Pure Ratios (Just Intonation)

Pure ratios are the foundation of RMT Compose. They represent musical intervals as exact fractions derived from the natural harmonic series.

What Are Pure Ratios?

When a string vibrates, it produces a fundamental frequency plus overtones at integer multiples:

Harmonic	Multiple	Note (if fundamental = C)
1st	1×	C
2nd	2×	C (octave)
3rd	3×	G
4th	4×	C (two octaves)
5th	5×	E
6th	6×	G
7th	7×	B♭ (slightly flat)
8th	8×	C (three octaves)

Pure ratios capture these natural relationships.

Common Intervals

Perfect Consonances

Interval	Ratio	Decimal	Sound
Unison	1/1	1.000	Same pitch
Octave	2/1	2.000	Same note, higher
Perfect fifth	3/2	1.500	Very stable
Perfect fourth	4/3	1.333	Stable

Imperfect Consonances

Interval	Ratio	Decimal	Sound
Major third	5/4	1.250	Bright, happy
Minor third	6/5	1.200	Dark, sad
Major sixth	5/3	1.667	Warm
Minor sixth	8/5	1.600	Melancholic

Seconds and Sevenths

Interval	Ratio	Decimal	Sound
Major second	9/8	1.125	Whole step
Minor second	16/15	1.067	Half step
Major seventh	15/8	1.875	Tension
Minor seventh	7/4	1.750	Bluesy

Using Ratios in RMT Compose

Expression Syntax

// Perfect fifth above BaseNote
module.baseNote.getVariable('frequency').mul(new Fraction(3, 2))

// Major third below (divide instead of multiply)
module.baseNote.getVariable('frequency').div(new Fraction(5, 4))

// Chain intervals: fifth + third = major seventh
module.baseNote.getVariable('frequency')
  .mul(new Fraction(3, 2))
  .mul(new Fraction(5, 4))
// = 3/2 × 5/4 = 15/8

Building Scales

Major Scale (Just Intonation)

Degree	Ratio	Interval from root
1 (Do)	1/1	Unison
2 (Re)	9/8	Major second
3 (Mi)	5/4	Major third
4 (Fa)	4/3	Perfect fourth
5 (Sol)	3/2	Perfect fifth
6 (La)	5/3	Major sixth
7 (Ti)	15/8	Major seventh
8 (Do)	2/1	Octave

Minor Scale (Just Intonation)

Degree	Ratio	Interval from root
1	1/1	Unison
2	9/8	Major second
3	6/5	Minor third
4	4/3	Perfect fourth
5	3/2	Perfect fifth
6	8/5	Minor sixth
7	9/5	Minor seventh
8	2/1	Octave

Building Chords

Major Triad

Note	Ratio	Interval
Root	1/1	-
Third	5/4	Major third
Fifth	3/2	Perfect fifth

root.frequency = module.baseNote.getVariable('frequency')
third.frequency = module.baseNote.getVariable('frequency').mul(new Fraction(5, 4))
fifth.frequency = module.baseNote.getVariable('frequency').mul(new Fraction(3, 2))

Minor Triad

Note	Ratio	Interval
Root	1/1	-
Third	6/5	Minor third
Fifth	3/2	Perfect fifth

Why Pure Ratios?

Advantages

1. Resonance: Pure intervals align with the overtone series, creating clear, ringing sounds
2. Exactness: No rounding or approximation - 3/2 is exactly 3/2
3. Mathematical elegance: Operations stay exact (3/2 × 5/4 = 15/8)
4. Flexibility: Any ratio can be expressed

Limitations

1. Wolf intervals: Some keys sound worse than others
2. Modulation: Changing keys can sound jarring
3. Instrument compatibility: Standard instruments are tuned to 12-TET

Comparison with Equal Temperament

Interval	Just Ratio	Just Decimal	12-TET Decimal	Difference
Perfect fifth	3/2	1.5000	1.4983	-0.11%
Major third	5/4	1.2500	1.2599	+0.79%
Minor third	6/5	1.2000	1.1892	-0.90%

The differences are audible! Pure thirds sound "sweeter" than 12-TET thirds.

Tips for Using Pure Ratios

1. Start with fifths and thirds - They're the most consonant
2. Listen carefully - Pure intervals have a distinct quality
3. Combine thoughtfully - Not all ratios combine well
4. Use octave transposition - Multiply/divide by 2 to shift octaves
5. Experiment freely - RMT makes it easy to try unusual ratios

Extended Just Intonation

Beyond the basic intervals, you can explore:

Ratio	Approximate Interval
7/6	Septimal minor third
7/5	Tritone (septimal)
11/8	Undecimal fourth
13/8	Tridecimal sixth

These intervals from the 7th, 11th, and 13th harmonics create unique, "otherworldly" sounds not found in Western music.

Next Steps

• Learn about Equal Temperament for an alternative approach
• Try the Build a Major Scale tutorial
• Explore the Expression Syntax Reference