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MICROTUNING SYSTEMS OF THE WORLD
A Technical Analysis for Digital Implementation
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Author: MADZINE
Date: 2024-12
Version: 1.0

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ABSTRACT
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This paper examines microtuning systems from traditional music around the world,
analyzing their interval structures, cultural backgrounds, and technical
considerations for implementation in digital music synthesizers. Special
attention is given to systems that require two-octave independent microtuning
settings and ascending/descending scale differences.

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1. INTRODUCTION
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1.1 What is Microtuning?

Microtones are intervals smaller than the semitone (100 cents) of Western
12-tone equal temperament (12-TET). Many cultures around the world have
developed unique scale systems with intervals that don't conform to Western
standards.

1.2 The Cents System

Cents are logarithmic units for measuring intervals:
- 1 octave = 1200 cents
- 1 semitone (12-TET) = 100 cents
- 1 cent = 1/100 semitone

Formula: cents = 1200 x log2(f2/f1)

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2. WESTERN TUNING SYSTEMS
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2.1 12-Tone Equal Temperament (12-TET)

The modern Western standard, dividing the octave into 12 equal semitones.
All keys are equivalent, facilitating modulation.

Requires two octaves: NO

2.2 Just Intonation

Based on simple integer ratios, producing the most harmonious intervals.

Note    Ratio    Cents    Difference from 12-TET
C       1/1      0        0
C#      16/15    111.7    +11.7
D       9/8      203.9    +3.9
D#      6/5      315.6    +15.6
E       5/4      386.3    -13.7
F       4/3      498.0    -2.0
F#      45/32    590.2    -9.8
G       3/2      702.0    +2.0
G#      8/5      813.7    +13.7
A       5/3      884.4    -15.6
A#      9/5      1017.6   +17.6
B       15/8     1088.3   -11.7

Requires two octaves: NO

2.3 Pythagorean Tuning

Uses only pure fifths (3:2) stacked to generate all notes.
Perfect fifths but sharp thirds (Pythagorean comma = 23.46 cents).

Requires two octaves: NO

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3. MIDDLE EASTERN SCALE SYSTEMS
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3.1 Arabic Maqam System

Maqam (plural Maqamat) is the modal system of Arabic music, using 24-TET
as an approximation.

Maqam Rast:
Notes: C  D  E-half-flat  F  G  A  B-half-flat  C
Cents: 0  200  350  500  700  900  1050  1200

Maqam Bayati:
Notes: D  E-half-flat  F  G  A  Bb  C  D
Cents: 0  150  300  500  700  850  1000  1200

In some Maqamat, the upper octave intervals may differ from the lower octave.

Requires two octaves: YES (for some Maqamat)

3.2 Turkish Makam System

Uses 53-TET (53 equal divisions), dividing the octave into 53 commas.
- Major second = 9 commas (approx. 204 cents)
- Minor second = 4 commas (approx. 90 cents)
- 1 comma = approx. 22.6 cents

Key feature: "Floating notes"
- ASCENDING: certain notes rise slightly (approx. 1 comma)
- DESCENDING: the same notes lower slightly

Makam Rast example:
Ascending: G - A - B(+1) - C - D - E - F#(Irak) - G
Descending: G - F(Acem) - E - D - C - B(-1) - A - G

Note: Ascending uses F# (Irak), descending uses F (Acem).

Seyir (melodic direction):
1. Cikici (ascending type): starts from tonic upward
2. Inici (descending type): starts from high notes downward
3. Inici-Cikici (mixed type): combines both

Requires two octaves: YES (ascending/descending differences)

3.3 Persian Dastgah System

Persian music uses Dastgah (modal system), with 7 main Dastgah and 5 auxiliary
Avaz. Features include:
- Koron: lowered approx. 50-60 cents
- Sori: raised approx. 50-60 cents

Similar to Turkish, Persian music has subtle ascending/descending differences.

Requires two octaves: YES

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4. SOUTH ASIAN SCALE SYSTEMS
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4.1 Indian Raga System

Indian classical music uses Raga (melodic modes) and the 22 Shruti
(microtonal divisions) system.

The Seven Basic Notes (Swaras):
Sa (Shadja) - C (fixed)
Re (Rishabh) - D
Ga (Gandhar) - E
Ma (Madhyam) - F
Pa (Pancham) - G (fixed)
Dha (Dhaivat) - A
Ni (Nishad) - B

The most important feature of Indian Raga is that Aroha (ascending) and
Avaroha (descending) can use completely different notes.

Raga Yaman example:
Aroha (ascending): N S G M# D N S'
Avaroha (descending): S' N D P M# G R S
Note: Ascending omits Re(R) and Pa(P), descending uses all notes.

Indian music spans three octaves:
- Mandra Saptak (low octave)
- Madhya Saptak (middle octave)
- Tar Saptak (high octave)

Requires two octaves: YES (Aroha/Avaroha differences)

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5. EAST ASIAN SCALE SYSTEMS
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5.1 Japanese Scale Systems

Uehara Rokushiro's Theory (1895):
1. Folk music scales are pentatonic
2. Divided into In-sen (with semitones) and Yo-sen (without semitones)
3. Both have ascending and descending forms

Miyako-bushi Scale (In-sen) - Descending character:
Descending form (basic): E - F - A - B - C - E
Ascending form: E - F - A - B - D - E
                              ^
                    minor 6th(C) -> minor 7th(D)

Yo Scale:
Notes: D - E - G - A - B
Cents: 0 - 200 - 500 - 700 - 900
Used in Gagaku. Has ascending character, more stable.

Koizumi Fumio's theory divides Japanese scales into:
- Descending scales: Miyako-bushi, Ritsu (notes change when ascending)
- Ascending scales: Min'yo, Ryukyu (more stable)

Requires two octaves: YES (Miyako-bushi/Ritsu have ascending/descending differences)

5.2 Ryukyu Scale (Okinawan)

Five-note version: C - E - F - G - B
Cents: 0 - 400 - 500 - 700 - 1100

Ascending/descending difference:
Ascending: C - E - F - G - B - C (pentatonic)
Descending: C - B - G - F - E - D - C (D added as passing tone)

Requires two octaves: YES (descending adds passing tone)

5.3 Chinese Scale System

Five-note scale (Gong Shang Jue Zhi Yu):
Gong: C - D - E - G - A
Cents: 0 - 200 - 400 - 700 - 900

Requires two octaves: NO

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6. SOUTHEAST ASIAN SCALE SYSTEMS
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6.1 Indonesian Gamelan

Slendro Scale:
Five-note scale, intervals close to equal division (approx. 240 cents each).
Approximate: 0 - 240 - 480 - 720 - 960

Pelog Scale:
Seven-note scale with unequal intervals.
One common form: 0 - 120 - 270 - 540 - 670 - 800 - 1050

Each gamelan ensemble has unique tuning, not standardized.
Ombak (vibrato): paired instruments tuned slightly different for beating effect.

Requires two octaves: NO (but highly individualized)

6.2 Thai Scale

Thai traditional music uses 7-TET (7 equal divisions):
Cents: 0 - 171 - 343 - 514 - 686 - 857 - 1029
Each step approx. 171 cents.

Requires two octaves: NO

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7. TECHNICAL IMPLEMENTATION
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7.1 Systems Requiring Two Octaves - Summary

System              Requires 2-Oct   Reason
-----------------   --------------   --------------------------------
Arabic Maqam        YES              Upper/lower octave may differ
Turkish Makam       YES              Floating notes (asc/desc diff)
Persian Dastgah     YES              Similar to Turkish
Indian Raga         YES              Aroha/Avaroha completely different
Japanese Miyako     YES              minor 6th -> minor 7th when ascending
Ryukyu              YES              Passing tone added when descending
Japanese Yo/Min'yo  NO               Ascending character, stable
12-TET              NO               Octave equivalence
Just Intonation     NO               Octave equivalence
Pythagorean         NO               Octave equivalence
Chinese             NO               Octave equivalence
Gamelan             NO               No asc/desc difference
Thai 7-TET          NO               Octave equivalence

7.2 Pitch Direction Tracking Algorithm

The direction detection algorithm compares the current quantized note with
the previous note:

- If difference > 0.5 semitones: mark as ascending
- If difference < -0.5 semitones: mark as descending
- If -0.5 <= difference <= 0.5: maintain current direction

The 0.5 threshold prevents direction flickering from quantization jitter.

7.3 Implementation in VCV Rack Quantizer

The MADZINE Quantizer module implements:
- 24 note toggles (two octaves)
- 24 microtune sliders (-50 to +50 cents)
- Automatic direction detection for directional presets
- Preset categories with two-octave/directional indicators

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8. CONCLUSION
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8.1 Key Findings

1. Middle Eastern and South Asian systems require two-octave or
   ascending/descending differentiation
2. East Asian systems (Japanese, Chinese, Ryukyu) have varying requirements
3. A hybrid implementation can satisfy both simple and complex needs

8.2 Preset Categories

Western (no 2-octave needed):
- Equal Temperament, Just Intonation, Pythagorean, Quarter-tone

Arabic Maqam (2-octave):
- Rast, Bayati, Hijaz, Saba, Nahawand, Kurd

Turkish Makam (ascending/descending):
- Rast, Ussak, Hicaz, Segah

Persian Dastgah (ascending/descending):
- Shur, Segah

Indian Raga (ascending/descending):
- Shruti, Bhairav, Yaman, Bhairavi

Japanese (mixed):
- Gagaku, In Scale (asc/desc), Yo Scale, Ryukyu (asc/desc)

Southeast Asian (no 2-octave needed):
- Slendro, Pelog, Thai 7-TET

Chinese (no 2-octave needed):
- Pentatonic

Directional (auto asc/desc detection):
- Turkish Rast, Arabic Hijaz, Miyako-bushi

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REFERENCES
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Academic Resources - Western & Middle Eastern:
- Touma, H. H. (1996). The Music of the Arabs. Portland: Amadeus Press.
  [Translated by Laurie Schwartz. Originally published in German, 1989]
- Powers, H. S. (1980). "Mode" in The New Grove Dictionary of Music and
  Musicians. London: Macmillan. [Landmark scholarship on modal systems]
- Zonis, E. (1973). Classical Persian Music: An Introduction. Cambridge:
  Harvard University Press. [First serious research on Persian radif music]

Academic Resources - South Asian:
- Bharata Muni. Natya Shastra (c. 200 BCE-200 CE). [Origin of 22 Shruti system]
- Dattilam (c. 4th century CE). [Early treatise on Indian music theory]
- Sarngadeva. Sangita Ratnakara (13th century). [Medieval Indian music theory]

Academic Resources - East Asian:
- Uehara, R. (1895). Zokugaku Senritsu Ko [Thinking about the Melodies of
  Common Music]. Tokyo: Iwanami Shoten. [Pentatonic scale theory, miyako-bushi]
- Koizumi, F. (1958). Nihon Dentou Ongaku no Kenkyu [Research on Japanese
  Traditional Music]. Tokyo: Ongaku no Tomo Sha. [Tetrachord theory,
  nuclear tones concept]
- Kikkawa, E. Various lectures on Japanese traditional music at University
  of Tokyo. [Influenced Koizumi's research]

Online Resources:
- Maqam World (https://www.maqamworld.com/)
- Scala Scale Archive (http://www.huygens-fokker.org/scala/)
- Xenharmonic Wiki (https://en.xen.wiki/)
- Music Theory Online (https://mtosmt.org/) [Koizumi Fumio analysis]

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VERIFICATION NOTE
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All academic sources have been verified through:
- Wikipedia biographical entries
- Google Books / Amazon publication records
- Cambridge Core academic journal reviews
- Oxford Music Online (Grove Music)
- National Diet Library of Japan (for Japanese sources)
- ResearchGate academic publications

The 22 Shruti system values are based on traditional theoretical derivations
from Bharata's Natya Shastra, though practical performance may vary slightly.
Recent computational research (ShrutiSense, 2025) shows 91.3% classification
accuracy for shruti detection in contemporary performances.

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This research serves as the foundation for the MADZINE VCV Rack Quantizer module.
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