The "Fundamental Shape" of "Two-Chords"

Dave Hilbert

I'm troubled by Tymoczko's claim that the Möbius strip models the "fundamental shape" of (representatives of) dyad (set) classes---as if this claim is akin to some objective and revelatory mathematical truth---when the resulting topology of the strip is simply a function of an a priori decision to privilege (near) maximal parsimony (as well as octave equivalence and EDO systems) while sketching the space. Privileging T₆ relations, for example, uncovers the topology of a torus, which suggests there's nothing objectively "fundamental" at all about dyadic space beyond how we perforce construct and organize that space prior to modeling it. The fact that theorists have historically privileged parsimonious relations is an insufficient defense to the general indictment. That might make it the fundamental shape with respect to historical approaches to musical modeling---a description I would embrace wholeheartedly---but that doesn't make it the fundamental shape equivalent to some universal law concerning the way (representatives of) dyad classes are organized. The parsimony-strip relationship is merely one reification of a number of possible models.

[And briefly to the issue of parsimony: Why privilege semitone movement when most of the musical events we experience (e.g., melodic and harmonic motion, scales, Schenkerian Urlinie and Ursatz structures, etc.) involve [02] dyads? If there exists a disconnect between “closeness” in perceived musical syntax and a "fundamental" model (e.g., Lewin’s S function, the neo-Riemannian <L> transform acting on major triads, etc.), shouldn't the model be revised?]

The claim reminds me of the bias inherent in the well-known Texas Sharpshooter fallacy, where a Texan "fires his gun at the side of a barn, paints a bullseye around the bullet hole, and claims to be a sharpshooter." Within that context, then, I offer the following narrative: 

1. We assume a dyadic model should be maximally parsimonious because that’s the traditional way modern theorists have modeled musical objects—from the conformist Tonnetz and the Power Towers to the Chicken-Wire Torus, Weitzmann regions, and Cube Dance. 


2. We manipulate a sketch of the space based on the assumption of (1) until we achieve maximal P_{1,0} parsimony along most axes.


3. We realize this structure could be modeled (only) by a Möbius strip.


4. We proclaim to have discovered the “fundamental shape of two chords.”

That progression offers quite the non sequitur, though it does seem representative of the bias that plagues many efforts in music theory (and mathematical music theory, in particular) to provide objective models of musical passages. But I'll leave that criticism for another post.  

     

Thomas Noll

Dear Colleagues,

I do not intend to go into the details of this criticism. I just feel that it would be good to add an epistemological prolegomenon. As far as I'm aware, the orbifold-approach to the study of voice leading is joint work by Clifton Callender, Ian Quinn and Dmitri Tymoczko. It is an a priori decision of this approach to model musical tones in terms of a real valued pitch height parameter. And to model voicings the authors use a separate copy of the real number space for each voice. The semitone is not privileged a priorily, though. You may likewise model the yowling of two wolfs in this space. The various factorisations of this space with respect to octave equivalence, voice permutation, transposition etc. provide an offer to the musical analyst. That these factorizations yield tori, Moebious strips etc. are irrevocable mathematical facts. And we may regard these facts as useful extensions of our shared music-theoretical knowledge. To what extend it is analytically insightful to trace a certain musical passage in the dyad-space is a different question.

Thomas

Thomas Noll

Departament de Teoria, Composició i Direcció

Escola Superior de Música de Catalunya

c/ Padilla 155, edifici L'Auditori

08013 Barcelona

Email: thomas.mamuth@gmail.com

Thomas Noll

I would like to add a thought to my own last sentence:

To what extend it is analytically insightful to trace a certain musical passage in the dyad-space is a different question.

The potential to gain analytical insights about a given voice leading with an orbifold analysis lies in the comparison of several analyses on the various levels of the orbifold hierarchy. I remember that Joe Straus demonstrated this quite convincingly at the 'John Clough'-Memorial Symposium, Chicago 2005 as a respondent to the presentation by Callender, Quinn and Tymoczko. It is implied by the method that the voice leading parsimony may increase from top levels to bottom levels (i.e. from pitch vectors to pitch class vectors to pitch class sets to transposition classes to chord classes ). So it is analytically interesting to see where in the multi-level analysis a noticable change in the voice leading distance happens.

However, it is perhaps less productive to negotiate about an ontological state of a particular orbifold, such as the dyad space.

Thomas

Dave Hilbert

The claim being made is that the Möbius strip is the fundamental shape of the dyad (set) class---as if it's somehow synonymous with the natural, inviolable laws of physics---yet this topological result is a function of the privileged design of the space, a design that will change based on our modeling priorities.

I've given an easy counterexample.

If it helps, call it "a topological shape based on what we theorists think are important considerations with respect to modeling pitch(-class) space, " but it's not, and cannot possibly be, "the (Platonic) fundamental shape" of an arranged space that's wholly dependent upon the priorities of the arranger(s) (or even a general consensus within the academic community).

We don't speak of "fundamental Klein-bottle topologies" in Tristan, even though they're in there, because we understand (if we're being intellectually honest) that we're arranging (i.e., hearing, imag(in)ing, etc.) Wagner's music in a way that allows those structures to emerge.

Marcel de Velde

The idea that a Moebius strip represents the musical interval space is completely totally incorrect.

It assumes enharmonic equivalence!

If enharmonic equivalence is a real thing in our perception of music then most of music theory is wrong.

There would be no difference between a minor second and an augmented prime. There would be no difference between a dominant 7th chord and a German sixth chord, the would be no cross relations, etc etc. Yet we all clearly know that we hear the difference, even when tuned the same.

The Moebius strip is nothing more than a mathematical trick of rolling up a 12-tone per octave piano keyboard. And the only suitable "theory" to base upon it would be things like "atonal theory".



It is also not the same as a tonnetz btw, which does make the enharmonic distinction (yet it does not display the full interval spectrum as some wrongly think as it doesn't do octaves and fourths etc, nor was it meant to, but that's another story).

Jason Yust

Tymoczko is perfectly clear in his book that the dyad space is modeling voice leading of dyads assuming octave and permutational equivalence. If you think he is saying something about "natural inviolate laws of physics" you are really not reading it very carefully. If you don't think that voice leading is interesting or important, fine, then make your case. But you undercut your credibility when you strawman someone in a way that makes it clear you haven't made half an attempt to understand them in the first place.

Dave Hilbert

I'll begin by apologizing to the list on Jason's behalf for his rude, condescending, and bullying comment. I don't believe such discourse should have a place in academic circles, even if you disagree passionately about a position, and the tired, prosaic "straw man" indictment is made all the more listless by the fact that it seems Mr. Yust didn't bother to read my comments carefully or review the content in the links I provided. I was warned by professionals in the field about getting involved with the SMT list, and now I believe I'm starting to understand why. 

I will, however, address the single point that was raised.

The issue of parsimony goes hand in hand with voice-leading considerations, and I made that connection abundantly clear in my comments. But that still does nothing to vitiate my general criticism of the implied description of the Möbius strip as a Platonic topological shape for (representatives of) dyad classes. Different voice-leading priorities yield different topologies, and that point remains undisputed. The specific motivation for organizing the space—whether it's an aRb relation or a voice-leading concern or a die roll—is irrelevant, and it does nothing to eliminate the problem I'm raising.

As an aside, I do apologize to the rest of the list if my comments were construed as an attack on Mr. Tymoczko. I didn't mean to imply he was being purposefully deceitful in his work, and I don't believe at all that he was being intentionally disingenuous. (Thus, the use of the word "we" in my "narrative.") I use his example as a general admonition to be wary of the creeping bias that, at times, attempts to insert itself into all of our intellectual pursuits.        

   

Thomas Noll

 Dear Colleages,

I fear we are talking at cross-purposes here.The Moebius-strip model of the dyad space is part of the orbifold approach to voice-leading. Is the reason for Mr. Hyatt's discomfort with the Moebius strip in effect a discomfort with the presuppositions of the orbifold approach? It would be good to clarify this from the outset.

Let us look at a concrete example in detail and trace the interaction of music-theoretical decisions and mathematical facts. I propose to look at the two-voice progression in the right hand of the first one and half measures of Beethoven's Piano Sonata No. 26 in E-flat major, Op. 81a, "Les Adieux".

The upper voice traverses a descending major third G4 - F4 - Eb4 in stepwise motion, while the lower voice leaps through a minor sixth Eb4 - Bb3 - G3.

(1) In order to follow Callender, Quinn and Tymoczoko in their reasoning one has to interpret these notes in terms of real-valued pitches. This involves a (normative, decriptive or ad-hoc) principle to attach real numbers to each note and to assume that the numbers inbetween these values are virtually meaningful as well. It is not a matter of course to subscribe to this presupposition.

(2) The dyad space could be interesting for this example if we would like to give consideration to the fact that the two different two-note-pairs (Eb4, G4) and (G3, Eb4) should be identified at a more abstract music-theoretical level of desription.

A meaningful discussion about the role of the Moebius-Strip and the music-theoretical meaning of its non-orientability can only start(!) if we are interested to take (1) and (2) into consideration at all.

Thomas Noll

The idea that a Moebius strip represents the musical interval space is completely totally incorrect.

Right! But who on earth pursues this idea? I think this discussion is on dyad space (i.e. the space pitch class (multi)sets with two elements).

 

 

Thomas Noll

If enharmonic equivalence is a real thing in our perception of music then most of music theory is wrong.

Despite of its uncouth formulation this statement represents a potentially legitimate element of criticism of the presuppositions  of the orbifold approach to voice leading. I tend to interpret this criticism as a demand for a refinenemt of the initial encoding of the voices, namely in terms of notes rather than pitches. That would be an interesting project!

But then it find it quite counterproductive to attack the Moebius-strip model of the dyad space which is an indisputable(!!!) consequence of the orbifold approach. How can we know the geometry of the enriched dyad space in a refined orbifold approach before we have done the work modeling and investigating it?

 

Marcel de Velde

Hello Thomas,

Why would the dyad pitch space be any different from the pitch space for triads or monophonic melodies etc?

Sure you may model it differently geometrically but the actual resulting pitch space would be the same. The Mobius strip used for dyad space is a repeating structure in that the infinite chain of fifths and octaves will no longer be infinite but must close resulting in enharmonic equivalence / an EDO / equal temperament with additional octave equivalence. One could use 53-tone equal temperament instead of 12-tone equal temperament to hide the enharmonic equivalence in a more remote corner but it would still be there.

Btw, as for using the Moebius strip to purely model a pitch space without necessarily only dyads is pretty old. Here a video explaining the concept: https://youtu.be/8bgvRvh88-w but it's been around much longer, not sure how long exactly. But it's complete nonsense of course, just like the Mobius dyad space. (edit: I'm mistaken here, this is again a video of the dyad space mobius strip. One could make an octave equavalent and enharmonic equivalent space a represent it on a Mobius strip or torus though.)



I don't know if this will work but I'll try to upload a picture of what is really the musical pitch space:

https://imgur.com/G7sAiaS This extends indefinatly in all directions depending on the music. There are other ways to arrange the intervals and come to a true interval space but a Mobius strip isn't one of them.

And it's easy to get practically close with a mobius strip for an octave equivalent pitch space and dyad space. Like I said before, model it on 53-tet. This would for almost all single compositions written so far in practice give a pitch space matching our brains musical interpretation and only give a theoretical problem in the larger scheme of things where 53-tet enharmonic equivalence kicks in.

But then not only will the Mobius strip grow considerably. It says nothing about which pitch to actually write, for instance a G# or an Ab etc. It basically says nothing about how music actually functions. All one could do is display the pitches written by a composer who does have this musical insight, hope for the best he wrote them correctly (not always the case, and for certain music styles most often NOT the case. I was just looking at the score for Rhapsody in Blue by Gershwin, wow what a spelling mess such huge errors right from the start) and then watch them being displayed. To actually know if the intervals displayed are correct one needs actual music theory, in order to write these intervals inc dyads one needs actual music theory. And our music theory says you need to understand the musical context to do a proper analysis. What's the tonic, where do the pitches lead to, are they a form of nonchord tone, what's the harmonic rhythm, etc etc etc. This all also goes for making music with only dyads, and the Mobius strip has nothing to do with any of this.

I'm suprised this Mobius strip thing has come from a music theorist at Princeton (though that he was taught by "professors who preferred abstract and atonal music" may be relevant). He should have known better. I'm not that surprised it was published in Science, these people will see the pretty geometric shapes but know nothing about music. But I'm most surprised this seems to have gotten some traction amongst American music theorists. You should all know better! This Mobius strip nonsense is nothing but a pretty looking gimick which has nothing to do with actual music and our perception of music. It's not just a small step back, it's a HUGE step back from regular tonal theory. And no, it's not even half useful. I can't find a single sensible sentence in that article, and when he talks about how "Bach, Mozart, and Beethoven, composers had become experts at crafting harmonious pieces that fully exploited these central regions of musical orbifolds" while HE CAN'T EVEN TELL APART major and minor intervals with augmented and diminished intervals!!

And then there's more poetry like this: "Tymoczko plays Chopin’s “Prelude in E” through the speakers of his laptop as the computer mon itor displays a three-dimensional projection of a four-dimensional orbifold. With each new chord, a ball moves through a latticework of points on the screen. Repeatedly the prelude returns to a particular point in the lattice — representing a diminished seventh chord — from which it branches first to a chord on the immediate right and then to a chord on the immediate left. But at other points in the prelude, the ball moves freely along the lattice, a kind of improvisation more commonly associated with modern music. “Composers in the 19th century had an intuitive understanding of the bizarre geometry of musical chord space,” Tymoczko says. “In fact, they had a better feel for non-Euclidean, higher-dimensional spaces than did their mathematical contemporaries.”".

I mean.. wow.. I can't find the words. Shame, facepalm, disbelief this is actually being taken serious are some things that come to mind. The only thing missing here is some A432 Hz universe thing to make the pseudoscience complete.



-Marcel de Velde

Marcel de Velde

One other thing.

Even if one were to use a large enough Mobius strip with enharmonic difference to a certain degree, let's say based on 53-tet. Then there's still a problem of octave equivalence and that one can't see any distinction in this space which is the higher and which is the lower voice. While there is no full description of this and while we're used to using inversions, these often do not function the same. Take the example of the cadential 6/4 which behaves very differently from the tonic chord, etc. Even in a 53-tet mobius strip these would be the exact same chord. Or more simply, a fourth above the bass which is often very dissonant and heard as wishing to resolve to the third is exactly the same dyad as a consonant completely stable perfect fifth in this Mobius strip.

To say that the Mobius strip models the fundamental shape of the dyads is simply false. It is in fact a complete mismatch to how our brain treats dyads or chords and completely disregards actual musical interpretation of the brain in favor of some fantasy geometrical shape. It is instead a very "lossy" representation, a form of non musical compression from which the original cannot be recovered.

Thomas Noll

Why would the dyad pitch space be any different from the pitch space for triads or monophonic melodies etc?

Thanks for sharing the link to this video. It don't quite get, what you regard as the "nonsense"-part. The author of this video did a fine didactical job to explain what I thought I should make clear to you.

The dyad space is the result of two projections: From the real plane of pitch pairs (two separate voices) one gets first the torus of octave classes and through the neglection of the identity of the voices one gets the Moebius strip of dyads. 

The good thing of the orbifold approach is that it combines these three spaces in a hierarchy of projections And each voice-leading of two voices can be studied in all three spaces at the same time. There is no reason to play them of against each other. The idea is rather to integrate information from all levels.

NB: For triads you would start with 3 dimensions instead of two, and you would go through an analogous process. But the resulting orbifolds are different.

Your criticism of the enharmonic identification motivates the introduction of yet richer space of note-pairs above the pitch-pairs. But that would be the content of a later posting.

 

Thomas Noll

This Mobius strip nonsense is nothing but a pretty looking gimick which has nothing to do with actual music and our perception of music. It's not just a small step back, it's a HUGE step back from regular tonal theory.

I think you are mistaken. First of all, this polemic is not the type of criticism which I would expect in a scientific ciscourse. Secondly I think that these orbifold-hierarchies which start from powers of the pitch height space can be suitably refined into model which starts from powers of a continuous note space. 

One should not throw out the baby with the bath water...

Marcel de Velde

Dear Thomas,

I do not see this as a scientific discourse but as debunking pseudo science. I do hope you see that my strong criticism is only towards the theory, not towards you or other people. I take particular issues with mathematical music theories that disregard musical knowledge. Perhaps this is because I've been researching microtonal music for more than a decade and I see literally thousands of people completely lost there in mathematical pseudo science not producing any worthy musical output anymore. This Mobius strip theory isn't as bad as the misconceptions out there in the microtonal / just intonation world but it is taking a step in that direction. A direction which I feel strongly students of music are not well served with.

I'm glad you see that the enharmonic equivalence in the Mobius strip dyad space is a serious error. And I hope that you will also see that the "permutation equivalence" which we could also call the "inversion equivalence" is another serious error. The very act of turning the torus into the mobius strip strips out what are explained as "duplicates". These are not to be seen as duplicates but as an essential differentiation between lower voice and higher voice. One cannot say in music that C-F with C in the bass is always equivalent to F-C with F in the bass. One cannot talk musically meaningfully about voice leading without the distinction between a dissonant fourth and a consonant fifth, this has been known for hundreds of years already. The Mobius strip dyad theory throws this knowledge away like it's meaningless, again very much against all sane music theories. And parallel fourth become the same as parallel fifths in the Mobius strip dyad theory, etc. This is not the correct way to go about it at all of course.

So how to fix these things. First of all, clearly turning the torus into the Mobius strip was a very bad idea. One needs to turn it back into a torus so we can make the musical distinction again between lower and higher voice. Then one needs to remove enharmonic equivalence, this can be done by exapnding the torus to 53-tet enharmonic equivalence to hide the equivalence problems, but much better still this should be done by just leaving out any type of enharmonic equivalence and simply leaving the ends open. We now have a flat piece of paper folded round only once for octave equivalence. Would this be sufficient to display all aspects of an idealized / uncomplicated voice leading? Well, not if one wishes to make the distinction between contrapuntal motion. For that one would need to get rid of octave equivalence as well and one would end up with a flat 2 dimensional surface for the dyad space. And would this be enough to display real world voice leading? No it wouldn't by itself as it doesn't display rhythm relevant to the analysis of voice leading, rhythm needed to make a proper analysis of the intervals and whether they function as form of nonchord tone or as a chord tone (this can give enharmonic different interpretation even, as a passing tone or appoggiatura relative to a C major chord would be for instance a D# leading to E, while when belonging to two seperate chords Cminor and Cmajor it would be an Eb moving to an E of the next chord, etc.) See how every "innovation" of the Mobius strip dyad space is basically based on bad musical "science"? There is no baby to save, only dirty bathwater..

-Marcel de Velde

Thomas Noll

Dear Marcel,

There is one source of misunderstanding, where you ignore the intense signals which I intended to emit. You criticise the acts of projection (such as from the torus to the Moebious strip as acts of neglections). As if the theorist gives up the domain of the projection in favor of the codomain - the factor space. But this is by no means(!!!) the idea of the orbifold approach. And I emphasized this twice (it is the main argument of my second posting and I repeated this in my last posting:

The good thing of the orbifold approach is that it combines these three spaces in a hierarchy of projections And each voice-leading of two voices can be studied in all three spaces at the same time. There is no reason to play them of against each other. The idea is rather to integrate information from all levels.

It is as if you criticise people to use the seven weekdays from monday till sunday as an illigimate neglection of the calendar. People use both, of course. Each wednesday is different from each other wednesday, but wednesdays also share common features in the organisation of our everyday life, which makes it useful to use the weekdays.

Further I don't think that you paraphrase my attitude towards enharmonic equivalence correctly:

I'm glad you see that the enharmonic equivalence in the Mobius strip dyad space is a serious error.

Enharmonic equivalence is the results of a projection as well. And to look at the codomain of that projection is not an error at all. It is a legitimate simplification and offers valuable insights. My argument is rather, that it is worthwhile to study a refined note space as the domain from where the pitch space emerges.

best

Thomas

Marcel de Velde

Dear Thomas,

I don't think there's a misunderstanding. I simply fail to see the musical insight gained from seeing for instance augmented sixths lumped together with major seconds / ninths, etc. And I don't think your analogy is very fitting. Weeks and different calenders etc are valid agreements between people, but the core of music theory is not an agreement, it is a discription of a subconscious interpretation of the human brain of music. And by what we already know about how this works we can tell that the Mobius strip dyad space does not fit our perception of music or more specifically that of voice leading. It's as simple as that.

But thank you for our exchange. I think I've made my point and others are free to disagree of course and continue to pursue applications and refinements of Mobius strip dyad theories. In any case I wish you the best of luck.

-Marcel de Velde

Thomas Noll

Dear Marcel,

thanks for the good wishes. You say:

 I simply fail to see the musical insight gained from seeing for instance augmented sixths lumped together with major seconds / ninths, etc.

Looking forward to see your analysis of Chopin's E-minor prelude. Measure 2 will give you a hard nut to crack: The same two notes form a minor seventh from the view point of counterpoint and an augmented sixth from the view point of harmony. Good Luck!

Thomas

Marcel de Velde

 

Looking forward to see your analysis of Chopin's E-minor prelude. Measure 2 will give you a hard nut to crack: The same two notes form a minor seventh from the view point of counterpoint and an augmented sixth from the view point of harmony.

I'm assuming you mean the minor seventh in measure 4? I'm tempted to say that Eb is spelled correctly, I think I hear that B in measure 3 as acting as a true suspension and that the real chord implied is not a dom7 (with major third D#) but a diminished seventh chord instead which comes when the B resolves to C. So no augmented sixth implied harmonically either in measure 4.

Not always are things easy to analyze (nor do I think our music theory is complete and fully correct), but this does not imply that a minor seventh is the same thing as an augmented sixth (or major second, diminished third, double diminished octave, etc as I argued before).

-Marcel de Velde

Thomas Noll

I'm assuming you mean the minor seventh in measure 4? I'm tempted to say that Eb is spelled correctly, I think I hear that B in measure 3 as acting as a true suspension and that the real chord implied is not a dom7 (with major third D#) but a diminished seventh chord instead which comes when the B resolves to C. So no augmented sixth implied harmonically either in measure 4

 Oops. I think we both mean the chord at the downbeat of measure 3. On the one hand I agree with Carl Schachter that there is a minor seventh F-Eb which contrapunctually resolves into a (major) sixth as part of a 7-6 chain in the left hand.. On the other hand - taking the fourth voice in the soprano into consideration as well - the harmony can be interpreted as a systematic denial to resolve the major thirds of dominants into their associated tonics. Instead these thirds are lowered and the chords are turned into minor-dominant seventh chords or (as in the case of measure 3) into half-diminished chords. So in (nearly) the same way as the root position Dominant-Seventh chord E7 on the downbeat of measure 4 becomes a minor seventh chord Em7 on the 3rd beat (which then together with the diminished chord on the 4th beat forms a II V progression in an implied D-Minor) is the augmented sixth chord F-A-D#-B on the downbeat of meaasure 3 a dominant chord in the main key E-minor which is turned into an (inverted) half-diminished chord F-A-D-B which becomes the II of a II - V progression in an implied A-Minor.

In this sense I argue that the question of correct spelling is subordinate. The contrapuntal Eb can be regarded to be more prominent than the harmonic  D#. But my argument is that Chopin managed to create a bifurcation of the two meanings of one and the same note. This makes it desirable

(1) to have the diminished second d2 available in the theory and

(2) to have a means to ignore it.

Thomas

Marcel de Velde

 

the harmony can be interpreted as a systematic denial to resolve the major thirds of dominants into their associated tonics. Instead these thirds are lowered and the chords are turned into minor-dominant seventh chords or (as in the case of measure 3) into half-diminished chords.

I don't agree with this. The major third is never established, instead is it made clear it is not a dominant seventh chord and neither is there a major third. There is uncertainty though before in this example the context makes clear what is what. This you find in almost all music to varying degrees. Clear examples of uncertainty are for instance passing tones or various other forms of nonchord tones. If we play the melody C-Db-Eb over a C minor chord the Db can just as wel be a C#. Not expected in western music, here we will draw expectation of diatonic movement from the rest of the western composition, but other than that there's actually nothing to prevent us from interpreting it as a C# and we can harmonize this melody more fully to either make clear it's a Db or a C#. This uncertainty is in some part inherent in music, if context is given this takes precedent, but it also has to do with the tuning system used. In 12tet the Db and C# are tuned the same, but if we were to tune them according to the pure chain of perfect fifths of 3/2 then the C# is about 23.5 cents higher than the Db and with an instrument capable of making this pitch difference heard clearly then the tuning difference is enough to indicate the difference between a Db and a C# in the above example. Similarly, if tuned as such the Eb of the Chopin piece is also heard as an Eb from the start, before the suspended B resolves to the C and before the diminished 7th chord resolves as such. Though context remains stronger, if we were to tune it as an Eb and then treat it otherwise as a major third of V it will be heard as an out of tune D#. These kinds of things are at play for the uncertainty of what interval is what. This is not even limited to the enharmonic difference btw, one can even in certain situations play a tone a semitone off from what is expected and hear it as a really really bad out of tune note. It's the most bad sounding thing one can play. But in any case, never does a tone have two functions at once to my knowledge. Uncertainty for whatever reason does not mean it is actually fulfilling both roles.

Btw, I think the diminished second is an actual interval one can use musically. It is harder to do so on a 12-tone equal tempered instrument as one has to make the interval perfectly clear through harmony alone, though this is possible. But quite easy to do in extended Pythagorean tuning. I also belief this is the interval used in several ancient Greek modes about which theorists have speculated a great deal many centuries ago, One tetrachord goes for instance F-C#-Db-C. If one wishes to ignore the enharmonic difference for whatever reason it is of course possible to do so, simplest thing in the world.

Perhaps we are getting mostly off-topic now though.. My apologies if anybody is bothered by this.

-Marcel de Velde

Dave Hilbert

 The Moebius-strip model of the dyad space is part of the orbifold approach to voice-leading.

Exactly. So, presupposing an orbifold approach (or a maximal-parsimony approach or a dissimilarity approach or prime-generator approach or a stochastic approach or a [whatever] approach) underscores my point. (Privileging a fabricated conformist system, as Marcel described, is only one of the outstanding problems.) Bias drives the lattice model, which then gives birth to the misconception of "the fundamental shape" that's reified by the Möbius strip. It is this compromised intellectual process to which I object---though it should be noted, too, that in voicing my concern, I'm merely renewing a very familiar objection to bias that any student of the history of music theory (and, particularly, that of the twentieth century) will immediately recognize.

What is perhaps more interesting (and more troubling) is the palpable deference to Platonism in DT's larger conceit, an idea confirmed by various comments published in the articles, and I'm rather surprised theorists haven't already voiced objections to this claim. (If they have, I'd love to be redirected to those publications.) Other than the physics of sound, within which I include the most consonant monochord divisions, I doubt very seriously one can speak of "Platonism" in music, and I suspect at least some members of the list would agree. (Let's define our terms: Platonism meaning the assertion of "ideal forms as an absolute and eternal reality of which the phenomena of the world are an imperfect and transitory reflection.")

The notion of the fundamental shape of "two-chords" makes a direct appeal to Platonic discovery, which is what galvanizes general interest in the paper and, in my opinion, explains its publication in Science. This is a problem not only because the paper fails to uncover Platonic "Truths" about musical space but also because it's indicative of a certain level of self-consciousness within various "mathemusical" theoretical circles. There exists a very real and present danger concerning predilections toward hijacking mathematical hieroglyphics (and in some cases, real mathematics) in an effort to legitimize music theory (and music-theoretic ideas) as a more substantive and objective (read: "less artsy") intellectual pursuit. The critical tipping point leads to crisis when theorists conflate the "Truths" inherent in the objective mathematical tools that are used for truths about the musical objects they're investigating. 

Such is the case here.    

And for what it's worth, I think Lewin would agree, not that he should be the principal adjudicator of all things music theory, but there's a reason he doesn't speak of his p-forms as "the" fundamental transformational network of Stockhausen's Klavierstück III. He recognized his approach was contingent upon several presuppositions and intellectual prejudices that belied any attempt to capture objective Truth (e.g., the commutative structure of his J-function; the subjective nature of his hearing that, as Lewin took the time to mention in the very first paragraph, differed from that of Jonathan Harvey; a conformist system that allowed him to speak of "0/6 complexes"; etc.). Lewin even goes so far as to reject explicitly the notion of Platonic abstraction when he admits he does not

...propose the "logic" of my decision algorithm...as a general rule for the analysis of set structure in pertinent music....The "logic" of the catalogue...does not reflect the application of some universal rule; rather it reflects a certain consistency in my hearing and thinking about that particular piece.... (Musical Form and Transformation, fn. 4, p. 22)    

This is the template to which mathematical approaches to music theory (and musical analysis) should adhere, and it is unfortunate DT's paper runs afoul of this ideal, even if, unlike Lewin, he's not offering a targeted analysis of a single composition. It is certainly not the only paper to do so, but theorists who count themselves among the members of the wider academic community and care about its reputation should hope it's the last.         

Dave Hilbert

I decided to revisit this thread and reread the comments, and I think one thing we can do is finally dismiss the indictment of using a straw-man argument in adjudicating the OP. From Tymoczko's paper in Science:

I now describe the geometry of musical chords. An ordered sequence of n pitches can be represented as a point in R^n. Directed line segments in this space represent voice leadings. A measure of voice-leading size assigns lengths to these line segments....To model an ordered sequence of n pitch classes, form the quotient space (R/12Z)^n, also known as the n-torus T^n. To model unordered n-note chords of pitch classes, identify all points (x_1, x_2,...,x_n) and (x_s(1), x_s(2),...,x_s(n)), where s is any permutation. The result is the global-quotient orbifold T^n/S_n, the n-torus T^n modulo the symmetric group S_n.

It should be clear, even by a cursory reading of the above passage, that the geometry of the quotient orbifold is induced by a predetermined precondition of (maximal) parsimony, a feature reified by the directed edges whose "lengths" represent voice-leading distances in R^n. "Points" of unordered sets of pitch classes will perforce be proximate to other "points" of unordered sets of pitch classes whose distances involve minimal voice-leading perturbations. The Möbius strip (figure 2) emerges from the decision to privilege parsimonious voice-leading principles in organizing the "points" in R^2.

It is this predetermined implementation of parsimonious voice leading in constructing the orbifold to which I objected in my OP because it leads, unfortunately, to indefensible claims of Platonic musical structures that do not exist. The Möbius strip is as much the "fundamental shape" of dyads as the dictionary is the "fundamental shape" of the English language.