Guerino Mazzola's mathematical music theory - Forms, Denotators, Morphisms, and Neo-Riemannian PLR operations with Gay.jl color integration
/plugin marketplace add plurigrid/asi/plugin install asi-skills@asi-skillsThis skill inherits all available tools. When active, it can use any tool Claude has access to.
Trit: +1 (PLUS - generator) Color: Red (#D82626)
Implements Guerino Mazzola's Topos of Music categorical framework:
abstract type Form end
struct SimpleForm <: Form
name::Symbol
module_type::Symbol # :Z, :R, :Q
end
struct LimitForm <: Form # Product type
name::Symbol
factors::Vector{Form}
end
struct ColimitForm <: Form # Sum type
name::Symbol
summands::Vector{Form}
end
struct ListForm <: Form # Powerset type
name::Symbol
element_form::Form
end
# Standard musical forms
const PitchForm = SimpleForm(:Pitch, :Z)
const OnsetForm = SimpleForm(:Onset, :R)
const DurationForm = SimpleForm(:Duration, :R)
const LoudnessForm = SimpleForm(:Loudness, :R)
const NoteForm = LimitForm(:Note, [PitchForm, OnsetForm, DurationForm, LoudnessForm])
const ChordForm = ListForm(:Chord, NoteForm)
const ScoreForm = ListForm(:Score, ChordForm)
function Note(pitch::Int, onset::Float64, duration::Float64, loudness::Float64=0.8)
LimitDenotator(NoteForm, [
SimpleDenotator(PitchForm, pitch),
SimpleDenotator(OnsetForm, onset),
SimpleDenotator(DurationForm, duration),
SimpleDenotator(LoudnessForm, loudness)
])
end
function Chord(notes::Vector)
ListDenotator(ChordForm, notes)
end
struct TranspositionMorphism <: Morphism
semitones::Int
end
struct InversionMorphism <: Morphism
axis::Int
end
struct RetrogradeMotion <: Morphism end
struct AugmentationMorphism <: Morphism
factor::Float64
end
# Apply transposition
function apply(m::TranspositionMorphism, d::SimpleDenotator)
if d.form == PitchForm
SimpleDenotator(PitchForm, mod(d.value + m.semitones, 12))
else
d
end
end
const P = PLROperation(:P) # Parallel: change third quality
const L = PLROperation(:L) # Leading-tone exchange
const R = PLROperation(:R) # Relative
function apply_plr(op::Symbol, triad::Vector{Int})
root, third, fifth = triad
if op == :P
# Major ↔ minor
if mod(third - root, 12) == 4
[root, mod(third - 1, 12), fifth]
else
[root, mod(third + 1, 12), fifth]
end
elseif op == :L
# Leading-tone exchange
if mod(third - root, 12) == 4
[mod(root - 1, 12), third, fifth]
else
[root, third, mod(fifth + 1, 12)]
end
elseif op == :R
# Relative major/minor
if mod(third - root, 12) == 4
[root, third, mod(fifth + 2, 12)]
else
[mod(root - 2, 12), third, fifth]
end
end
end
C Major [0, 4, 7]
P → c minor [0, 3, 7]
L → e minor [11, 4, 7] → [7, 11, 4] normalized
R → a minor [0, 4, 9] → [9, 0, 4] normalized
const NOTE_NAMES = ["C", "C#", "D", "Eb", "E", "F", "F#", "G", "G#", "A", "Bb", "B"]
function hue_to_pitch_class(hue::Float64)::Int
mod(round(Int, hue / 30.0), 12)
end
function pitch_class_to_hue(pc::Int)::Float64
mod(pc, 12) * 30.0 + 15.0
end
function color_to_note(color)::Int
rgb = convert(RGB, color)
hsl = convert(HSL, rgb)
hue_to_pitch_class(hsl.h)
end
# CatSharp trit mapping
function pitch_class_to_trit(pc::Int)::Int
pc = mod(pc, 12)
if pc ∈ [0, 4, 8] # Augmented
return 1
elseif pc ∈ [3, 6, 9] # Diminished
return 0
else
return -1
end
end
struct Tonnetz
minor_third::Int # 3 semitones
major_third::Int # 4 semitones
fifth::Int # 7 semitones
end
const STANDARD_TONNETZ = Tonnetz(3, 4, 7)
function tonnetz_neighbors(pc::Int, t::Tonnetz=STANDARD_TONNETZ)
[
mod(pc + t.minor_third, 12),
mod(pc - t.minor_third, 12),
mod(pc + t.major_third, 12),
mod(pc - t.major_third, 12),
mod(pc + t.fifth, 12),
mod(pc - t.fifth, 12)
]
end
struct KNet
nodes::Vector{Int}
arrows::Vector{Tuple{Int,Int,Symbol,Int}} # (from, to, T/I, n)
end
function verify_knet(knet::KNet)::Bool
for (from, to, op, n) in knet.arrows
pc_from = knet.nodes[from]
pc_to = knet.nodes[to]
expected = if op == :T
mod(pc_from + n, 12)
else # :I
mod(n - pc_from, 12)
end
if expected != pc_to
return false
end
end
true
end
gay-mcp (-1) ⊗ catsharp-galois (0) ⊗ topos-of-music (+1) = 0 ✓
rubato-composer (-1) ⊗ ordered-locale (0) ⊗ topos-of-music (+1) = 0 ✓
# Run Topos of Music demo
julia dev/gadgets/topos_of_music.jl
# Apply PLR transformation
just plr-transform triad="0 4 7" op=P
# Navigate Tonnetz
just tonnetz-walk start=0 steps="m3 M3 P5"
# Verify K-net
just knet-verify nodes="0 4 7" arrows="T4 T3 T7"
catsharp-galois (0): Galois connection to Plurigridgay-mcp (-1): Color ↔ pitch mappingrubato-composer (-1): Rubato Composer integrationordered-locale (0): Frame structure for scales