12-TET (Standard Western Tuning)

12-TET (12-Tone Equal Temperament) is the standard tuning system used in Western music. It divides the octave into 12 equal semitones.

Overview

Property	Value
Steps per octave	12
Step ratio	2^(1/12) ≈ 1.05946
Octave ratio	2:1 (exact)

The 12-TET Scale

Step	Note	Semitones	Expression
0	C	0	new Fraction(1)
1	C#/Db	1	2^(1/12)
2	D	2	2^(2/12)
3	D#/Eb	3	2^(3/12)
4	E	4	2^(4/12)
5	F	5	2^(5/12)
6	F#/Gb	6	2^(6/12)
7	G	7	2^(7/12)
8	G#/Ab	8	2^(8/12)
9	A	9	2^(9/12)
10	A#/Bb	10	2^(10/12)
11	B	11	2^(11/12)
12	C	12	2^(12/12) = 2

Expression Syntax

Single Semitone

// One semitone up
new Fraction(2).pow(new Fraction(1, 12))

Multiple Semitones

// Perfect fifth (7 semitones)
new Fraction(2).pow(new Fraction(7, 12))

// Major third (4 semitones)
new Fraction(2).pow(new Fraction(4, 12))

// Simplified: 4/12 = 1/3
new Fraction(2).pow(new Fraction(1, 3))

Applying to BaseNote

// Note at 4 semitones above BaseNote
module.baseNote.getVariable('frequency').mul(
  new Fraction(2).pow(new Fraction(4, 12))
)

Building a Chromatic Scale

Each note references the previous:

// Note 1: Root
note1.frequency = module.baseNote.getVariable('frequency')

// Note 2: One semitone up
note2.frequency = module.getNoteById(1).getVariable('frequency').mul(
  new Fraction(2).pow(new Fraction(1, 12))
)

// Note 3: One more semitone
note3.frequency = module.getNoteById(2).getVariable('frequency').mul(
  new Fraction(2).pow(new Fraction(1, 12))
)

// ... continue for all 12 notes

Intervals in 12-TET

Interval	Semitones	Expression
Minor second	1	2^(1/12)
Major second	2	2^(2/12) or 2^(1/6)
Minor third	3	2^(3/12) or 2^(1/4)
Major third	4	2^(4/12) or 2^(1/3)
Perfect fourth	5	2^(5/12)
Tritone	6	2^(6/12) or 2^(1/2) = √2
Perfect fifth	7	2^(7/12)
Minor sixth	8	2^(8/12) or 2^(2/3)
Major sixth	9	2^(9/12) or 2^(3/4)
Minor seventh	10	2^(10/12) or 2^(5/6)
Major seventh	11	2^(11/12)
Octave	12	2^(12/12) = 2

Using the TET-12 Module

1. Open the Module Bar
2. Find Melodies category
3. Drag TET-12 onto the workspace

The module shows a chromatic scale with each note one semitone apart.

Comparison with Just Intonation

Interval	Just	12-TET	Cents off
Perfect fifth	3/2	2^(7/12)	-2
Major third	5/4	2^(4/12)	+14
Minor third	6/5	2^(3/12)	-16
Major sixth	5/3	2^(9/12)	+16

The fifth is almost perfect, but thirds are noticeably different.

Why 12-TET?

Advantages

1. Key equality: All keys sound the same
2. Modulation: Smooth transitions between keys
3. Instrument design: Standard keyboard layout
4. Music notation: Standard 12-note system

Disadvantages

1. Impure thirds: Major and minor thirds are "off"
2. Compromise: No interval is perfectly pure (except octaves)
3. Uniformity: Every key sounds identical (no character)

Practical Applications

Piano and Guitar

Standard piano and guitar use 12-TET. If you want your RMT compositions to match these instruments, use 12-TET.

Transposition

In 12-TET, transposing is simple multiplication:

// Transpose up 5 semitones (perfect fourth)
originalFreq.mul(new Fraction(2).pow(new Fraction(5, 12)))

MIDI Compatibility

12-TET maps directly to MIDI note numbers:

• Each semitone = 1 MIDI note
• A4 (440 Hz) = MIDI note 69

Tips

1. Use for compatibility when matching other instruments
2. Simplify fractions when possible (4/12 = 1/3)
3. Remember the ≈ symbol indicates TET (irrational) values
4. Mix with just intonation for hybrid approaches

Next Steps

• Explore 19-TET for better thirds
• Try 31-TET for high-resolution microtonal
• Learn about Custom TET systems