Doodles: Tuning Systems

Wed, 08 Apr 2026 17:53:13 -0700

Some prelimiaries…

%pip install tabulate

import math
import itertools
from IPython.display import display, HTML, display_html, display_markdown, 
    Markdown
from tabulate import tabulate

type Hz = float
type Note = int

A_440_Note = 69
A_440_Hz = 440.0

def note_class(n: Note) -> tuple[int, int]:
    octaves, steps = divmod(n, 12)
    return octaves - 1, steps

def note_name(n: Note) -> str:
    # MIDI note 24 is C1
    note_names = ['C', 'C♯/D♭', 'D', 'D♯/E♭', 'E', 'F', 'F♯/G♭', 'G', 'G♯/A♭',
        'A', 'A♯/B♭', 'B'] octaves, steps = note_class(n)
    return f"{note_names[steps]}{octaves}"    

assert note_name(24) == 'C1'

12-TET Tempering

12-tone Equal Temperament divides an octave into a twelve logarithmically-equal parts. This is convenient because it has a very compact closed-form expression.

Music on

Doodles: Tuning Systems

12-TET Tempering