Process journal of learning by Attila Vajda

moments as morphisms -- music ≅ mathematics?

I am constructing the functor F : Music ⟶ Math,

𝓕 : 𝓜𝓾𝓼𝓲𝓬 ⟶ 𝓜𝓪𝓽𝓱

and I wonder what structure preserving maps (e.g. functors or homomorphisms) exist from music ⟶ mathematics, or

M ⟶ 𝓜,

and if

M ≅ 𝓜 ?

I think of M and 𝓜 as arrows in an arrow only category and ⟶ as structure preserving maps between them.

I found for example a proof in mathematics seems similar to ii-V-I in tension ⊣ relaxation:

proof : premises ⟶ conclusion ⟹ (functor) ii-V-I: subdominant ⟶ dominant ⟶ tonic

Example: ii-V-I : Dm7 ⟶ G7 ⟶ Cmaj7

is a mapping from subdominant to tonic, like

proof is a mapping Γ ⊢ φ.

I also noticed polyrhythms 2:3, 3:4, 4:5 compose associatively like arrows f, g, h in a category.

Is there literature on this?

I’m learning—feedback welcome!

I am already fascinated out by things like C is an identity arrow C! And moving from C to D is an arrow C ⟶ D, and you can play C-E as a chord (which is I don't know what structure yet), and then D-F, C-E ⟶ D-F. So, there are things like Cmaj7 ⟶ G7 is maybe C ⊣ G, adjoint functor? I don't yet know. Like inhale ⊣ exhale, ∀ ⊣ ∃, global ⊣ local, etc Or proof as a morphism analogous to ii-V-I, and music having internal logic and a subobject classifier Ω, to me this stuff is resource of fun and wonder!

Then, there are things like what makes Cmaj7 tonic, what is the logic so it sounds resolved? Which perhaps have to do with intersections of shapes of truth, maybe logic as geometry.

So maybe music is a category, and mathematics is a category, and they are isomorphic to each other! Sitting at the piano exploring this stuff is like playing mathematics. For example, what I mentioned in the post with polyrhythm: play 2/4 with left hand, 3/4 with right, so 2:3. Then you can switch to 3:4, play 3/4 with left, and 4/4 with right, which is a structure preserving map also.

Then there is the combinatorics of variable components, reflections, symmetry, time. It sounds good! Or the constructive approach to compostion, it is easy, in my experience, to learn / compose simple ideas with the same structural approach I learn through mathematics exploration.

G : Definitions ⟶ AbstractStructure

H : Definitions ⟶ PracticalUse

η : G ⟶ H

𝓜𝓾𝓼𝓲𝓬 𝓜𝓪𝓽𝓱

pitch ⟶ frequency (ℝ⁺) interval ⟶ difference (Δx ∈ ℤ) chord ⟶ tuple in vector space key ⟶ affine transform inversion ⟶ group automorphism canon ⟶ morphism composition

-- Define categories: Music and Math structure MusicObj where name : String

structure MathObj where desc : String

-- Example Functor structure MusicToMathFunctor where onObj : MusicObj → MathObj onHom : (f : MusicObj → MusicObj) → (MathObj → MathObj)`

∀p ∈ Pitches, ∃f ∈ ℝ⁺, Frequency(f) ∧ MapsTo(p, f) ∀c ∈ Chords, ∃v ∈ ℝⁿ, VectorForm(c, v)

Harmony(x) ↔ ∃a,b,c. Interval(a,b) ∧ Interval(b,c) ∧ Relation(a,b,c) Canon(f) ↔ ∃g. f ∘ g = g ∘ f

I don't know what else to improve, there is yet no supportive feedback on how to edit posts.

def charityReframe (statement : Criticism) : PositiveInterpretation := assume goodIntent : true

∀t, mindful(t)

Oh this is interesting, look:

Light /
on off

That's A + B

Here is another cool diagram:

● │ 🎨 (coproduct / sum) /
● ●

⨅ ~ ∀ ~ Π ~ × ⨆ ~ ∃ ~ ∑ ~ +

a skankin’ functor 🏀🎶 ska🎷 (ℤ₄, perm, sync) rhythm cycles, permutations, syncopation jazz🌱 (𝕂, improv⟶, sync) 🎷 expression, structure, freedom

f : A ⟶ Ω in jazz = musical intuition-as-logic

Ω_jazz

structure MusicalPulse := (rhythmicAngle : ℝ) -- phase offset, e.g. syncopation ∈ [0,2π] (tempo : ℝ) -- beats per minute (feel : String) -- swing, straight, laid-back, etc.

Music as algebraic types:

def Note := String -- e.g. "C", "D", "E" ... def Scale := List Note

def C_major : Scale := ["C", "E", "G"]

def select_note : Unit → Note := sorry -- pick a note def build_chord : List Note → Scale := id -- identity for example

Map := 🧭 := ⟶ := f := morphism := structure := music := metaphor Vamp ↻

C₁ ⟶ C₂ ⟶ ... ⟶ Cₙ ↑ ↓ └──────loop───────┘

structure Vamp (C : Type) := (chords : List C) (repeat : Bool := true)

🪐

𝓕 : 🎵⟶𝓜 ≅ ⟨ii⟶V⟶I⟩ ≈ Γ⊢φ ⟶ proof ≈ cadence 🐚 def cadence : Prem ⟶ Conc := ii ≫ V ≫ I assoc: 🥁:2:3⊗3:4⊗4:5 ∼ f∘g∘h in 𝒞⛵️

structure PluckVar := (tech : String) -- e.g. "thumb", "hybrid", "harmonic" attack : ℝ -- 0.0 = soft, 1.0 = sharp angle : ℝ -- pluck angle dynamic : ℝ -- volume fretPress : ℝ -- pressure 0.0–1.0 slur : Bool

Oh this is musical also:

chant : Breath → Calm humming : Breath → Calm 🎵 Ska + bounce = rhythm of regeneration!

structure JazzTopos := (Objs : Type) (Ω : Objs → Type) -- truth-space depends on object (provable : Π {X}, Ω X → Prop) (paths : Π {X Y}, (X ⟶ Y) → Ω X → Ω Y)

Coltrane dreamed Ayler, perceived a future internal logic before it was audible, intuition ahead of formal system

a topos can contain more than is expressible with current morphisms (proofs/paths)

φ ∈ Ω, but no f : ⊤ ⟶ φ the truth is there, the proof isn't yet.

🎵 the tone is correct in the dream but not reachable in today’s harmonic grammar.

Eric Dolphy plays against the structure in a different internal logic pulling in truth-values from another Ω

band’s Ω₁ = tonal logic, diatonic constraints Dolphy’s Ω₂ = chromatic, expanded modal, even bird-call logic.

there may be no morphism Ω₁ ⟶ Ω₂ — yet they coexist, braided by intuition and listening.

Miles’s gestural logic is transformative contextual proof-building reframes the space so that

f : ⊤ ⟶ φ_wrong becomes f' : ⊤ ⟶ φ_new, where φ_new` now is in Ω the space bent, and the logic evolved.

mindfulness reframing:

re-label experience context morphs; logic shifts negative becomes generative.

a dynamic topos the logic of the situation emerges from your stance.

∀ “wrong note” is a potential for growth in a more flexible system. ∀ missed proof may be true in another topos. ∀ failure is data for transformation, not judgment.