > configuration spaces. I'm not sure if you're denying this, or are > confused about this, or are wondering about its relevance, or what ...

I'm saying I don't know what a configuration space is, but if you don't put harmonic distance in I don't expect you'll get harmonic distance out. Harmonic distance is a generalization of Tenney Harmonic Distance, which James Tenney introduced in the paper I haven't read that the lattice you saw came from. It's a taxi cab distance on a harmonic lattice where the length of an axis depends on the logarithm of the prime number. That means the harmonic distance of a ratio n:d is log(n*d).

It can mean all kinds of things, but has something to do with the harmonic complexity of an interval which in turn has something to do with its dissonance.

>> > For example: if you look at the 12 diatonic scales in the
>> > 7-dimensional space, they are naturally ordered by voice-leading
>> > proximity into a circular chain, where each scale shares 6 notes with
>> > its neighbors. This is just the circle of fifths.
>>
>> That isn't something that falls out naturally from the geometry.
> > I don't know why you say that. The relations I'm talking about do > fall out of the geometry: just take the 12 familiar diatonic scales > and plot them in 7-dimensional scale space. You'll see that they do > indeed, as I claimed, form a circular chain. They are connected by > short line segments representing voice leading in which only one note > moves, and it moves by one semitone. Check out "Scale Networks and > Debussy" which depicts a portion of 7-note scale space, as a cubic > lattice, and shows some of the nearby scales.

They fall out of diatonic scales. They don't fall out of the geometry unless you artificially select diatonic scales. You can get a circle of any interval you like by choosing the right scales in the right equal temperaments.

>> Diatonic scales are very unusual musical objects in that they're related
>> by fifths in this way. In general terms, the optimal interval for
>> transposing an MOS scale is the generator.
> > > Sort of. Actually the 12 familiar diatonic scales would be linked by > very efficient voice leading even if they were not strictly speaking > generated. Here's a thought experiment: take the 12 familiar > diatonic scales and retune each one as you like. For instance, tune > C major using pythagorean tuning; tune G major using just intonation; > tune D major using meantone temperament; and so forth. The scales > are still going to be linked by very smooth voice leading. And > they're no longer all generated -- nor are they, strictly speaking > transpositionally related!

How about we tune them to 7 note equal temperament? Then they'll be linked by perfectly smooth voice leading by transpositions by the fifth of 7 note equal temperament. But they'll be linked by equally smoogh voice leadings by transpositions of any other interval within the scale. It's nice that you allow a bit of leeway for mistuning but closeness to a generated scale must be important.

> The business about their being "generated" is a bit of a distractor. > What's important is that diatonic scales are nearly even, and hence > cluster around the center of the orbifold, and hence are linked by > efficient voice leading.

The business about them being generated is exactly what tells you that they can be transposed by the generator so that they share all but one note. The fact that they're MOSs tells you that the note they don't share will move by a scale step. The fact that they're maximally even tells you that they're proper, which may be good for voice leading.

>>Incidentally, relating scales by the notes they have in common is
>> similar to Rothenberg's efficiency. I think his work is more closely
>> related to yours than tuning lattices are, although it's not what this
>> thread's about. He's generally in favor among tuning theorists but
>> ignored by academics. The diatonic set theorists are slowly reinventing
>> his ideas. But don't let that put you off.
> > I'm certainly curious about this -- who is he, exactly? What are the > important papers?

Gene answered this.

>> What's this "diatonic metric"? It looks suspiciously like a distance in
>> a lattice defined on scale steps.
> > Any scale can be used to define a metric on (continuous) circular > pitch space such that it's adjacencies are one "unit" apart. (Cf. > the supplementary materials of my Science paper, near the end of the > second-to-last section on maximal evenness.) This is what I'm > talking about.

So pitch space is stretched to get the tones and semitones of a particular diatonic scale to be equally far apart?

>> > BTW, one thing I think is cool is that you needn't take your space of
>> > notes to be a torsor of your group of intervals. It's perfectly
>> > possible, for instance, to let the group of real numbers act as
>> > intervals on circular pitch space. This allows you to group together
>> > octave-related pitches while still distinguishing motion by an
>> > ascending octave from motion by a descending octave.
>>
>> Does it? How does that work?
> > > You can just take an interval to be a line segment (unidirectional > path) in pitch class space. So write
> > C-(+12)->C
> > to refer to the path that moves a complete clockwise circumference > (representing an ascending octave) around pitch class space. Or
> > C-(-12)->C
> > For the one-octave descending path.
> > The point is that the intervals act on the space, but not freely or > faithfully. This can be useful analytically.

Is this like walking round a circular path, and getting back where you started but knowing you've been somewhere?

> If you want to see how this works check out the "circular pitch class > space" window in Chord Geometries -- watch how pitch intervals are > represented by paths, which may wind one or more times around the > circle.

I can't find "circular pitch class space" but there is "Circular Space". And it isn't very enlightening. It's what I'd expect modulo-octave pitch space to look like.

Graham

🔗Carl Lumma <ekin@lumma.org>

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>I'm saying I don't know what a configuration space is

I just mean that a point in my orbifold represents a configuration of points on the ordinary log-frequency line (where we treat two configurations as "the same" if they are related by common musical equivalences). The point {C, G}, for instance, that you have two pitches and one of them is some C (in some octave) and the other is some G (in some octave).

>, but if you don't
>put harmonic distance in I don't expect you'll get harmonic distance
>out. Harmonic distance is a generalization of Tenney Harmonic Distance,
>which James Tenney introduced in the paper I haven't read that the
>lattice you saw came from. It's a taxi cab distance on a harmonic
>lattice where the length of an axis depends on the logarithm of the
>prime number. That means the harmonic distance of a ratio n:d is log(n*d).
>
>It can mean all kinds of things, but has something to do with the
>harmonic complexity of an interval which in turn has something to do
>with its dissonance.

OK, this notion of "harmonic distance" I probably have very little to say about. I can talk about "harmonic distance" between chords and so forth. But this issue of interval complexity is outside of my purview.

>They fall out of diatonic scales. They don't fall out of the geometry
>unless you artificially select diatonic scales. You can get a circle of
>any interval you like by choosing the right scales in the right equal
>temperaments.

At 5:15 PM -0700 7/16/06, Carl Lumma wrote:
>Graham's point may be, the geometry falls out of the scale, not
>the other way around.

Sorry, yes, I agree. My orbifolds don't single out the diatonic scales for you. You have to do that yourself. Once you do, their magical relationships are manifested by the geometry.

>How about we tune them to 7 note equal temperament? Then they'll be
>linked by perfectly smooth voice leading by transpositions by the fifth
>of 7 note equal temperament.

These voice leadings won't be independent -- all voices will move in the same direction by the same amount.

> But they'll be linked by equally smoogh
>voice leadings by transpositions of any other interval within the scale.
> It's nice that you allow a bit of leeway for mistuning but closeness
>to a generated scale must be important.

Only insofar as the perfectly even chord is always generated. Chords that are close to generated chords, such as (C, D, E) will have smooth voice leadings to one of their inversions, but not necessarily to their transpositions. In this sense, generatedness isn't crucial for efficient voice leading.

>The business about them being generated is exactly what tells you that
>they can be transposed by the generator so that they share all but one
>note.

Yes. However, the voice leading need not be small: (C, D, E)->(C, D, Bb) involves 6 semitones of motion.

>The fact that they're MOSs tells you that the note they don't
>share will move by a scale step.

I'm still not sure I get the "MOS." But the one scale-step motion implies that they're maximally even (and not transpositionally symmetrical) relative to the scale. I suspect this is just what MOS means.

>So pitch space is stretched to get the tones and semitones of a
>particular diatonic scale to be equally far apart?

Yup. This is what is happening when "do, a deer" becomes "re, a drop."

>Is this like walking round a circular path, and getting back where you
>started but knowing you've been somewhere?

Exactly!

> > If you want to see how this works check out the "circular pitch class
>> space" window in Chord Geometries -- watch how pitch intervals are
> > represented by paths, which may wind one or more times around the
>> circle.
>
>I can't find "circular pitch class space" but there is "Circular Space".
> And it isn't very enlightening. It's what I'd expect modulo-octave
>pitch space to look like.

You're right, it's nothing to write home about. The only interesting thing is that pitch intervals are represented by paths -- you can walk around the circle more than once, when you move by an octave or more. And voice leadings are represented by collections of paths. This shows you how the group of real numbers acts on the circular space -- you have more than one interval linking any two notes.

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
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Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Graham Breed <gbreed@gmail.com>

Carl Lumma wrote:

>>The business about their being "generated" is a bit of a distractor. >>What's important is that diatonic scales are nearly even, and hence >>cluster around the center of the orbifold, and hence are linked by >>efficient voice leading.
> > > So equal temperaments are optimal from a voice-leading perspective...

There's a trivial voice-leading from an equal temperament to its counterpart moved by any interval of the scale. That means none of the notes move, not that they all move by that interval. If you move the scale by a JI consonance instead, the minimal amount each voice has to move is the error in that interval in the given equal temperament. So in a sense, voice leading tells you about how good an equal temperament is. The problem is that if all the voices move by this minimal amount then they all move in the same direction.

If you take an MOS, and move it by a JI interval, some notes can move by the error of the interval in a temperament including that MOS. But other notes have to move by a larger step. The more complex the interval is in the temperament, the more notes move by the larger step. So the voice-leading distance depends on both the errors and complexities of the consonances, which makes it a badness measure.

It should work for higher rank temperaments as well. Or even well temperaments. It'd be interesting to see what scales do come out as optimal. I don't know if it would *mean* anything, but interesting to see nevertheless. My initial thought was that equal temperaments would end up as optimal because they're so simple. But if you want voices to move independently you might get something lumpier coming out.

It'd be nice if inconsistent temperaments entailed voice crossings but I don't think it works like that.

Graham

🔗Graham Breed <gbreed@gmail.com>

Dmitri Tymoczko wrote:

>>How about we tune them to 7 note equal temperament? Then they'll be
>>linked by perfectly smooth voice leading by transpositions by the fifth
>>of 7 note equal temperament.
> > These voice leadings won't be independent -- all voices will move in > the same direction by the same amount.

I mean you transpose the scale by an interval in circular pitch space to get another set of intervals. Not that all the voices move by the same interval. (Although they will, where the interval is a unison.)

>> But they'll be linked by equally smoogh
>>voice leadings by transpositions of any other interval within the scale.
>> It's nice that you allow a bit of leeway for mistuning but closeness
>>to a generated scale must be important.
> > Only insofar as the perfectly even chord is always generated. Chords > that are close to generated chords, such as (C, D, E) will have > smooth voice leadings to one of their inversions, but not necessarily > to their transpositions. In this sense, generatedness isn't crucial > for efficient voice leading.

Okay.

>>The business about them being generated is exactly what tells you that
>>they can be transposed by the generator so that they share all but one
>>note.
> > Yes. However, the voice leading need not be small: (C, D, E)->(C, D, > Bb) involves 6 semitones of motion.
> >>The fact that they're MOSs tells you that the note they don't
>>share will move by a scale step.
> > I'm still not sure I get the "MOS." But the one scale-step motion > implies that they're maximally even (and not transpositionally > symmetrical) relative to the scale. I suspect this is just what MOS > means.

"MOS" is also "Myhill's property", "distributional evenness", "well formed", or other things in the literature, near enough. Maximally even scales are a subset of MOS. A maximally even scale is made up of no more than two different sizes of scale step, and the larger can't be more than twice the size of the smaller. For a general MOS the ratio of the sizes of the two scale steps can be as big as you like so long as the pattern of scale steps is perserved.

Because maximally even scales are guaranteed to be proper they are relevant for generalized diatonics as well as chords for voice leading. But a maximally even scale has to be a subset of an equal temperament and this isn't a property of proper MOS scales in general. You can still relate a family of MOS scales to a maximally even archetype. They all have the same initial ordering, and so all the other properties Rothenberg derives from it. They can also be described with the same rank 2 temperament mapping.

>>Is this like walking round a circular path, and getting back where you
>>started but knowing you've been somewhere?
> > > Exactly!

<snip>

> You're right, it's nothing to write home about. The only interesting > thing is that pitch intervals are represented by paths -- you can > walk around the circle more than once, when you move by an octave or > more. And voice leadings are represented by collections of paths. > This shows you how the group of real numbers acts on the circular > space -- you have more than one interval linking any two notes.

Octave equivalence is a niggling detail with some of the temperament finding algorithms. It's possible to frame the problem in octave equivalent terms, and to an extent understand the solution the same way. But you have to make the octaves explicit for the intermediate calculations or they get very ugly.

The problem is that division in modulo space has more than one answer. In terms of paths, you could have traveled around the octave any number of times before you reached a perfect fifth. If you then divide those paths into two equal parts they end up at a different interval depending on whether they went round the octave and even or odd number of times. Perhaps it will help to think this way, I don't know, but it's at least something to think about.

Graham

🔗djwolf_frankfurt <djwolf_frankfurt@yahoo.com>

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

> > So equal temperaments are optimal from a voice-leading perspective...
>
>There's a trivial voice-leading from an equal temperament to its
>counterpart moved by any interval of the scale. That means none of the
>notes move, not that they all move by that interval.

Yes, this is right. If you have an equal temperament, then the minimal voice leading between that temperament and its transposition is either 0 (if you're transposing by an interval in the scale), or completely parallel (if not).

So, take the equal temperament {C, E, G#}. There are trivial voice leadings to its T0, T4 and T8 forms. However, the voice leading to every other temperament is completely parallel. For example, the minimal voice leading from {C, E, G#} to its T1, T5, and T9 forms is:

(C, E, G#)->(Db, F, A).

In which all voices move upward by semitone.

> If you move the
>scale by a JI consonance instead, the minimal amount each voice has to
>move is the error in that interval in the given equal temperament. So
>in a sense, voice leading tells you about how good an equal temperament
>is. The problem is that if all the voices move by this minimal amount
>then they all move in the same direction.

Right -- interesting idea. Voice leadings can tell you something about the "error" of a temperament. Of course, for a general temperament you'll need to look at a collection of voice leadings -- transposition by perfect fifth, major third, etc.

>If you take an MOS, and move it by a JI interval, some notes can move by
>the error of the interval in a temperament including that MOS. But
>other notes have to move by a larger step. The more complex the
>interval is in the temperament, the more notes move by the larger step.
> So the voice-leading distance depends on both the errors and
>complexities of the consonances, which makes it a badness measure.

Though, I must say, I have a hard time feeling that the four minor thirds in the diatonic scale are somehow "bad." The variety of interval sizes is also what makes the scale cool. So one might need to think a little about various competing desires including variety.

>My initial thought was that equal temperaments would
>end up as optimal because they're so simple.

Sorry, I might've misunderstood you before. Your intuition here is completely correct. For any transposition Tx, an n-note equal temperament has the smallest possible voice leading to its Tx form. Any other temperament will have a larger voice leading to that Tx form ... (This is in the next-to-last section of the supplementary materials of my paper.)

>It'd be nice if inconsistent temperaments entailed voice crossings but I
>don't think it works like that.

Yeah, well the minimal voice leading is never going to involve crossings. You might be able to get crossings if you specify that common tones be retained. For instance if you want a voice leading from C major to Gb major that retains common tones then you will have crossings.

DT
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WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

I suppose, reflecting on my earlier post, that an equal temperament is always going to minimize the total error, for any interval. This is kind of interesting -- people often talk about homogeneity as the prime value of an equal temperament, but overall error-minimization may also be relevant.

>"MOS" is also "Myhill's property", "distributional evenness", "well
>formed", or other things in the literature, near enough.

Well, herein lies my confusion. The subtle differences between these three properties are relevant to what you can say about their voice leading.

>Maximally even
>scales are a subset of MOS.

Is this really right? I had thought that an MOS scale had to be generated, in which case the octatonic scale is ME but not MOS. That's what's said here:

http://en.wikipedia.org/wiki/MOS_scale

> A maximally even scale is made up of no
>more than two different sizes of scale step, and the larger can't be
>more than twice the size of the smaller. For a general MOS the ratio of
>the sizes of the two scale steps can be as big as you like so long as
>the pattern of scale steps is perserved.

If a MOS scale has to be generated, then it is a slightly more general version of "well formed." It's what Toussaint, et al, refer to as an "Erdos deep" scale, since Erdos discusses a higher-dimensional analogue of the property.

If an MOS scale doesn't have to be generated, then it's a slightly more general property, and includes the ME sets, as well as uneven sets like {C, C#, F#, G}.

BTW, from a voice leading perspective it matters that the step sizes (as well as all the other scalar interval sizes) are close together. So concepts like ME and WF are more directly relevant.

DT
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WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>If I may now be permitted a possibly outrageous assertion, the
>consistancy, efficiency and comfort of fingering for each recognized
>chord type and each voice leading between chords in a given mapping on
>a generalized and open-ended keyboard* is as good a measure of the
>quality of voice leading in that temperament as any other.

I doubt this is going to work.

1) The keyboard itself doesn't tell you what you need to know. If two fingers move by 2 keys, is that a greater or lesser distance than if one finger moves by 4 keys? A measure of voice leading size answers questions like this. Just saying "look at a keyboard" doesn't. All of this is discussed extensively in the first four sections of the "supplementary materials" of my paper.

2) Second, voice leading occurs on non-keyboard instruments. It's not at all clear why we should use keyboard comfort to analyze voice leading in a string quartet, or chorus or indeed any non-keyboard instrument.

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗djwolf_frankfurt <djwolf_frankfurt@yahoo.com>

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:
>
> >If I may now be permitted a possibly outrageous assertion, the
> >consistancy, efficiency and comfort of fingering for each recognized
> >chord type and each voice leading between chords in a given mapping on
> >a generalized and open-ended keyboard* is as good a measure of the
> >quality of voice leading in that temperament as any other.
>
> I doubt this is going to work.
>
> 1) The keyboard itself doesn't tell you what you need to know. If
> two fingers move by 2 keys, is that a greater or lesser distance than
> if one finger moves by 4 keys? A measure of voice leading size
> answers questions like this.

I did not say that we wouldn't pay attention to the distances traveled
by individual fingers. In fact.... (continued below)

Just saying "look at a keyboard"
> doesn't. All of this is discussed extensively in the first four
> sections of the "supplementary materials" of my paper.
>
> 2) Second, voice leading occurs on non-keyboard instruments. It's
> not at all clear why we should use keyboard comfort to analyze voice
> leading in a string quartet, or chorus or indeed any non-keyboard
> instrument.
>

...you need not think of the generalized keyboard as a real, existing,
instrument at all (there are very few of them anyway, so this
shouldn't be a problem). You can throw away the metaphors of fingering
and handposition altogether if you will and substitute number and
distance of moves. The generalized keyboard is just a spatial
description of a scale.

Incidentally, my original draft of this post included a caveat about
the traditional harmony teacher's bias against keyboard voicing in
favor of choral voicing. Voice leading can just as well be represented
on a keyboard if the spacing is open or gapped or closed, and if you
want to notate voice crossings, then notate them.

It occurs to me that you may not have understand my reference here to
generalized keyboards, and I apologize if I was not clear. See Diagram
1 of Wilson's article: http://www.anaphoria.com/xen3b.PDF (this is one
of Wilson's articles sent by pony to Chalmers from his ranch in
Chihuahua).

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>I did not say that we wouldn't pay attention to the distances traveled
>by individual fingers. In fact.... (continued below)

>...you need not think of the generalized keyboard as a real, existing,
>instrument at all (there are very few of them anyway, so this
>shouldn't be a problem). You can throw away the metaphors of fingering
>and handposition altogether if you will and substitute number and
>distance of moves. The generalized keyboard is just a spatial
>description of a scale.

OK, so in your previous post you talked about "efficiency and comfort of fingering." This led me to think you were talking about physical motion. But let's throw away the fingering metaphor.

Any spatial representation of a scale is going to represent a voice leading as a collection of distances: "C moves up one chromatic step to C#, E doesn't move and remains fixed, and G moves up two chromatic steps to A."

Or: "E moves up one diatonic step to F, C doesn't move and remains fixed, and G moves up one diatonic step to A."

Or you might have some other set of distances, derived perhaps from the 2D layouts in the Wilson article you sent me.

This is the data that any method of measuring voice leading size has to begin with. The question that any measure has to answer is: how do we compare two *sets* of scale-distances? In other words, how do we compare the following:

A. a voice leading in which one note moves up by semitone, one note is held constant, and one note moves up by two semitones; or
B. a voice leading in which two notes are held constant, and one note moves by three semitones.

Different measures of voice leading size deliver different answers: some say these two are equal, some say the first is bigger; some say the second is bigger. Do you just add up the distances? Or do you square the distances, add them, and take the square root? Or maybe we should just take the largest distance moved by any individual voice! All of these remain options, regardless of what keyboard you're representing things on.

The keyboard metaphor does not yet add any content whatseover. It might give you a different conception of the distance that each individual voice moves, but it does not tell you how to compare sets of distances to one another.

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗Graham Breed <gbreed@gmail.com>

Dmitri Tymoczko wrote:
> I suppose, reflecting on my earlier post, that an equal temperament > is always going to minimize the total error, for any interval. This > is kind of interesting -- people often talk about homogeneity as the > prime value of an equal temperament, but overall error-minimization > may also be relevant.

You must have additional assumptions here, like an interval has to be transposed to each scale step, you're minimizing the worst error in such an interval, and there's a fixed number of notes.

>>"MOS" is also "Myhill's property", "distributional evenness", "well
>>formed", or other things in the literature, near enough.
> > Well, herein lies my confusion. The subtle differences between these > three properties are relevant to what you can say about their voice > leading.

The subtle difference is whether or not the period has to equal the octave. The trouble is that the authors who come up with these terms aren't always clear about it. The current thinking is that an MOS needn't have an octave period, although it does for all published examples, and from what I remember the last word from Erv is that both generators should be consonances. Myhill's property does require an octave period. Distributional evenness probably doesn't. Carey and Clampitt, who defined well formed scales, said "By interval of periodicity we mean an interval whose two boundary pitches are functionally equivalent. Normally, the octave is the interval of
periodicity" whatever that's supposed to mean. Then they say non-degenerate (unequal) well formed scales are the same thing as Myhill's property.

>>Maximally even
>>scales are a subset of MOS.
> > Is this really right? I had thought that an MOS scale had to be > generated, in which case the octatonic scale is ME but not MOS. > That's what's said here:
> > http://en.wikipedia.org/wiki/MOS_scale

That's a good example of a vague document that doesn't address the issue. (It's also incorrect to say the term MOS is unpublished -- it's used in Xenharmonikon and John Chalmers' book. Perhaps I'll edit that.)

An octatonic scale is an MOS with a period of a minor third.

>> A maximally even scale is made up of no
>>more than two different sizes of scale step, and the larger can't be
>>more than twice the size of the smaller. For a general MOS the ratio of
>>the sizes of the two scale steps can be as big as you like so long as
>>the pattern of scale steps is perserved.
> > If a MOS scale has to be generated, then it is a slightly more > general version of "well formed." It's what Toussaint, et al, refer > to as an "Erdos deep" scale, since Erdos discusses a > higher-dimensional analogue of the property.

He does? Another term to add to the mix then.

> If an MOS scale doesn't have to be generated, then it's a slightly > more general property, and includes the ME sets, as well as uneven > sets like {C, C#, F#, G}.

An MOS does have to be generated, but that set is an MOS in 12-equal with a period of a half-octave and a generator of a semitone.

> BTW, from a voice leading perspective it matters that the step sizes > (as well as all the other scalar interval sizes) are close together. > So concepts like ME and WF are more directly relevant.

Propriety does that, even if it wasn't Rothenberg's motivation. Agmon proved that ME implies propriety, and I worked out a simpler proof along with some other proofs of obvious things:

http://x31eq.com/proof.html

You can tell if an MOS is proper by looking at the ratio of its step sizes.

Graham

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Carl Lumma <ekin@lumma.org>

[Daniel Wolf wrote...]
>If I may now be permitted a possibly outrageous assertion, the
>consistancy, efficiency and comfort of fingering for each recognized
>chord type and each voice leading between chords in a given mapping on
>a generalized and open-ended keyboard* is as good a measure of the
>quality of voice leading in that temperament as any other.

I'm not so sure about this. Unfortunately, fingers are not all
the same length, and on the Scalatron, for example, this means minor
triads are fingered differently than major ones... there's a host
of asymmetries introduced by the peculiarities of the hand....

>Moreover,
>the keyboard mapping makes some additional voice leading distinctions
>clear: first, between moves in which at least one finger doesn't
>move, and those in which the whole hand moves.

If your goal is determining the difficulty of playing something
on a keyboard, this makes sense. If the goal is difficulty of
singing, I don't think it's as applicable. If the goal is a broader
theoretical one (as suggested in Dmitri's work), I'm not sure
it's applicable either. But examples win the day.

>*This is _not_ true on a Halberstadt keyboard, which is neither
>generalized nor open-ended.

I've always though it'd be neat if one could show a difference in
the voice leading of Bach's keyboard works vs. his ensemble ones.
Was he just writing stuff that happened to be possible at the
keyboard (or is anything possible given enough determination)? Or
was he writing keyboard-possible music for choirs and such, because
he was a 'keyboard composer'? Or, was he so good that he made sure
his keyboard works were keyboard-playable, while going nuts with his
ensemble music? We know he transcribed ensemble pieces for keyboard
and vice versa, but on balance.......

-Carl

🔗Carl Lumma <ekin@lumma.org>

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

🔗Carl Lumma <ekin@lumma.org>

It seems to me the main benefit of using a bosanquet lattice
over a standard harmonic lattice that's been tilted to agree
with pitch height (like...

http://www.redshift.com/~dcanright/eps4ji/harmela2.htm

...) is that octaves are properly represented without having
to resort to things like the grey offset lattice above.

-C.

At 12:58 PM 7/17/2006, you wrote:
>BTW, it seems to me that if we're measuring voice-leading distances
>in pitch space, we want to obey the following elementary principles:
>
>1) As you make a glissando from pitch x to pitch y, you are
>continuously getting farther from x and closer to y.
>
>2) Distances in pitch space should be invariant under transposition
>in log frequency space.
>
>These principles determine a unique measure of musical distance, up
>to a simple multiplicative constant. Most lattice-based measures
>violate the first principle. For this reason they strike me as
>better-suited for investigating harmonic features than voice leading
>distances.
>
>This is basically why I don't think it's worth fussing with, e.g.
>Bosanquet lattices, hexagonal keyboards, etc. If we want to satisfy
>these principles, we're stuck with log-frequency space.
>
>DT
>--
>WARNING: Princeton Email is currently very unreliable. If you need
>to reach me quickly, you should call me.
>
>Dmitri Tymoczko
>Assistant Professor of Music, Princeton University
>Radcliffe Institute for Advanced Study
>34 Concord Ave.
>Cambridge, MA 02138
>FAX: (617) 495 8136
>http://music.princeton.edu/~dmitri
>
>
>
>
>Yahoo! Groups Links
>
>
>
>

🔗Graham Breed <gbreed@gmail.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Graham Breed <gbreed@gmail.com>

🔗Carl Lumma <ekin@lumma.org>

>>>>>A maximally even scale is made up of no
>>>>>more than two different sizes of scale step, and the larger can't be
>>>>>more than twice the size of the smaller.
>>>>
>>>>Not that it matters, but I don't believe this is technically
>>>>correct. They simply must be consecutive steps of an ET.
>>>
>>>Technically, yes, you could say that an equal temperament is a maximally
>>>even scale with a small step size of zero. Otherwise, how do you plan
>>>to solve
>>>
>>>x - y = 1
>>>
>>>for x/y > 2 with positive integers?
>>
>>
>> I haven't a clue what you're talking about. The definition
>> of ME is what I said. Read the paper.
>
>I don't have the paper. If you do, can you please quote it instead of
>talking riddles?

I thought you were around here a few weeks ago when all this
was mentioned.

This is from Clough, Engebretsen, Kochavi "Scales, Sets, and
Interval Cycles: A Taxonomy", which cites Wilson's letter to
Chalmers, incidentally.

""
well-formed - G-set where each generating interval
spans a constant number of scale steps.

Myhill property - Each generic interval comes in
two specific sizes.

distributionally even - Each generic interval comes in
either one or two specific sizes.

maximally even - Each generic interval comes in either
one integer size or two consecutive integer sizes.
""

-Carl

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

> > I suppose, reflecting on my earlier post, that an equal temperament
>> is always going to minimize the total error, for any interval. This
>> is kind of interesting -- people often talk about homogeneity as the
>> prime value of an equal temperament, but overall error-minimization
>> may also be relevant.
>
>You must have additional assumptions here, like an interval has to be
>transposed to each scale step, you're minimizing the worst error in such
>an interval, and there's a fixed number of notes.

Yes, you need additional assumptions -- you're looking at finite scales and asking for there to be a *unique* interval of type X above every note in the scale. That is, for every scale degree p, you want there to be another scale degree q, such that q is to be treated as the scale's representation of "the interval of type X above p" and it does not play that role for any other scale degree r.

Interestingly, you don't have to assume that you want to minimize the worst error -- you can minimize the sum of the errors, or the sum of the squared errors, or many many other plausible measures of error. In each case, equal temperament does best.

>The subtle difference is whether or not the period has to equal the
>octave. The trouble is that the authors who come up with these terms
>aren't always clear about it.

This is why you guys ought to try to publish your stuff once in a while! ;-)

>An octatonic scale is an MOS with a period of a minor third.

I still think that the MOS scales are not going to contain all the ME scales -- because the following set is ME in 24 tone equal temperament:

{0, 2, 4, 5, 7, 9, 11, 12, 14, 16, 17, 19, 21, 23} (two octaves of the standard major scale).

But it's not generated, no matter what you take the periodicity to be. (Its two halves are individually generated, but the whole isn't.)

>The current thinking is that an MOS
>needn't have an octave period, although it does for all published
>examples, and from what I remember the last word from Erv is that both
>generators should be consonances.

Personally, I'd recommend cleaning up the terminological mess as follows:

A well-formed scale ("MOS" scale) has scalar intervals that come in at most two sizes. An ME scale is simply an MOS scale that is as even as possible, relative to some "chromatic" scale in which it is embedded. (We use the scale-step metric to judge evenness here.)

Well-formed scales, thus defined, come in three categories:

1. "Obviously generated" scales -- a sequence of n notes each x units above its predecessor, where nx < the octave. Example: {C, C#, D}.
2. "Non-obviously generated" scales -- as above, but here nx is greater than an octave. (Formula to follow.) Example: {C, D, G}
3. Transpositionally degenerate scales: either of the first two types, multiply embedded within an octave in a transpositionally symmetrical way.
Example 3a: {C, C#, D, F#, G, Ab}.
Example 3b: {0, 2, 7, 12, 14, 19} in 24tet.

Then you have the very nice, clean result that ME scale is just the most even MOS scale in a given temperament. Furthermore, you have the non-obvious proof that the most even set of any cardinality is always MOS.

By the way, I was wrong: Toussaint's "Erdos deep" scales are just generated scales.

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗djwolf_frankfurt <djwolf_frankfurt@yahoo.com>

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:
>
>
> This is why you guys ought to try to publish your stuff once in a
while! ;-)
>

Dmitri, there's a long, hard, history to this. While it's true that
some people working in tuning do not write in a standard academic
format, there have been several who have and the work has simply been
rejected.

Walter O'Connell, for example, could not get published in the US, but
was published in _die Reihe_, a journal effectively sealed with the
PNM kiss of academic death. Chalmers's _Divisions of the Tetrachord_
was published by a composers' publishing cooperative. In the 60s and
70s, professional music theory in the US meant either history of
theory, Schenker, or Sets, so tuning was a marginal concern (the same
went for non-Shenkerian tonal analysis, although this was not the case
elsewhere). Even mentioning an interest in tuning in a college
admissions interview was a ticket to the exit door in most schools
back then. Before email, the alternative had been setting up our own
journals. Xenharmonikon has been around since the early 70's, and a
few University libraries have always subscribed -- Berkeley, Wesleyan
-- I was editor of XH for a short time and sent, at my own expense,
complete sets of back issues to several University libraries.
Princeton was among those that simply trashed them. Oh well, my
student loan was only 9%...

But the establishment of the original Mills-based tuning list (in '94
or '95?) and the cluster of lists around it has created a lively
scene, with a changing cast of characters with different interests and
abilities. (In my experience, the list closest to it in spirit is the
Waste.org Pynchon list). There is substantial theoretical work which
has been done here; and I would identify papers by Erlich and Jonathan
Walker, materials online by Paul Hahn (on consistancy, completeness
and diameter), Schulter, Smith, Lumma, and Breed, the Scala program by
Op de Coul, de Laubenfel's adaptive tuning project, and the work of
some others who were more active earlier on -- Chalmers, Sethares,
Alves, Polansky -- or have not been on the list directly -- Wilson,
Tenney, Barlow --, all of whom have published more conventionally.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:

> 1. Composers use scalar transposition all the time: "do, a dear"
> becomes "re, a drop." This process is absolutely ubiquitous in
> Western music, and it is considered a kind of transposition
> (translation). For it to be so, you need to use the scale-based
> metric. This "scale step" metric is used everywhere, in every tonal
> piece.

"Must" is a strong claim, and in this case false, I think.

Suppose we notate the notes of meantone in the seven nominals plus
octaves plus accidentals notation. It is possible to mechanically
translate this to and from notations such as 2^a f^b, where f is the
flat fifth, or other meantone notations. We can now define a "scalar"
modal shift operator Sn on all the notes of meantone, by adding n
sharps or (if n is negative) -n flats in the correct manner to the
nominals. The notes can also be notated in this way, and the result,
if you wished, could be converted back to the former system, by moving
the extra sharps or flats from the nominals back to the sharps/flats
counter.

This may strike you as a completely artificial proceedure, but once we
have decided on octave equivalence, which makes octaves special, it
really isn't. When we do that many rank two temperaments become octave
+generator temperaments (if the octave can be used as one of the
generators) and then you get MOS for it. You can define the same sorts
of modal shift operators in this more general case. If this is too
artifical, I suggest that at least it is *less* artificial than
scale-based metrics.

Once again, this is a kind of broken symmetry, from the pint of view
of the <7 11 16| val modal shifts are identity mappings. If we notate
meantone using <7 11 16| and another val, all of the changes take
place with respect to the values of the other val.

> 2. This perspective allows you to find interesting parallelism of
> musical structures. For instance, suppose you made a voice leading
> map of diatonic trichords (measuring voice leading in scale steps).
> You'd find it was basically isomorphic to the voice leading map of
> familiar near-diatonic scales (measuring voice leading in chromatic
> scale steps). See Appendix I of "Scale Networks and Debussy."

This, I think, is clearly something that generalizes also, but I don't
think you need to introduce unusual metrics to do it.

> >I don't think Lewin made any assumption that a musical space was
> >principal, by the way.
>
> He unquestionably did. Look at the definition of a Generalized
> Interval System in "Generalized Musical Intervals and
> Transformations."

Sorry, I was relying on my evidently unreliable memory. Our library
seems to have lost its copy of Lewin.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:
>
> BTW, it seems to me that if we're measuring voice-leading distances
> in pitch space, we want to obey the following elementary principles:
>
> 1) As you make a glissando from pitch x to pitch y, you are
> continuously getting farther from x and closer to y.
>
> 2) Distances in pitch space should be invariant under transposition
> in log frequency space.

We don't, however, need to measure voice leading distance in pitch
space (or pitch class space, which would be another possibility.) We
can measure voice leading with respect to one or more equal
temperament vals, which are only approximately logarithmic mappings.
This has its uses.

> These principles determine a unique measure of musical distance, up
> to a simple multiplicative constant. Most lattice-based measures
> violate the first principle. For this reason they strike me as
> better-suited for investigating harmonic features than voice leading
> distances.

The real trick is to do both at the same time.

> This is basically why I don't think it's worth fussing with, e.g.
> Bosanquet lattices, hexagonal keyboards, etc. If we want to satisfy
> these principles, we're stuck with log-frequency space.

And if we want to consider harmonic features also, we may not be
well-served simply by paying attention to cents and nothing else.

🔗Carl Lumma <ekin@lumma.org>

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗Graham Breed <gbreed@gmail.com>

Dmitri Tymoczko wrote:
>> > I suppose, reflecting on my earlier post, that an equal temperament
>>
>>> is always going to minimize the total error, for any interval. This
>>> is kind of interesting -- people often talk about homogeneity as the
>>> prime value of an equal temperament, but overall error-minimization
>>> may also be relevant.
>>
>>You must have additional assumptions here, like an interval has to be
>>transposed to each scale step, you're minimizing the worst error in such
>>an interval, and there's a fixed number of notes.
> > Yes, you need additional assumptions -- you're looking at finite > scales and asking for there to be a *unique* interval of type X above > every note in the scale. That is, for every scale degree p, you want > there to be another scale degree q, such that q is to be treated as > the scale's representation of "the interval of type X above p" and it > does not play that role for any other scale degree r.

The value of an equal temperament is that you can take an interval and transpose it to any other degree of a fixed scale without it getting any worse. Historically, homogeneity seems to have been a *bad* thing because it takes away key character. But with more transposition, musicians drifted towards tunings where the worst intervals weren't as bad.

> Interestingly, you don't have to assume that you want to minimize the > worst error -- you can minimize the sum of the errors, or the sum of > the squared errors, or many many other plausible measures of error. > In each case, equal temperament does best.

In general, the average deviation for an MOS doesn't depend on the generator. That means the sum of errors is also constant provided all tempered intervals are in the same direction of where they should be. The sum of squared errors looks more like the worst error, and so it favours equal temperament. But there's nothing stopping you taking the sum of square root errors to favour lumpy scales.

>>The subtle difference is whether or not the period has to equal the
>>octave. The trouble is that the authors who come up with these terms
>>aren't always clear about it.
> > This is why you guys ought to try to publish your stuff once in a while! ;-)

Paul Erlich's forthcoming Xenharmonikon article covers temperaments where the period doesn't equal an octave. Unfortunately, he wrote that when he believed that an MOS required an octave period. He's also "dubious" about whether such things are linear temperaments.

>>An octatonic scale is an MOS with a period of a minor third.
> > > I still think that the MOS scales are not going to contain all the ME > scales -- because the following set is ME in 24 tone equal > temperament:
> > {0, 2, 4, 5, 7, 9, 11, 12, 14, 16, 17, 19, 21, 23} (two octaves of > the standard major scale).
> > But it's not generated, no matter what you take the periodicity to > be. (Its two halves are individually generated, but the whole isn't.)

I don't see a problem. It's an MOS with a half-octave periodicity and a generator of 5 steps.

> Personally, I'd recommend cleaning up the terminological mess as follows:
>
> A well-formed scale ("MOS" scale) has scalar intervals that come in > at most two sizes. An ME scale is simply an MOS scale that is as > even as possible, relative to some "chromatic" scale in which it is > embedded. (We use the scale-step metric to judge evenness here.)

I much prefer the definition of a "b from a" ME scale as

p(n) = floor(na/b)

because it doesn't require you to count intervals or identify a period and generator. So it's the scale that's as even as possible and happens to be an MOS.

I notice your ME definition in voiceleading.pdf doesn't require b<=a, so it's possible to get the whole of a-equal with duplicates.

> Well-formed scales, thus defined, come in three categories:
> > 1. "Obviously generated" scales -- a sequence of n notes each x units > above its predecessor, where nx < the octave. Example: {C, C#, D}.

That might be useful because some important scales like the decimal scale in miracle are of this type. Another interesting special case is where you only have one generator step per period. That's true of the octatonic scale, and also temperaments like the 58 note MOS of mystery with two parallel scales of 29-equal.

> 2. "Non-obviously generated" scales -- as above, but here nx is > greater than an octave. (Formula to follow.) Example: {C, D, G}
> 3. Transpositionally degenerate scales: either of the first two > types, multiply embedded within an octave in a transpositionally > symmetrical way.
> Example 3a: {C, C#, D, F#, G, Ab}.
> Example 3b: {0, 2, 7, 12, 14, 19} in 24tet.

The trouble is, the "degenerate" in "degenerate well formed" doesn't mean this, although I think I thought it did once. Best to say "an MOS where the period isn't an octave". (It should be obvious that you're defining "MOS" so that the period doesn't have to be an octave.)

> Then you have the very nice, clean result that ME scale is just the > most even MOS scale in a given temperament. Furthermore, you have > the non-obvious proof that the most even set of any cardinality is > always MOS.

Yes. Given equal temperament.

Graham

🔗Graham Breed <gbreed@gmail.com>

Carl Lumma wrote:
>>>>>>A maximally even scale is made up of no >>>>>>more than two different sizes of scale step, and the larger can't be >>>>>>more than twice the size of the smaller.
>>>>>
>>>>>Not that it matters, but I don't believe this is technically
>>>>>correct. They simply must be consecutive steps of an ET.
>>>>
>>>>Technically, yes, you could say that an equal temperament is a maximally >>>>even scale with a small step size of zero. Otherwise, how do you plan >>>>to solve
>>>>
>>>>x - y = 1
>>>>
>>>>for x/y > 2 with positive integers?
>>>
>>>
>>>I haven't a clue what you're talking about. The definition
>>>of ME is what I said. Read the paper.
>>
>>I don't have the paper. If you do, can you please quote it instead of >>talking riddles?
> > I thought you were around here a few weeks ago when all this
> was mentioned.

Yes, but I didn't notice anything that contradicted my definition, which I got from Agmon.

> This is from Clough, Engebretsen, Kochavi "Scales, Sets, and
> Interval Cycles: A Taxonomy", which cites Wilson's letter to
> Chalmers, incidentally.

It's not the original paper so it doesn't have the canonical definition.

> ""
> well-formed - G-set where each generating interval
> spans a constant number of scale steps.
> > Myhill property - Each generic interval comes in
> two specific sizes.
> > distributionally even - Each generic interval comes in
> either one or two specific sizes.
> > maximally even - Each generic interval comes in either
> one integer size or two consecutive integer sizes.
> ""

So what? That's not what you said before, and it's obviously the same as my definition of

p(n) = floor(na/b)

for 0 <= b < a that I use for the proofs page. I even prove that it's implied by my definition. I don't prove the opposite but I'm guessing somebody has.

So, let's go back to what you said wasn't technically correct

>>>>>>A maximally even scale is made up of no
>>>>>>more than two different sizes of scale step, and the larger can't be
>>>>>>more than twice the size of the smaller.

No more than two is correct. It can be one. The second part only applies where the two steps have different sizes, so

x - y = 1

for diatonic steps of x and y chromatic steps. I take it that's what "consecutive integer sizes" means. The larger being no more than twice the size of the smaller means

x <= 2y

x/y <= 2

If y is zero this doesn't work, as I said before. You can ponder the definitions to decide if it would still count as maximally even. That aside, substitute the formalization of *your* preferred definition to get

(y+1)/y <= 2
1 + 1/y <= 2
1/y <= 1
1 <= y

which is true for all positive integer values of y. So what I said is correct. What's the problem?

Graham

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

> > This is why you guys ought to try to publish your stuff once in a
>while! ;-)
>>
>
>Dmitri, there's a long, hard, history to this. While it's true that
>some people working in tuning do not write in a standard academic
>format, there have been several who have and the work has simply been
>rejected.

I know. I was being rueful. Academic music theorists are often insular and small minded, and the problem was a lot worse a few decades ago.

For what it's worth, my own first paper ("Consecutive Semitone Constraint") was rejected 3 times, the last time so rudely that I gave up on the idea of publishing it. One day, out of the blue I got an email from a theorist at Eastman to whom I'd sent the paper years before -- he was cleaning his office, found the paper at the bottom of a large pile, and before throwing it away, decided to read it. He liked it, and asked whether I was still interested in publishing it. And that's how I became a composer/theorist rather than just a composer.

Two further thoughts:

1) For a term like "MOS" to be really useful, we have to know exactly what it means. My sense from the list is that the meaning of the term isn't fixed. Does it include all the ME sets? Does it include the octatonic scale? Does it only apply to generated sets? Etc. Everyone has a slightly different idea. Publication or no, these issues need to be settled.

2) It seems like the tuning community is using Wikipedia as an alternative to publication. I have some qualms about this -- the music articles in Wikipedia have a heavy tuning-list slant, and, by emphasizing terms like "MOS scale," they often give a misleading impression about the centrality of these terms to music scholarship.

Please keep in mind that I like you people, and respect you, and think you should get your due recognition. I think it would be good to try to bring the tuning community closer to the academic theory community. For example, someone should write a careful, accurate history of the intersection between tuning theory and more academic Clough-related scale work. I suspect you could get the thing published, given the more open-minded state of the theory community today. I'd certainly be happy to read such a history, and to make suggestions about where to send it and how to pitch it.

The more mathematically inclined among you should be aware of the new Journal of Mathematics and Music, and of semi-regular "music and math" sessions being run at the AMS. Perhaps this would be a place where you could present some of your work.

DT
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🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>Suppose we notate the notes of meantone in the seven nominals plus
>octaves plus accidentals notation. It is possible to mechanically
>translate this to and from notations such as 2^a f^b, where f is the
>flat fifth, or other meantone notations. We can now define a "scalar"
>modal shift operator Sn on all the notes of meantone, by adding n
>sharps or (if n is negative) -n flats in the correct manner to the
>nominals. The notes can also be notated in this way, and the result,
>if you wished, could be converted back to the former system, by moving
>the extra sharps or flats from the nominals back to the sharps/flats
>counter.
>
>This may strike you as a completely artificial proceedure,

Yes that it does. Furthermore, I doubt you'd really hold onto this position if you thought about it for a while.

You can turn all sorts of contortions to avoid talking about "scale steps" if you want -- but whatever you do, if it is to reflect musical practice, will be equivalent to talking about translation relative to a scale-step metric. Furthermore, the latter description much more closely reflects the way composers talk and think.

Composers use scalar transposition all the time -- they're constantly taking a musical pattern (say, a triad) and shifting it up some number of scale steps. They say, e.g. "let's take this up a minor third," or "here, Palestrina shifts the headmotive up a step, from Dorian to Phrygian." The language here is patently that of scale degrees, and of translations with regard to the scale-degree metric.

Furthermore, let's not forget that composers from Bach to Debussy and beyond often shift a melodic pattern from one scale to another! Again, this has an extremely natural description in the perspective I advocate: you're taking a single pattern of intervals and presenting them in the context of multiple scales.

>but once we
>have decided on octave equivalence, which makes octaves special, it
>really isn't. When we do that many rank two temperaments become octave
>+generator temperaments (if the octave can be used as one of the
>generators) and then you get MOS for it. You can define the same sorts
>of modal shift operators in this more general case. If this is too
>artifical, I suggest that at least it is *less* artificial than
>scale-based metrics.

It much less naturally maps on to the kind of concepts musicians are thinking about. Furthermore, if it reflects what musicians actually do, then it is formally equivalent to the scale-degree metric described above.

>Once again, this is a kind of broken symmetry, from the pint of view
>of the <7 11 16| val modal shifts are identity mappings. If we notate
>meantone using <7 11 16| and another val, all of the changes take
>place with respect to the values of the other val.

Sorry, I have no idea what you're talking about here -- remember, I'm an outsider.

DT
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Dmitri Tymoczko
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🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

> >Personally, I'd recommend cleaning up the terminological mess as follows:
>>
>>A well-formed scale ("MOS" scale) has scalar intervals that come in
>>at most two sizes.
>
>That's DE.

Yes, I realized that after I sent the post. The one issue I have here is that "Distributionally Even" is a terrible name for the term -- the point is that DE scales can be very even or very uneven. My mind recoils at the thought that {C, C#, D} is even in any way!

More generally, my point was that the DE property (which I'd like to call something else) is the deep and robust one -- it's embedded very naturally in the geometry of the orbifolds, and it maps very naturally onto the classifications and terminology associated with maximal evenness. It's easiest to think of MOS/well-formedness as a special case of DE.

> >An ME scale is simply an MOS scale that is as
>>even as possible, relative to some "chromatic" scale in which it is
>>embedded. (We use the scale-step metric to judge evenness here.)
>
>Vague. And ME should be deprecated in favor of DE.

No, it's not vague. "Evenness" can be very precisely defined in terms of voice leading distance from the nearest equal-tempered/EDO chord.

And ME and DE are very different things -- they have different voice leading properties. For example, in any equal tempered scale, the maximally even set will have the minimal voice leading to all its transpositions. This is not true of the DE sets.

The real issue here is that there are two properties: evenness (which is a continuous quality, not a binary yes/no kind of quality), and the special intervallic structures associated with the DE sets. The ME sets are the DE sets that maximize this quantity, evenness. The nontrivial thing is that they also maximize evenness even when you look at the non-DE sets.

> >Toussaint's "Erdos deep" scales are just
> >generated scales.
>
>Sweet jesus. :)

Yeah, it's a bit much -- but Toussaint has a very nice paper relating some common music-theory questions to mathematical work.

DT
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Dmitri Tymoczko
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🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>We don't, however, need to measure voice leading distance in pitch
>space (or pitch class space, which would be another possibility.) We
>can measure voice leading with respect to one or more equal
>temperament vals, which are only approximately logarithmic mappings.
>This has its uses.

I'm sure, and I'd be curious to know what they are. However, I suspect that we much more frequently and commonly need to measure distances in pitch space -- especially when voice leading is our concern.

> > These principles determine a unique measure of musical distance, up
>> to a simple multiplicative constant. Most lattice-based measures
>> violate the first principle. For this reason they strike me as
>> better-suited for investigating harmonic features than voice leading
>> distances.
>
>The real trick is to do both at the same time.

Well, sure -- no argument here. We need to think about music in many ways, often at the same time!

DT
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Dmitri Tymoczko
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Cambridge, MA 02138
FAX: (617) 495 8136
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🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

> > Interestingly, you don't have to assume that you want to minimize the
>> worst error -- you can minimize the sum of the errors, or the sum of
>> the squared errors, or many many other plausible measures of error.
>> In each case, equal temperament does best.
>
>In general, the average deviation for an MOS doesn't depend on the
>generator. That means the sum of errors is also constant provided all
>tempered intervals are in the same direction of where they should be.
>The sum of squared errors looks more like the worst error, and so it
>favours equal temperament.

I don't think this has much to do with MOS or intervals of generation or consonance per se.

Suppose I want a five note scale such that every note has a unique other note that is as close as possible to being a tritone above it. I want to minimize the total error in one of a very number of general ways. What's the best I can do? A five note equal-tempered scale. For many measures of error, the five-note equal tempered scale will beat other five-note generated collections (or MOS collections, or what have you) such as {C, C#, D, D#, E}.

The same goes for any interval -- quarter tone, just fifth, whatever.

>But there's nothing stopping you taking the
>sum of square root errors to favour lumpy scales.

You're right that I'm requiring a measure that satisfies what I call the "distribution constraint." These measures are the ones for which "voice crossings" never reduce the size of a voice leading. For what it's worth, that seems like it could be relevant here.

> {0, 2, 4, 5, 7, 9, 11, 12, 14, 16, 17, 19, 21, 23} (two octaves of
> > the standard major scale).
>>
>> But it's not generated, no matter what you take the periodicity to
>> be. (Its two halves are individually generated, but the whole isn't.)
>
>I don't see a problem. It's an MOS with a half-octave periodicity and a
>generator of 5 steps.

You can do it this way -- this requires you to first "wrap around" the half octave (going from 5 to 0 to 7 to 2, etc.) so that the half octave acts like a modulus. Then, having generated half the pattern in this way, you change your mind and transpose the whole thing up a half-octave to fill out the actual modulus.

The point is that during the derivation you need two separate generators, and two separate moduli. One generator is the interval 5, the other is the interval 12. One modulus is the interval 12, the other is 24. For this reason I am uncomfortable talking about "the" modulus, or "the" generator of the scale {0, 2, 4, 5, 7, 9, 11, 12, 14, 16, 17, 19, 21, 23}.

I'm just saying that the terminological shoe is pinching a bit uncomfortably here.

> > A well-formed scale ("MOS" scale) has scalar intervals that come in
>> at most two sizes. An ME scale is simply an MOS scale that is as
>> even as possible, relative to some "chromatic" scale in which it is
>> embedded. (We use the scale-step metric to judge evenness here.)
>
>I much prefer the definition of a "b from a" ME scale as
>
>p(n) = floor(na/b)
>
>because it doesn't require you to count intervals or identify a period
>and generator. So it's the scale that's as even as possible and happens
>to be an MOS.

Sure, fine -- whatever you like.

>I notice your ME definition in voiceleading.pdf doesn't require b<=a, so
>it's possible to get the whole of a-equal with duplicates.

Yup, correct.

>The trouble is, the "degenerate" in "degenerate well formed" doesn't
>mean this, although I think I thought it did once. Best to say "an MOS
>where the period isn't an octave". (It should be obvious that you're
>defining "MOS" so that the period doesn't have to be an octave.)

Well I don't like this for the reasons expressed above.

> > Then you have the very nice, clean result that ME scale is just the
>> most even MOS scale in a given temperament. Furthermore, you have
>> the non-obvious proof that the most even set of any cardinality is
>> always MOS.
>
>Yes. Given equal temperament.

Or the "scale step" metric which converts any scale to an equal temperament.

DT
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Cambridge, MA 02138
FAX: (617) 495 8136
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🔗Carl Lumma <ekin@lumma.org>

>> This is from Clough, Engebretsen, Kochavi "Scales, Sets, and
>> Interval Cycles: A Taxonomy", which cites Wilson's letter to
>> Chalmers, incidentally.
>
>It's not the original paper so it doesn't have the canonical definition.

It's a paper from the originator of the concept that specifically
addresses the question of defining these things. It's the paper
Clough himself pointed us to when this came up.

>> ""
>> well-formed - G-set where each generating interval
>> spans a constant number of scale steps.
>>
>> Myhill property - Each generic interval comes in
>> two specific sizes.
>>
>> distributionally even - Each generic interval comes in
>> either one or two specific sizes.
>>
>> maximally even - Each generic interval comes in either
>> one integer size or two consecutive integer sizes.
>> ""
>
>So what? That's not what you said before,

Oh no?

>So, let's go back to what you said wasn't technically correct
>
> >>>>>>A maximally even scale is made up of no
> >>>>>>more than two different sizes of scale step, and the larger
> >>>>>>can't be more than twice the size of the smaller.
>
>No more than two is correct. It can be one. The second part only
>applies where the two steps have different sizes, so
>
>x - y = 1
>
>for diatonic steps of x and y chromatic steps. I take it that's what
>"consecutive integer sizes" means.

It means they're integers. There's nothing in the text I called
incorrect that wouldn't apply to a scale with steps of 113.459 cents
and 173.678 cents. It also means they're consecutive. Scales with
L=5 and s=3 aren't ME.

-Carl

🔗Kraig Grady <kraiggrady@anaphoria.com>

pardon me if i am being dense on this.
Voice leading traditionally is based on 4 voices not 3.
In practice at least of ones these notes is doubled in another octave and moves in a direction different from it octavean pair

Dmitri Tymoczko wrote:
>
> > > Interestingly, you don't have to assume that you want to minimize the
> >> worst error -- you can minimize the sum of the errors, or the sum of
> >> the squared errors, or many many other plausible measures of error.
> >> In each case, equal temperament does best.
> >
> >In general, the average deviation for an MOS doesn't depend on the
> >generator. That means the sum of errors is also constant provided all
> >tempered intervals are in the same direction of where they should be.
> >The sum of squared errors looks more like the worst error, and so it
> >favours equal temperament.
>
> I don't think this has much to do with MOS or intervals of generation
> or consonance per se.
>
> Suppose I want a five note scale such that every note has a unique
> other note that is as close as possible to being a tritone above it.
> I want to minimize the total error in one of a very number of general
> ways. What's the best I can do? A five note equal-tempered scale.
> For many measures of error, the five-note equal tempered scale will
> beat other five-note generated collections (or MOS collections, or
> what have you) such as {C, C#, D, D#, E}.
>
> The same goes for any interval -- quarter tone, just fifth, whatever.
>
> >But there's nothing stopping you taking the
> >sum of square root errors to favour lumpy scales.
>
> You're right that I'm requiring a measure that satisfies what I call
> the "distribution constraint." These measures are the ones for which
> "voice crossings" never reduce the size of a voice leading. For what
> it's worth, that seems like it could be relevant here.
>
> > {0, 2, 4, 5, 7, 9, 11, 12, 14, 16, 17, 19, 21, 23} (two octaves of
> > > the standard major scale).
> >>
> >> But it's not generated, no matter what you take the periodicity to
> >> be. (Its two halves are individually generated, but the whole isn't.)
> >
> >I don't see a problem. It's an MOS with a half-octave periodicity and a
> >generator of 5 steps.
>
> You can do it this way -- this requires you to first "wrap around"
> the half octave (going from 5 to 0 to 7 to 2, etc.) so that the half
> octave acts like a modulus. Then, having generated half the pattern
> in this way, you change your mind and transpose the whole thing up a
> half-octave to fill out the actual modulus.
>
> The point is that during the derivation you need two separate
> generators, and two separate moduli. One generator is the interval
> 5, the other is the interval 12. One modulus is the interval 12, the
> other is 24. For this reason I am uncomfortable talking about "the"
> modulus, or "the" generator of the scale {0, 2, 4, 5, 7, 9, 11, 12,
> 14, 16, 17, 19, 21, 23}.
>
> I'm just saying that the terminological shoe is pinching a bit
> uncomfortably here.
>
> > > A well-formed scale ("MOS" scale) has scalar intervals that come in
> >> at most two sizes. An ME scale is simply an MOS scale that is as
> >> even as possible, relative to some "chromatic" scale in which it is
> >> embedded. (We use the scale-step metric to judge evenness here.)
> >
> >I much prefer the definition of a "b from a" ME scale as
> >
> >p(n) = floor(na/b)
> >
> >because it doesn't require you to count intervals or identify a period
> >and generator. So it's the scale that's as even as possible and happens
> >to be an MOS.
>
> Sure, fine -- whatever you like.
>
> >I notice your ME definition in voiceleading.pdf doesn't require b<=a, so
> >it's possible to get the whole of a-equal with duplicates.
>
> Yup, correct.
>
> >The trouble is, the "degenerate" in "degenerate well formed" doesn't
> >mean this, although I think I thought it did once. Best to say "an MOS
> >where the period isn't an octave". (It should be obvious that you're
> >defining "MOS" so that the period doesn't have to be an octave.)
>
> Well I don't like this for the reasons expressed above.
>
> > > Then you have the very nice, clean result that ME scale is just the
> >> most even MOS scale in a given temperament. Furthermore, you have
> >> the non-obvious proof that the most even set of any cardinality is
> >> always MOS.
> >
> >Yes. Given equal temperament.
>
> Or the "scale step" metric which converts any scale to an equal > temperament.
>
> DT
> -- > WARNING: Princeton Email is currently very unreliable. If you need
> to reach me quickly, you should call me.
>
> Dmitri Tymoczko
> Assistant Professor of Music, Princeton University
> Radcliffe Institute for Advanced Study
> 34 Concord Ave.
> Cambridge, MA 02138
> FAX: (617) 495 8136
> http://music.princeton.edu/~dmitri <http://music.princeton.edu/%7Edmitri>
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗Carl Lumma <ekin@lumma.org>

>Two further thoughts:
>1) For a term like "MOS" to be really useful, we have to know
>exactly what it means.

That's why I think we should stop using it, unless we're talking
about something in a Wilson paper. After a day with Wilson at
his house in 1998, a bunch of correspondence, then his lecture
at Microfest 2001 which was supposed to be about 3-D projections
of 6-D space (using Zome models he'd built with the kit I'd given
him) but which instead ended up being about why precise terminology
is something he'd never aspire to......

>Does it include the octatonic scale?

Kraig's tried to ask him on several occasions, but my sense is
that such questions are a bit foreign to him.

>2) It seems like the tuning community is using Wikipedia as an
>alternative to publication. I have some qualms about this -- the
>music articles in Wikipedia have a heavy tuning-list slant, and, by
>emphasizing terms like "MOS scale," they often give a misleading
>impression about the centrality of these terms to music scholarship.

At first we stayed away because of No Original Research. But it
seemed like other people were writing the articles anyway, and
doing it badly, so...

Music theory articles have a tuning-list slant? Wow, that isn't
the sense I've had. But then again, I think of most music theory
as badly needing tuning generalization, so I'm probably not the
best person to judge this.

>The more mathematically inclined among you should be aware of the new
>Journal of Mathematics and Music, and of semi-regular "music and
>math" sessions being run at the AMS. Perhaps this would be a place
>where you could present some of your work.

Gene and Paul E. both need to be published. For a while there was
a project here to jointly write a paper, but we weren't able to
arrive at a consensus on, well, much of anything.

-Carl

🔗Carl Lumma <ekin@lumma.org>

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:

> For what it's worth, my own first paper ("Consecutive Semitone
> Constraint") was rejected 3 times, the last time so rudely that I
> gave up on the idea of publishing it. One day, out of the blue I got
> an email from a theorist at Eastman to whom I'd sent the paper years
> before -- he was cleaning his office, found the paper at the bottom
> of a large pile, and before throwing it away, decided to read it. He
> liked it, and asked whether I was still interested in publishing it.
> And that's how I became a composer/theorist rather than just a
> composer.

Wow. I have a story very much like that. I wrote "Transformation and
Invariance in Music Theory" over twenty years ago, when the ideas in
it were possibly a little ahead of their time, and in any case were
framed in language requiring a lot of algebra. It got bounced. Long
afterwards, I got email from someone inquiring about it; it seems he
had a copy which had been passed from hand to hand and much
mimeographed, resulting from my having deposited a copy with the Just
Intonation Network I imagine.

> Two further thoughts:
>
> 1) For a term like "MOS" to be really useful, we have to know
> exactly what it means. My sense from the list is that the meaning of
> the term isn't fixed.

It used to have a clear, fixed meaning around here until Paul Erlich
got into one of his correct terminology frenzies. If I were going to
define it, I would say that it is a quasiperiodic scale with two step
sizes, resulting from iterating a generating interval g modulo a
period interval P. That is, it is S = {P^a g^b| b is an integer in the
range n1 <= b <= n2 and S has two step sizes}. The canonical example
would be {2^a f^b| -1<=b<=5} where f is a meantone fifth.

> 2) It seems like the tuning community is using Wikipedia as an
> alternative to publication.

That and web pages.

I have some qualms about this -- the
> music articles in Wikipedia have a heavy tuning-list slant, and, by
> emphasizing terms like "MOS scale," they often give a misleading
> impression about the centrality of these terms to music scholarship.

On the other hand, we've been somewhat forced into it by correcting
the bias of not mentioning the originators of a concept and requiring
that statements be made in full generality.

> Please keep in mind that I like you people, and respect you, and
> think you should get your due recognition. I think it would be good
> to try to bring the tuning community closer to the academic theory
> community. For example, someone should write a careful, accurate
> history of the intersection between tuning theory and more academic
> Clough-related scale work.

I've been working (should work harder) on a paper on the "new
paradigm/middle way" that originated here. But I haven't thought of
what we could do re scale theory. I tried to bring the terminology
into line with standard mathematics terminology here, which also makes
many proofs and definitions more perspicuous:

http://66.98.148.43/~xenharmo/quasi.htm

Beyond that, there is a large amount of scale theory in the sense of
more or less practical scale construction which has been going on.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:

> >Once again, this is a kind of broken symmetry, from the pint of view
> >of the <7 11 16| val modal shifts are identity mappings. If we notate
> >meantone using <7 11 16| and another val, all of the changes take
> >place with respect to the values of the other val.
>
> Sorry, I have no idea what you're talking about here -- remember, I'm
> an outsider.

Sorry, I was somehow under the delusion you'd looked at my web site.

A homomorphic mapping from the p-pimit, for example from the 5-limit
group 2^a 3^b 5^c, can be denoted by a "bra" vector such as <7 11 16|.
My word for such things is "val", since they are Z-linear combinations
of p-adic valuations from one point of view. By sheer obnoxious
persistence, I've gotten that term widely understood around here. It
is a fundamental notion, and I think needs a word for it.

The dual notation, a "ket" vector, would represent 81/80 = 2^(-4)3^4
5^(-1) by |-4 4 -1>. You then get the usual bra-ket braket or braket
product. The notation is due to Dirac:

http://en.wikipedia.org/wiki/Bra-ket_notation

except of course Dirac had in mind something beyond abelian groups.

The point here is that given a modal transformation q' = Sn(q), we
have h7(q') = h7(q) if h7 = <7 11 16|. This strikes me as a much more
insightful way of viewing the question. Of course in general, given a
scale one can define a modal transformation relative to that scale,
which could (though it is hardly required) involve constructing a
scale metric. I don't see what added understanding you get from the
scale metric.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:

Suppose I want a seven note scale such that every note has a unique
other note which is as close as possible to being a third above it. I
want to minimize the total error in one of a small number of general
ways. What's the best I can do?

NOT a seven note equal scale. The correct answer is the diatonic
scale. You get a similar answer for five-note scales if you decide you
want every note to have either a fifth or a minor sixth above it.

A five note equal-tempered scale.
> For many measures of error, the five-note equal tempered scale will
> beat other five-note generated collections (or MOS collections, or
> what have you) such as {C, C#, D, D#, E}.

And for other measures of error, it does very poorly. It's all a
matter of what you are looking for.

> You can do it this way -- this requires you to first "wrap around"
> the half octave (going from 5 to 0 to 7 to 2, etc.) so that the half
> octave acts like a modulus. Then, having generated half the pattern
> in this way, you change your mind and transpose the whole thing up a
> half-octave to fill out the actual modulus.

I don't think this is the best approach. I'd do what mentioned before,
which is P^a g^b with "b" constrained to lie between certain values.
Scales are often easier to deal with if you simply accept they are
quasiperiodic functions and stop worrying so much about reducing to
the octave.

🔗monz <monz@tonalsoft.com>

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

>
> > 2) It seems like the tuning community is using Wikipedia as an
> > alternative to publication.
>
> That and web pages.

Particularly my Tonalsoft Encyclopedia
(... which unfortunately is down at the moment)

> [Dmitri Tymoczko]
> > Please keep in mind that I like you people, and respect you,
> > and think you should get your due recognition. I think it
> > would be good to try to bring the tuning community closer
> > to the academic theory community. For example, someone
> > should write a careful, accurate history of the intersection
> > between tuning theory and more academic Clough-related scale
> > work.
>
> I've been working (should work harder) on a paper on the
> "new paradigm/middle way" that originated here. But I haven't
> thought of what we could do re scale theory. I tried to bring
> the terminology into line with standard mathematics terminology
> here, which also makes many proofs and definitions more
> perspicuous:
>
> http://66.98.148.43/~xenharmo/quasi.htm
>
> Beyond that, there is a large amount of scale theory in the
> sense of more or less practical scale construction which
> has been going on.

There's also that paper that Graham Breed recently wrote,
about the new paradigm. Can someone dig up the link?

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Graham Breed <gbreed@gmail.com>

Carl Lumma wrote:
>>>This is from Clough, Engebretsen, Kochavi "Scales, Sets, and
>>>Interval Cycles: A Taxonomy", which cites Wilson's letter to
>>>Chalmers, incidentally.
>>
>>It's not the original paper so it doesn't have the canonical definition.
> > It's a paper from the originator of the concept that specifically
> addresses the question of defining these things. It's the paper
> Clough himself pointed us to when this came up.

Does it contradict the definitions I, Agmon, and Tymoczko use?

>>>maximally even - Each generic interval comes in either
>>>one integer size or two consecutive integer sizes.
>>>""
>>
>>So what? That's not what you said before,
> > Oh no?

You didn't say "either one ... or".

>>So, let's go back to what you said wasn't technically correct
>>
>>
>>>>>>>>A maximally even scale is made up of no
>>>>>>>>more than two different sizes of scale step, and the larger
>>>>>>>>can't be more than twice the size of the smaller.
>>
>>No more than two is correct. It can be one. The second part only >>applies where the two steps have different sizes, so
>>
>>x - y = 1
>>
>>for diatonic steps of x and y chromatic steps. I take it that's what >>"consecutive integer sizes" means.
> > It means they're integers. There's nothing in the text I called
> incorrect that wouldn't apply to a scale with steps of 113.459 cents
> and 173.678 cents. It also means they're consecutive. Scales with
> L=5 and s=3 aren't ME.

So what?

Graham

🔗Graham Breed <gbreed@gmail.com>

Dmitri Tymoczko wrote:
>> > Interestingly, you don't have to assume that you want to minimize the
>>
>>> worst error -- you can minimize the sum of the errors, or the sum of
>>> the squared errors, or many many other plausible measures of error.
>>> In each case, equal temperament does best.
>>
>>In general, the average deviation for an MOS doesn't depend on the
>>generator. That means the sum of errors is also constant provided all
>>tempered intervals are in the same direction of where they should be.
>>The sum of squared errors looks more like the worst error, and so it
>>favours equal temperament.
> > I don't think this has much to do with MOS or intervals of generation > or consonance per se.

I think you're right.

> Suppose I want a five note scale such that every note has a unique > other note that is as close as possible to being a tritone above it. > I want to minimize the total error in one of a very number of general > ways. What's the best I can do? A five note equal-tempered scale. > For many measures of error, the five-note equal tempered scale will > beat other five-note generated collections (or MOS collections, or > what have you) such as {C, C#, D, D#, E}.

One day I'll write a paper on why 5-equal is the Scale of Nature and we should all be writing music in it.

>>I don't see a problem. It's an MOS with a half-octave periodicity and a
>>generator of 5 steps.
> > You can do it this way -- this requires you to first "wrap around" > the half octave (going from 5 to 0 to 7 to 2, etc.) so that the half > octave acts like a modulus. Then, having generated half the pattern > in this way, you change your mind and transpose the whole thing up a > half-octave to fill out the actual modulus.

It requires the scale to be periodic about the half-octave.

> The point is that during the derivation you need two separate > generators, and two separate moduli. One generator is the interval > 5, the other is the interval 12. One modulus is the interval 12, the > other is 24. For this reason I am uncomfortable talking about "the" > modulus, or "the" generator of the scale {0, 2, 4, 5, 7, 9, 11, 12, > 14, 16, 17, 19, 21, 23}.

You need two separate generators. You don't need any moduli.

> I'm just saying that the terminological shoe is pinching a bit > uncomfortably here.

We have to look at these scales because they represent important rank 2 temperaments. As a result we don't usually think in octave-equivalent space because it gets too messy. That isn't such a sacrifice because it doesn't properly model the music-theoretical concept of octave equivalence anyway.

I wrote up some of the background to this earlier in the year:

http://x31eq.com/paradigm.html

It's supposed to be understandable by music professors, so you can test that ;)

>>I notice your ME definition in voiceleading.pdf doesn't require b<=a, so
>>it's possible to get the whole of a-equal with duplicates.
> > Yup, correct.

In that case you'll get some pathological scales that aren't proper, if it makes sense to apply propriety to them at all. For example, 9 from 7 is

0, 0, 1, 2, 3, 3, 4, 5, 6, 7

7-3=4 can span 4 diatonic steps, but 3-0=3 can span 5.

Graham

🔗monz <monz@tonalsoft.com>

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗djwolf_frankfurt <djwolf_frankfurt@yahoo.com>

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗monz <monz@tonalsoft.com>

--- In tuning-math@yahoogroups.com, "djwolf_frankfurt"
<djwolf_frankfurt@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> 3/~xenharmo/quasi.htm
> > >
> > > Beyond that, there is a large amount of scale theory in the
> > > sense of more or less practical scale construction which
> > > has been going on.
> >
> >
> > There's also that paper that Graham Breed recently wrote,
> > about the new paradigm. Can someone dig up the link?
> >
>
> Has there been a good write-up of the grand tuning system
> classification scenario? Unfortunately my article in RevConMus was
> done before this work, so my treatment of scale classification was
> left in the scale-tree era (which is okay since the article was
> dedicated to Wilson and the scale tree does remains useful for
> locating related MOS).
>
> I didn't follow the work closely as I was working on my
> "Gender Garapan and Counterpoint" project at the time, but
> I followed closely enough to recognize that the result is
> impressive and useful.

Well, fortuitously for me, Graham just put the link into
another message he wrote to Dmitri:

http://x31eq.com/paradigm.html

I think this and Paul Erlich's "Middle Path" paper are two
of the most significant documents i've seen on tuning in
the last couple of years.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >
> > Gene Ward Smith wrote:
> >
> > > I've been working (should work harder) on a paper on the
> > > "new paradigm/middle way" that originated here. But I
> > > haven't thought of
> >
> > What are you planning to cover?
>
> The idea is simply to be as mathematical as I like when
> developing much of the usual stuff. Things I play down such
> as Grassmannians get their due.

Yay!

I've decided to resume my return-to-school to study math.
So i look forward to eventually understanding all this stuff
and incorporating it into Tonescape.

The essential idea is to allow the user to create
sonic topologies, which map simultaneously to tuning systems
and to the computer screen, and then to manipulate the
topologies and tuning systems by manipulating the display
on the computer screen -- i.e., compose, Tonescape-style.

Anyway, go for it -- give full rein to your mathematical
ideas about tunings, and let's start analyzing microtonal
compositions according to your ideas. Let the shower of
academic papers begin ... i agree with Dmitri that we
need them.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

> > I'm just saying that the terminological shoe is pinching a bit
>> uncomfortably here.
>
>We have to look at these scales because they represent important rank 2
>temperaments. As a result we don't usually think in octave-equivalent
>space because it gets too messy. That isn't such a sacrifice because it
>doesn't properly model the music-theoretical concept of octave
>equivalence anyway.

I'm not really arguing that you shouldn't think about the objects, just that it might be nice to describe them a bit differently. For instance "all scalar intervals come in at most two flavors" does the job quite nicely. You can then say that these scales are all "generated" -- if you take a slightly expanded view of "generated."

>I wrote up some of the background to this earlier in the year:
>
>http://x31eq.com/paradigm.html
>
>It's supposed to be understandable by music professors, so you can test
>that ;)

Looks perfectly comprehensible. I'll have a closer look, and maybe make some comments, over the next few weeks.

> >>I notice your ME definition in voiceleading.pdf doesn't require b<=a, so
>>>it's possible to get the whole of a-equal with duplicates.
>>
>> Yup, correct.
>
>In that case you'll get some pathological scales that aren't proper, if
>it makes sense to apply propriety to them at all. For example, 9 from 7 is
>
>0, 0, 1, 2, 3, 3, 4, 5, 6, 7
>
>7-3=4 can span 4 diatonic steps, but 3-0=3 can span 5.

I think there's a mistake here -- your scale has 10 notes and is not maximally even. You either mean:

0 0 1 2 3 3 4 5 6

0 0 1 2 3 4 4 5 6 7

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>[MOS] used to have a clear, fixed meaning around here until Paul Erlich
>got into one of his correct terminology frenzies. If I were going to
>define it, I would say that it is a quasiperiodic scale with two step
>sizes, resulting from iterating a generating interval g modulo a
>period interval P. That is, it is S = {P^a g^b| b is an integer in the
>range n1 <= b <= n2 and S has two step sizes}. The canonical example
>would be {2^a f^b| -1<=b<=5} where f is a meantone fifth.

Seems to me you can just use "S has two step sizes" as your definition; the rest of it can be derived from that.

Again these DE/MOS scales have a very nice representation in the orbifolds; they can actually be used as basis vectors so you can write arbitrary chords as sums of MOS scales.

>A homomorphic mapping from the p-pimit, for example from the 5-limit
>group 2^a 3^b 5^c, can be denoted by a "bra" vector such as <7 11 16|.
>My word for such things is "val", since they are Z-linear combinations
>of p-adic valuations from one point of view.

I got it -- the "val" is just the linear function that maps lattice points into scale degree space.

But if the lattice vector is the "monz," then the whole notation is "valmonz," which is certainly less euphonous than "bra-ket." It'd be better if the "val" were the "joe" or something. Or the "monz" were the "u" vector.

>The point here is that given a modal transformation q' = Sn(q), we
>have h7(q') = h7(q) if h7 = <7 11 16|. This strikes me as a much more
>insightful way of viewing the question. Of course in general, given a
>scale one can define a modal transformation relative to that scale,
>which could (though it is hardly required) involve constructing a
>scale metric. I don't see what added understanding you get from the
>scale metric.

Gene, I really think you're being stubborn here. The point is that the musical term "scale step" (or "generic interval") *is* a unit of distance. This term is built into our musical practice. And the understanding that it is a unit of distance is also pretty directly reflected in our practice with the term.

In fact, you use it -- and it's mathematical equivalents -- on your website all over the place!

Furthermore, not every musically interesting or useful scale is going to have an (informative) description using vals and monzes. I can just choose five arbitrary frequencies within the octave to serve as my scale. Describing this with bra-ket notation strikes me as overkill in a big way.

I'll propose a test: let's imagine a situation where a composer plays a single set of scalar intervals in the context of several scales of differing cardinaility -- first the chromatic scale, then the diatonic scale, then a made-up (octave repeating) scale consisting of five arbitrarily chosen frequencies -- say {C, D#, E, A, A quartertone sharp, B}.

I have my description: the same series of scalar intervals (distances-relative-to-the scale!) is repeated in the context of three different scales.

Why don't you tell me how you'd describe the situation?

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗Carl Lumma <ekin@lumma.org>

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>Suppose I want a seven note scale such that every note has a unique
>other note which is as close as possible to being a third above it. I
>want to minimize the total error in one of a small number of general
>ways. What's the best I can do?
>
>NOT a seven note equal scale. The correct answer is the diatonic
>scale. You get a similar answer for five-note scales if you decide you
>want every note to have either a fifth or a minor sixth above it.

Yes -- if you get to choose multiple intervals (two "kinds of thirds"), then equal temperament no longer minimizes the error. Hopefully, I never claimed otherwise.

If you have to choose a single interval, then equal temperaments do minimize what I consider to be very natural measures of total error.

> A five note equal-tempered scale.
>> For many measures of error, the five-note equal tempered scale will
>> beat other five-note generated collections (or MOS collections, or
>> what have you) such as {C, C#, D, D#, E}.
>
>And for other measures of error, it does very poorly. It's all a
>matter of what you are looking for.

Which are we disagreeing about?

1. The mathematics of adding up the individual errors?
2. What your ideal is?

I think the disagreement is about #2 -- as you say, we can take the ideal to be that the scale has "either a major third or minor third," or just a major third, or many other things. Not all of these choices will have the result that equal temperaments come out ideal.

Is there really a disagreement about #1? Are you saying we should do something other than add up the individual errors, or add up their squares and take the square root, or take the largest, or ... Because here I think you probably do what what you do with these individual errors to be constrained by the very general principles I describe as "the distribution constraint."

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗djwolf_frankfurt <djwolf_frankfurt@yahoo.com>

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:

>
> Well, fortuitously for me, Graham just put the link into
> another message he wrote to Dmitri:
>
> http://x31eq.com/paradigm.html
>

Graham --

There are some excellent things here. Just a few comments:

(1) using "do re mi" carries its own ambiguities, I would just say,
for better or worse, "diatonic scale". (In the paragraph about
Srutis, you
are right about the possibility of describing Indian musics in terms
of small variations from a 12tet chromatic, but Do Re Mi can't be
compared with Srgam because in the Indian solfege the internal terms
in each tetrachord can be raised or lowered so the scales don't have
a single fixed (however coarsely) shape as does the diatonic.)

(2) I would date a 7-out-12 paradigm both more specifically and
narrowly. In the 14th and 15th century music I've been working with
recently, a 12-tone chromatic plays no role at all, the paradigm is
instead something more modest: two diatonic collections, one with a
hard and the other with a soft b, with extra tones added as ficta (an
eb below the bb or an f# above the b) or chromatic neighboring tones.
On the other hand, there were times in the meantone era when the tonal
resources were more than twelve tones.

(2) the definition of meantone has to be a bit clearer. At a minimum,
one should indicate why the tones are mean, and what interval they are
the mean of, and specify that in 1/4 comma meantone (which many of us
simply call meantone), the third is just and the fifths are tempered
so that 4 of them divide a 5:2 equally. All of the meantone-like
tunings then share the premise that the major third (specifically: all
the thirds that are notated as major thirds) is the same as the sum of
four fifths, mod the octave. I would also emphasize that classical
meantone was not neccessarily limited to an octave with twelve tones,
but the sequence of meantone fifths could be continued indefinitely,
and meantone keyboards with 14-16 tones were not unknown.

(3) it is not clear to me that "one unavoidable problem with the
paradigm is that it makes things more complex than they were before"
is necessarily true. What is true is that the scheme is inclusive so
that a diversity of scale types are accounted for, and that
comprehensiveness may be better characterized as unfamiliar than complex.

(4) I would also drop the paradigm discussion. You have the outline
for a strong presentation on its own terms and, if you don't mind my
advice, the scholarly world doesn't need another paper beginning with
Kuhn unless you're really prepared to handle the inevitable arguments.
Why not simply say that "this is a new approach to scales
(period)"? Also consider the addition of a clear statement
explaining concisely why this approach is useful and does things that
prior approaches cannot do.

Once again, the page is excellent and I have learned quite a bit.

DJW

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:
>
> >[MOS] used to have a clear, fixed meaning around here until Paul Erlich
> >got into one of his correct terminology frenzies. If I were going to
> >define it, I would say that it is a quasiperiodic scale with two step
> >sizes, resulting from iterating a generating interval g modulo a
> >period interval P. That is, it is S = {P^a g^b| b is an integer in the
> >range n1 <= b <= n2 and S has two step sizes}. The canonical example
> >would be {2^a f^b| -1<=b<=5} where f is a meantone fifth.
>
> Seems to me you can just use "S has two step sizes" as your
> definition; the rest of it can be derived from that.

No, MOS is a stronger condition. Consider the "melodic minor" scale;
this takes the chain of fifths for C major diatonic, from F to B, and
substitutes Eb for E. The result is no longer a chain of fifths, and
hence no longer a MOS, yet it satisfies the condition of having only
two step sizes.

Still, it's a strong condition. If we want a strictly proper meantone
scale with only two step sizes, we get only the diatonic, harmonic
minor, and major locrian scales. The combination of the two conditions
might be worth further exploration.

> Again these DE/MOS scales have a very nice representation in the
> orbifolds; they can actually be used as basis vectors so you can
> write arbitrary chords as sums of MOS scales.

How can I write an arbitrary chord of n notes in terms of a sum of MOS
scales with seven notes?

> >A homomorphic mapping from the p-pimit, for example from the 5-limit
> >group 2^a 3^b 5^c, can be denoted by a "bra" vector such as <7 11 16|.
> >My word for such things is "val", since they are Z-linear combinations
> >of p-adic valuations from one point of view.
>
> I got it -- the "val" is just the linear function that maps lattice
> points into scale degree space.

Well, I would emphasize the group homomorphism; we don't need to
assume anything about lattices or scale degres. In mathspeak, they are
members of Hom(Np, Z), the group of homomorphisms from the p-limit
group Np to Z. The p-adic valuations are all vals on Np; for example,
if q is a ratio of odd numbers, then v2(2^a q) = a is the 2-adic
valuation. If we restrict it to just positive rationals and the
5-limit, we can write it as <1 0 0|.

> But if the lattice vector is the "monz," then the whole notation is
> "valmonz," which is certainly less euphonous than "bra-ket." It'd be
> better if the "val" were the "joe" or something. Or the "monz" were
> the "u" vector.

I named the "monz" after Joe, actually, more or less by accident. Now,
if I had named it after myself, it would be the valjean, and you could
talk about the use of the Gene valjean in music theory. A chance missed.

> Gene, I really think you're being stubborn here. The point is that
> the musical term "scale step" (or "generic interval") *is* a unit of
> distance.

But this point of view is too limited to cover the matter; we don't
just want modal transformations, but scale-switching transformations
generally. If you take the five meantone scales I mentioned as having
a circle of thirds, and use every mode, you can take a piece in two
part harmony diatonic which emphasizes thirds and translate it to 35
different versions, all of which would make harmonic sense.

> This term is built into our musical practice.

Not really. Modal transformations are far less important to common
practice music that major/minor, and minor includes natural, melodic,
and harmonic minor.

And the
> understanding that it is a unit of distance is also pretty directly
> reflected in our practice with the term.

It's a unit of distance which the <7 11| meantone val, or the <7 11
16| JI val, perfectly captures, but which your scale method is at too
low a level of abstraction to capture.

> In fact, you use it -- and it's mathematical equivalents -- on your
> website all over the place!

I discuss an idea like it in the page on quasiperiodic scales,
certainly. Given any quasiperiodic scale, the corresponding
transformation is definable, and may or may not be musically interesting.

> Furthermore, not every musically interesting or useful scale is going
> to have an (informative) description using vals and monzes. I can
> just choose five arbitrary frequencies within the octave to serve as
> my scale. Describing this with bra-ket notation strikes me as
> overkill in a big way.

Bras and kets were not introduced to discuss scales, but regular
temperaments. One point is that they do so in a language so general
that not only are scales not being directly addessed, no assumption is
being made that octaves have a special status. Given bras or kets, you
can define the corresponding exterior algebra, and that's where the
action is.

> Why don't you tell me how you'd describe the situation?

I'd describe it as not really being about scales at all. Simply look
at the motive in question: C, D#, E, A, A and analyze it on its own
terms. To me, the first basic question is whether the fact that it is
spelled in this way is intended literally, in which case I want to
analyze it as meantone, or whether it is simply shorthand for
something in 12 equal, in which case I want to analyze it in terms of
12 equal.

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

> > Seems to me you can just use "S has two step sizes" as your
>> definition; the rest of it can be derived from that.
>
>No, MOS is a stronger condition.

Sorry, what I meant was: "each of S's scalar intervals come in at most two sizes."

>Still, it's a strong condition. If we want a strictly proper meantone
>scale with only two step sizes, we get only the diatonic, harmonic
>minor, and major locrian scales. The combination of the two conditions
>might be worth further exploration.

I assume you mean melodic minor, not harmonic minor -- since the harmonic minor has three step sizes.

And what exactly do you mean by calling these scales "meantone" scales? You mean, you're imagining them embedded in a meantone 12-notes-to-the octave chromatic scale? Or what?

Why call the acoustic scale (melodic minor ascending) "meantone" rather than equal tempered, or anything else?

> > Again these DE/MOS scales have a very nice representation in the
>> orbifolds; they can actually be used as basis vectors so you can
>> write arbitrary chords as sums of MOS scales.
>
>How can I write an arbitrary chord of n notes in terms of a sum of MOS
>scales with seven notes?

No, you can write an arbitrary n-note chord as a sum of MOS scales with n notes. The MOS scales form basis vectors.

> > Gene, I really think you're being stubborn here. The point is that
>> the musical term "scale step" (or "generic interval") *is* a unit of
>> distance.
>
>But this point of view is too limited to cover the matter; we don't
>just want modal transformations, but scale-switching transformations
>generally.

What's the problem? I have a musical pattern specified as a sequence of scale-step motions. For instance: "do, a deer; ray, a drop" is: (+1, +1, -1, +1 + 1). Now, you give me any scale, and I can play the pattern in it. The motive is represented as a series of within-scale distances.

Octave equivalence makes no difference -- you can use the same pattern of scalar distances in non-octave repeating scales and in octave repeating scales. Just go up one scale step when it says "go up one scale step."

>Not really. Modal transformations are far less important to common
>practice music that major/minor, and minor includes natural, melodic,
>and harmonic minor.

Personally, I'm interested in Renaissance music, jazz, Debussy, and the music yet to be written. That's why I want a notion of scale that's sufficiently general.

Modal transformations are certainly found throughout classical music -- off the top of my head, I can think of an example from the Bach D-minor 2-part invention where the (harmonic minor) motive gets transformed into the diatonic scale, and then into melodic minor scale (where it appears as a weird mode of the scale -- check it out); an example from the Symphonie Fantastique where a diatonic pattern gets shifted to the chromatic scale; and example from a Chopin Mazurka where a tune gets shifted from a diatonic scale to a mode of the harmonic minor. I could go on ...

> > Furthermore, not every musically interesting or useful scale is going
>> to have an (informative) description using vals and monzes. I can
>> just choose five arbitrary frequencies within the octave to serve as
>> my scale. Describing this with bra-ket notation strikes me as
>> overkill in a big way.
>
>Bras and kets were not introduced to discuss scales, but regular
>temperaments.

Which is why it is so odd that you think they can replace scalar terminology!

It looks to me like you misunderstood the question I asked earlier, so let me try it again. Let's imagine a situation where a single musical pattern -- for clarity, let's say "do, a deer, re a drop" is presented in the context of a number of scales: first chromatic, then diatonic, then a made-up (octave repeating) scale consisting of five arbitrarily chosen frequencies -- say {C, D#, E, A, A quartertone sharp, B}.

How would you describe this?

You wrote:

>I'd describe it as not really being about scales at all. Simply look
>at the motive in question: C, D#, E, A, A and analyze it on its own
>terms.

This makes no sense to me: the motive is "do, a deer, re a drop" -- but it exists in a variety of scales. Perhaps no one form is primary. Here, the motive "on its own terms" simply *is* the pattern of scalar distances: (+1, +1, -1, +1 + 1). These distances can be manifested in a variety of scales.

> To me, the first basic question is whether the fact that it is
>spelled in this way is intended literally, in which case I want to
>analyze it as meantone, or whether it is simply shorthand for
>something in 12 equal, in which case I want to analyze it in terms of
>12 equal.

This fetish for notation strikes me as unhealthy. You can find this kind of thing happening in Chopin and Bach, who no doubt improvised on keyboards in a variety of temperaments (and played the same pieces in a variety of temperaments). Speaking personally, I often improvise with no idea how I'd notate what I'm playing. Music is about sound, not graphical squiggles.

If I may venture a hypothesis. What's going on here is that you're interested in your "new paradigm," where scales are represented using bra-ket notation. Because of your commitment to this paradigm, you're trying to do entirely without familiar and very natural notions like "scale step." But in fact this is very difficult to do: your notation -- and paradigm -- may be useful for certain questions of tuning, but it doesn't provide a very useful vocabulary for thinking about scales.

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗monz <monz@tonalsoft.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:

> I assume you mean melodic minor, not harmonic minor -- since the
> harmonic minor has three step sizes.

Right, sorry.

> And what exactly do you mean by calling these scales "meantone"
> scales? You mean, you're imagining them embedded in a meantone
> 12-notes-to-the octave chromatic scale? Or what?

No, I mean they are defined in any meantone system. It really makes
more sense to look at them in 19 or 31 than 12, where they are only
proper and not strictly proper.

> Why call the acoustic scale (melodic minor ascending) "meantone"
> rather than equal tempered, or anything else?

Because looking at it in terms of 12 equal temperament is both
ahistorical and simply not the best way of understanding the scale in
any case.

If you stick it into 31 equal, you get a lot of choices of generator
you could use define it, but it has the property of being what I was
calling "unambiguously meantone", meaning that it is more compact (by
quite a margin, actually) in terms of meantone generators than any
others. Defined in terms of meantone, it is translatable into other
meantone tunings. This has the practical consequence that a piece
written in melodic minor can be played in any meantone tuning. The
whole system of diatonicity, with major and the various versions of
minor, are all best understood in meantone. That is, after all, their
historical context, but it also is the context where they simply make
the most objective sense.

> >But this point of view is too limited to cover the matter; we don't
> >just want modal transformations, but scale-switching transformations
> >generally.
>
> What's the problem? I have a musical pattern specified as a sequence
> of scale-step motions. For instance: "do, a deer; ray, a drop" is:
> (+1, +1, -1, +1 + 1). Now, you give me any scale, and I can play the
> pattern in it. The motive is represented as a series of within-scale
> distances.

Two points: this is considerably more general than what I thought you
were saying, and it still does not require intoducing metrics, unless
you want to call the number of scale steps between two notes of a
quasiperiodic scale a "metric". It's the KISS principle I have in mind
here, which suggests you not introduce bells and whistles unless you
can do something as a result.

> >Bras and kets were not introduced to discuss scales, but regular
> >temperaments.
>
> Which is why it is so odd that you think they can replace scalar
terminology!

It would depend on the context. In general, no; in the case we were
talking about, the invariance property seemed to me to be the heart of
the matter.

> This makes no sense to me: the motive is "do, a deer, re a drop" --
> but it exists in a variety of scales. Perhaps no one form is
> primary. Here, the motive "on its own terms" simply *is* the pattern
> of scalar distances: (+1, +1, -1, +1 + 1). These distances can be
> manifested in a variety of scales.

If you are asking how do I interpret the abstract "scalar" (can we
find another word?) distance (+1 +1 -1 +1 +1), the answer is that in
that case, that's how I would interpret it. Given any base point on
any quasiperiodic scale, and you can turn it into actual notes. The
results will differ wildly.

> This fetish for notation strikes me as unhealthy. You can find this
> kind of thing happening in Chopin and Bach, who no doubt improvised
> on keyboards in a variety of temperaments (and played the same pieces
> in a variety of temperaments). Speaking personally, I often
> improvise with no idea how I'd notate what I'm playing. Music is
> about sound, not graphical squiggles.

There's more involved than a fetish for notation. One hobby of mine is
lifting pieces from 12-et (a tuning I find mildly annoying) to
meantone, which is much better on my nerves. In that case what you are
doing could from one point of view be described as trying to find a
consistent spelling.

> If I may venture a hypothesis. What's going on here is that you're
> interested in your "new paradigm," where scales are represented using
> bra-ket notation.

No, because the new paradigm is on a more abstract level; the concern
is not with scales, but with regular temperaments.

Because of your commitment to this paradigm,
> you're trying to do entirely without familiar and very natural
> notions like "scale step."

In that case, why did I go to the trouble of writing my page on
quasiperidic scales, which I think puts scale steps into the correct
conceptual framework?

But in fact this is very difficult to do:
> your notation -- and paradigm -- may be useful for certain questions
> of tuning, but it doesn't provide a very useful vocabulary for
> thinking about scales.

Scales are a very big field and they can be approached from many
directions. One approach is to first define a regular tuning; the
simplest way to do that is to either use an equal temperament or else
just intonation, but scales in terms of regular temperaments more
generally are also of interest. I've found in practical terms that
scales invovling tempering out 225/224 and nothing else are often
excellent value for the money.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Kraig Grady <kraiggrady@anaphoria.com>

I guess we could also point out that MOS is the archetype for constant structures which can be seen as a chain of a generator that is variable within certain limits but the resultant has any interval when it occurs subtended by the same number of steps

Carl Lumma wrote:
>
> Gene wrote...
> >> >[MOS] used to have a clear, fixed meaning around here until Paul > Erlich
> >> >got into one of his correct terminology frenzies. If I were going to
> >> >define it, I would say that it is a quasiperiodic scale with two step
> >> >sizes, resulting from iterating a generating interval g modulo a
> >> >period interval P. That is, it is S = {P^a g^b| b is an integer in the
> >> >range n1 <= b <= n2 and S has two step sizes}. The canonical example
> >> >would be {2^a f^b| -1<=b<=5} where f is a meantone fifth.
> >>
> >> Seems to me you can just use "S has two step sizes" as your
> >> definition; the rest of it can be derived from that.
> >
> >No, MOS is a stronger condition. Consider the "melodic minor" scale;
> >this takes the chain of fifths for C major diatonic, from F to B, and
> >substitutes Eb for E. The result is no longer a chain of fifths, and
> >hence no longer a MOS, yet it satisfies the condition of having only
> >two step sizes.
>
> No, there are three types of 5th.
>
> -Carl
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗djwolf_frankfurt <djwolf_frankfurt@yahoo.com>

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

> > And what exactly do you mean by calling these scales "meantone"
>> scales? You mean, you're imagining them embedded in a meantone
>> 12-notes-to-the octave chromatic scale? Or what?
>
>No, I mean they are defined in any meantone system. It really makes
>more sense to look at them in 19 or 31 than 12, where they are only
>proper and not strictly proper.

I'm curious -- could you explain further?

You can define them the way I do in "Scale Networks and Debussy," where you basically use the voice leading motions that give rise to the circle of fifths, but in the wrong order.

> > Why call the acoustic scale (melodic minor ascending) "meantone"
>> rather than equal tempered, or anything else?
>
>Because looking at it in terms of 12 equal temperament is both
>ahistorical and simply not the best way of understanding the scale in
>any case.

Don't know about ahistorical. For the music I love most -- Debussy and jazz, for example, the scale is a thoroughly equal tempered animal.

I'm also curious about "simply not the best way of understanding the scale in any case." I'm an equal-tempered guy, playing the scale at my equal tempered piano. Explain to me how I should think of the scale. (I'm not being a smart ass, I'm genuinely curious.)

>If you stick it into 31 equal, you get a lot of choices of generator
>you could use define it, but it has the property of being what I was
>calling "unambiguously meantone", meaning that it is more compact (by
>quite a margin, actually) in terms of meantone generators than any
>others. Defined in terms of meantone, it is translatable into other
>meantone tunings. This has the practical consequence that a piece
>written in melodic minor can be played in any meantone tuning. The
>whole system of diatonicity, with major and the various versions of
>minor, are all best understood in meantone. That is, after all, their
>historical context, but it also is the context where they simply make
>the most objective sense.

Sorry, can you spell this out more? You're saying that on some particular tuning lattice, the melodic minor scale is particularly compact, and this is very enlightening? Just run me through the details.

>Two points: this is considerably more general than what I thought you
>were saying, and it still does not require intoducing metrics, unless
>you want to call the number of scale steps between two notes of a
>quasiperiodic scale a "metric".

That is *EXACTLY* what I'm doing. And in fact the scale step metric is indeed a metric. It is non-negative, symmetric, obeys the triangle inequality, etc.

If you want to be cute you can extend the metric over all of pitch (or pitch class space), but this is just a party trick. However, in C major, it's kind of fun to think of C# as "half a step" above C, just as E quarter tone sharp is "half a step" above E.

>It's the KISS principle I have in mind
>here, which suggests you not introduce bells and whistles unless you
>can do something as a result.

There are no bells and whistles to introduce. The "scale step distances" are precisely what is conserved when you play a musical pattern first in one scale and then in another. Or when you transpose something diatonically.

It's hardly introducing crazed, extra machinery to note that transposition is translation, and translation occurs relative to some metric or other.

>If you are asking how do I interpret the abstract "scalar" (can we
>find another word?) distance (+1 +1 -1 +1 +1), the answer is that in
>that case, that's how I would interpret it. Given any base point on
>any quasiperiodic scale, and you can turn it into actual notes. The
>results will differ wildly.

Which is what I've been saying for half a dozen emails now. "Scalar distance" is a kind of "distance." Yeesh!

>There's more involved than a fetish for notation. One hobby of mine is
>lifting pieces from 12-et (a tuning I find mildly annoying) to
>meantone, which is much better on my nerves. In that case what you are
>doing could from one point of view be described as trying to find a
>consistent spelling.

Well, I'd be curious to hear some of the translations.

I imagine there are going to be lots of cases -- in even moderately chromatic music -- where there simply is no consistent spelling. You're going to have to make a sacrifice somewhere.

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗monz <monz@tonalsoft.com>

--- In tuning-math@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> doesn't Euler also?

My goodness, no.

Euler certainly deserves the credit for the prime-factor
exponent *concept*, but he just as certainly never advocated
using them as part of the musical notation. For him, notes
were still written with the plain old A,B,Cs and #/b.

For those interested, here's a paper about Euler's music-theory
which i translated from French and made into a webpage:

http://sonic-arts.org/monzo/euler/euler-en.htm

-monz
http://tonalsoft.com
Tonescape microtonal music software

> djwolf_frankfurt wrote:
> >
> > --- In tuning-math@yahoogroups.com
> > <mailto:tuning-math%40yahoogroups.com>, "Gene Ward Smith"
> > <genewardsmith@> wrote:
> > >
> > > --- In tuning-math@yahoogroups.com
> > <mailto:tuning-math%40yahoogroups.com>, "monz" <monz@> wrote:
> > >
> > > > He wanted a convenient name for the concept,
> > > > and i was apparently the first theorist to suggest
> > > > the use of the prime-factor exponents in series as a
> > > > part of actual musical notation:
> > >
> >
> > Tony Conrad was notating with prime factors in the early
> > sixties; La Monte Young got the notation from him.

🔗Kraig Grady <kraiggrady@anaphoria.com>

so i the distinction is using it for actual notation. i see

monz wrote:
>
> --- In tuning-math@yahoogroups.com > <mailto:tuning-math%40yahoogroups.com>, Kraig Grady <kraiggrady@...> > wrote:
> >
> > doesn't Euler also?
>
> My goodness, no.
>
> Euler certainly deserves the credit for the prime-factor
> exponent *concept*, but he just as certainly never advocated
> using them as part of the musical notation. For him, notes
> were still written with the plain old A,B,Cs and #/b.
>
> For those interested, here's a paper about Euler's music-theory
> which i translated from French and made into a webpage:
>
> http://sonic-arts.org/monzo/euler/euler-en.htm > <http://sonic-arts.org/monzo/euler/euler-en.htm>
>
> -monz
> http://tonalsoft.com <http://tonalsoft.com>
> Tonescape microtonal music software
>
> > djwolf_frankfurt wrote:
> > >
> > > --- In tuning-math@yahoogroups.com > <mailto:tuning-math%40yahoogroups.com>
> > > <mailto:tuning-math%40yahoogroups.com>, "Gene Ward Smith"
> > > <genewardsmith@> wrote:
> > > >
> > > > --- In tuning-math@yahoogroups.com > <mailto:tuning-math%40yahoogroups.com>
> > > <mailto:tuning-math%40yahoogroups.com>, "monz" <monz@> wrote:
> > > >
> > > > > He wanted a convenient name for the concept,
> > > > > and i was apparently the first theorist to suggest
> > > > > the use of the prime-factor exponents in series as a
> > > > > part of actual musical notation:
> > > >
> > >
> > > Tony Conrad was notating with prime factors in the early
> > > sixties; La Monte Young got the notation from him.
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗monz <monz@tonalsoft.com>

Hi Dmitri,

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:
>
> > > And what exactly do you mean by calling these scales "meantone"
> >> scales? You mean, you're imagining them embedded in a meantone
> >> 12-notes-to-the octave chromatic scale? Or what?
> >
> >No, I mean they are defined in any meantone system. It really makes
> >more sense to look at them in 19 or 31 than 12, where they are only
> >proper and not strictly proper.
>
> I'm curious -- could you explain further?

I don't know if this will help, but here goes:

http://tonalsoft.com/enc/p/proper.aspx

> > > Why call the acoustic scale (melodic minor ascending)
> > > "meantone" rather than equal tempered, or anything else?
> >
> > Because looking at it in terms of 12 equal temperament
> > is both ahistorical and simply not the best way of
> > understanding the scale in any case.
>
> Don't know about ahistorical. For the music I love most
> -- Debussy and jazz, for example, the scale is a thoroughly
> equal tempered animal.

I'd agree with you that for both Debussy and jazz, the
harmonic basis is the 12-edo tuning and the scales derived
from it.

However, Gene is absolutely correct in that all of the
diatonic scales arose first within a pythagorean (3-limit JI)
context -- in "modern" music-theory, this ocurred between
c.800 AD and c.1450 -- then very shortly after the shift to
5-limit JI, that context changed to meantone in any of its
varieties -- this ocurred starting in the mid-to-late 1400s.

It just so happens that 12-edo is a member of the meantone
family (among many others), which is precisely why composers
were able to adapt to it during the 1800s.

Thus, the entire "common-practice" diatonic harmony is
founded upon tunings of the meantone family.

It is precisely the fact that 12-edo also belongs to many
other tuning families, that enabled composers like Debussy
and those working in jazz were able to develop entirely new
harmonic techniques that are impossible in a meantone context
for any meantone other than 12-edo.

> I'm also curious about "simply not the best way of
> understanding the scale in any case." I'm an equal-tempered
> guy, playing the scale at my equal tempered piano. Explain
> to me how I should think of the scale. (I'm not being a
> smart ass, I'm genuinely curious.)

Based on what i just wrote above, you should be able
to recognize the historical value of understanding Eurocentric
harmonic practice in terms of the entire meantone family first,
and the specific 12-edo instance of meantone second.

I probably should point out here that i personally consider
the recent development of the concept of tuning "families"
to be perhaps the most significant thing to come out of
the internet tuning lists.

http://tonalsoft.com/enc/f/family.aspx

Gene can answer all the rest ...
i just felt like jumping in here. :)

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Kraig Grady <kraiggrady@anaphoria.com>

didn't Debussy state he thought in 36 pitches to the octave?

monz wrote:
>
> Hi Dmitri,
>
> --- In tuning-math@yahoogroups.com > <mailto:tuning-math%40yahoogroups.com>, Dmitri Tymoczko <dmitri@...> > wrote:
> >
> > > > And what exactly do you mean by calling these scales "meantone"
> > >> scales? You mean, you're imagining them embedded in a meantone
> > >> 12-notes-to-the octave chromatic scale? Or what?
> > >
> > >No, I mean they are defined in any meantone system. It really makes
> > >more sense to look at them in 19 or 31 than 12, where they are only
> > >proper and not strictly proper.
> >
> > I'm curious -- could you explain further?
>
> I don't know if this will help, but here goes:
>
> http://tonalsoft.com/enc/p/proper.aspx > <http://tonalsoft.com/enc/p/proper.aspx>
>
> > > > Why call the acoustic scale (melodic minor ascending)
> > > > "meantone" rather than equal tempered, or anything else?
> > >
> > > Because looking at it in terms of 12 equal temperament
> > > is both ahistorical and simply not the best way of
> > > understanding the scale in any case.
> >
> > Don't know about ahistorical. For the music I love most
> > -- Debussy and jazz, for example, the scale is a thoroughly
> > equal tempered animal.
>
> I'd agree with you that for both Debussy and jazz, the
> harmonic basis is the 12-edo tuning and the scales derived
> from it.
>
> However, Gene is absolutely correct in that all of the
> diatonic scales arose first within a pythagorean (3-limit JI)
> context -- in "modern" music-theory, this ocurred between
> c.800 AD and c.1450 -- then very shortly after the shift to
> 5-limit JI, that context changed to meantone in any of its
> varieties -- this ocurred starting in the mid-to-late 1400s.
>
> It just so happens that 12-edo is a member of the meantone
> family (among many others), which is precisely why composers
> were able to adapt to it during the 1800s.
>
> Thus, the entire "common-practice" diatonic harmony is
> founded upon tunings of the meantone family.
>
> It is precisely the fact that 12-edo also belongs to many
> other tuning families, that enabled composers like Debussy
> and those working in jazz were able to develop entirely new
> harmonic techniques that are impossible in a meantone context
> for any meantone other than 12-edo.
>
> > I'm also curious about "simply not the best way of
> > understanding the scale in any case." I'm an equal-tempered
> > guy, playing the scale at my equal tempered piano. Explain
> > to me how I should think of the scale. (I'm not being a
> > smart ass, I'm genuinely curious.)
>
> Based on what i just wrote above, you should be able
> to recognize the historical value of understanding Eurocentric
> harmonic practice in terms of the entire meantone family first,
> and the specific 12-edo instance of meantone second.
>
> I probably should point out here that i personally consider
> the recent development of the concept of tuning "families"
> to be perhaps the most significant thing to come out of
> the internet tuning lists.
>
> http://tonalsoft.com/enc/f/family.aspx > <http://tonalsoft.com/enc/f/family.aspx>
>
> Gene can answer all the rest ...
> i just felt like jumping in here. :)
>
> -monz
> http://tonalsoft.com <http://tonalsoft.com>
> Tonescape microtonal music software
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:

> >No, I mean they are defined in any meantone system. It really makes
> >more sense to look at them in 19 or 31 than 12, where they are only
> >proper and not strictly proper.
>
> I'm curious -- could you explain further?

In trying to deal with a 7-note scale in 12-et, you are pushing its
ability to separate interval classes to the limit, simply because of
the proportion between the number of notes. In the diatonic scale, for
instance, the interval class for the fourth contains six perfect
fourths and one augmeted fourth, and the interval class for the fifth
six perfect fifth and one diminished fifth. In meantone tunings where
the fifth is flatter than 700 cents, the augmented fourth, the true
tritone, is flatter than the diminished fifth. But 12-et is an
extreme, very sharp meantone tuning where they are not distinguished,
and hence the diatonic scale is only proper, but not strictly proper,
as the interval classes overlap on a boundry. The other scales I
mentioned have even more boundry overlap.

> Don't know about ahistorical. For the music I love most -- Debussy
> and jazz, for example, the scale is a thoroughly equal tempered
> animal.

Debussy and jazz don't have much to do with diatonic theory, but
instead bring to the fore other aspects of 12-et. However, jazziness
is certainly capable of being captured in other tunings. 22 notes to
the octave seems to be a good choice; you might check out the examples
here:

http://en.wikipedia.org/wiki/22_equal_temperament#Musical_examples

While I'm hardly Mr Cool Jazz Guy, I think there is something jazzy
about my music sometimes, as for example this piece in 46-et:

http://www.xenharmony.org/ogg/gene/maz/GeneSmith_Chromosounds_mazrend0.mp3

> I'm also curious about "simply not the best way of understanding the
> scale in any case." I'm an equal-tempered guy, playing the scale at
> my equal tempered piano. Explain to me how I should think of the
> scale. (I'm not being a smart ass, I'm genuinely curious.)

The whole major/minor system which is one of the mainstays of common
pratice music arose in the early modern era, at a period when meantone
was the way things other than guitars and lutes were tuned. All of the
scales arise in a natural way theoretically, without even addressing
this history, simply by comparison of proper scales which are common
between the meantone systems of 12, 19 or 31 notes. To attempt to
hammer it into 12-et only system is ashistorical and to me clearly
misguided.

> >If you stick it into 31 equal, you get a lot of choices of generator
> >you could use define it, but it has the property of being what I was
> >calling "unambiguously meantone", meaning that it is more compact (by
> >quite a margin, actually) in terms of meantone generators than any
> >others.

> Sorry, can you spell this out more? You're saying that on some
> particular tuning lattice, the melodic minor scale is particularly
> compact, and this is very enlightening? Just run me through the
> details.

I wouldn't worry much about the lattice aspect, because we are only
looking at pitch classes along a single line of generators--a rank one
group, isomorphic as a group to Z. Since 31 is a prime number, any n
scale steps 0 < n < 31 will define a generator relatively prime to 31.
If we take the notes of the scale s[i] in terms of scale steps up to
31, we may convert them into generator steps by looking at s[i]/n mod
31. This can be more or less compact simply in the sense that the
maximum minus the minimum value can be greater or less. Doing this to
the diatonic scale, melodic minor, harmonic minor etc. scales shows
they are more compactly expressed with a meantone generator (13 or 18
in this case) than any other. Hence, they are in a clearly defined
sense intrinsically meantone.

> That is *EXACTLY* what I'm doing. And in fact the scale step metric
> is indeed a metric. It is non-negative, symmetric, obeys the
> triangle inequality, etc.

Yes, well, but you will confuse peoplem when you use the word "metric"
here unless you make it clear you mean "metric" in the sense of graph
theory, since normally people have in mind something else--even though
a metric in the graph theory sense satisfies the axioms of a metric space.

> If you want to be cute you can extend the metric over all of pitch
> (or pitch class space), but this is just a party trick.

I thought you were insisting on the party trick, hence my objection.

> Which is what I've been saying for half a dozen emails now. "Scalar
> distance" is a kind of "distance." Yeesh!

"Scalar distance" is not a standard term, I hope?

> I imagine there are going to be lots of cases -- in even moderately
> chromatic music -- where there simply is no consistent spelling.
> You're going to have to make a sacrifice somewhere.

I responded to the claim that Wagner could not be put into meantone by
trying. It seemed to me that sometimes this didn't work very well, and
other times it did:

http://www.xenharmony.org/wagmean.htm

Here are some midi files in case you have a slow connection:

http://www.xenharmony.org/meanmidi.htm

The 21st century still has ultraconservatives:

http://www.xenharmony.org/mmm.htm

From what I've heard of yourm music I'm not too sanguine about
converting it to meantone, but if you have some conservative music
tucked awayn in a drawer it might be possible.

🔗Graham Breed <gbreed@gmail.com>

Dmitri Tymoczko wrote:
>> > I'm just saying that the terminological shoe is pinching a bit
>>
>>> uncomfortably here.
>>
>>We have to look at these scales because they represent important rank 2
>>temperaments. As a result we don't usually think in octave-equivalent
>>space because it gets too messy. That isn't such a sacrifice because it
>>doesn't properly model the music-theoretical concept of octave
>>equivalence anyway.
> > I'm not really arguing that you shouldn't think about the objects, > just that it might be nice to describe them a bit differently. For > instance "all scalar intervals come in at most two flavors" does the > job quite nicely. You can then say that these scales are all > "generated" -- if you take a slightly expanded view of "generated."

What way of describing them are you objecting to? I can't be sure by tracing back the thread. Two sizes of intervals isn't an obviously useful property, or one that it's easy to see that a scale has, unless you know that it's a property of all MOS scales. The concept of MOS has been around for over 30 years and it's quite flexible enough to cover periods other than the octave (and Kraig, who's in contact with Erv, is happy that it's always been that flexible).

MOS scales are generated in the sense that a free abelian group is generated. Regular temperaments happen to be free abelian groups. Rank 2 temperaments always have two generators, by definition. There are a lot of cases where it's useful to fix one of those generators so that it equally divides the octave (or generalization thereof) and when we do that we call it the "period" of the temperament. Once you've got a canonical period and generator it makes sense to describe the MOS scales that work well with the temperament accordingly.

There are other ways of defining temperaments. Two archetypal equal temperaments is one that I'm fond of. In that case, you can describe an MOS scale as having the same number of notes as either of the defining equal temperaments, or their sum. And you can define the pattern of notes as a generalization of maximally even sets, or by making the large and small intervals be as evenly spaced as possible. Whatever you do the result is an MOS.

>>I wrote up some of the background to this earlier in the year:
>>
>>http://x31eq.com/paradigm.html
>>
>>It's supposed to be understandable by music professors, so you can test
>>that ;)
> > Looks perfectly comprehensible. I'll have a closer look, and maybe > make some comments, over the next few weeks.

That's good! I've been reading your papers as well.

>> >>I notice your ME definition in voiceleading.pdf doesn't require b<=a, so
>>
>>>>it's possible to get the whole of a-equal with duplicates.
>>>
>>> Yup, correct.
>>
>>In that case you'll get some pathological scales that aren't proper, if
>>it makes sense to apply propriety to them at all. For example, 9 from 7 is
>>
>>0, 0, 1, 2, 3, 3, 4, 5, 6, 7
>>
>>7-3=4 can span 4 diatonic steps, but 3-0=3 can span 5.
> > > I think there's a mistake here -- your scale has 10 notes and is not > maximally even. You either mean:
> > 0 0 1 2 3 3 4 5 6

That's it. No mistake, I wrote it including the octave.

Graham

🔗Graham Breed <gbreed@gmail.com>

djwolf_frankfurt wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
> > >>Well, fortuitously for me, Graham just put the link into
>>another message he wrote to Dmitri:
>>
>>http://x31eq.com/paradigm.html
>>
> > > Graham --
> > There are some excellent things here. Just a few comments:

I wasn't planning to change it any more, but if something's wrong or difficult to understand ...

> (1) using "do re mi" carries its own ambiguities, I would just say,
> for better or worse, "diatonic scale". (In the paragraph about
> Srutis, you
> are right about the possibility of describing Indian musics in terms
> of small variations from a 12tet chromatic, but Do Re Mi can't be
> compared with Srgam because in the Indian solfege the internal terms
> in each tetrachord can be raised or lowered so the scales don't have
> a single fixed (however coarsely) shape as does the diatonic.) Yes, it's ambiguous, which means I can pin it on a lot of different traditions. It's also widely known phrase. Much more so than "diatonic" because Hollywood musicals have a much wider reach than music theory.

> (2) I would date a 7-out-12 paradigm both more specifically and
> narrowly. In the 14th and 15th century music I've been working with
> recently, a 12-tone chromatic plays no role at all, the paradigm is
> instead something more modest: two diatonic collections, one with a
> hard and the other with a soft b, with extra tones added as ficta (an
> eb below the bb or an f# above the b) or chromatic neighboring tones.
> On the other hand, there were times in the meantone era when the tonal
> resources were more than twelve tones.

You can certainly do that, but it's the kind of detail I tried to avoid. Although I do mention more than 12 notes under "extended meantone".

> (2) the definition of meantone has to be a bit clearer. At a minimum,
> one should indicate why the tones are mean, and what interval they are
> the mean of, and specify that in 1/4 comma meantone (which many of us
> simply call meantone), the third is just and the fifths are tempered
> so that 4 of them divide a 5:2 equally. All of the meantone-like
> tunings then share the premise that the major third (specifically: all
> the thirds that are notated as major thirds) is the same as the sum of
> four fifths, mod the octave. I would also emphasize that classical
> meantone was not neccessarily limited to an octave with twelve tones,
> but the sequence of meantone fifths could be continued indefinitely,
> and meantone keyboards with 14-16 tones were not unknown.

I'll have a look at the definition, but I don't want to get into specific tunings. Most people don't think in meantone any more anyway.

> (3) it is not clear to me that "one unavoidable problem with the
> paradigm is that it makes things more complex than they were before"
> is necessarily true. What is true is that the scheme is inclusive so
> that a diversity of scale types are accounted for, and that
> comprehensiveness may be better characterized as unfamiliar than complex.

A diversity of scale types is more complex than one scale type. How can you avoid that?

> (4) I would also drop the paradigm discussion. You have the outline
> for a strong presentation on its own terms and, if you don't mind my
> advice, the scholarly world doesn't need another paper beginning with
> Kuhn unless you're really prepared to handle the inevitable arguments.
> Why not simply say that "this is a new approach to scales
> (period)"? Also consider the addition of a clear statement
> explaining concisely why this approach is useful and does things that
> prior approaches cannot do.

Well, that's the whole conceit the piece is based on. I'm not going to change it now. And the point is that it isn't a new approach, but a fundamentally new way of thinking. And that it can't do anything other paradigms can't, but it does make some things easier to think about. And because it's a new way of thinking it's difficult for outsiders to understand, or see the value of, and there's no way of convincing them by logical argument. I haven't actually read Kuhn but I think this is the kind of thing he talks about.

> Once again, the page is excellent and I have learned quite a bit.

That's good! Thanks for reading!

Graham

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>In trying to deal with a 7-note scale in 12-et, you are pushing its
>ability to separate interval classes to the limit, simply because of
>the proportion between the number of notes. In the diatonic scale, for
>instance, the interval class for the fourth contains six perfect
>fourths and one augmeted fourth, and the interval class for the fifth
>six perfect fifth and one diminished fifth. In meantone tunings where
>the fifth is flatter than 700 cents, the augmented fourth, the true
>tritone, is flatter than the diminished fifth. But 12-et is an
>extreme, very sharp meantone tuning where they are not distinguished,
>and hence the diatonic scale is only proper, but not strictly proper,
>as the interval classes overlap on a boundry. The other scales I
>mentioned have even more boundry overlap.

I see. So is the idea is that "properness" (or strict properness) is an important enough property that you want to use a temperament according to which common scales are strictly proper? And this will allow you, in the harmonic minor scale for instance, to distinguish the augmented second from the minor third?

That's definitely a very interesting idea -- if any of you decide to write an article about the intersections between academic theory and tuning-theory, you should include something about that. To my knowledge, the academic theory discussions of properness (including Jay Rahn, who uses "contradiction" and "ambiguity") tend to focus on the equal tempered case.

>Debussy and jazz don't have much to do with diatonic theory, but
>instead bring to the fore other aspects of 12-et.

Well, I don't know about this. It depends on how you define "diatonic theory." I think you can make a case -- as I do in "The Consecutive Semitone Constraint ..." and "Scale Networks and Debussy" -- that Debussy's music has a lot to do with scale theory. In particular, I think Debussy was quite consciously involved in the business of finding new scales. And the scales he was interested in were ones that resembled the diatonic.

So, is that "diatonic theory?" I'd say so, in some general sense.

>The whole major/minor system which is one of the mainstays of common
>pratice music arose in the early modern era, at a period when meantone
>was the way things other than guitars and lutes were tuned.

It arose in that era, but survived into an era where equal temperaments became increasingly common. And from very early on -- definitely early 1800s, but probably much earlier -- you find composers starting to take advantage of the possibilities of equal temperament. As you say, Wagner often doesn't sound so hot in meantone.

So the situation here is really complex -- equal temperament was definitely a late development, but an important one. And the transition to equal temperament was relatively smooth. So to insist on a meantone approach to familiar scales strikes me as just as odd as insisting on an equal-tempered approach. In some important sense, the diatonic scale can be both an equal-tempered and an unequal-tempered entity.

> Doing this to
>the diatonic scale, melodic minor, harmonic minor etc. scales shows
>they are more compactly expressed with a meantone generator (13 or 18
>in this case) than any other. Hence, they are in a clearly defined
>sense intrinsically meantone.

There's a really interesting observation in here. And, interestingly enough, it's (roughly) the *same* observation as the following:

"The common scales (melodic minor and harmonic minor/major) are the ones you get if you start with a diatonic scale and shift one note by one semitone in a way that does not change the size of the scale's thirds."

The relation between meantone-compactness and third-preservation (in the context of single-semitone voice leading) is not at all obvious, at least to me. It's basically spelled out in Appendix I of "Scale Networks and Debussy."

So would you say that melodic minor and major major/minor are "fundamentally meantone?" Or "fundamentally near diatonic?" I prefer the latter, for obvious reasons.

>"Scalar distance" is not a standard term, I hope?

I use it. I also use the terms "scalar interval" and "chromatic interval" where Clough uses "generic interval" and "specific interval." The reason I use these terms is precisely to emphasize that, in the two cases, you're measuring distance relative to two different scales.

>From what I've heard of yourm music I'm not too sanguine about
>converting it to meantone, but if you have some conservative music
>tucked awayn in a drawer it might be possible.

Well, in grad. school most people thought that all of my music was conservative!

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>Two sizes of intervals isn't an obviously
>useful property,

Oh, I think it's a reasonably useful property. In general, you want scalar transposition to be something like chromatic transposition. The two-sizes-of-interval property helps with this -- it makes the chords deform predictably under scalar transposition.

> or one that it's easy to see that a scale has, unless
>you know that it's a property of all MOS scales.

I agree its useful to know which scales have the two-sizes-of-interval property, and that there's a cute structural description.

As I said before, it's probably worth finding a better term than either MOS (which seems to be used in different ways, and anyway to be sort of Erv Wilson's private property, and anyway "moment of symmetry" doesn't make much sense) or DE (since we're really not talking about "evenness.")

BTW, from what I'm hearing recently, most peole seem to think MOS = DE, in which case the statement in Clough's article (claiming to have isolated DE scales for the first time) is wrong. Clough claims that MOS = WF (well-formed), but if the period is allowed to be smaller than an octave then MOS = DE.

I would propose "bi-intervallic" or "dual generated" as an alternative to MOS/DE.

>There are other ways of defining temperaments. Two archetypal equal
>temperaments is one that I'm fond of. In that case, you can describe an
>MOS scale as having the same number of notes as either of the defining
>equal temperaments, or their sum. And you can define the pattern of
>notes as a generalization of maximally even sets, or by making the large
>and small intervals be as evenly spaced as possible. Whatever you do
>the result is an MOS.

Don't quite follow, but am curious.

>You either mean:
> >
>> 0 0 1 2 3 3 4 5 6
>
>That's it. No mistake, I wrote it including the octave.

I get it now. Here are the scale's (ascending) scalar intervals:

0 1 1 1 0 1 1 1 1 (1 step)
1 2 2 1 1 2 2 2 1 (2 steps)
2 3 2 2 2 3 3 2 2 (3 steps)
3 3 3 3 3 4 3 3 3 (4 steps)

3 4 4 4 4 4 4 4 4 (5 steps)
4 5 5 5 4 5 5 5 4 (6 steps)
5 6 6 5 5 6 6 5 5 (7 steps)
6 7 6 6 6 7 6 6 6 (8 steps)

You're right that 4 steps and 5 steps have the same intervals, so the scale is not proper. I think this will always happen if you have more than one zero-step interval.

In any case, it's a cute point that "maximally even" doesn't necessarily imply "proper" in the case of multisets.

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗alternativetuning <alternativetuning@yahoo.com>

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:
>
>
> It arose in that era, but survived into an era where equal
> temperaments became increasingly common. And from very early on --
> definitely early 1800s, but probably much earlier -- you find
> composers starting to take advantage of the possibilities of equal
> temperament. As you say, Wagner often doesn't sound so hot in
> meantone.
>
> So the situation here is really complex -- equal temperament was
> definitely a late development, but an important one. And the
> transition to equal temperament was relatively smooth. So to insist
> on a meantone approach to familiar scales strikes me as just as odd
> as insisting on an equal-tempered approach. In some important sense,
> the diatonic scale can be both an equal-tempered and an
> unequal-tempered entity.
>

Equal temperament can be usefully thought of as one of the
meantone-like temperaments (in which the Major third is the sum of
four fifths; 12tet can also be heard as 1/11-comma temperament), but
it's more difficult to do the opposite. (ET, arguably evolved via 1/5
and 1/6-comma temperaments and well-temperaments, to the point where
for late Haydn and early Beethoven, Kirnberger's circulating
temperament was, for all intents and purposes, understood as equal
temperament. Suddenly, in the decade after Mozart's death, Haydn
starts regularly using keys that had been rare and no longer took the
good-in-meantone keys as a limit (a restriction which holds for
Mozart's catalog). Bach's use of a circulating temperament represents
an earlier cul-de-sac in this development without direct acceptance by
following generations of composers.)

djw

🔗alternativetuning <alternativetuning@yahoo.com>

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...>
> As I said before, it's probably worth finding a better term than
> either MOS (which seems to be used in different ways, and anyway to
> be sort of Erv Wilson's private property, and anyway "moment of
> symmetry" doesn't make much sense) or DE (since we're really not
> talking about "evenness.")
>

If you reconstruct the way in which Wilson initially observed moments
of symmetry, as those moments in the reiteration of a generating
interval where the melodic symmetry along the series of generators,
each subtending the same number of scale degrees is apparent.

For example, with a generator of 5/12 of an octave, we have an MOS at

two tones, each fourth (and the remainder) subtending 0 tones:

C F

three tones, each fourth (and the remainder) subtending 0 tones:

G C F

four tones, no symmetry

five tones, each fourth subtending 1 tone:

A c D f G a C d F

six tones, no symmetry

seven tones, each fourth subtending 2 tones:

Bc d Ef g A bc D ef G a bC d eF

etc.

Wilson's scale tree is just a catalogue of these MOS indexed by the
size of the generator. An MOS is associated with every pair of
relatively prime numbers.

DJW

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗Carl Lumma <ekin@lumma.org>

🔗monz <monz@tonalsoft.com>

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> However, Gene is absolutely correct in that all of the
> diatonic scales arose first within a pythagorean (3-limit JI)
> context -- in "modern" music-theory, this ocurred between
> c.800 AD and c.1450 -- then very shortly after the shift to
> 5-limit JI, that context changed to meantone in any of its
> varieties -- this ocurred starting in the mid-to-late 1400s.

Actually, Gene never even mentioned pythagorean
-- i put that in myself for historical accuracy.

I originally put "meantone" in the new subject heading here,
but "meantone" really is not even the correct name to apply
to the basis of the diatonic scales, because meantone itself
works the same way as pythagorean -- that is, as a linear
(or perhaps i should say "rank-2" instead?) tuning based on
2 generators: one which is the equivalence-interval (the
2:1 ratio or "octave") and the ~3:2 ratio or "perfect-5th".

So what would be the name which covers this quality shared
by both pythagorean and meantone? Is that "regular temperament"?
Somehow it seems to me that that name is not specific enough.

This in fact is exactly why the European tuning paradigm
was able to shift so easily from pythagorean to meantone.
All that needed to change was the actual tuning of the
perfect-5th generator, everything else could stay the
same.

Of course, the fact that all meantones shift the emphasis
from concordance of the 5th to concordance of the
major-3rd, means that harmonic practice did in fact
change quite drastically. But aside from that, other
voice-leading considerations were able to remain
essentially unchanged from pythagorean practice.

> It just so happens that 12-edo is a member of the meantone
> family (among many others), which is precisely why composers
> were able to adapt to it during the 1800s.
>
> Thus, the entire "common-practice" diatonic harmony is
> founded upon tunings of the meantone family.
>
> It is precisely the fact that 12-edo also belongs to many
> other tuning families, that enabled composers like Debussy
> and those working in jazz were able to develop entirely new
> harmonic techniques that are impossible in a meantone context
> for any meantone other than 12-edo.

It was after the large-scale adoption of 12-edo,
with its ability both to manage meantone and to
provide a doorway into other tuning families, that
things really started to change in a big way.
This pretty much began with Beethoven c.1800 and,
i would say, reached its culmination in Schoenberg
and jazz c.1925-1950.

(Of course there was much music after this which
continued -- and still continues -- to explore
the non-meantone aspects of 12-edo ... i'm just
generalizing to keep the argument simple.)

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Carl Lumma <ekin@lumma.org>

>So what would be the name which covers this quality shared
>by both pythagorean and meantone? Is that "regular temperament"?
>Somehow it seems to me that that name is not specific enough.

Pythagorean isn't a temperament, unless you're treating its
triads as consonances.

The two scales share a lot of properties. One of them isn't
Rothenberg propriety, though if you use Lumma propriety they're
close.

Probably the most significant commonalities is that they're
both 7 notes, and both highly omnitetrachordal -- they have
strong symmetry at the fifth.

Rothenberg says scales are identified by their rank order
matrices. As you can see (hit Show Message Options -> Use
Fixed Width Font if you're viewing this on the web)...

1/1 : 2 4 5 8 10 12 13
9/8 : 2 3 5 8 10 11 13
81/64 : 1 3 5 8 9 11 13
4/3 : 2 4 7 8 10 12 13
3/2 : 2 4 5 8 10 11 13
27/16 : 2 3 5 8 9 11 13
243/128: 1 3 5 6 9 11 13

1/1 : 2 4 5 8 10 12 13
194.0 : 2 3 5 8 10 11 13
388.0 : 1 3 5 8 9 11 13
503.0 : 2 4 6 8 10 12 13
697.0 : 2 4 5 8 10 11 13
891.0 : 2 3 5 8 9 11 13
1085.0 : 1 3 5 7 9 11 13

...they're similar.

-Carl

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

> > >"Scalar distance" is not a standard term, I hope?
>>
>> I use it. I also use the terms "scalar interval" and "chromatic
>> interval" where Clough uses "generic interval" and "specific
>> interval." >
>It's overloading a term which is used in mathematics, physics, and
>computer science with another meaning. People trained in one of those
>areas may tend to have a Pavlovian response that the word means
>something having nothing to do with musical scales.

For god sake, Gene! This is really going to far.

Of course a scale step is a measure of distance! It obeys the mathematical requirements for being a metric, it corresponds to the way people talk. Just get over it already.

A: Let's transpose this song up.
B: OK, how far?
C: Two semitones.

I'm not going to discuss this anymore, because it's simply pointless. You're making no arguments whatsoever. Let's just move on.

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:

> I see. So is the idea is that "properness" (or strict properness) is
> an important enough property that you want to use a temperament
> according to which common scales are strictly proper?

I came at it the other way around. Unless you put conditions on
scales, you will be buried alive under the alternatives. One condition
which seems to have a clear justification is propriety. One thing you
might note, incidently, is that the transformations by scale step
we've been talking about make the most sense when the scales are proper.

So, simply starting out to survey seven-note proper scales in 12, 19,
and 31, what do you find? In 12, all the proper scales are of diatonic
type. In 19, all but one are. In 31, there are many possibilities,
though that number is vastly reduced by your suggestion of requiring
only two sizes of scale steps. But among these possibilities, there
are a number which are of meantone type, and these include the usual
suspects already identified from 12 and 19.

And this will
> allow you, in the harmonic minor scale for instance, to distinguish
> the augmented second from the minor third?

Yes, in any meantone tuning flatter than 12-et (which is a very
special, extreme case of meantone) the augmented second is a different
interval than the minor third. In septimal meantone, it serves as a
7/6, and in the optimal meantone tuning range that becomes very close
to its value.

> >Debussy and jazz don't have much to do with diatonic theory, but
> >instead bring to the fore other aspects of 12-et.
>
> Well, I don't know about this. It depends on how you define
> "diatonic theory." I think you can make a case -- as I do in "The
> Consecutive Semitone Constraint ..." and "Scale Networks and Debussy"
> -- that Debussy's music has a lot to do with scale theory.

It's true that "diatonic set theory" means a lot more than "diatonic",
but that's a kind of abuse of language.

> >The whole major/minor system which is one of the mainstays of common
> >pratice music arose in the early modern era, at a period when meantone
> >was the way things other than guitars and lutes were tuned.
>
> It arose in that era, but survived into an era where equal
> temperaments became increasingly common. And from very early on --
> definitely early 1800s, but probably much earlier -- you find
> composers starting to take advantage of the possibilities of equal
> temperament.

I don't know of any earlier examples; I'd be interested to see them.

> So the situation here is really complex -- equal temperament was
> definitely a late development, but an important one. And the
> transition to equal temperament was relatively smooth. So to insist
> on a meantone approach to familiar scales strikes me as just as odd
> as insisting on an equal-tempered approach.

The familiar scales of the diatonic system are not the same as the
octatonic, or Liszt Prometheus, or other non-meantone scales, and did
not arise in the same way or in the same period of time. It makes
sense to me to analyze things as they are, not as they came to be
regarded at a much later period.

In some important sense,
> the diatonic scale can be both an equal-tempered and an
> unequal-tempered entity.

It can be an equal-tempered entity in a multitude of ways, and that is
the key fact which tells you it cannot be confined to one equal
temperament. It is at least as valid (I would say historically and in
terms of tuning accuracy more valid) to regard it as a scale of 31
equal than 12 equal.

> "The common scales (melodic minor and harmonic minor/major) are the
> ones you get if you start with a diatonic scale and shift one note by
> one semitone in a way that does not change the size of the scale's
> thirds."

That's almost true. There is a proper scale I've termed the diminished
diatonic (since it didn't seem to have a name) which also has this
property. It is not a part of traditional music theory and is not even
possible to play in 12-equal: C Db E Fb G Ab Bb

In a standard meantone keyboard tuning for 12 notes, you can't play
this, as E is +4 generator steps and Fb is -8 generator steps, and
they don't both appear on the same keyboard, in any transposition. On
split-note keyboards you could play it but I've never heard of such a
thing.

But if you leave that off, you are right. The properties of being a
proper seven-note scale with a circle of thirds defines the diatonic
system in the more general sense, including harmonic major and all the
various species of minor. This seems to go back to the first discovery
of the diatonic scale, in ancient Mesopotamia. While we are sorely
lacking in musical examples, we do have tuning instructions which seem
to define a diatonic scale with a full circle of thirds, and a chain
of six fifths.

http://www.kingmixers.com/Terp.html

There is arguably a continuous tradition here, from the Middle East to
Greece to Rome to Medieval Europe to Renaissance Europe, where the
scale is, it seems, rediscovered. Medieval practices involving
parallel thirds such as gymel may have played a role, and it could be
that this is what the Sumerians were doing. It's a little late to check.

> So would you say that melodic minor and major major/minor are
> "fundamentally meantone?" Or "fundamentally near diatonic?" I
> prefer the latter, for obvious reasons.

I would say both are clearly true, and also that they belong to a
diatonic scale system in a wider sense.

> Well, in grad. school most people thought that all of my music was
> conservative!

I believe it, but the people on that web page are *seriously*
conservative, especially Peppercorn.

🔗monz <monz@tonalsoft.com>

Hi Dmitri,

--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:

> To my knowledge, the academic theory discussions of
> properness (including Jay Rahn, who uses "contradiction"
> and "ambiguity") tend to focus on the equal tempered case.

The academic theory discussions of *everything* concerning
music tend to focus on the equal tempered case!

That's something that is finally changing slowly,
but *very* slowly.

> So the situation here is really complex -- equal temperament
> was definitely a late development, but an important one.
> And the transition to equal temperament was relatively smooth.
> So to insist on a meantone approach to familiar scales strikes
> me as just as odd as insisting on an equal-tempered approach.
> In some important sense, the diatonic scale can be both an
> equal-tempered and an unequal-tempered entity.

You're obviously using the terms "equal temepered" and
"equal temperament" to refer specifically to 12-edo, which
is extremely common among almost all music-theorists,
especially those in academia.

This is unfortunate, because 19-edo and 31-edo are also
both equal-temperaments, and as Gene pointed out, both
also belong to the meantone family, along with 12-edo.
And both were discussed, advocated, and used by theorists
and composers as early as 1558.

19-edo, Guillaume Costeley 1558:
http://en.wikipedia.org/wiki/19_equal_temperament

19-edo, Salinas 1577:
http://tonalsoft.com/enc/number/19edo.aspx

31-edo, Lemme Rossi 1666:
http://en.wikipedia.org/wiki/Lemme_Rossi

31-edo, Christiaan Huygens 1691:
http://www.xs4all.nl/~huygensf/doc/lettre.html

(Huygens's own personal notes from 1661 contain a comparison
of 1/4-comma meantone with 31-edo.)

Composers of the late Renaissance and early Baroque
knew of the use of 12-edo for fretted string instruments
(primarily lutes and guitars), but never expected any
tuning other than meantone for all other instruments.

During the later Baroque, the well-temperaments became
popular on keyboards, which limited the pitch universe
to an unequal 12-note set, and paved the way for the
general adoption of 12-edo in the early 1800s.

Unless by "meantone" one means specifically only
1/4-comma meantone, it is incorrect to speak of "meantone"
and "equal temperament" as if they are two different
types of tunings:

* some tunings in the meantone family are equal-temperaments
and others are not;

* some equal-temperaments belong to the meantone family
and others do not.

And as i wrote in another post, meantone itself was a
later development -- historical priority for the diatonic
scales belongs to pythagorean tuning.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗monz <monz@tonalsoft.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
> <snip ...> The properties of being a proper seven-note scale
> with a circle of thirds defines the diatonic system in the
> more general sense, including harmonic major and all the
> various species of minor. This seems to go back to the
> first discovery of the diatonic scale, in ancient Mesopotamia.
> While we are sorely lacking in musical examples, we do have
> tuning instructions which seem to define a diatonic scale
> with a full circle of thirds, and a chain of six fifths.
>
> http://www.kingmixers.com/Terp.html

At the end of this webpage:

http://www.sonic-arts.org/monzo/babylonian/hurrian/monzh6.htm

i put a description of the Babylonian "tuning tablet"
(CBS-10996), which Franklin describes in his Chapter 6.

I interpret the tuning tablet to be more than just a
set of tuning instructions, because i made a MIDI rendition
of it and it sounds to me like it could be a lyre etude.
The link in my Hurrian Hymn webpage is now broken, but
the piece is here:

/tuning-math/files/monz/cbs10996.mid

> There is arguably a continuous tradition here, from the
> Middle East to Greece to Rome to Medieval Europe to
> Renaissance Europe, where the scale is, it seems, rediscovered.
> Medieval practices involving parallel thirds such as gymel
> may have played a role, and it could be that this is what
> the Sumerians were doing. It's a little late to check.

But you have one tuning theorist here who totally agrees
with all of this!

It's also worth noting that the ancient Greek diatonic genus
was not necessarily in pythagorean tuning, despite the
fact that the usual description of it was pythagorean. See:

http://tonalsoft.com/enc/d/diatonic-genus.aspx

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

🔗djwolf_frankfurt <djwolf@snafu.de>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗monz <monz@tonalsoft.com>

--- In tuning-math@yahoogroups.com, "djwolf_frankfurt" <djwolf@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> >
>
> >
> > And as i wrote in another post, meantone itself was a
> > later development -- historical priority for the diatonic
> > scales belongs to pythagorean tuning.
> >
>
> Priority yes, but meantone developed as a solution for
> instruments in the immediate context of a substantial
> vocal repertoire in which pythagorean thirds were no
> longer acceptable.
>
> It is hard to imagine that the first musicians with whom
> we definitively associate use of meantone came to meantone
> directly from a pythagorean environment, but were used to
> vocal polyphony in which just thirds were featured.

We have direct evidence, from Theinred of Dover in the 1100s
and Walter Odington in the early 1300s, that singers in
England were using 5-limit major and minor 3rds. And the
documents show that this trend spread very quickly to Paris,
and thence to Italy and other parts of the continent.

In 1318, Marchetto of Padua wrote about the division of the
whole-tone into 5 parts, various pairs of which formed his
3 different types of semitones and a diesis.

It's inevitable that working within these more subtly-tuned
musical environments, problems of commatic shift and drift
would crop up. Thus, i'm inclined to believe that musicians
dabbled in meantone from as early as c.1300 if only for
practical reasons.

Certainly, _a capella_ vocal music from after this time
would have been performed in some form of dynamic adaptive-JI,
so that listeners would hear what essentially seemed to be
5-limit JI. This is, as you point out, in stark contrast
to the pythagorean tradition which had existed in Europe
before then.

Mark Lindley has shown that in the late-1300s/early-1400s,
keyboard compositions were written in which a 12-note
tuning was used, tuned as a pythagorean chain from Gb to B,
but with the Gb and Db being treated as the 5-limit F# and C#
a skhisma (~2 cents) higher, and the Ab and Eb being used
either as pythagorean flats or as 5-limit G# and D#.
Thus, a wide variety of pythagorean and quasi-5-limit-JI
intervals was available.

http://tonalsoft.com/enc/s/schismic-tuning.aspx

Bartolomeo Ramos (sometimes spelled Ramis) wrote the first
treatise which described monochord division of a 5-limit JI
tuning, in 1482. Arnold Schlick described what appears to
be 1/4-meantone in 1511. Pietro Aron described it beyond
doubt in 1529.

-monz
http://tonalsoft.com
Tonescape microtonal music software