Interval Exploration

A systematic workflow for understanding and experimenting with musical intervals using RMT Compose.

What Are Intervals?

An interval is the distance between two pitches. In RMT Compose, intervals are expressed as frequency ratios:

Interval	Ratio	Cents	Description
Unison	1/1	0	Same pitch
Minor Second	16/15	112	Half step
Major Second	9/8	204	Whole step
Minor Third	6/5	316	Sad quality
Major Third	5/4	386	Happy quality
Perfect Fourth	4/3	498	Open sound
Tritone	45/32	590	Tense, unstable
Perfect Fifth	3/2	702	Strong consonance
Minor Sixth	8/5	814	Somewhat dark
Major Sixth	5/3	884	Bright
Minor Seventh	9/5	1018	Jazzy tension
Major Seventh	15/8	1088	Leading tone
Octave	2/1	1200	Same note, higher

Setting Up an Interval Lab

Step 1: Create the Reference Note

// Note 1: Reference pitch
frequency: base.f
startTime: 0
duration: 2

Step 2: Create the Interval Note

// Note 2: Interval above reference
frequency: [1].f * (3/2)  // Perfect fifth
startTime: [1].t
duration: [1].d

Step 3: Experiment

Change the ratio in Note 2 to hear different intervals. Both notes play simultaneously for direct comparison.

Legacy JavaScript syntax

// Note 1: Reference pitch
frequency: module.baseNote.getVariable('frequency')
startTime: new Fraction(0)
duration: new Fraction(2)

// Note 2: Interval above reference
frequency: module.getNoteById(1).getVariable('frequency').mul(new Fraction(3, 2))  // Perfect fifth
startTime: module.getNoteById(1).getVariable('startTime')
duration: module.getNoteById(1).getVariable('duration')

Workflow: Systematic Interval Study

Phase 1: Perfect Consonances

Start with the most stable intervals:

// Unison
* (1/1)

// Octave
* 2

// Perfect Fifth
* (3/2)

// Perfect Fourth
* (4/3)

Listen for: Clarity, lack of beating, stability

Phase 2: Imperfect Consonances

Move to pleasing but less stable intervals:

// Major Third
* (5/4)

// Minor Third
* (6/5)

// Major Sixth
* (5/3)

// Minor Sixth
* (8/5)

Listen for: Warmth, color, character differences

Phase 3: Dissonances

Explore tension-creating intervals:

// Major Second
* (9/8)

// Minor Second
* (16/15)

// Major Seventh
* (15/8)

// Minor Seventh
* (9/5)

// Tritone
* (45/32)

Listen for: Tension, desire to resolve, roughness

Comparing Pure vs Tempered

Setup

Create two interval notes:

// Note 2: Pure major third (5/4)
frequency: [1].f * (5/4)

// Note 3: 12-TET major third (4 semitones)
frequency: [1].f * 2^(4/12)

Legacy JavaScript syntax

// Note 2: Pure major third (5/4)
frequency: module.getNoteById(1).getVariable('frequency').mul(new Fraction(5, 4))

// Note 3: 12-TET major third (4 semitones)
frequency: module.getNoteById(1).getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(4, 12)))

Listen Carefully

1. Play Note 1 + Note 2 (pure third) - smooth, beatless
2. Play Note 1 + Note 3 (tempered third) - subtle beating

The difference is ~14 cents, audible on sustained tones.

Interval Inversion

Every interval has an inversion that completes the octave:

Interval	Ratio	Inversion	Ratio
Minor 2nd	16/15	Major 7th	15/8
Major 2nd	9/8	Minor 7th	16/9
Minor 3rd	6/5	Major 6th	5/3
Major 3rd	5/4	Minor 6th	8/5
Perfect 4th	4/3	Perfect 5th	3/2
Tritone	45/32	Tritone	64/45

Exploring Inversions

// Original: Major third above
frequency: [1].f * (5/4)

// Inversion: Minor sixth below (same pitch class, lower octave)
frequency: [1].f / (8/5)

Legacy JavaScript syntax

// Original: Major third above
frequency: module.getNoteById(1).getVariable('frequency').mul(new Fraction(5, 4))

// Inversion: Minor sixth below (same pitch class, lower octave)
frequency: module.getNoteById(1).getVariable('frequency').div(new Fraction(8, 5))

Compound Intervals

Intervals larger than an octave:

// Minor 9th (octave + minor 2nd)
* (32/15)

// Major 9th (octave + major 2nd)
* (9/4)

// Minor 10th (octave + minor 3rd)
* (12/5)

// Major 10th (octave + major 3rd)
* (5/2)

// Perfect 11th (octave + perfect 4th)
* (8/3)

// Perfect 12th (octave + perfect 5th)
* 3

Interval Chains

Build scales by stacking intervals:

Pythagorean Tuning (Stacking Fifths)

// Stack perfect fifths, reduce to one octave

// C (root)
* (1/1)

// G (fifth)
* (3/2)

// D (brought down an octave: 3/2 × 3/2 ÷ 2 = 9/8)
* (9/8)

// A (9/8 × 3/2 = 27/16)
* (27/16)

// E (27/16 × 3/2 ÷ 2 = 81/64)
* (81/64)

Third-Based Tuning (5-Limit)

Use ratios with factors of 2, 3, and 5:

// Major scale using just thirds

// C (root)
* (1/1)

// D (major second: 9/8)
* (9/8)

// E (major third: 5/4)
* (5/4)

// F (perfect fourth: 4/3)
* (4/3)

// G (perfect fifth: 3/2)
* (3/2)

// A (major sixth: 5/3)
* (5/3)

// B (major seventh: 15/8)
* (15/8)

Hearing the Harmonic Series

The harmonic series contains all pure intervals:

// Fundamental
* 1   // 440 Hz

// 2nd harmonic (octave)
* 2   // 880 Hz

// 3rd harmonic (octave + fifth)
* 3   // 1320 Hz

// 4th harmonic (two octaves)
* 4   // 1760 Hz

// 5th harmonic (two octaves + major third)
* 5   // 2200 Hz

// 6th harmonic (two octaves + fifth)
* 6   // 2640 Hz

// 7th harmonic (two octaves + minor seventh - slightly flat)
* 7   // 3080 Hz

Reducing to One Octave

Bring harmonics into the same octave by dividing by powers of 2 until the ratio is between 1 and 2:

// 3rd harmonic → fifth: 3 ÷ 2 = 3/2
* (3/2)

// 5th harmonic → major third: 5 ÷ 4 = 5/4
* (5/4)

// 7th harmonic → harmonic seventh: 7 ÷ 4 = 7/4
* (7/4)

Interval Quality Exploration

Consonance vs Dissonance

Create a progression from most consonant to most dissonant:

// Most consonant
* 1        // Unison
* 2        // Octave
* (3/2)    // Fifth
* (4/3)    // Fourth
* (5/4)    // Major third
* (6/5)    // Minor third

// More dissonant
* (9/8)    // Major second
* (16/15)  // Minor second
* (45/32)  // Tritone

Character Comparison

Compare intervals with similar sizes but different qualities:

// Major vs Minor Third
* (5/4)    // Major: bright
* (6/5)    // Minor: dark

// Major vs Minor Second
* (9/8)    // Major: open
* (16/15)  // Minor: tight

// Major vs Minor Seventh
* (15/8)   // Major: leading
* (9/5)    // Minor: bluesy

Using the Module Bar for Intervals

Tip: The Module Bar's "Drop at:" toggle speeds up interval exploration:

• Start mode: Stack interval notes at the same time for harmonic comparison (chords)
• End mode: Chain intervals sequentially for melodic comparison (sequences)

When you've saved interval modules to your library, you can quickly drop them onto your workspace to build chords or progressions.

Saving Your Discoveries

Create an Interval Module Library

Save each interval as a module:

1. Build the two-note interval
2. Save as "Interval - [Name] ([Ratio])"
3. Organize in an "Intervals" category

Example Naming

• "Interval - Perfect Fifth (3/2)"
• "Interval - Major Third (5/4)"
• "Interval - Harmonic Seventh (7/4)"

Exercises

Exercise 1: Identify by Sound

1. Create all 12 intervals in separate modules
2. Close your eyes, load a random one
3. Try to identify the interval

Exercise 2: Build a Chord

1. Choose a root note
2. Add intervals to build: Major, Minor, Diminished, Augmented triads
3. Listen to how interval combinations create chord quality

Exercise 3: Compare TET Systems

1. Build the same interval in pure, 12-TET, 19-TET, and 31-TET
2. Play each version
3. Note which TET best approximates the pure interval

Next Steps

• Microtonal Experiments - Apply intervals to microtonal music
• Chaining Notes - Build complex interval relationships
• Tuning Systems - Deeper tuning theory