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Appendix B: the OPTIC hierarchy

Lessons 1 to 4 climb a ladder without naming it. Lesson 1 turns a fretboard position into a pitch and a pitch into a pitch class. Lesson 2 turns a set of pitch classes into a scale and compares scales that are transpositions of each other. Lesson 4 identifies sets related by transposition and inversion and calls the result a set class. Each of those steps forgets something on purpose, and the order in which you forget them is a hierarchy that music theory has a name for.

Callender, Quinn and Tymoczko call the five equivalences OPTIC, in “Generalized Voice-Leading Spaces” (Science, 2008):

equivalence two things are the same when they differ only by…
O octave moving notes by whole octaves
P permutation the order of the notes
T transposition moving everything by the same interval
I inversion turning the intervals upside down
C cardinality doubling a note that is already there

They are independent: you can apply any subset, and each subset names a different musical object. “Chord” is usually OPC. “Scale type” is OPTC. “Set class” is OPTIC, all five.

Terminal window
dotnet run --project code/music-theory-ga/GaTheory -c Release -- l10

The climb

Section titled “The climb”

Start at the bottom, with something you can actually do with your hands: the open C chord, x32010.

== One fingering, climbed rung by rung
rung                       the object                         what it forgets
a fingering                x32010                             nothing: strings, frets, muted strings
the pitches                C3 E3 G3 C4 E4                     which string each note was played on
- O, octave                0 4 7 0 4                          the octave of each note
- P, permutation           0 0 4 4 7                          the order of the notes
- C, cardinality           0 4 7                              doubled notes
- T, transposition         0 4 7                              which key it is in
- I, inversion             (037)                              major against minor

Read the third column downwards and you have the whole appendix. A fingering knows everything. By the top, all that survives is the shape (037) — a note, a note three semitones up, a note four semitones above that, in some key, either way up.

Note what is not on the ladder. The first step, from x32010 to C3 E3 G3 C4 E4, is not an OPTIC equivalence at all: it is the instrument. Two different fingerings that sound the same five pitches are the same music and different guitar. OPTIC has nothing to say about that difference, which is exactly why a program that works on set classes cannot tell you where to put your fingers.

What each rung throws away

Section titled “What each rung throws away”
== What each rung identifies
rung                       distinct objects     of the C major triad
fingerings, frets 0 to 4   63                   ways to play these three pitch classes
pitch multisets            5                    notes sounding in x32010
the pitch-class set        1                    0 4 7
- T: transposition class   1                    one of the 12 major triads
- I: set class             1                    one of the 24 major and minor triads

Sixty-three ways to play those three pitch classes on the first five frets, and the ladder’s first three rungs flatten all of them into one object. That is the point of an equivalence, and it is also the cost: everything a guitarist cares about lives below the rung where the ladder starts.

== How many objects there are at each rung
rung                           course   GA       check
pitch-class sets (P, O, C)     4096     4096     ok
transposition classes (+T)     352      352      ok
set classes (+I)               224      224      ok
cardinalities (+C)             13       13       ok

4096 subsets of twelve pitch classes; 352 of them up to transposition; 224 up to transposition and inversion. GA’s PitchClassSet, TranspositionClass and SetClass agree with the course on all three counts, which is the appendix’s way of saying the hierarchy is really there in the code.

Two chords at a time

Section titled “Two chords at a time”
== The same climb for four chords
chord          pitch classes      transposition class / set class
C              0 4 7              0 4 7  /  (037)
A minor        0 4 9              0 3 7  /  (037)
G              2 7 E              0 4 7  /  (037)
F              0 5 9              0 4 7  /  (037)
C and G are different chords, the same transposition class, the same set class.
C and A minor are different transposition classes and the same set class: I joins them.

C and G are different chords that meet at the T rung. C and A minor are different transposition classes that meet at the I rung — which is the formal version of the fact, in lesson 5, that a major key and its relative minor share a set of notes. F and G are two more major triads: the same rung, the same object.

GA’s type at each rung

Section titled “GA’s type at each rung”
== GA's types, one per rung
rung                       course                     GA                         check
- O, P, C: a set           id 145: 0 4 7              id 145: 0 4 7              ok
cardinality                3                          3                          ok
- I: prime form            (037)                      (037)                      ok
interval-class vector      <0 0 1 1 1 0>              <0 0 1 1 1 0>              ok
GA's SetClass for this set: SetClass[3 (Tritonic)-<0 0 1 1 1 0>/137]
its Forte number: 3-11
  • PitchClassSet is the object after O, P and C: a 12-bit id, no octaves, no order, no doubling. Lesson 7’s key-finding bug is a direct consequence — a PitchClassSet cannot know which note is the tonic, because “which note is first” is precisely what P threw away.
  • TranspositionClass adds T: 352 items.
  • SetClass adds I: 224 items, each with a prime form, a cardinality and an interval-class vector.
  • Cardinality is C, kept as a property rather than a quotient.

Two rungs have no type, and both absences show up elsewhere in this course. There is no object between a pitch and a pitch class — no “pitch with a spelling” that survives a chord — which is lesson 3’s spelling problem. And there is nothing below PitchClassSet that keeps the order of the notes, which is lesson 7’s.

GA’s OPTIC-K embedding schema

Section titled “GA’s OPTIC-K embedding schema”

GA uses the same vocabulary for machine learning. Its embedding schema, documented in .agent/skills/optic-k-schema-guardian/SKILL.md and implemented in Common/GA.Business.ML/Embeddings/EmbeddingSchema.cs, is 216 dimensions split into named partitions. Two of them are this ladder, cut in half:

Partition Dimensions Weight Purpose, in GA’s words
STRUCTURE 6-29 0.45 “Pitch-class set invariants (O+P+T+I). Core musical identity.”
MORPHOLOGY 30-53 0.25 “Physical fretboard realization (geometry/fingering).”

STRUCTURE is the top of the ladder: the four equivalences, the object that survives them, weighted highest because two voicings of the same set class really are the same harmony. MORPHOLOGY is everything the ladder discarded on the way up: which string, which fret, which finger — the step from x32010 to C3 E3 G3 C4 E4 that OPTIC does not model.

That split is the right one, and it is worth saying why. A search that used only STRUCTURE would return the C major triad played anywhere, including places no hand reaches. A search that used only MORPHOLOGY would return shapes that look alike and sound unrelated. The two partitions are the two halves of this appendix, given weights.

To verify: the dimension ranges and weights above are read from the skill document at the pinned commit, whose front matter calls the schema v1.4; the course program does not build GA.Business.ML, so nothing here is checked by running it, unlike the tables above.

Climbing the ladder by hand

Section titled “Climbing the ladder by hand”

Two sites let you do every rung of this appendix with a mouse, and both are worth an evening.

Ian Ring’s scale finder is a twelve-bead bracelet, exactly the diagram of lesson 2. Click the beads to build a set; the page names it and links to its page. Its three buttons are three rungs of the ladder:

  • Rotate up and Rotate down apply T, one semitone at a time. Rotate twelve times and you are back where you started: that orbit is the transposition class.
  • Reflect applies I. If reflecting gives you the set you already had, the set is inversionally symmetric and its set class contains 12 sets rather than 24 — the diatonic collection, the whole-tone scale and the octatonic scale all behave this way, which is lesson 2’s section on symmetry.
  • The scale’s number is the 12-bit id of lesson 2, and each scale’s page gives its modes, its interval vector and whether it is a Z-relation partner, which is lesson 4.

It is the fastest way to check a claim in this course: the counts in lesson 2 and the vectors in lesson 4 can be confirmed one set at a time on Ring’s pages, and that is how several of them were.

Harmonious approaches the same material from the other end. It calls itself “a map of all chromatic-cluster-free hexatonic, heptatonic, and octotonic scales and modes and their compatible chords (Levine 1995) in twelve-tone equal temperament”, and it is organised by key signature and by chord rather than by set number. Where Ring gives you the quotient — one page per scale, transpositions collapsed — Harmonious gives you the fibre: this key, these chords, these substitutions. Between them they are the two directions of this appendix, and neither is a replacement for the other:

Ian Ring Harmonious
Unit a set class, keyless a key and its chords
Rungs it shows T and I, as buttons below the ladder: voicings and chord–scale pairs
Best for checking a count or a vector finding a substitution you can play
Relation to GA same 12-bit numbering same chord–scale idea as lessons 6 and 7

GA sits between the two: SetClass is Ring’s object, Voicing and Fretboard are closer to Harmonious’s, and the OPTIC-K schema above is a deliberate attempt to hold both in one vector.

What to remember

Section titled “What to remember”
  • OPTIC is five independent things to forget, not one operation. Naming which ones you have applied tells you what your type can and cannot answer.
  • Every bug in Appendix C’s defects 8, 9, 12 and 19 is a rung being climbed and then a question asked that needed the discarded information: a spelling after O, a tonic after P, a count after C.
  • The instrument is below the bottom rung. OPTIC says nothing about fingering, which is why GA needs a second partition for it.

Sources

Section titled “Sources”