Exploring Intervals
An interval is a frequency ratio. RMT Compose ships 46 of them as one-drag modules, each carrying its ratio, its cents and its limit family. This is a workflow for hearing them all.
Set up an interval lab
Two notes, sounding together, one of which you retune.
| Note | frequency | startTime | duration |
|---|---|---|---|
| 1 (reference) | base.f | base.t | beat(base) * 4 |
| 2 (the interval) | [1].f * (3/2) | [1].t | [1].d |
Create both from the ADD NOTE / SILENCE section of a widget. Note 1 comes from the BaseNote's widget — it has no At Start / At End toggle, and its button reads Create. Note 2 comes from note 1's widget: pick Note, pick At Start so it stacks, then Create Note. To try a different interval, edit note 2's frequency in its Raw: field and press Save.
Note 2 is defined against note 1, so it follows note 1 anywhere. Note 1 is your movable reference.
Legacy JavaScript syntax
// Note 2: a perfect fifth above note 1
frequency: module.getNoteById(1).getVariable('frequency').mul(new Fraction(3, 2))
startTime: module.getNoteById(1).getVariable('startTime')
duration: module.getNoteById(1).getVariable('duration')Before you listen: turn reverb off
Reverb is on by default. It blurs the beating that tells you an interval is out of tune. Open the Settings panel from the top-bar gear → Audio → switch off Enable reverb.
Then shift-click (or long-press) Play to loop, and let the dyad sustain while you listen.
The shipped intervals
Every ratio below exists as a module. Search the library bar's magnifier by ratio (5/4), by family (7-limit) or by tag (comma) to find one instantly.
3-limit (Pythagorean) — 11 modules
Prime factors 2 and 3 only. Everything is built from stacked fifths.
| Name | Ratio | Cents |
|---|---|---|
| Unison | 1/1 | 0 |
| Major 2nd | 9/8 | 203.91 |
| Pythagorean minor 3rd | 32/27 | 294.14 |
| Pythagorean major 3rd | 81/64 | 407.82 |
| Perfect 4th | 4/3 | 498.05 |
| Pythagorean tritone | 729/512 | 611.73 |
| Perfect 5th | 3/2 | 701.96 |
| Pythagorean major 6th | 27/16 | 905.87 |
| Pythagorean minor 7th | 16/9 | 996.09 |
| Pythagorean major 7th | 243/128 | 1109.78 |
| Octave | 2/1 | 1200 |
5-limit (classic just) — 10 modules
Add the prime 5, and thirds become sweet.
| Name | Ratio | Cents |
|---|---|---|
| Just minor 2nd | 16/15 | 111.73 |
| Minor whole tone | 10/9 | 182.40 |
| Just minor 3rd | 6/5 | 315.64 |
| Just major 3rd | 5/4 | 386.31 |
| Just augmented 4th | 45/32 | 590.22 |
| Just diminished 5th | 64/45 | 609.78 |
| Just minor 6th | 8/5 | 813.69 |
| Just major 6th | 5/3 | 884.36 |
| Just minor 7th | 9/5 | 1017.60 |
| Just major 7th | 15/8 | 1088.27 |
7-limit (septimal) — 10 modules
The seventh harmonic. Bluesy, barbershop, "locked".
| Name | Ratio | Cents |
|---|---|---|
| Septimal whole tone | 8/7 | 231.17 |
| Septimal minor 3rd | 7/6 | 266.87 |
| Septimal major 3rd | 9/7 | 435.08 |
| Septimal tritone | 7/5 | 582.51 |
| Septimal tritone (wide) | 10/7 | 617.49 |
| Septimal narrow 5th | 32/21 | 729.22 |
| Septimal minor 6th | 14/9 | 764.92 |
| Septimal major 6th | 12/7 | 933.13 |
| Harmonic 7th | 7/4 | 968.83 |
| Septimal major 7th | 63/32 | 1172.74 |
Higher limits (11–23) — 9 modules
Neutral intervals — genuinely between major and minor.
| Name | Ratio | Cents |
|---|---|---|
| 17th harmonic | 17/16 | 104.96 |
| Tridecimal neutral 2nd | 13/12 | 138.57 |
| 19th harmonic | 19/16 | 297.51 |
| Undecimal neutral 3rd | 11/9 | 347.41 |
| Undecimal tritone | 11/8 | 551.32 |
| 23rd harmonic | 23/16 | 628.27 |
| Tridecimal diminished 5th | 13/9 | 636.62 |
| Tridecimal neutral 6th | 13/8 | 840.53 |
| Undecimal neutral 7th | 11/6 | 1049.36 |
Commas — 6 modules
The tiny gaps that make tuning systems necessary.
| Name | Ratio | Cents |
|---|---|---|
| Schisma | 32805/32768 | 1.95 |
| Diaschisma | 2048/2025 | 19.55 |
| Syntonic comma | 81/80 | 21.51 |
| Pythagorean comma | 531441/524288 | 23.46 |
| Septimal comma | 64/63 | 27.26 |
| Enharmonic diesis | 128/125 | 41.06 |
A systematic pass
Phase 1 — perfect consonances
Drag each onto your reference note, in Start mode so it stacks:
Unison (1/1) · Octave (2/1) · Perfect 5th (3/2) · Perfect 4th (4/3)
Listen for: no beating at all. These are the intervals where the partials line up.
Phase 2 — imperfect consonances
Just major 3rd (5/4) · Just minor 3rd (6/5) · Just major 6th (5/3) · Just minor 6th (8/5)
Listen for: warmth and colour. Still smooth, but with character.
Phase 3 — dissonances
Major 2nd (9/8) · Just minor 2nd (16/15) · Just major 7th (15/8) · Just minor 7th (9/5) · Just augmented 4th (45/32)
Listen for: roughness, and the pull toward resolution.
Phase 4 — the septimal world
Septimal minor 3rd (7/6) · Septimal tritone (7/5) · Harmonic 7th (7/4)
Listen for: the 7/4 in particular. Against a 12-TET minor seventh it is 31 cents flat, and it locks in a way the tempered one never does.
Pure vs tempered
Add a third note and compare the same interval two ways.
| Note | frequency | Meaning |
|---|---|---|
| 1 | base.f | reference |
| 2 | [1].f * (5/4) | pure major third — 386.3¢ |
| 3 | [1].f * 2^(4/12) | 12-TET major third — 400¢ |
Play 1+2: smooth, beatless. Play 1+3: a slow shimmer. The gap is 13.7 cents, and on a sustained tone it is unmistakable.
Note 3 will be crosshatched and its value shown with a ≈ prefix — that is the app telling you 2^(4/12) is irrational and has been approximated. See Understanding SymbolicPower.
Interval inversion
An interval and its inversion sum to an octave: multiply the two ratios and you get 2/1.
| Interval | Ratio | Inversion | Ratio | Product |
|---|---|---|---|---|
| Just minor 2nd | 16/15 | Just major 7th | 15/8 | 2/1 |
| Major 2nd | 9/8 | Pythagorean minor 7th | 16/9 | 2/1 |
| Just minor 3rd | 6/5 | Just major 6th | 5/3 | 2/1 |
| Just major 3rd | 5/4 | Just minor 6th | 8/5 | 2/1 |
| Perfect 4th | 4/3 | Perfect 5th | 3/2 | 2/1 |
| Just augmented 4th | 45/32 | Just diminished 5th | 64/45 | 2/1 |
Two different minor sevenths, both real
The inversion of the 9/8 major second is 16/9 (Pythagorean minor 7th, 996.09¢). The 5-limit minor seventh is 9/5 (1017.60¢). They differ by a syntonic comma (81/80), and both ship. Which one you want depends on how you got there — 16/9 if you inverted a whole tone, 9/5 if you stacked a minor third on a fifth.
To sound an inversion below the reference, divide:
[1].f / (8/5) # a just minor sixth below note 1Compound intervals
Beyond the octave. Multiply the simple ratio by 2:
| Compound | Ratio | Built from |
|---|---|---|
| Minor 9th | 32/15 | 16/15 × 2 |
| Major 9th | 9/4 | 9/8 × 2 |
| Minor 10th | 12/5 | 6/5 × 2 |
| Major 10th | 5/2 | 5/4 × 2 |
| Perfect 11th | 8/3 | 4/3 × 2 |
| Perfect 12th | 3/1 | 3/2 × 2 |
[1].f * (9/4) # a major ninth above note 1Or drag the Major 2nd module on and press the note's ▲ arrow once — the default arrow interval is ×2, so it becomes a major ninth. The arrows fold the factor into the coefficient rather than stacking multipliers, so ▲ then ▼ returns you to exactly where you were.
The harmonic series
Every pure interval hides in the harmonic series. Multiply the fundamental by an integer:
| Harmonic | Expression | Interval above the fundamental |
|---|---|---|
| 1 | base.f | fundamental |
| 2 | base.f * 2 | octave |
| 3 | base.f * 3 | octave + fifth |
| 4 | base.f * 4 | two octaves |
| 5 | base.f * 5 | two octaves + major 3rd |
| 6 | base.f * 6 | two octaves + fifth |
| 7 | base.f * 7 | two octaves + harmonic 7th |
Build all seven as notes stacked on the same start time and you have a harmonic stack you can hear as a single timbre.
Reducing to one octave
Divide by powers of 2 until the ratio sits between 1 and 2 — which is where the simple ratios come from:
| Harmonic | Reduced | Name |
|---|---|---|
| 3 | 3/2 | Perfect 5th |
| 5 | 5/4 | Just major 3rd |
| 7 | 7/4 | Harmonic 7th |
| 9 | 9/8 | Major 2nd |
| 11 | 11/8 | Undecimal tritone |
| 13 | 13/8 | Tridecimal neutral 6th |
That table is the odd-harmonic logic behind the shipped Base-3 chord (3:5:7:9) and Base-5 chord (5:7:9:11), and behind the 81-note Tesla scale, which walks the odd harmonics 9, 11, 13 … 169 over 9.
Interval chains
Pythagorean: stack fifths
Each note is a fifth above the previous, folded back down an octave when it overshoots:
| Note | frequency | Absolute ratio |
|---|---|---|
| 1 | base.f | 1/1 |
| 2 | [1].f * (3/2) | 3/2 |
| 3 | [2].f * (3/2) / 2 | 9/8 |
| 4 | [3].f * (3/2) | 27/16 |
| 5 | [4].f * (3/2) / 2 | 81/64 |
Because each note is written against the previous one, retuning note 1 shifts the entire chain.
Stack twelve fifths and you land a Pythagorean comma (531441/524288, 23.46¢) above seven octaves — not on them. That gap is why equal temperament exists, and it is one of the six comma modules.
5-limit: a just major scale
All degrees against the base:
| Degree | frequency |
|---|---|
| C | base.f |
| D | base.f * (9/8) |
| E | base.f * (5/4) |
| F | base.f * (4/3) |
| G | base.f * (3/2) |
| A | base.f * (5/3) |
| B | base.f * (15/8) |
| C′ | base.f * 2 |
Every note hangs off the base, so changing base.f transposes the whole scale.
Save what you learn
Don't rebuild the interval library — it already exists. Do save the combinations you find.
Select the notes of a chord or a chain you like (shift-drag a marquee), then press Copy to Modules in the group widget. It lands in the library's Custom section, rooted at its earliest note, with the internal dependency tree intact — so dropping it on any note transposes the whole thing correctly.
Exercises
1 — Identify by sound. Collapse every section except Intervals. Drag a tile onto your reference note without reading its label, play, and guess. The tooltip (Perfect 5th (3/2, 702¢)) gives you the answer.
2 — Build the four triads. On one root, stack:
| Triad | Third | Fifth |
|---|---|---|
| Major | (5/4) | (3/2) |
| Minor | (6/5) | (3/2) |
| Diminished | (6/5) | (7/5) |
| Augmented | (5/4) | (25/16) |
Then compare with the shipped Major, Minor, Diminished and Augmented chord modules — those are the exact ratios they use.
3 — Rank the temperaments. Build a major third four ways on the same root: (5/4), 2^(4/12), 2^(6/19), 2^(10/31). Play each against the root. Rank them by how close to beatless they sound, then check your ranking against the cents: 386.3, 400, 378.9, 387.1.
Next
- Microtonal Experiments — structured listening
- Microtonal Composition — write in these tunings
- Ratios — the theory