Yahoo Tuning Groups Ultimate Backup tuning Tuning Lattices (II)

I'd also like to make a few remarks concerning James Tenney's graph, which Daniel Wolf posted on the web.

1) If we think of this space as continuous, then the X axis rather hard to interpret -- moving left or right doesn't change the pitch that's being represented in any way. So for some (but not all) musical purposes, the line at the center encodes all the needed information. Neither Tenney nor I invented this line, which is just log-frequency space.
Typically, a (continuous) n-dimensional tuning lattice can be written as the product of a single line that represents log-frequency space and an (n-1) dimensional space whose meaning is a hard to interpret in the above sense. My sense is that this redundancy often arises only when you try to make the tuning lattices continuous -- for instance in a discrete 2 x 3 tuning grid, every vertex corresponds to a distinct pitch, because the generators are relatively prime.

2) As Daniel Wolf observes, any chord of pitches can be represented as a set of points on the central Y axis of Tenney's graph, and that voice leading distances can be represented by how far the points have to move to get from one configuration to another. This is just to say that we can represent chords of pitches as unordered collections of points in log-frequency space, and that voice leading distance is represented by the distance between one configuration and another.
The problem is that this representation doesn't perspicuously capture the natural music-theoretical idea that chords containing the pitch classes are similar objects: for instance {C4, E4, G4} and {C4, G4, E5} are represented by distinct configurations of points on the central axis, whereas -- for many (but not all) music-theoretical purposes -- it is useful to think of them as instances of the same object, the C major chord.

3) To this end, it is useful to represent chords as sets of unordered points on a circle, so that the C major chord is {C, E, G}, or {12, 4, 7} on the circumference of an ordinary clock face. The problem with this representation is that it doesn't clearly depict the relation between a chord's internal structure and it's voice leading possibilities.
(Again, one could "Tenney-ify" the circle by adding additional dimensions, and then one could use these dimensions to draw tuning lattices of various sorts. From the standpoint of voice leading we don't need to do this, but for thinking about tuning it might well be useful.)

4) Given 3), it's useful to look at the configuration space of unordered points on a circle -- i.e. the geometrical spaces in which each point represents a *configuration* of points on the circle. These are the orbifolds I discuss in my paper. In these configuration spaces a point represents a chord, and a line segment represents a voice leading between two chords.
Happily, when you understand these orbifolds, the connection between a chord's internal structure and its voice leading capabilities becomes perfectly clear. The musical properties of "evenness" and "unevenness" simply correspond to the distance from the center of the orbifold; transpositional symmetry is represented by one set of discrete symmetries of the orbifold; and inversional symmetry is represented by another set of reflectional symmetries of the orbifold.

I don't want to make any grand claims for my work. In some sense, it's quite surprising that these spaces weren't discovered 100 years ago. But I do think that this is relatively new territory, and rather different from the discrete lattices familiar from tuning theory.

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'd also like to make a few remarks concerning James Tenney's graph, > which Daniel Wolf posted on the web.

It's nice to see you've joined the bear pit!

> 1) If we think of this space as continuous, then the X axis rather > hard to interpret -- moving left or right doesn't change the pitch > that's being represented in any way. So for some (but not all) > musical purposes, the line at the center encodes all the needed > information. Neither Tenney nor I invented this line, which is just > log-frequency space.

Yes.

> Typically, a (continuous) n-dimensional tuning lattice can be > written as the product of a single line that represents log-frequency > space and an (n-1) dimensional space whose meaning is a hard to > interpret in the above sense. My sense is that this redundancy often > arises only when you try to make the tuning lattices continuous -- > for instance in a discrete 2 x 3 tuning grid, every vertex > corresponds to a distinct pitch, because the generators are > relatively prime.

A lattice, as I understand the algebraic definition, is a regular array of points embedded in some continuous space. It's probably best to think of it in that way for now.

> 2) As Daniel Wolf observes, any chord of pitches can be represented > as a set of points on the central Y axis of Tenney's graph, and that > voice leading distances can be represented by how far the points have > to move to get from one configuration to another. This is just to > say that we can represent chords of pitches as unordered collections > of points in log-frequency space, and that voice leading distance is > represented by the distance between one configuration and another.

You could also consider the lattice distance as part of the voice leading distance. It may be putting the cart before the horse, but intervals in the diatonic scale are usually favoured melodically. Minimizing steps on the spiral of fifths helps you to avoid them. If you only consider log-frequency space, and round everything to the nearest semitone, you lose this distinction.

> The problem is that this representation doesn't perspicuously > capture the natural music-theoretical idea that chords containing the > pitch classes are similar objects: for instance {C4, E4, G4} and {C4, > G4, E5} are represented by distinct configurations of points on the > central axis, whereas -- for many (but not all) music-theoretical > purposes -- it is useful to think of them as instances of the same > object, the C major chord.

Harmonic lattices can be octave-equivalent if you want them to be. They're usually drawn with the factors two of removed, and lines joining basic consonances in a hexagonal arrangement. You can also roll up JI lattices to represent temperaments. Any equal temperament is the surface of a torus in this model which is superficially similar to your work.

> 3) To this end, it is useful to represent chords as sets of unordered > points on a circle, so that the C major chord is {C, E, G}, or {12, > 4, 7} on the circumference of an ordinary clock face. The problem > with this representation is that it doesn't clearly depict the > relation between a chord's internal structure and it's voice leading > possibilities.
> (Again, one could "Tenney-ify" the circle by adding > additional dimensions, and then one could use these dimensions to > draw tuning lattices of various sorts. From the standpoint of voice > leading we don't need to do this, but for thinking about tuning it > might well be useful.)

The most obvious application is adaptive temperament, because it's another optimization problem and picks up where voice leading leaves off.

> 4) Given 3), it's useful to look at the configuration space of > unordered points on a circle -- i.e. the geometrical spaces in which > each point represents a *configuration* of points on the circle. > These are the orbifolds I discuss in my paper. In these > configuration spaces a point represents a chord, and a line segment > represents a voice leading between two chords.
> Happily, when you understand these orbifolds, the connection > between a chord's internal structure and its voice leading > capabilities becomes perfectly clear. The musical properties of > "evenness" and "unevenness" simply correspond to the distance from > the center of the orbifold; transpositional symmetry is represented > by one set of discrete symmetries of the orbifold; and inversional > symmetry is represented by another set of reflectional symmetries of > the orbifold.
> > I don't want to make any grand claims for my work. In some sense, > it's quite surprising that these spaces weren't discovered 100 years > ago. But I do think that this is relatively new territory, and > rather different from the discrete lattices familiar from tuning > theory.

If tuning theoery and voice leading both work geometrically, perhaps a unified geometry would handle harmonization and the like.

We use other geometric ideas as well. Regular temperaments are generated by a small set of intervals, and properties such as the overall error in a temperament are continuous functions of the sizes of those intervals. You can also plot regular temperaments in a space of errors of prime intervals, which is analagous to the harmonic lattices. One point represents just intonation, and the nearer a temperament gets to this point the more accurately it represents just intonation. Another idea is that all tunings related by a uniform scale stretch are equivalent, which is a kind of projective geometry. In this space, each mapping of an equal temperament corresponds to a distinct point and you can find an equal temperament mapping arbitrarily close to any point in the space.

Graham

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>A lattice, as I understand the algebraic definition, is a regular array
>of points embedded in some continuous space. It's probably best to
>think of it in that way for now.

Well, I would say that not all lattices can be embedded into a meaningful continuous geometry. (Cf., for instance, the definition here: http://mathworld.wolfram.com/Lattice.html.) The interesting thing is that the lattices in tuning theory are of this type: when you try to embed them in a continuous space, you find that only one dimension contains meaningful information. This is kind of cool and interesting, in my opinion.

>You could also consider the lattice distance as part of the voice
>leading distance. It may be putting the cart before the horse, but
>intervals in the diatonic scale are usually favoured melodically.
>Minimizing steps on the spiral of fifths helps you to avoid them. If
>you only consider log-frequency space, and round everything to the
>nearest semitone, you lose this distinction.

I can't imagine measuring voice leading on the lattice. On a 2x3 lattice, the distance C4->G4 would be 1, while the distance C4->D4 would be 3 (up two perfect fifths, and down one octave). So anyone minimizing lattice distance would prefer lots of voices moving by perfect fifth to voices moving by step. This might be useful for some purposes, but not voice leading, where steps are smaller than fifths.

If you want to study diatonic voice leading, the simplest thing to do is look at the diatonic lattice in my orbifolds. My first post in the list described how to get the ChordGeometries program to show this lattice. This is how I'd treat adaptive tuning, too.

Also, I don't round to the nearest semitone. If one voice moves by a quarter-tone, then you've moved 0.5 semitones! No rounding, ever.

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:
>>A lattice, as I understand the algebraic definition, is a regular array
>>of points embedded in some continuous space. It's probably best to
>>think of it in that way for now.
> > > Well, I would say that not all lattices can be embedded into a > meaningful continuous geometry. (Cf., for instance, the definition > here: http://mathworld.wolfram.com/Lattice.html.) The interesting > thing is that the lattices in tuning theory are of this type: when > you try to embed them in a continuous space, you find that only one > dimension contains meaningful information. This is kind of cool and > interesting, in my opinion.

I never did understand that definition. Do you mean tuning theory lattices are embedded into a meaningful continuous geometry, aren't, or can be? Are you using a mathematical definition of "meaningful"?

>>You could also consider the lattice distance as part of the voice
>>leading distance. It may be putting the cart before the horse, but
>>intervals in the diatonic scale are usually favoured melodically.
>>Minimizing steps on the spiral of fifths helps you to avoid them. If
>>you only consider log-frequency space, and round everything to the
>>nearest semitone, you lose this distinction.
> > I can't imagine measuring voice leading on the lattice. On a 2x3 > lattice, the distance C4->G4 would be 1, while the distance C4->D4 > would be 3 (up two perfect fifths, and down one octave). So anyone > minimizing lattice distance would prefer lots of voices moving by > perfect fifth to voices moving by step. This might be useful for > some purposes, but not voice leading, where steps are smaller than > fifths.

If you used a harmonic lattice with no accounting for interval size then, yes, that would be a pretty stupid thing to do. So don't do it. (And why mix octaves with perfect fifths for anything?) Mixing interval sizes with the spiral of fifths, like I said before, might help. Otherwise, some lattice of tones and diatonic semitones that discourages subtraction, maybe.

> If you want to study diatonic voice leading, the simplest thing to do > is look at the diatonic lattice in my orbifolds. My first post in > the list described how to get the ChordGeometries program to show > this lattice. This is how I'd treat adaptive tuning, too.

I've done that, although I haven't worked out what it means yet. I don't think I want to study diatonic voice leading. What does this have to do with adaptive tuning?

> Also, I don't round to the nearest semitone. If one voice moves by a > quarter-tone, then you've moved 0.5 semitones! No rounding, ever.

The instructions for the software say "Pitch classes are represented by integers. 0 = C , 1 = Cs/Df, 2 =D, 3 = Ds/Ef, and so on." That certainly makes it look like you're rounding to the nearest semitone.

Graham

🔗Carl Lumma <clumma@yahoo.com>

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>Do you mean tuning theory
>lattices are embedded into a meaningful continuous geometry, aren't, or
>can be? Are you using a mathematical definition of "meaningful"?

What I think is that many of the standard tuning lattices cannot be embedded into a continuous geometry. If you look at the Tenney image that Daniel Wolf posted a little while ago, and you imagine the space to be continuous, then moving horizontally does not change the pitch. It's a real challenge to provide this horizontal axis with a mathematical meaning.

I don't have a technical definition of "meaningful" in mind here. I just mean to be asking a straightforward question: what does the horizontal axis represent, if we imagine the space to be continuous? I can't answer this question, but maybe someone can.

> > If you want to study diatonic voice leading, the simplest thing to do
>> is look at the diatonic lattice in my orbifolds. My first post in
>> the list described how to get the ChordGeometries program to show
>> this lattice. This is how I'd treat adaptive tuning, too.
>
>I've done that, although I haven't worked out what it means yet. I
>don't think I want to study diatonic voice leading. What does this have
>to do with adaptive tuning?

I think it's probably possible to think of adaptive tuning as a voice leading problem. I haven't thought much about this, though. One natural issue is that suppose you have some chord C which you want to adaptively retune. One thing you could do is retune C in such a way as to minimize the distance between the old (out of tune) chord C and the new retuned variant of the chord. This is essentially a voice leading problem. There's actually some fancy algorithms for doing this speedily.

>The instructions for the software say "Pitch classes are represented by
>integers. 0 = C , 1 = Cs/Df, 2 =D, 3 = Ds/Ef, and so on." That
>certainly makes it look like you're rounding to the nearest semitone.

Oh yeah, in the software, not in the theory. This is because I couldn't figure out a good, general way to draw the non-integer labels, and thought the whole thing would take a lot more time to program. If I can get the Princeton people to rewrite the program, I'll try to ask them to fix this.

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 <clumma@yahoo.com>

> I think it's probably possible to think of adaptive tuning as
> a voice leading problem. I haven't thought much about this,
> though. One natural issue is that suppose you have some chord
> C which you want to adaptively retune. One thing you could do
> is retune C in such a way as to minimize the distance between
> the old (out of tune) chord C and the new retuned variant of
> the chord. This is essentially a voice leading problem.
> There's actually some fancy algorithms for doing this speedily.

One approach is as follows. In pure JI, there is no voice
leading problem -- that is, we mean the composer intends JI.
The problem arises if the composer expects certain commas
to vanish. Simply put those commas in the kernel of a
temperament, tune the *melodic* intervals of the music in
that temperament, but tune the *harmonic* intervals in JI.
You get an adaptive tuning solution that's optimal under whatever
criteria you optimized the tuning of the temperament (I think).

This can perhaps be further improved by tuning the attacks
of all pitches to the temperament, and then bending them
into JI some number of miliseconds later. Some evidence
suggests that perception of melody is more sensitive to attacks,
and perception of dissonance more sensitive to duration.
In any case, this is apparently what happens quite
naturally in human chamber performances (Manuel has a cite).

-Carl

🔗Graham Breed <gbreed@gmail.com>

Carl Lumma wrote:
>>You could also consider the lattice distance as part of the voice >>leading distance. It may be putting the cart before the horse, but >>intervals in the diatonic scale are usually favoured melodically.
> > > I'm not sure that's always true. In barbershop, tiny intervals
> are regularly heard in the voice leading. Leading tones are
> favored sharp (word has it) in a variety of settings and I think
> this is just a fuzzy rule -- there's no specific target above
> the diatonic leading tone.

Dimitry's model is supposed to work with different metrics, so it can handle different styles. Pure vertical consonances are very important in barbershop and so you can set your parameters accordingly. Then again, a metric that gave intervals with a small pitch difference a large voice leading distance could well blow up the geometry anyway.

Another interesting thing about barbershop is that it uses 7-limit harmony. In meantone terms that means that some augmented and diminished intervals become more consonant. So applying the appropriate lattice would give you the right melodic rules.

What I said about the cart and horse is that the rule about avoiding augmented and diminished intervals in voices (which I'm sure exists) is more about making the score readable than anything to do with interval sizes. So you could think about voice leading in terms of pitch height and then sort out the spelling later. This is probably how some composers worked but I'd guess not all. The only way to be sure is to try both and see how well they work.

Graham

🔗Graham Breed <gbreed@gmail.com>

Carl Lumma wrote:
>>I think it's probably possible to think of adaptive tuning as
>>a voice leading problem. I haven't thought much about this,
>>though. One natural issue is that suppose you have some chord
>>C which you want to adaptively retune. One thing you could do
>>is retune C in such a way as to minimize the distance between
>>the old (out of tune) chord C and the new retuned variant of
>>the chord. This is essentially a voice leading problem.
>>There's actually some fancy algorithms for doing this speedily.
> > One approach is as follows. In pure JI, there is no voice
> leading problem -- that is, we mean the composer intends JI.
> The problem arises if the composer expects certain commas
> to vanish. Simply put those commas in the kernel of a
> temperament, tune the *melodic* intervals of the music in
> that temperament, but tune the *harmonic* intervals in JI.
> You get an adaptive tuning solution that's optimal under whatever
> criteria you optimized the tuning of the temperament (I think).

You can even set impossible targets for the melodic intervals. Or contextual things like leading tones being narrowed and consecutive intervals being clearly distinguished.

> This can perhaps be further improved by tuning the attacks
> of all pitches to the temperament, and then bending them
> into JI some number of miliseconds later. Some evidence
> suggests that perception of melody is more sensitive to attacks,
> and perception of dissonance more sensitive to duration.
> In any case, this is apparently what happens quite
> naturally in human chamber performances (Manuel has a cite).

That's something a synthesizer could implement if it's told the ideal harmonic and melodic pitches. The problem I'm thinking of is minimizing pitch shift and pitch drift of a large number of notes while maximizing consonance. It's something John deLaubenfels was working on, and making good progress with. But he had a classical model of springs that may or may not translate to a shortest-path geometric problem. I think he used chord templates as well. A more general algorithm would work with an arbitrary regular temperament class and an arbitrary consonance function, which is where the lattice might be useful.

Graham

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

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

> 1) If we think of this space as continuous, then the X axis rather
> hard to interpret -- moving left or right doesn't change the pitch
> that's being represented in any way. So for some (but not all)
> musical purposes, the line at the center encodes all the needed
> information. Neither Tenney nor I invented this line, which is just
> log-frequency space.

I think it's helpful to view the line as representing a linear
functional, with the space being what it maps to.

I think this is mixing apples and oranges. You have a tuning lattice,
which lives inside of R^n, and you have a linear functional which
takes elements of R^n, including lattice points, and maps them to a
one-dimensional space. You can then glue them together, but what does
that get you?

> 2) As Daniel Wolf observes, any chord of pitches can be represented
> as a set of points on the central Y axis of Tenney's graph, and that
> voice leading distances can be represented by how far the points have
> to move to get from one configuration to another. This is just to
> say that we can represent chords of pitches as unordered collections
> of points in log-frequency space, and that voice leading distance is
> represented by the distance between one configuration and another.

I think the simplest baseline approach is to assume the voices are
distinguished and that octaves are also distinguished. This removes
the complexities involved in considering orbifolds, and allows you to
work in a real normed vector space, which is a great advantage. I
don't quite understand why you want to work with sets of notes or
multisets of notes rather than tuples.

> Happily, when you understand these orbifolds, the connection
> between a chord's internal structure and its voice leading
> capabilities becomes perfectly clear. The musical properties of
> "evenness" and "unevenness" simply correspond to the distance from
> the center of the orbifold; transpositional symmetry is represented
> by one set of discrete symmetries of the orbifold; and inversional
> symmetry is represented by another set of reflectional symmetries of
> the orbifold.

Though maybe this helps explain it.

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

Ah.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:
>
> >A lattice, as I understand the algebraic definition, is a regular array
> >of points embedded in some continuous space. It's probably best to
> >think of it in that way for now.
>
> Well, I would say that not all lattices can be embedded into a
> meaningful continuous geometry. (Cf., for instance, the definition
> here: http://mathworld.wolfram.com/Lattice.html.)

Unfortunately, what this means is that there are two entirely
different mathematical uses of the word "lattice" in mathematical
English. One of them, which is the one we are using, is "lattice" in
the sense, or in a slightly more general sense, of Conway and Sloan's
"Sphere Packing, Lattices and Groups". These lattices are discrete
subgroups of a real normed vector space.

Exactly. Another distinction which is very important and which the
lattice capures is equal or unequal, as applied either to pitches or
to pitch classes. Two chords sharing a note, or sharing an interval,
is a fundamental relationsip very relevant to this question. Moreover,
the lattice can be subjected to commatic equivalencies, and pitches or
pitch classes glued together, and the same basis distinction between
the same and not the same applied.

> I can't imagine measuring voice leading on the lattice. On a 2x3
> lattice, the distance C4->G4 would be 1, while the distance C4->D4
> would be 3 (up two perfect fifths, and down one octave). So anyone
> minimizing lattice distance would prefer lots of voices moving by
> perfect fifth to voices moving by step. This might be useful for
> some purposes, but not voice leading, where steps are smaller than
> fifths.

One thing which is quite relevant to that, and which I've actually
often made use of for compositional purposes, is the extistence of
linear mappings which are appoximately equal to the pitch map: equal
temperament mappings for small numbers of notes to the octave. Given a
chord with lattice information for each note, and an approximate
tuning map (what I call a "val") which is eg 4-equal or 5-equal, it is
possible to add voice-leading relevant information to the tuning
lattice mix.

> If you want to study diatonic voice leading, the simplest thing to do
> is look at the diatonic lattice in my orbifolds.

I think orbifolds are by their very nature (generalizations of
manifolds which have singular loci) are not going to be a simple way
of looking at things. I think to the extent possible, working in R^n
would be better, and then save the orbifolds for when you want to
elaborate the theory.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

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

> What I think is that many of the standard tuning lattices cannot be
> embedded into a continuous geometry.

As far as I've seen, they all can be.

> I don't have a technical definition of "meaningful" in mind here. I
> just mean to be asking a straightforward question: what does the
> horizontal axis represent, if we imagine the space to be continuous?
> I can't answer this question, but maybe someone can.

I'm not sure what you are looking at, but the meanig of the axes would
depend on the lattice. Some completely different lattices, which
therefore have completely different meanings, can be found on my web site:

http://www.xenharmony.org/sevlat.htm
http://www.xenharmony.org/kees.htm
http://www.xenharmony.org/top.htm
http://www.xenharmony.org/hahn.htm
http://www.xenharmony.org/bosanquet.html

http://66.98.148.43/~xenharmo/sevlat.htm
http://66.98.148.43/~xenharmo/kees.htm
http://66.98.148.43/~xenharmo/top.htm
http://66.98.148.43/~xenharmo/hahn.htm
http://66.98.148.43/~xenharmo/bosanquet.html

This certainly does not exhaust the possible uses of lattices in
modeling music, but does give some idea of the scope of the idea. All
of the lattices, incidentally, have precise mathematical definitions.

> I think it's probably possible to think of adaptive tuning as a voice
> leading problem. I haven't thought much about this, though. One
> natural issue is that suppose you have some chord C which you want to
> adaptively retune. One thing you could do is retune C in such a way
> as to minimize the distance between the old (out of tune) chord C and
> the new retuned variant of the chord. This is essentially a voice
> leading problem. There's actually some fancy algorithms for doing
> this speedily.

Of course, you need to locate the chords in a context with other chords.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗yahya_melb <yahya@melbpc.org.au>

Hallo all,

--- In tuning@yahoogroups.com, Carl Lumma wrote:
[snip]
> This can perhaps be further improved by tuning the attacks
> of all pitches to the temperament, and then bending them
> into JI some number of miliseconds later. Some evidence
> suggests that perception of melody is more sensitive to attacks,
> and perception of dissonance more sensitive to duration.
> In any case, this is apparently what happens quite
> naturally in human chamber performances (Manuel has a cite).

This is really interesting! It might acccount for some of
the "pitchiness" in performances - accompanied vocal as well as
purely instrumental. Can you, Carl, or perhaps Manuel, give me a
reference for this?

Graham Breed replied:
> That's something a synthesizer could implement if it's told
> the ideal harmonic and melodic pitches. ...

And, presumably, the envelope of the pitch slide? (Is "portamento"
the correct term here?)

By "envelope" I mean here the correct time scales for holding the
attack (melodic) pitch, sliding to the sustain (harmonic) pitch, and
sustaining the latter.

Should these time scales be given in absolute or percentage terms?

How fast should that slide be?

> ... The problem I'm
> thinking of is minimizing pitch shift and pitch drift of a large
> number of notes while maximizing consonance. It's something John
> deLaubenfels was working on, and making good progress with. But
> he had a classical model of springs that may or may not translate
> to a shortest-path geometric problem. I think he used chord
> templates as well. A more general algorithm would work with an
> arbitrary regular temperament class and an arbitrary consonance
> function, which is where the lattice might be useful.

Graham, the problem you pose sounds decidedly non-trivial!

What constraints and assumptions do you think are relevant;
what are the boundary conditions?

Regards,
Yahya

🔗Carl Lumma <clumma@yahoo.com>

> --- In tuning@yahoogroups.com, Carl Lumma wrote:
> [snip]
> > This can perhaps be further improved by tuning the attacks
> > of all pitches to the temperament, and then bending them
> > into JI some number of miliseconds later. Some evidence
> > suggests that perception of melody is more sensitive to attacks,
> > and perception of dissonance more sensitive to duration.
> > In any case, this is apparently what happens quite
> > naturally in human chamber performances (Manuel has a cite).
>
> This is really interesting! It might acccount for some of
> the "pitchiness" in performances - accompanied vocal as well as
> purely instrumental. Can you, Carl, or perhaps Manuel, give me
> a reference for this?

I realized after I posted this that I should probably clarify.
The way I wrote it, you might think performers hit first an
accurately-tempered note, and then an accurate JI note. That
may be the case, but Manuel's cite doesn't say this (I don't
think). It just says the part about first hitting an initial
pitch and then gliding to a secondary pitch over the course
of some time measured in ms. This is due, I believe, to the
feedback performers use to achieve the impressive intonation
accuracy they (sometimes) do.

> Should these time scales be given in absolute or percentage terms?

Absolute, according to the cite IIRC.

> How fast should that slide be?

The distance divided by the absolute time. :)

-Carl

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

Some responses to Gene Ward Smith:

>I think this is mixing apples and oranges. You have a tuning lattice,
>which lives inside of R^n, and you have a linear functional which
>takes elements of R^n, including lattice points, and maps them to a
>one-dimensional space. You can then glue them together, but what does
>that get you?

I'm not sure this is worth discussing further, since I think we both understand the essential mathematical situation. Clearly, there is a sense in which the lattices can be embedded within R^n. You can just stipulate that they are.

However, when you do this, it's not obvious that the continuous space that embeds the lattice is particularly a meaningful one. The points on the lattice uniquely represent pitches, and yet points in the continuous space do not uniquely represent pitches. I gave an example of this with regard to the diagram posted by Daniel Wolf.

>I think the simplest baseline approach is to assume the voices are
>distinguished and that octaves are also distinguished. This removes
>the complexities involved in considering orbifolds, and allows you to
>work in a real normed vector space, which is a great advantage. I
>don't quite understand why you want to work with sets of notes or
>multisets of notes rather than tuples.

The use of the orbifolds allows you to represent musical concepts ("chord," "major triad") in a natural and efficient geometrical manner. It therefore allows you to answer natural musical questions really easily. For example: what's the most efficient voice leading from a root-position major triad to a first inversion major triad? Or: why are there so many major-third-related major triads in Schubert?

> > If you want to study diatonic voice leading, the simplest thing to do
>> is look at the diatonic lattice in my orbifolds. >
>I think orbifolds are by their very nature (generalizations of
>manifolds which have singular loci) are not going to be a simple way
>of looking at things. I think to the extent possible, working in R^n
>would be better, and then save the orbifolds for when you want to
>elaborate the theory.

Oh, trust me -- they're really not so bad! The shapes are really quite intuitive, and the singularities are not at all difficult. And once you start thinking in terms of the orbifolds, a lot of musically interesting relationships become clear.

For instance: the smoothest voice leading between two transpositionally-related two-note chords tends to occur when the transposition is 1/2 of an octave. Thus (C, G) is close to (C#, F#). Similarly, the smoothest voice leading between two transpositionally related three-note chords tends to occur when the transposition is 1/3 of an octave. Thus (C, E, G) is close to (B, E, G#). [I'm ordering chords to make the relevant voice leading clear, here.] Similarly, the smoothest voice leading between two transpositionally related four-note chords tends to occur when the transposition is ... 1/4 of an octave! (C, E, G, Bb) is close to (C#, E, G, A). And so on: for an n-note chord, you tend to get smooth voice leading at a transposition of 1/n of an octave.

This is not at all obvious in R^n, but is totally obvious in the orbifolds. Once you understand the general principle it can actually be useful when composing, or analyzing music. For instance, it helps you understand why there are all those major-third-related triads in Schubert!

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>

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

> I'm not sure this is worth discussing further, since I think we both
> understand the essential mathematical situation. Clearly, there is a
> sense in which the lattices can be embedded within R^n. You can just
> stipulate that they are.

If they aren't, they are free abelian groups, not lattices.

> However, when you do this, it's not obvious that the continuous space
> that embeds the lattice is particularly a meaningful one. The points
> on the lattice uniquely represent pitches, and yet points in the
> continuous space do not uniquely represent pitches.

That's correct, but I don't think that too much should be made of it.
A (Euclidean) lattice is equivalent to a finitely generated free
abelian group, together with a quadratic form (or equivalently,
bilinear form.) The group can simply be Z^n. Hence, to say something
is a lattice gives extra information in the form of a quadratic
form--which defines a geometric distance. More general lattices, not
in a Euclidean space, can still be regarded as putting a metric
structure onto an abelian group, and it is that step which is the key
element.

> The use of the orbifolds allows you to represent musical concepts
> ("chord," "major triad") in a natural and efficient geometrical
> manner. It therefore allows you to answer natural musical questions
> really easily. For example: what's the most efficient voice leading
> from a root-position major triad to a first inversion major triad?
> Or: why are there so many major-third-related major triads in
> Schubert?

It's a nice concept, and I don't want to disparage it. It's just my
feeling that as much as possible ought to be done without invoking
orbifolds, and clearly a great deal can be. Orbifolds are something
for the later chapters in the book, so to speak.

> This is not at all obvious in R^n, but is totally obvious in the
> orbifolds. Once you understand the general principle it can actually
> be useful when composing, or analyzing music. For instance, it helps
> you understand why there are all those major-third-related triads in
> Schubert!

They have a characteristic Schubertian sound which I think justifies
them in itself; the major third, after all, is a five-limit consonance
and the relation is formally like dominant/tonic.

🔗Carl Lumma <clumma@yahoo.com>

🔗yahya_melb <yahya@melbpc.org.au>

Thanks, Carl! That is certainly clearer.

You keep saying "Manuel's cite", but have no reference, though?

Regards,
Yahya

--- In tuning@yahoogroups.com, "Carl Lumma" wrote:
>
> > --- In tuning@yahoogroups.com, Carl Lumma wrote:
> > [snip]
> > > This can perhaps be further improved by tuning the attacks
> > > of all pitches to the temperament, and then bending them
> > > into JI some number of miliseconds later. Some evidence
> > > suggests that perception of melody is more sensitive to
attacks,
> > > and perception of dissonance more sensitive to duration.
> > > In any case, this is apparently what happens quite
> > > naturally in human chamber performances (Manuel has a cite).
> >
> > This is really interesting! It might acccount for some of
> > the "pitchiness" in performances - accompanied vocal as well as
> > purely instrumental. Can you, Carl, or perhaps Manuel, give me
> > a reference for this?
>
> I realized after I posted this that I should probably clarify.
> The way I wrote it, you might think performers hit first an
> accurately-tempered note, and then an accurate JI note. That
> may be the case, but Manuel's cite doesn't say this (I don't
> think). It just says the part about first hitting an initial
> pitch and then gliding to a secondary pitch over the course
> of some time measured in ms. This is due, I believe, to the
> feedback performers use to achieve the impressive intonation
> accuracy they (sometimes) do.
>
> > Should these time scales be given in absolute or percentage
> > terms?
>
> Absolute, according to the cite IIRC.
>
> > How fast should that slide be?
>
> The distance divided by the absolute time. :)
... O ... K!

🔗Graham Breed <gbreed@gmail.com>

yahya_melb wrote:

>>... The problem I'm
>>thinking of is minimizing pitch shift and pitch drift of a large
>>number of notes while maximizing consonance. It's something John
>>deLaubenfels was working on, and making good progress with. But
>>he had a classical model of springs that may or may not translate
>>to a shortest-path geometric problem. I think he used chord
>>templates as well. A more general algorithm would work with an >>arbitrary regular temperament class and an arbitrary consonance >>function, which is where the lattice might be useful.
> > Graham, the problem you pose sounds decidedly non-trivial! If it were trivial there'd already be an open source reference.

> What constraints and assumptions do you think are relevant; > what are the boundary conditions?

As I have a physics background, I naturally think of it as an energy minimization. You could set it so that the further a note gets from its tempered pitch, or the more a melodic interval gets from its ideal value, or the lower a chord scores on whatever consonance function you use, the more energy the system has. You could control pitch drift by averages rather than individual notes.

It might also work as a linear program. For that, you could constrain the overall pitch drift to be zero, and give each note a fixed tuning range. Then optimize relative to your melodic and harmonic ideals. I'll write something about linear programming on tuning-math.

Graham

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>That's correct, but I don't think that too much should be made of it.
>A (Euclidean) lattice is equivalent to a finitely generated free
>abelian group, together with a quadratic form (or equivalently,
>bilinear form.) The group can simply be Z^n. Hence, to say something
>is a lattice gives extra information in the form of a quadratic
>form--which defines a geometric distance. More general lattices, not
>in a Euclidean space, can still be regarded as putting a metric
>structure onto an abelian group, and it is that step which is the key
>element.

How do we know that the Euclidean metric is the appropriate one? And why say that any particular metric is intrinsic to the various tuning lattices? Personally, I'd be very careful about asserting that any one particular metric (or even any metric at all) does the (musical, theoretical) work we want it to do. And I'd certainly place more emphasis on the Abelian group structure (which strikes me as reasonably objective and well-grounded) than on any purported metric that is attached to the group (which strikes me as likely to be simply made up).

In my part of the world, music theorists have had a bad habit of simply *declaring* that this or that metric is the appropriate one for measuring voice leading. This has always struck me as lazy -- how do we know which metric (if any) Schubert was using when thinking about voice leading? How do we know that listeners' responses are consistent with any metric at all?

For this reason, I'm very careful -- in the supplementary materials of my paper -- to think about "voice leading distance" in a way that is well-grounded in musical practice. What I find is that musical practice doesn't determine a specific metric, but it does suggest general principles about measuring distance, and these general principles allow me to do all the work I want.

Here, I would suggest that it's better to think of the tuning lattices as groups -- or really, torsors -- with various metrical structures as optional, extra, add-ons.

> > This is not at all obvious in R^n, but is totally obvious in the
>> orbifolds. Once you understand the general principle it can actually
>> be useful when composing, or analyzing music. For instance, it helps
>> you understand why there are all those major-third-related triads in
>> Schubert!
>
>They have a characteristic Schubertian sound which I think justifies
>them in itself; the major third, after all, is a five-limit consonance
>and the relation is formally like dominant/tonic.

So Schubert used the progressions because they have a characteristic Schubertian sound? ;-)

Formally, the progression (C, E, G)->(B, E, G#)->(C, E, G) does indeed resemble a dominant: you have a leading tone rising to the tonic, and a chromatic upper neighbor to scale degree 5, as in the resolution of a diminished seventh chord. But this resemblance lies in the voice leading properties of the chord ...

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:
>>That's correct, but I don't think that too much should be made of it.
>>A (Euclidean) lattice is equivalent to a finitely generated free
>>abelian group, together with a quadratic form (or equivalently,
>>bilinear form.) The group can simply be Z^n. Hence, to say something
>>is a lattice gives extra information in the form of a quadratic
>>form--which defines a geometric distance. More general lattices, not
>>in a Euclidean space, can still be regarded as putting a metric
>>structure onto an abelian group, and it is that step which is the key
>>element.
> > > How do we know that the Euclidean metric is the appropriate one? And > why say that any particular metric is intrinsic to the various tuning > lattices? Personally, I'd be very careful about asserting that any > one particular metric (or even any metric at all) does the (musical, > theoretical) work we want it to do. And I'd certainly place more > emphasis on the Abelian group structure (which strikes me as > reasonably objective and well-grounded) than on any purported metric > that is attached to the group (which strikes me as likely to be > simply made up).

Abelian groups are good. I'd worry if they were objectively good because music's a subjective discipline. But they do capture some important, subjective qualities of musical intervals. They main thing they don't have is an ordering by pitch height. I think that lattices can provide this, and so they're the most appropriate abstraction for scales with discrete pitches -- in itself an ethnocentric assumption, I suppose.

Normally, when Gene talks about lattices, I think he assumes the norm measures harmonic distance. That's a lot woolier, but it's also one of the main things you get from a group of intervals that you don't get from a continuous, one-dimensional pitch space. So if you're going to all the trouble of using groups you probably have some idea of a harmonic complexity to justify it. A bigger problem is that harmonic complexity should probably be a function of chords rather than intervals but intervallic complexity can be a useful approximation.

The group itself is still flexible. There are always things you can do in a regular temperament that don't add up in the just intonation it approximates. There are inharmonic timbres that won't work as a group based on prime ratios. Octave equivalence is context dependent. In some of these cases a metric is a simple way to tie down the group.

> In my part of the world, music theorists have had a bad habit of > simply *declaring* that this or that metric is the appropriate one > for measuring voice leading. This has always struck me as lazy -- > how do we know which metric (if any) Schubert was using when thinking > about voice leading? How do we know that listeners' responses are > consistent with any metric at all?

You know by doing analyses of the music Schubert left us, or experiments on his listeners. As your paper's being hyped as putting music theory on a scientific footing I hope we'll be seeing more of these kind of considerations.

> For this reason, I'm very careful -- in the supplementary materials > of my paper -- to think about "voice leading distance" in a way that > is well-grounded in musical practice. What I find is that musical > practice doesn't determine a specific metric, but it does suggest > general principles about measuring distance, and these general > principles allow me to do all the work I want.

And a good thing it is too. I hope you don't expect us to disagree on this point.

I do note that the Mathworld definition of "Geometry" does imply a metric. If you're allowed to talk about geometry without a specific metric I hope you'll allow us to talk about lattices under the same conditions. Lattice theory looks like a pretty good structure to talk about lattices *without* specifying the metric.

> Here, I would suggest that it's better to think of the tuning > lattices as groups -- or really, torsors -- with various metrical > structures as optional, extra, add-ons.

How is that different to thinking of them as different kinds of lattices? The only practical difference I can see is that thinking of them as lattices means you can use lattice theory. So it's a practical choice and one not worth arguing in the abstract.

Nevertheless, I can see that some groups of musical intervals don't have a homomorphism into pitch-height space. Adaptive and well temperaments, for example. So groups are about the right level for me.

Graham

🔗Graham Breed <gbreed@gmail.com>

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>They main thing
>they [groups] don't have is an ordering by pitch height. I think >that lattices
>can provide this, and so they're the most appropriate abstraction for
>scales with discrete pitches -- in itself an ethnocentric assumption, I
>suppose.

Good point! You need something else besides the group structure -- a function that maps each group element into a distance in continuous pitch space. I think this is what Gene was talking about when he talked about functionals.

>Normally, when Gene talks about lattices, I think he assumes the norm
>measures harmonic distance. That's a lot woolier, but it's also one of
>the main things you get from a group of intervals that you don't get
>from a continuous, one-dimensional pitch space.

Well, yes and no. (Note, by the way: a continuous pitch space is going to have a continuous group of intervals associated with it.) 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.

The orbifolds I talk about in my paper are just configuration spaces of one-dimensional pitch and pitch classes. Each point in one of my orbifolds represents a configuration of points on the one dimensional spaces. It's a very cool fact that many common tonal notions of "harmonic distance" fall out of these spaces automatically.

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.

An exactly analogous structure exists in three-dimensional space. If you use the diatonic metric on that space, you find that voice leading proximity groups the seven diatonic triads into a circular chain whose members share two notes with each of their neighbors. In C major:

C - a -F - d - b dim - G - e - [C]

The analogy between this "circle of triads" and the traditional "circle of fifth-related major scales" is very, very deep. (I talk about this in Appendix I of my "Scale Networks and Debussy," on my website.)

It's quite possible to construct a very useful notion of tonal harmonic distance out of these two structures. Again, I think it's really interesting to ask whether these continuous orbifold structures can do some of the work we've traditionally used tuning lattices to do.

>As your paper's being hyped as putting music theory
>on a scientific footing I hope we'll be seeing more of these kind of
>considerations.

Personally, I would reject any hype of this sort! I think the paper is noteworthy because it provides a new and interesting set of models for thinking about music. I don't think of myself as doing anything different, methodologically, from what theorists have always done.

>You may be wondering what a torsor is. I chased it down to here:
>
>http://math.ucr.edu/home/baez/torsors.html
>
>and the link near the bottom. It looks like if intervals are a group
>then notes are a torsor.

This site also discusses torsors in the context of music:

http://www.math.ucr.edu/home/baez/week234.html

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.

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 <clumma@yahoo.com>

DT 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.

Can you give an example? Harmonic distance around here
tends to mean something that corresponds reasonably well to
pyschoacoustic dissonance.

> It's a very cool fact that many common tonal notions of
> "harmonic distance" fall out of these spaces automatically.

It looks to me like you've put those notions in, by assuming
12, octave equivalence, etc.

> It's quite possible to construct a very useful notion of tonal
> harmonic distance out of these two structures.

Tonal harmonic distance and pyschoacoustic harmonic distance
are two different things, but I think plain graph distance
on the A2, A3, etc. lattices we typically use here is pretty
good at the former.

> Again, I think it's
> really interesting to ask whether these continuous orbifold
> structures can do some of the work we've traditionally used
> tuning lattices to do.

I'd love to see what happens with other scales. Paul Erlich's
decatonic scales in 22 (or, more generally, the "pajara"
rank 2 temperament) offer novel tetradic voice-leading
possibilities.

-Carl

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

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

> How do we know that the Euclidean metric is the appropriate one?

It might not be. If you look at my web pages, you'll find quite a
collection of Euclidean and non-Euclidean lattices.

And
> why say that any particular metric is intrinsic to the various tuning
> lattices?

When people draw diagrams, or assume certain symmetries, that has
implications for the metric. If we assume the 5-limit Tonnetz,
diagrammed symmetrically as a hexagonal lattice, is intended literally
then (assuming literal entails Euclidean, and it is after all a
picture) we are forced to the conclusion that the lattice is A2, and
the quadratic form up to a scaling factor is Q(3^a 5^b) = a^2+ab+b^2.
If we don't want to assume Euclidean spaces, and would rather use what
we call the "Hahn norm" around here, you still get a very similar,
albeit non-Euclidean, lattice.

Personally, I'd be very careful about asserting that any
> one particular metric (or even any metric at all) does the (musical,
> theoretical) work we want it to do. And I'd certainly place more
> emphasis on the Abelian group structure (which strikes me as
> reasonably objective and well-grounded) than on any purported metric
> that is attached to the group (which strikes me as likely to be
> simply made up).

Well, certainly we do place more emphasis on the Euclidean group
stucture, and on its homomorphic mappings, than on lattices.

> In my part of the world, music theorists have had a bad habit of
> simply *declaring* that this or that metric is the appropriate one
> for measuring voice leading. This has always struck me as lazy --
> how do we know which metric (if any) Schubert was using when thinking
> about voice leading? How do we know that listeners' responses are
> consistent with any metric at all?

Doesn't this question cut to the heart of your own work?

> Here, I would suggest that it's better to think of the tuning
> lattices as groups -- or really, torsors -- with various metrical
> structures as optional, extra, add-ons.

Have you been reading John Baez?

> Formally, the progression (C, E, G)->(B, E, G#)->(C, E, G) does
> indeed resemble a dominant: you have a leading tone rising to the
> tonic, and a chromatic upper neighbor to scale degree 5, as in the
> resolution of a diminished seventh chord. But this resemblance lies
> in the voice leading properties of the chord ...

I'm not convinced you can reduce it to voice-leading; in fact I don't
think it is true. That the chords are harmonically connected is
extremely important.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Dmitri Tymoczko wrote:
>
> > Here, I would suggest that it's better to think of the tuning
> > lattices as groups -- or really, torsors -- with various metrical
> > structures as optional, extra, add-ons.
>
> You may be wondering what a torsor is. I chased it down to here:
>
> http://math.ucr.edu/home/baez/torsors.html
>
> and the link near the bottom. It looks like if intervals are a group
> then notes are a torsor.

In various contexts, there are words and phrases for a group acting
faithfully on a set, and in particular for a group acting on itself.
Regular representation, faithful permutation representation,
homogenous space, being some of the main terms, which are by no means
the same in definition but which might be applied to some of the same
things. In music theory, David Lewin introduced the term "musical
space". A few weeks back, physicist John Baez introduced another of
these terms, so far mostly used by physicists, into the mix of musical
terminology: "torsor": a torsor is another word for homogenous space,
or homogenous space with trivial point stabilzer. It is a set on which
a group G acts freely and transitively.

This is a cute word, and I am wondering if music theorists have taken
it up, or are going to take it up. However, a more boring-sounding
term may really be more relevant, namely "regular representation."
This is a group acting on itself, and that's the typical situation we
have in music theory with pitch and pitch class groups.

http://en.wikipedia.org/wiki/Regular_representation
http://en.wikipedia.org/wiki/Homogeneous_space
http://en.wikipedia.org/wiki/Principal_homogeneous_space

Incidentally, if Dmitri were willing to move this conversation to
tuning-math, it would make a lot of sense. We normally can't get away
with talking about torsors or orbifolds on this group.

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

By the way, I'm happy to move over to tuning-math. I joined this list because I noticed that there was some discussion of my paper on it, and that some people were claiming (falsely, in my view) that my geometrical models were old hat in the tuning community.

>Can you give an example? Harmonic distance around here
>tends to mean something that corresponds reasonably well to
>pyschoacoustic dissonance.

Oh, well I may be misunderstanding. The way I would tend to interpret "harmonic distance" is that it's trying to answer questions like: "why does a G major triad seem 'close' to a C major triad, while an F#-major triad seems 'far' from a C major triad?" Or: "where do the norms of traditional tonal chord-progression come from?"

The answers certainly involve a heavy dose of psychoacoustics. But they also, I think, involve voice leading. And personally, I think the voice leading explanation may be a little more general than the standard tuning-theory lattices ...

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>

>How do we know that listeners' responses are
> > consistent with any metric at all?
>
>Doesn't this question cut to the heart of your own work?

I would say "no." This is because in my own work (see the first four sections of the online materials) I take great care only to assume a metric that satisfies certain general features. I'm pretty confident that my assumptions -- for instance, that "voice crossings" don't decrease the size of a voice leading -- are consistent with the way composers thing.

> > Formally, the progression (C, E, G)->(B, E, G#)->(C, E, G) does
>> indeed resemble a dominant: you have a leading tone rising to the
>> tonic, and a chromatic upper neighbor to scale degree 5, as in the
>> resolution of a diminished seventh chord. But this resemblance lies
>> in the voice leading properties of the chord ...
>
>I'm not convinced you can reduce it to voice-leading; in fact I don't
>think it is true. That the chords are harmonically connected is
>extremely important.

I'm not sure I understand how you guys on this list use the term "harmonic." What does "harmonically connected" mean to you? The fact that the E major chord shares a common tone with a C major chord? Because there are plenty of dominant substitutions that don't share common tones. For example, (Db, F, Ab, B)->(C, E, G, C) can act as a T-D progression, as can (B, D#, G#)->(C, E, G).

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 <clumma@yahoo.com>

> >Can you give an example? Harmonic distance around here
> >tends to mean something that corresponds reasonably well to
> >pyschoacoustic dissonance.
>
> Oh, well I may be misunderstanding. The way I would tend to
> interpret "harmonic distance" is that it's trying to answer
> questions like: "why does a G major triad seem 'close' to a
> C major triad, while an F#-major triad seems 'far' from a
> C major triad?" Or: "where do the norms of traditional tonal
> chord-progression come from?"

Why don't we call it tonal distance for the purposes of
the tuning lists (or feel free to suggest something else)?
It certainly is an interesting question, which we sometimes
discuss around here.

> The answers certainly involve a heavy dose of psychoacoustics.
> But they also, I think, involve voice leading. And personally,
> I think the voice leading explanation may be a little more
> general than the standard tuning-theory lattices ...

The first answer I think is common tones, which effect
voice leading as well as other things (like the harmonic
template thing I mentioned, in the case of consonant
chords). Also in the case of consonsonat chords, sharing
a common tone means not only that one voice leading step
can be zero, but that at least one will itself be a
consonance.

Voice leading distance alone seems to fail. C-E-G ->
C#-E#-G# has a total motion of 3 semitones and a max
motion of 1 semitone, yet these two chords are fairly
distant unless we're talking flamenco.

Anyway, I would like very much to call your attention to
this page

http://www.lumma.org/tuning/erlich/

In particular, have you read

http://www.lumma.org/tuning/erlich/erlich-decatonic.pdf

And heard

http://www.lumma.org/tuning/erlich/decatonic-swing.mp3

-Carl

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>Voice leading distance alone seems to fail. C-E-G ->
>C#-E#-G# has a total motion of 3 semitones and a max
>motion of 1 semitone, yet these two chords are fairly
>distant unless we're talking flamenco.

There are more sophisticated ways of doing this. This is what I was trying to say in my previous email.

For instance you can have a two-tiered hierarchical model, where the lowest level involves voice leading within some scale, and where the higher level involves scale-to-scale voice leadings. (Thinking of a scale as itself a chord.) As a very simple example, let's define harmonic distance as the number of one-step within-scale voice leading motions plus the number of single-semtitone scale-to-scale voice leading motions. Let's further stipulate that scale-to-scale voice leading motions should only involve pitches not contained in the current chord. This is equivalent to requiring that modulations involve a pivot chord.

So let's see how long it takes to get from C-E-G, understood as the tonic of C major, to C#-E#-G# under these circumstances.

CHORD SCALE
C-E-G (C major)
B-E-G (C major)
B-E-G (G major)
B-E-G (E minor)
B-D#-F# (E minor -- counts as two steps)
B-D#-F# (E major)
B-D#-G# (E major)
B-D#-G# (F# melodic minor ascending)
C#-E#-G# (F# melodic minor ascending -- counts as two steps)

So that's 11 steps, which is pretty far.

Obviously, one can refine this very simplistic method of calculating harmonic distance. But such models can be pretty firmly based in voice leading, and it strikes me as a reasonable approximation to traditional ideas about modulation.

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>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > > For instance, it helps
> > > you understand why there are all those major-third-related
> > > triads in Schubert!
> >
> > They have a characteristic Schubertian sound which I think
> > justifies them in itself;
>
> I was going to ask, "As opposed to other composers?"
>

... actually Haydn was the first to use this type of relation, I think
in Op.74 or so. Beethoven caught on with the 'Waldstein' sonata but
I'm not sure it really became part of his vocabulary. Arguably
Schubert didn't care very much for the usual tonic-dominant contrasts
and found something that fitted his preferred mood (or mode) swings.

Now, I can't remember which late Schubert sonata has a long section
which oscillates back and forth between tonalities of C and B, but
that seems quite relevant to the CEG / C#E#G# voiceleading...

C-E-G -> Db-F-Ab is perfectly within the scale of F harmonic minor if
you disregard parallel fifths. (Do voice-leadings have to obey
traditional rules of counterpoint?) Now how is this necessarily
different from C-E-G -> C#-E#-G#? How does the voice-leading apparatus
deal with enharmonics?

~~~T~~~

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>... actually Haydn was the first to use this type of relation, I think
>in Op.74 or so. Beethoven caught on with the 'Waldstein' sonata but
>I'm not sure it really became part of his vocabulary. Arguably
>Schubert didn't care very much for the usual tonic-dominant contrasts
>and found something that fitted his preferred mood (or mode) swings.

Yeah, the major-third switches were used to relate key areas before Schubert. In Schubert you start to see local juxtapositions of major-third related triads much more frequently -- for instance in the G major quartet he moves directly from D to F# to Bb.

>C-E-G -> Db-F-Ab is perfectly within the scale of F harmonic minor if
>you disregard parallel fifths. (Do voice-leadings have to obey
>traditional rules of counterpoint?) Now how is this necessarily
>different from C-E-G -> C#-E#-G#? How does the voice-leading apparatus
>deal with enharmonics?

As long as you tune "Ab" differently from "G#" I can represent them in my continuous orbifolds. If they're two names for the same pitch, then I can't distinguish them. However, the kind of "harmonic distance" measure I outlined previously will distinguish enharmonically equivalent pitches -- as you point out, its quicker to get from C-E-G to Db-F-Ab then from C-E-G to C#-E#-G#.

The closeness between C-E-G and Db-F-Ab, in the F harmonic minor is also exploited in the Schubert song "The Young Nun," where he uses it to modulate -- via a single pivot chord -- from F minor to F# minor.

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 <clumma@yahoo.com>

> > > > For instance, it helps
> > > > you understand why there are all those major-third-related
> > > > triads in Schubert!
> > >
> > > They have a characteristic Schubertian sound which I think
> > > justifies them in itself;
> >
> > I was going to ask, "As opposed to other composers?"
>
> ... actually Haydn was the first to use this type of relation,

Don't you think it probably appears in the Ars subtilior, and
in the North German toccatas of the early baroque?

> C-E-G -> Db-F-Ab is perfectly within the scale of F harmonic
> minor

But it's relatively rare in music.

> (Do voice-leadings have to obey
> traditional rules of counterpoint?)

No. Dmitri's using voice leading to study a broader range
of forms.

> Now how is this necessarily
> different from C-E-G -> C#-E#-G#? How does the voice-leading
> apparatus deal with enharmonics?

Depends on the tuning. Dmitri hasn't shown examples of
tunings other than 12-tET, but his methods can possible
handle arbitrary tunings.

-Carl

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:
>
> Yeah, the major-third switches were used to relate key areas before
> Schubert.

... and probably one can find chords with roots a major third distant
juxtaposed in much earlier music (ie pre-Haydn), but they weren't
anything to do with key areas.

> (...) How does the voice-leading apparatus deal with enharmonics?
>
> As long as you tune "Ab" differently from "G#" I can represent them
> in my continuous orbifolds. If they're two names for the same pitch,
> then I can't distinguish them. However, the kind of "harmonic
> distance" measure I outlined previously will distinguish
> enharmonically equivalent pitches -- as you point out, its quicker to
> get from C-E-G to Db-F-Ab then from C-E-G to C#-E#-G#.

... hum, if enharmonics are equivalent, as in today's dominant 12-tone
thinking, then "harmonic distance" is a purely formal measure,
depending only on notation? Then I don't think it can be much use, or
at least one would have to rewrite Haydn's journeys from E flat major
to B major or E major with six or seven flats to make then respect the
notion.

If the price of giving up harmonic distance is to accept C-E-G to
C#-E#-G# as good voiceleading, I'm more than willing to pay. Classical
music is scattered with examples of parallel major triads (Gerontius
and Eroica I coda for two) and the slight puzzle is not why they are
allowed, but why they make such a curious impression in such places.

To change subject, if C#-F and Db-F *are* slightly different pitches
as in meantone, then they will look a little bit different in the
orbifold. But so close together that one will scarcely be able to tell
just by looking at the dots which one is a 5-limit consonance and
which is dissonant (by 'classical' standards).

My impression is that the orbifolds are not particularly adapted to
analyse details of intonation. That is not really a surprise, since
one can learn classical harmony and counterpoint without thinking hard
about tuning, by rounding everything to the nearest semitone and
leaving musicians or piano tuners to sort it out. Nothing about
classical counterpoint required one unique tuning.

For various reasons the orbifolds seem neatest in equal temperament,
due to symmetries of the tuning, therefore a logical next step would
be to look at voice leading in 19EDO, 31EDO etc. or even
neo-Pythagorean equal temperaments.

Now these 'other' EDOs are mostly prime, so you can't divide the
octave equally: every chord must be asymmetrical. Suppig's Labyrinthus
Musicus of 1722 (a composition written specially for 31EDO) tries to
make a feature out of using a diminished 7th as a pivot chord, which
it can only be if one note is tweaked by a diesis. In other words he
uses the voiceleading:

B-D-F-Ab to B-D-F-G#

which is the minimal distance between two chords in 31EDO!

~~~T~~~

🔗Dmitri Tymoczko <dmitri@Princeton.EDU>

>... hum, if enharmonics are equivalent, as in today's dominant 12-tone
>thinking, then "harmonic distance" is a purely formal measure,
>depending only on notation? Then I don't think it can be much use, or
>at least one would have to rewrite Haydn's journeys from E flat major
>to B major or E major with six or seven flats to make then respect the
>notion.

Well, I think you can get a lot of mileage out of the local musical context. Usually, in tonal music, we're hearing relative to some scale. Even if C#-F and Db-F were acoustically the same (as in an equal temperament), they're usually found in very different situations. For instance, we might hear C#-F in a D minor context, so we're expecting two notes between them, whereas Db-F would be more common in F minor, where there's just one note between them.

Given these local contexts, we're going to be thinking and hearing in terms of scale steps. These "scale step contexts" can be represented as (unequal) lattices built in the orbifolds. Modulation represents a shift-of-lattice. These lattice-contexts express what's different between Db and C#.

>To change subject, if C#-F and Db-F *are* slightly different pitches
>as in meantone, then they will look a little bit different in the
>orbifold. But so close together that one will scarcely be able to tell
>just by looking at the dots which one is a 5-limit consonance and
>which is dissonant (by 'classical' standards).

Well, you could zoom in very close!

But I see your point. The musical difference between C#-F and Db-F is greater than any intonational difference will be. I guess I think that's as it should be, since what creates the important musical difference is actually the context, rather than the dyads in isolation.

>My impression is that the orbifolds are not particularly adapted to
>analyse details of intonation. That is not really a surprise, since
>one can learn classical harmony and counterpoint without thinking hard
>about tuning, by rounding everything to the nearest semitone and
>leaving musicians or piano tuners to sort it out. Nothing about
>classical counterpoint required one unique tuning.

Yes -- though over on tuning-math I've been exploring the idea that voice leading considerations might suggest some other reasons why equal temperaments might be desirable -- roughly, they allow you to minimize the tuning's "total error" for every interval.

But it's important to say that I didn't develop these orbifolds to think about tuning, and it may well be that they're not the optimal tool for that job.

>If the price of giving up harmonic distance is to accept C-E-G to
>C#-E#-G# as good voiceleading, I'm more than willing to pay. Classical
>music is scattered with examples of parallel major triads (Gerontius
>and Eroica I coda for two) and the slight puzzle is not why they are
>allowed, but why they make such a curious impression in such places.

To me, there's not such a big issue here. C-E-G to C#-E#-G# is rare in classical music for at least two reasons:

1) The progression takes place in chromatic space, so it will tend to occur most frequently when composers are thinking about the chromatic scale as a primary object, rather than the diatonic. This starts to occur frequently in the mid-nineteenth century.

2) In the most efficient voiceleading between these chords, every voice moves in parallel.

So, although C-E-G to C#-E#-G# has the virtue of efficiency, there's an easy story to tell about why it doesn't happen more often.

>For various reasons the orbifolds seem neatest in equal temperament,
>due to symmetries of the tuning, therefore a logical next step would
>be to look at voice leading in 19EDO, 31EDO etc. or even
>neo-Pythagorean equal temperaments.

Well, they really are continuous spaces, so any scale is going to be represented by a lattice in the space. It doesn't matter whether this lattice is regular (as it is for an equal temperament), or unequal.

>Suppig's Labyrinthus
>Musicus of 1722 (a composition written specially for 31EDO) tries to
>make a feature out of using a diminished 7th as a pivot chord, which
>it can only be if one note is tweaked by a diesis. In other words he
>uses the voiceleading:
>
>B-D-F-Ab to B-D-F-G#
>
>which is the minimal distance between two chords in 31EDO!

Oh, that is extremely cool! Is there a recording/score available?

BTW, in a c-note equal temperament, the minimal voice leading between two n note chords related by transposition x is ||nx|| mod c. (That is, the "norm" (or size, or generalized absolute value) of nx modulo c.)

So, assuming the minor third is 8 semitones in 31tet, and the diminished seventh chord is an {0, 8, 16, 24} tetrachord, you get 8 * 4 = 32 = 1 mod 31, so diminished sevenths are related by single-semitone voice leading in 31tet.

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>

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

> But I see your point. The musical difference between C#-F and Db-F
> is greater than any intonational difference will be.

I don't see how you can conclude this, or even how to assign any clear
meaning to the claim. The septimal meantone interpretation of a
diminished fourth is 9/7, and in some meantone tunings it will be that
sharp or even sharper. In 31-et, it is 425.8 cents, far too sharp to
sound like a just major third, but still over nine cents shy of 9/7.
But in the range of the Wilson fifth it becomes a clear 9/7, and by
the time we get to 19-et it is up to 442 cents and doing double duty
as a 21/16.

I've had a lot of experience listing to what happens when you replace
the major third with 9/7 in common practice music from testing extreme
circulating temperaments and so forth, and while it's striking, it
doesn't really sound much like a normal triad. I wish I knew of any
discussion of the sound of diminished fourth triads in remote keys
from the meantone era, because musicians then certainly encountered them.

> Yes -- though over on tuning-math I've been exploring the idea that
> voice leading considerations might suggest some other reasons why
> equal temperaments might be desirable -- roughly, they allow you to
> minimize the tuning's "total error" for every interval.

Thye minimize something, but I don't think "total error" is a good way
to characterize what that is.

> To me, there's not such a big issue here. C-E-G to C#-E#-G# is rare
> in classical music for at least two reasons:

It's used a lot more in popular music, though.

🔗Graham Breed <gbreed@gmail.com>

Dmitri Tymoczko wrote:
>>... hum, if enharmonics are equivalent, as in today's dominant 12-tone
>>thinking, then "harmonic distance" is a purely formal measure,
>>depending only on notation? Then I don't think it can be much use, or
>>at least one would have to rewrite Haydn's journeys from E flat major
>>to B major or E major with six or seven flats to make then respect the
>>notion.
> > Well, I think you can get a lot of mileage out of the local musical > context. Usually, in tonal music, we're hearing relative to some > scale. Even if C#-F and Db-F were acoustically the same (as in an > equal temperament), they're usually found in very different > situations. For instance, we might hear C#-F in a D minor context, > so we're expecting two notes between them, whereas Db-F would be more > common in F minor, where there's just one note between them.

Yes, because C# is nearer to other notes of D minor on the spiral of fifths, but Db is nearer to other notes of F minor.

> Given these local contexts, we're going to be thinking and hearing in > terms of scale steps. These "scale step contexts" can be represented > as (unequal) lattices built in the orbifolds. Modulation represents > a shift-of-lattice. These lattice-contexts express what's different > between Db and C#.

Is that the simplest way of explaining it? An alternative would be to ensure that ascending or descending on the staff follows ascending or descending in semitone-measured pitch height. That's something you talk about in your key signatures paper, so make it more general and correct spelling is a matter of voice leading. It also means everything goes in the right direction for all the various meantone tunings.

An interesting test case is Gesualdo. A few years back I worked out a simple program to sort out the spelling of some of his more chromatic passages. One of the main rules is that he's very strict about minimizing chords on the spiral of fifths.

Graham

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:
>
> > then "harmonic distance" is a purely formal measure,
> >depending only on notation? Then I don't think it can be much use,
(...)

> Well, I think you can get a lot of mileage out of the local musical
> context. Usually, in tonal music, we're hearing relative to some
> scale.

I would say key rather than scale, for reasons to be explained.

> Even if C#-F and Db-F were acoustically the same (as in an
> equal temperament), they're usually found in very different
> situations. For instance, we might hear C#-F in a D minor context,
> so we're expecting two notes between them, whereas Db-F would be more
> common in F minor, where there's just one note between them.
>
> Given these local contexts, we're going to be thinking and hearing in
> terms of scale steps. These "scale step contexts" can be represented
> as (unequal) lattices built in the orbifolds. Modulation represents
> a shift-of-lattice. These lattice-contexts express what's different
> between Db and C#.

I'm almost convinced. My example of C-E-G to C#-E#-G# in 'normal'
Romantic harmony involved the keys of E or B, which do interpolate
some sort of 'harmonic distance' between the two chords. For Haydn Eb
to B modulation we have to say that it is really (functionally?) Cb
major, renotated for convenience.

> C-E-G to C#-E#-G# is rare
> in classical music for at least two reasons:
>
> 1) The progression takes place in chromatic space, so it will tend to
> occur most frequently when composers are thinking about the chromatic
> scale as a primary object, rather than the diatonic. This starts to
> occur frequently in the mid-nineteenth century.

I think tonality makes more sense as primary object here. If the
chromatic scale is primary then what sense can you make out of
modulation or harmonic distance? Actually what you have is not a
binary opposition 'this note is / is not in the scale' (which is
harmonic distance), rather a hierarchy, in which each note of the
chromatic scale is allowed but has a different function and degree of
'strangeness' in the tonality. So harmonic distance may be defined via
tonality and chord progressions, without reference to precisely
defined scales. Every note can occur in every tonality, it just has a
different relation to the tonal centre.

When Mozart writes F# E# F# E# F# in D major (K387) it is not making
the chromatic scale a primary object, nor using a scale or tonality of
F# minor, but simply decorating F# with a bit of foreign colour.

Going from isolated strange tones to entire strange chords, if writing
in E major we could use the chord of C major without preparation (cf
Haydn op.54 in C, Brahms sextet in G), despite its 'distance' (...
back with Schubertian third progressions, except as local colour
rather than structure).

> >Suppig's Labyrinthus Musicus of 1722 (...)
> >
> >B-D-F-Ab to B-D-F-G#
> >
> >which is the minimal distance between two chords in 31EDO!
>
> Oh, that is extremely cool! Is there a recording/score available?

The piece is published in facsimile by Diapason Press
http://diapason.xentonic.org/ttl/
although their manner of doing business is shrouded in perpetual
mystery. I borrowed a copy from Andreas Sparschuh.

There will never be a recording because the piece is long and of poor
musical quality, notable only for its microtonality. Now if you go to
19EDO the same progression is also minimal I think.

~~~T~~~