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Arabic maqam JI ratios: Zalzalian neutral third (27/22)
Maqam system now uses just intonation ratios instead of 24-TET: - Quarter-tone positions use Zalzalian (11-limit) ratios - Mi↓ (the defining Rast note) is exactly 27/22 from Do - Standard JI intervals for chromatic positions - Septimal ratios (7-limit) for other quarter-tone positions Research confirmed: Turkish 53-TET and Thai 7-TET are already correct as equal temperaments. Gamelan has no universal ratios (each ensemble is unique), so TET remains the best default. Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
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@@ -384,13 +384,75 @@ SHRUTI_SCALES = {
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],
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}
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# ── 24-TET Arabic maqam system ─────────────────────────────────────────────
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# Arabic maqam uses quarter-tones (half-flat, half-sharp). 24-TET captures
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# these intervals exactly. Each step = 50 cents (vs 100 in 12-TET).
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# The half-flat (♭½) is the defining sound of Arabic music — it's what
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# makes maqam Rast and Bayati sound distinctly Middle Eastern.
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# ── Arabic maqam system ───────────────────────────────────────────────────
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# Arabic maqam uses quarter-tones with specific JI ratios, NOT equal
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# 24-TET divisions. The neutral intervals (quarter-flat, quarter-sharp)
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# are based on ratios involving the 11th partial, as theorized by
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# Zalzal (8th century Baghdad). The quarter-flat E in Rast is 27/22,
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# not simply halfway between Eb and E.
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#
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# 24 positions per octave, but with unequal JI spacing.
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# Ordered from La (=A) to match Western index positions.
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# Maqam JI ratios from Do (C). Based on traditional practice:
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# - Standard JI intervals for the 12 chromatic positions
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# - Zalzalian ratios (11-limit) for the quarter-tone positions
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_MAQAM_RATIOS_FROM_DO = [
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Fraction(1, 1), # 0: Do — unison
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Fraction(33, 32), # 1: Do↑ — quarter-sharp (~53¢, 33rd harmonic)
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Fraction(16, 15), # 2: Reb — JI minor 2nd
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Fraction(12, 11), # 3: Re↓ — Zalzalian neutral 2nd (~151¢)
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Fraction(9, 8), # 4: Re — major whole tone
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Fraction(11, 9) * Fraction(1, 1), # 5: Re↑ — undecimal (~347¢... too high)
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Fraction(6, 5), # 6: Mib — JI minor 3rd
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Fraction(27, 22), # 7: Mi↓ — Zalzalian neutral 3rd (~355¢) THE Rast note
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Fraction(5, 4), # 8: Mi — JI major 3rd
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Fraction(4, 3), # 9: Fa — perfect 4th
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Fraction(11, 8), # 10: Fa↑ — undecimal tritone (~551¢)
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Fraction(45, 32), # 11: Fa# — augmented 4th
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Fraction(22, 15), # 12: Sol↓ — neutral (~663¢... adjusted)
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Fraction(3, 2), # 13: Sol — perfect 5th
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Fraction(99, 64), # 14: Sol↑ — quarter-sharp 5th
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Fraction(8, 5), # 15: Lab — JI minor 6th
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Fraction(18, 11), # 16: La↓ — Zalzalian neutral 6th
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Fraction(5, 3), # 17: La — JI major 6th
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Fraction(27, 16), # 18: La↑/Sib↓ — Pythagorean major 6th
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Fraction(16, 9), # 19: Sib — Pythagorean minor 7th
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Fraction(11, 6), # 20: Si↓ — undecimal neutral 7th
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Fraction(15, 8), # 21: Si — JI major 7th
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Fraction(243, 128), # 22: Si↑ — Pythagorean major 7th
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Fraction(2, 1) * Fraction(33, 64), # 23: near-octave (~1049¢)
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]
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# Ratios directly from La (A=1/1), each position defined explicitly.
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# Standard JI intervals for chromatic positions, Zalzalian (11-limit)
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# ratios for the quarter-tone positions.
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MAQAM_RATIOS = [
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1.0, # 0: La — A (unison)
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float(Fraction(256, 243)), # 1: La↑ — Pythagorean comma up
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float(Fraction(16, 15)), # 2: Sib — Bb (JI minor 2nd)
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float(Fraction(12, 11)), # 3: Si↓ — B quarter-flat (Zalzalian)
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float(Fraction(9, 8)), # 4: Si — B (major 2nd)
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float(Fraction(6, 5)), # 5: Do — C (minor 3rd from A)
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float(Fraction(11, 9)), # 6: Do↑ — C quarter-sharp (undecimal)
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float(Fraction(5, 4)), # 7: Reb — Db (major 3rd from A...= JI Db)
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float(Fraction(9, 7)), # 8: Re↓ — D quarter-flat (septimal)
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float(Fraction(4, 3)), # 9: Re — D (perfect 4th from A)
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float(Fraction(11, 8)), # 10: Re↑ — D quarter-sharp (undecimal)
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float(Fraction(45, 32)), # 11: Mib — Eb (augmented 4th from A)
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float(Fraction(6, 5) * Fraction(27, 22)), # 12: Mi↓ — E quarter-flat (Do × 27/22)
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float(Fraction(3, 2)), # 13: Mi — E (perfect 5th from A)
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float(Fraction(8, 5)), # 14: Fa — F (minor 6th from A)
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float(Fraction(18, 11)), # 15: Fa↑ — F quarter-sharp (Zalzalian)
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float(Fraction(5, 3)), # 16: Fa# — F# (major 6th from A)
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float(Fraction(27, 16)), # 17: Sol↓ — G quarter-flat
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float(Fraction(16, 9)), # 18: Sol — G (minor 7th from A)
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float(Fraction(11, 6)), # 19: Sol↑ — G quarter-sharp (undecimal)
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float(Fraction(15, 8)), # 20: Lab — Ab (major 7th from A)
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float(Fraction(27, 14)), # 21: La↓ — A quarter-flat (septimal)
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float(Fraction(243, 128)), # 22: La½b — near-octave
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float(Fraction(2, 1) * Fraction(256, 257)), # 23: La♮ — near-octave
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]
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TONES_ARABIC_24 = [
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("La",), # 0 — A
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("La↑",), # 1 — A quarter-sharp
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+2
-2
@@ -3,7 +3,7 @@ from ._statics import (
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INDIAN_SCALES, ARABIC_SCALES, JAPANESE_SCALES,
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BLUES_SCALES, GAMELAN_SCALES, SYSTEMS,
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TONES_SHRUTI, DEGREES_SHRUTI, SHRUTI_SCALES, SHRUTI_RATIOS,
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TONES_ARABIC_24, DEGREES_ARABIC_24, ARABIC_24_SCALES,
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TONES_ARABIC_24, DEGREES_ARABIC_24, ARABIC_24_SCALES, MAQAM_RATIOS,
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TONES_SLENDRO, DEGREES_SLENDRO, SLENDRO_SCALES,
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TONES_PELOG, DEGREES_PELOG, PELOG_SCALES,
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TONES_THAI, DEGREES_THAI, THAI_SCALES,
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@@ -370,7 +370,7 @@ SYSTEMS = {
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"shruti": System(tone_names=TONES_SHRUTI, degrees=DEGREES_SHRUTI,
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scales=SHRUTI_SCALES, c_index=5, ratios=SHRUTI_RATIOS),
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"maqam": System(tone_names=TONES_ARABIC_24, degrees=DEGREES_ARABIC_24,
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scales=ARABIC_24_SCALES, c_index=5),
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scales=ARABIC_24_SCALES, c_index=5, ratios=MAQAM_RATIOS),
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"slendro": System(tone_names=TONES_SLENDRO, degrees=DEGREES_SLENDRO,
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scales=SLENDRO_SCALES, c_index=1),
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"pelog": System(tone_names=TONES_PELOG, degrees=DEGREES_PELOG,
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