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

javascript

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

Step 2: Create the Interval Note

javascript

// 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')

Step 3: Experiment

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

Workflow: Systematic Interval Study

Phase 1: Perfect Consonances

Start with the most stable intervals:

javascript

// Unison
.mul(new Fraction(1, 1))

// Octave
.mul(new Fraction(2, 1))

// Perfect Fifth
.mul(new Fraction(3, 2))

// Perfect Fourth
.mul(new Fraction(4, 3))

Listen for: Clarity, lack of beating, stability

Phase 2: Imperfect Consonances

Move to pleasing but less stable intervals:

javascript

// Major Third
.mul(new Fraction(5, 4))

// Minor Third
.mul(new Fraction(6, 5))

// Major Sixth
.mul(new Fraction(5, 3))

// Minor Sixth
.mul(new Fraction(8, 5))

Listen for: Warmth, color, character differences

Phase 3: Dissonances

Explore tension-creating intervals:

javascript

// Major Second
.mul(new Fraction(9, 8))

// Minor Second
.mul(new Fraction(16, 15))

// Major Seventh
.mul(new Fraction(15, 8))

// Minor Seventh
.mul(new Fraction(9, 5))

// Tritone
.mul(new Fraction(45, 32))

Listen for: Tension, desire to resolve, roughness

Comparing Pure vs Tempered

Setup

Create two interval notes:

javascript

// 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

Play Note 1 + Note 2 (pure third) - smooth, beatless
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

javascript

// 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:

javascript

// Minor 9th (octave + minor 2nd)
.mul(new Fraction(32, 15))

// Major 9th (octave + major 2nd)
.mul(new Fraction(9, 4))

// Minor 10th (octave + minor 3rd)
.mul(new Fraction(12, 5))

// Major 10th (octave + major 3rd)
.mul(new Fraction(5, 2))

// Perfect 11th (octave + perfect 4th)
.mul(new Fraction(8, 3))

// Perfect 12th (octave + perfect 5th)
.mul(new Fraction(3, 1))

Interval Chains

Build scales by stacking intervals:

Pythagorean Tuning (Stacking Fifths)

javascript

// Stack perfect fifths, reduce to one octave

// C (root)
.mul(new Fraction(1, 1))

// G (fifth)
.mul(new Fraction(3, 2))

// D (brought down an octave: 3/2 × 3/2 ÷ 2 = 9/8)
.mul(new Fraction(9, 8))

// A (9/8 × 3/2 = 27/16)
.mul(new Fraction(27, 16))

// E (27/16 × 3/2 ÷ 2 = 81/64)
.mul(new Fraction(81, 64))

Third-Based Tuning (5-Limit)

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

javascript

// Major scale using just thirds

// C (root)
.mul(new Fraction(1, 1))

// D (major second: 9/8)
.mul(new Fraction(9, 8))

// E (major third: 5/4)
.mul(new Fraction(5, 4))

// F (perfect fourth: 4/3)
.mul(new Fraction(4, 3))

// G (perfect fifth: 3/2)
.mul(new Fraction(3, 2))

// A (major sixth: 5/3)
.mul(new Fraction(5, 3))

// B (major seventh: 15/8)
.mul(new Fraction(15, 8))

Hearing the Harmonic Series

The harmonic series contains all pure intervals:

javascript

// Fundamental
.mul(new Fraction(1, 1))   // 440 Hz

// 2nd harmonic (octave)
.mul(new Fraction(2, 1))   // 880 Hz

// 3rd harmonic (octave + fifth)
.mul(new Fraction(3, 1))   // 1320 Hz

// 4th harmonic (two octaves)
.mul(new Fraction(4, 1))   // 1760 Hz

// 5th harmonic (two octaves + major third)
.mul(new Fraction(5, 1))   // 2200 Hz

// 6th harmonic (two octaves + fifth)
.mul(new Fraction(6, 1))   // 2640 Hz

// 7th harmonic (two octaves + minor seventh - slightly flat)
.mul(new Fraction(7, 1))   // 3080 Hz

Reducing to One Octave

Bring harmonics into the same octave:

javascript

// 3rd harmonic → fifth: 3/2
.mul(new Fraction(3, 2))

// 5th harmonic → major third: 5/4
.mul(new Fraction(5, 4))

// 7th harmonic → harmonic seventh: 7/4
.mul(new Fraction(7, 4))

Interval Quality Exploration

Consonance vs Dissonance

Create a progression from most consonant to most dissonant:

javascript

// Most consonant
.mul(new Fraction(1, 1))   // Unison
.mul(new Fraction(2, 1))   // Octave
.mul(new Fraction(3, 2))   // Fifth
.mul(new Fraction(4, 3))   // Fourth
.mul(new Fraction(5, 4))   // Major third
.mul(new Fraction(6, 5))   // Minor third

// More dissonant
.mul(new Fraction(9, 8))   // Major second
.mul(new Fraction(16, 15)) // Minor second
.mul(new Fraction(45, 32)) // Tritone

Character Comparison

Compare intervals with similar sizes but different qualities:

javascript

// Major vs Minor Third
.mul(new Fraction(5, 4))   // Major: bright
.mul(new Fraction(6, 5))   // Minor: dark

// Major vs Minor Second
.mul(new Fraction(9, 8))   // Major: open
.mul(new Fraction(16, 15)) // Minor: tight

// Major vs Minor Seventh
.mul(new Fraction(15, 8))  // Major: leading
.mul(new Fraction(9, 5))   // Minor: bluesy

Saving Your Discoveries

Create an Interval Module Library

Save each interval as a module:

Build the two-note interval
Save as "Interval - [Name] ([Ratio])"
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

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

Exercise 2: Build a Chord

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

Exercise 3: Compare TET Systems

Build the same interval in pure, 12-TET, 19-TET, and 31-TET
Play each version
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

Interval Exploration ​

What Are Intervals? ​

Setting Up an Interval Lab ​

Step 1: Create the Reference Note ​

Step 2: Create the Interval Note ​

Step 3: Experiment ​

Workflow: Systematic Interval Study ​

Phase 1: Perfect Consonances ​

Phase 2: Imperfect Consonances ​

Phase 3: Dissonances ​

Comparing Pure vs Tempered ​

Setup ​

Listen Carefully ​

Interval Inversion ​

Exploring Inversions ​

Compound Intervals ​

Interval Chains ​

Pythagorean Tuning (Stacking Fifths) ​

Third-Based Tuning (5-Limit) ​

Hearing the Harmonic Series ​

Reducing to One Octave ​

Interval Quality Exploration ​

Consonance vs Dissonance ​

Character Comparison ​

Saving Your Discoveries ​

Create an Interval Module Library ​

Example Naming ​

Exercises ​

Exercise 1: Identify by Sound ​

Exercise 2: Build a Chord ​

Exercise 3: Compare TET Systems ​

Next Steps ​

Interval Exploration

What Are Intervals?

Setting Up an Interval Lab

Step 1: Create the Reference Note

Step 2: Create the Interval Note

Step 3: Experiment

Workflow: Systematic Interval Study

Phase 1: Perfect Consonances

Phase 2: Imperfect Consonances

Phase 3: Dissonances

Comparing Pure vs Tempered

Setup

Listen Carefully

Interval Inversion

Exploring Inversions

Compound Intervals

Interval Chains

Pythagorean Tuning (Stacking Fifths)

Third-Based Tuning (5-Limit)

Hearing the Harmonic Series

Reducing to One Octave

Interval Quality Exploration

Consonance vs Dissonance

Character Comparison

Saving Your Discoveries

Create an Interval Module Library

Example Naming

Exercises

Exercise 1: Identify by Sound

Exercise 2: Build a Chord

Exercise 3: Compare TET Systems

Next Steps