The Mathematical Foundation
Modern music relies on Equal Temperament (ET). Unlike ancient systems based on pure ratios, ET is a compromise that divides the octave into twelve perfectly equal steps.
The Key Advantage: Perfect Modulation. The distance between any two adjacent notes is identical (100 cents), allowing music to move freely between all keys.
Evolution of Scales
1. The Pentatonic (Start)
Anhemitonic: No half steps. Naturally melodic and "safe" because it lacks strong dissonance.
2. The Diatonic (Goal)
Asymmetric: 7 notes with maximal separation. The specific pattern of Whole and Half steps creates tonal direction (a sense of "home").
Scale Geometry
Interval Composition breakdown
The "Excluded Five" Algorithm
A diatonic scale is defined as much by what it omits as what it includes. In a 12-tone system, choosing 7 notes means strictly excluding 5.
HardHatPiano Logic: The validation engine doesn't check if you played the right notes. It checks if you avoided the wrong ones.
Deep Symmetry: Mirror Modes
The hidden palindromes of the keyboard.
Ionian
mirrors
Phrygian
Dorian
is
Palindromic
Mixolydian
mirrors
Aeolian
Lydian
mirrors
Locrian
"If we view the keyboard from the back (reversed), the geometry reveals these palindromic pairs."