31-TET

31-TET (31-Tone Equal Temperament) divides the octave into 31 equal steps. It provides excellent approximations to just intonation intervals and is considered one of the best meantone temperaments.

Overview

Property	Value
Steps per octave	31
Step ratio	2^(1/31) ≈ 1.02263
Octave ratio	2:1 (exact)

Why 31-TET?

31-TET was studied by Christiaan Huygens in the 17th century. It offers:

1. Excellent thirds: Nearly pure major and minor thirds
2. Good fifths: Close to just intonation
3. Septimal intervals: Good approximation of 7-limit ratios
4. High resolution: Fine pitch control for microtonal music

Expression Syntax

Single Step

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

Multiple Steps

// Major third (10 steps in 31-TET)
new Fraction(2).pow(new Fraction(10, 31))

// Perfect fifth (18 steps in 31-TET)
new Fraction(2).pow(new Fraction(18, 31))

Applying to BaseNote

// Note at 10 steps above BaseNote (major third)
module.baseNote.getVariable('frequency').mul(
  new Fraction(2).pow(new Fraction(10, 31))
)

Intervals in 31-TET

Interval	31-TET Steps	Just Ratio	Quality
Minor second	3	16/15	Excellent
Major second	5	9/8	Good
Minor third	8	6/5	Excellent
Major third	10	5/4	Excellent
Perfect fourth	13	4/3	Good
Tritone	15-16	7/5 or 10/7	Two options
Perfect fifth	18	3/2	Very good
Minor sixth	21	8/5	Excellent
Major sixth	23	5/3	Excellent
Septimal seventh	25	7/4	Very good
Minor seventh	26	9/5	Good
Major seventh	28	15/8	Good
Octave	31	2/1	Perfect

Comparison with Just Intonation

Interval	Just	31-TET	Cents off
Perfect fifth	3/2	2^(18/31)	-5.2
Major third	5/4	2^(10/31)	+0.8
Minor third	6/5	2^(8/31)	-5.9
Septimal seventh	7/4	2^(25/31)	+1.1

The thirds are remarkably close to pure!

Using the TET-31 Module

1. Open the Module Bar
2. Find Melodies category
3. Drag TET-31 onto the workspace

7-Limit Harmony

31-TET approximates 7-limit just intonation well:

Interval	Ratio	31-TET Steps
Septimal minor third	7/6	7
Septimal tritone	7/5	15
Septimal minor seventh	7/4	25

This enables "blue notes" and jazz-like harmony with mathematical precision.

Notation Systems

31-TET has several notation approaches:

Ups and Downs

Uses arrows to modify standard pitches:

• C, C^, C#, Cv, C##... (^ = up, v = down)

Half-Sharp/Half-Flat

Uses additional accidentals:

• C, C half-sharp, C#, C three-quarter-sharp...

Color Names

Based on just intonation approximations:

• "red" notes (5-limit: 5/4, 5/3)
• "blue" notes (7-limit: 7/4, 7/6)

Musical Applications

Meantone Repertoire

Historical music written for meantone temperament sounds excellent in 31-TET.

Extended Harmony

Jazz and contemporary music can use septimal intervals for:

• More consonant dominant sevenths
• "Natural" blue notes
• Extended chord voicings

Microtonal Composition

31 notes per octave enable:

• Subtle pitch variations
• Glissando-like melodic lines
• New harmonic possibilities

Challenges

Complexity

31 pitches per octave require careful organization.

Learning Curve

Musicians need significant retraining.

Instruments

Very few physical instruments support 31-TET. RMT Compose is an excellent tool for exploration.

Tips

1. Focus on triads first - Appreciate the pure thirds
2. Try septimal intervals - 7/4 and 7/6 have unique colors
3. Compare with 12-TET - The difference in thirds is dramatic
4. Use as a reference - Even if you don't compose in 31-TET, hearing pure intervals trains your ear

Example: Major Triad in 31-TET

// Root
root.frequency = module.baseNote.getVariable('frequency')

// Major third (10 steps)
third.frequency = module.baseNote.getVariable('frequency').mul(
  new Fraction(2).pow(new Fraction(10, 31))
)

// Perfect fifth (18 steps)
fifth.frequency = module.baseNote.getVariable('frequency').mul(
  new Fraction(2).pow(new Fraction(18, 31))
)

This triad sounds remarkably pure compared to 12-TET!

Next Steps

• Explore Bohlen-Pierce for a non-octave system
• Create your own system with Custom TET
• Return to Pure Ratios to compare with just intonation