Microtonal Composition

Explore music beyond the standard 12-note scale using RMT Compose's powerful ratio and equal temperament systems.

What is Microtonal Music?

Microtonal music uses intervals smaller than the standard semitone, or uses tuning systems that divide the octave differently than the familiar 12-tone equal temperament (12-TET).

RMT Compose excels at microtonal composition because it works with exact ratios rather than fixed pitch values.

Pure Intervals vs Equal Temperament

The Problem with 12-TET

Standard 12-TET divides the octave into 12 equal parts. Each semitone is exactly 2^(1/12) = ~1.0595. This creates slight deviations from pure ratios:

Interval	Pure Ratio	Pure Cents	12-TET Cents	Difference
Major Third	5/4	386.3	400	+13.7
Perfect Fifth	3/2	702.0	700	-2.0
Minor Seventh	7/4	968.8	1000	+31.2

Pure Ratios in RMT Compose

Use exact ratios for pure, beatless intervals:

// Pure major third (5:4)
base.f * (5/4)

// Pure perfect fifth (3:2)
base.f * (3/2)

// Pure harmonic seventh (7:4)
base.f * (7/4)

Legacy JavaScript syntax

javascript

// Pure major third (5:4)
module.baseNote.getVariable('frequency').mul(new Fraction(5, 4))

// Pure perfect fifth (3:2)
module.baseNote.getVariable('frequency').mul(new Fraction(3, 2))

// Pure harmonic seventh (7:4)
module.baseNote.getVariable('frequency').mul(new Fraction(7, 4))

Alternative Equal Temperaments

19-TET

Divides the octave into 19 equal parts. Better approximation of pure thirds than 12-TET.

// 19-TET semitone
2^(1/19)

// 19-TET major third (6 steps)
base.f * 2^(6/19)

// 19-TET perfect fifth (11 steps)
base.f * 2^(11/19)

Legacy JavaScript syntax

javascript

// 19-TET semitone
new Fraction(2).pow(new Fraction(1, 19))

// 19-TET major third (6 steps)
module.baseNote.getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(6, 19)))

// 19-TET perfect fifth (11 steps)
module.baseNote.getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(11, 19)))

31-TET

Divides the octave into 31 equal parts. Excellent approximation of many pure intervals.

// 31-TET semitone
2^(1/31)

// 31-TET major third (10 steps)
base.f * 2^(10/31)

// 31-TET perfect fifth (18 steps)
base.f * 2^(18/31)

Legacy JavaScript syntax

javascript

// 31-TET semitone
new Fraction(2).pow(new Fraction(1, 31))

// 31-TET major third (10 steps)
module.baseNote.getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(10, 31)))

// 31-TET perfect fifth (18 steps)
module.baseNote.getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(18, 31)))

24-TET (Quarter Tones)

Divides the octave into 24 equal parts, adding quarter tones between standard semitones.

// Quarter tone up from A
base.f * 2^(1/24)

// Standard semitone in 24-TET
base.f * 2^(2/24)

Legacy JavaScript syntax

javascript

// Quarter tone up from A
module.baseNote.getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(1, 24)))

// Standard semitone in 24-TET
module.baseNote.getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(2, 24)))

Bohlen-Pierce Scale

A unique non-octave scale based on the 3:1 ratio (tritave) divided into 13 equal parts.

// Bohlen-Pierce step
3^(1/13)

// Example: 5 BP steps above base
base.f * 3^(5/13)

Legacy JavaScript syntax

javascript

// Bohlen-Pierce step
new Fraction(3).pow(new Fraction(1, 13))

// Example: 5 BP steps above base
module.baseNote.getVariable('frequency')
  .mul(new Fraction(3).pow(new Fraction(5, 13)))

Why Bohlen-Pierce?

Built around 3:1 instead of 2:1
Better approximation of odd harmonics (3:1, 5:3, 7:3)
Completely different sound palette from octave-based scales

Just Intonation

Pure intervals derived from simple ratios. Build scales using only whole-number ratios.

5-Limit Just Intonation

Uses ratios with prime factors 2, 3, and 5 only:

javascript

// Just major scale
C:  1/1     // Unison
D:  9/8     // Major second
E:  5/4     // Major third
F:  4/3     // Perfect fourth
G:  3/2     // Perfect fifth
A:  5/3     // Major sixth
B:  15/8    // Major seventh
C': 2/1     // Octave

// E (major third)
base.f * (5/4)

// G (perfect fifth)
base.f * (3/2)

// A (major sixth)
base.f * (5/3)

Legacy JavaScript syntax

javascript

// E (major third)
module.baseNote.getVariable('frequency').mul(new Fraction(5, 4))

// G (perfect fifth)
module.baseNote.getVariable('frequency').mul(new Fraction(3, 2))

// A (major sixth)
module.baseNote.getVariable('frequency').mul(new Fraction(5, 3))

7-Limit Just Intonation

Adds the seventh harmonic for bluesy intervals:

// Harmonic seventh (flat seventh)
base.f * (7/4)

// Septimal minor third
base.f * (7/6)

// Septimal tritone
base.f * (7/5)

Legacy JavaScript syntax

javascript

// Harmonic seventh (flat seventh)
module.baseNote.getVariable('frequency').mul(new Fraction(7, 4))

// Septimal minor third
module.baseNote.getVariable('frequency').mul(new Fraction(7, 6))

// Septimal tritone
module.baseNote.getVariable('frequency').mul(new Fraction(7, 5))

Building a Microtonal Composition

Tip: Use the Module Bar's "Drop at:" toggle to speed up composition:
End mode: Chain notes sequentially (great for scales and melodies)
Start mode: Stack notes at the same time (great for chords)

Example: 19-TET Melody

// Note 1: Root
frequency: base.f
startTime: 0
duration: 1

// Note 2: 19-TET major second (3 steps)
frequency: [1].f * 2^(3/19)
startTime: [1].t + [1].d
duration: 1

// Note 3: 19-TET major third (6 steps)
frequency: [1].f * 2^(6/19)
startTime: [2].t + [2].d
duration: 1

// Note 4: 19-TET perfect fourth (8 steps)
frequency: [1].f * 2^(8/19)
startTime: [3].t + [3].d
duration: 1

Legacy JavaScript syntax

javascript

// Note 1: Root
frequency: module.baseNote.getVariable('frequency')
startTime: new Fraction(0)
duration: new Fraction(1)

// Note 2: 19-TET major second (3 steps)
frequency: module.getNoteById(1).getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(3, 19)))
startTime: module.getNoteById(1).getVariable('startTime')
  .add(module.getNoteById(1).getVariable('duration'))
duration: new Fraction(1)

// Note 3: 19-TET major third (6 steps)
frequency: module.getNoteById(1).getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(6, 19)))
startTime: module.getNoteById(2).getVariable('startTime')
  .add(module.getNoteById(2).getVariable('duration'))
duration: new Fraction(1)

// Note 4: 19-TET perfect fourth (8 steps)
frequency: module.getNoteById(1).getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(8, 19)))
startTime: module.getNoteById(3).getVariable('startTime')
  .add(module.getNoteById(3).getVariable('duration'))
duration: new Fraction(1)

Example: Just Intonation Chord

// Root
frequency: base.f
startTime: 0
duration: 2

// Pure major third
frequency: [1].f * (5/4)
startTime: [1].t
duration: [1].d

// Pure perfect fifth
frequency: [1].f * (3/2)
startTime: [1].t
duration: [1].d

// Harmonic seventh (for a dominant seventh chord)
frequency: [1].f * (7/4)
startTime: [1].t
duration: [1].d

Legacy JavaScript syntax

javascript

// Root
frequency: module.baseNote.getVariable('frequency')
startTime: new Fraction(0)
duration: new Fraction(2)

// Pure major third
frequency: module.getNoteById(1).getVariable('frequency')
  .mul(new Fraction(5, 4))
startTime: module.getNoteById(1).getVariable('startTime')
duration: module.getNoteById(1).getVariable('duration')

// Pure perfect fifth
frequency: module.getNoteById(1).getVariable('frequency')
  .mul(new Fraction(3, 2))
startTime: module.getNoteById(1).getVariable('startTime')
duration: module.getNoteById(1).getVariable('duration')

// Harmonic seventh (for a dominant seventh chord)
frequency: module.getNoteById(1).getVariable('frequency')
  .mul(new Fraction(7, 4))
startTime: module.getNoteById(1).getVariable('startTime')
duration: module.getNoteById(1).getVariable('duration')

The ≈ Symbol

When you use irrational values (like 2^(1/12)), RMT Compose displays ≈ before the frequency value:

≈ 466.16 Hz

This indicates the displayed value is an approximation of an algebraically exact value. The internal computation preserves full precision through the SymbolicPower system.

Comparing Tuning Systems

Hear the Difference

Create the same melody in different tunings:

12-TET version: Use 2^(N/12)
Just intonation version: Use pure ratios like 5/4, 3/2
19-TET version: Use 2^(N/19)

Compare the sound quality, especially on sustained chords.

Interval Comparison Table

Interval	Pure	12-TET	19-TET	31-TET
Major 2nd	9/8	2^(2/12)	2^(3/19)	2^(5/31)
Major 3rd	5/4	2^(4/12)	2^(6/19)	2^(10/31)
Perfect 4th	4/3	2^(5/12)	2^(8/19)	2^(13/31)
Perfect 5th	3/2	2^(7/12)	2^(11/19)	2^(18/31)
Major 6th	5/3	2^(9/12)	2^(14/19)	2^(23/31)

Tips for Microtonal Composition

1. Start with Familiar Structures

Build chords and scales you know, then substitute microtonal intervals.

2. Trust Your Ears

Numbers guide you, but the sound is what matters. Experiment!

3. Use Pure Intervals for Sustained Sounds

Irrational TET intervals can cause subtle beating on long notes. Pure ratios sound smoother.

4. Explore the Module Library

Load pre-built TET scales from the Module Bar to experiment quickly.

5. Document Your Discoveries

Save interesting microtonal modules with descriptive names for future use.

Next Steps

SymbolicPower Algebra - Deep dive into irrational number handling
Complex Dependencies - Build sophisticated note relationships
Microtonal Experiments - Practical experimentation workflow

Microtonal Composition ​

What is Microtonal Music? ​

Pure Intervals vs Equal Temperament ​

The Problem with 12-TET ​

Pure Ratios in RMT Compose ​

Alternative Equal Temperaments ​

19-TET ​

31-TET ​

24-TET (Quarter Tones) ​

Bohlen-Pierce Scale ​

Why Bohlen-Pierce? ​

Just Intonation ​

5-Limit Just Intonation ​

7-Limit Just Intonation ​

Building a Microtonal Composition ​

Example: 19-TET Melody ​

Example: Just Intonation Chord ​

The ≈ Symbol ​

Comparing Tuning Systems ​

Hear the Difference ​

Interval Comparison Table ​

Tips for Microtonal Composition ​

1. Start with Familiar Structures ​

2. Trust Your Ears ​

3. Use Pure Intervals for Sustained Sounds ​

4. Explore the Module Library ​

5. Document Your Discoveries ​

Next Steps ​

Microtonal Composition

What is Microtonal Music?

Pure Intervals vs Equal Temperament

The Problem with 12-TET

Pure Ratios in RMT Compose

Alternative Equal Temperaments

19-TET

31-TET

24-TET (Quarter Tones)

Bohlen-Pierce Scale

Why Bohlen-Pierce?

Just Intonation

5-Limit Just Intonation

7-Limit Just Intonation

Building a Microtonal Composition

Example: 19-TET Melody

Example: Just Intonation Chord

The ≈ Symbol

Comparing Tuning Systems

Hear the Difference

Interval Comparison Table

Tips for Microtonal Composition

1. Start with Familiar Structures

2. Trust Your Ears

3. Use Pure Intervals for Sustained Sounds

4. Explore the Module Library

5. Document Your Discoveries

Next Steps