Music Theory

Music Theory Reference

Reference material for the harmonic, tuning, and frequency concepts used throughout the Amplified Futures modules. Covers both practical lookup tables and the underlying physics and music theory.

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Contents

1. The Harmonic Series
2. V/OCT Standard and Frequency Reference
3. Partials → Intervals Table
4. Just Intonation vs Equal Temperament
5. Ptolemaic Just Intonation (12-note scale)
6. Odd Harmonic Series — StringMassCore HARM mode
7. Common Chords in V/OCT
8. Quick Cheat Sheet

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The Harmonic Series

When any pitched sound is produced — a plucked string, a blown pipe, an oscillator — the sound is not a single pure frequency. It is a superposition of multiple sine waves at integer multiples of the fundamental frequency. These are called partials or overtones.

A note at 110 Hz (A2) contains:

Partial	Frequency	Interval above fundamental
1 (fundamental)	110 Hz	—
2	220 Hz	Octave
3	330 Hz	Octave + perfect 5th
4	440 Hz	2 octaves
5	550 Hz	2 octaves + major 3rd (slightly flat)
6	660 Hz	2 octaves + perfect 5th
7	770 Hz	2 octaves + flat minor 7th (not in 12-TET)
8	880 Hz	3 octaves
…	…	…

Why this matters: The natural harmonic series defines which intervals sound consonant (partials align) or dissonant (partials clash). The tuning choices in HarmonicPressure and StringMassCore are grounded in this physics.

The series is infinite — but useful content is in partials 1–16

The amplitude of each partial typically decreases with harmonic number. In electronic synthesis with HarmonicPressure, every partial has equal amplitude — this is more like a bright buzzing tone than a natural acoustic instrument, which is part of the no-wave aesthetic.

Odd vs even harmonics

• Odd harmonics (1, 3, 5, 7, 9…): characteristic of square waves and some reed instruments. Sound brighter and more "hollow".
• Even harmonics (2, 4, 6, 8…): characteristic of triangle waves and bowed strings. Add octave doubling and "warmth".
• Both together: full harmonic content (saw wave, guitar).

StringMassCore's HARM mode uses odd harmonics only (sections at ratios 1, 3, 5, 7, 9, 11, 13, 15 — all odd).

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V/OCT Standard and Frequency Reference

V/OCT (volts per octave) is the standard pitch CV format in Eurorack/VCV Rack. The relationship is:

f = f₀ × 2^(V/oct)

where f₀ = 261.63 Hz (C4) at 0V.

Octave reference

Note	V/OCT	Frequency (Hz)
C0	−4.000	16.35
C1	−3.000	32.70
C2	−2.000	65.41
C3	−1.000	130.81
C4	0.000	261.63
C5	+1.000	523.25
C6	+2.000	1046.50
C7	+3.000	2093.00
C8	+4.000	4186.01

Chromatic semitone reference (one octave from C4)

Each semitone = 1/12 V ≈ 83.3 mV.

Note	Semitones from C4	V/OCT	Frequency (Hz)
C4	0	0.0000	261.63
C#4/Db4	1	0.0833	277.18
D4	2	0.1667	293.66
D#4/Eb4	3	0.2500	311.13
E4	4	0.3333	329.63
F4	5	0.4167	349.23
F#4/Gb4	6	0.5000	369.99
G4	7	0.5833	392.00
G#4/Ab4	8	0.6667	415.30
A4	9	0.7500	440.00
A#4/Bb4	10	0.8333	466.16
B4	11	0.9167	493.88
C5	12	1.0000	523.25

Reference frequencies

Standard pitch	Note	Frequency
Concert A	A4	440.00 Hz
Middle C	C4	261.63 Hz
Low bass	C2	65.41 Hz
Human voice lower	E2	82.41 Hz
DroneCore min pitch	A2	~82 Hz
DroneCore max pitch	E6	~1319 Hz

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Partials → Intervals Table

For HarmonicPressure: partial n at root pitch P gives output V/OCT = P + log₂(n).

For root = 0V (C4), the output pitches are:

Partial (n)	V/OCT output	Note (root C)	12-TET nearest	Cents deviation	Interval name	Consonance
1	0.000	C4	C	0¢	Unison	Perfect
2	1.000	C5	C	0¢	Octave	Perfect
3	1.585	G5	G	+2¢	Perfect 5th	Perfect
4	2.000	C6	C	0¢	Octave × 2	Perfect
5	2.322	E6	E	−14¢	Major 3rd	Consonant
6	2.585	G6	G	+2¢	Perfect 5th	Perfect
7	2.807	Bb6♭	Bb	−31¢	Septimal m7	Dissonant*
8	3.000	C7	C	0¢	Octave × 3	Perfect
9	3.170	D7	D	+4¢	Major 2nd	Mild
10	3.322	E7	E	−14¢	Major 3rd	Consonant
11	3.459	F+7	—	+49¢ above F	11th harmonic	No 12-TET equiv.
12	3.585	G7	G	+2¢	Perfect 5th	Perfect
13	3.700	Ab+7	—	+41¢ above Ab	13th harmonic	No 12-TET equiv.
14	3.807	Bb♭7	Bb	−31¢	Septimal m7	Dissonant*
15	3.907	B7	B	−12¢	Major 7th	Mild
16	4.000	C8	C	0¢	Octave × 4	Perfect

* The septimal minor 7th (7:4) is a "flat" Bb — 31 cents flatter than equal temperament. It is the characteristic interval of blues, jazz, and barbershop harmony. Acoustically, it produces zero beating with the fundamental. In EQUAL tuning mode, HarmonicPressure rounds it to the nearest 12-TET pitch.

Intervals 11 and 13 have no equivalent in 12-tone equal temperament. They fall between semitones and represent genuine microtonality. In JUST mode they appear at their exact acoustic frequencies; in EQUAL mode they are rounded to the nearest semitone, losing their character.

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Just Intonation vs Equal Temperament

Equal Temperament (12-TET)

The standard Western tuning system. Each octave is divided into 12 equal semitones by multiplying frequency by 2^(1/12) ≈ 1.05946 per step. All intervals except the octave are slightly out of tune with the harmonic series — but the error is consistent and all keys are equally usable.

Just Intonation (JI)

Intervals are defined as simple integer ratios — exactly the harmonic series. Intervals are acoustically pure: no beating between the harmonics of the two notes. But each JI scale is built from a specific root and different keys require retuning.

Comparison of common intervals

Interval	JI ratio	JI cents	12-TET cents	Difference
Perfect unison	1:1	0	0	0¢
Minor 2nd	16:15	112	100	+12¢
Major 2nd	9:8	204	200	+4¢
Minor 3rd	6:5	316	300	+16¢
Major 3rd	5:4	386	400	−14¢
Perfect 4th	4:3	498	500	−2¢
Augmented 4th	45:32	590	600	−10¢
Perfect 5th	3:2	702	700	+2¢
Minor 6th	8:5	814	800	+14¢
Major 6th	5:3	884	900	−16¢
Minor 7th	9:5	1018	1000	+18¢
Septimal m7	7:4	969	1000	−31¢
Major 7th	15:8	1088	1100	−12¢
Octave	2:1	1200	1200	0¢

The most audible differences are the major 3rd (14¢ flat in JI) and the septimal m7 (31¢ flat). These are the tensions you hear when HarmonicPressure outputs partial 5 or 7.

Why use JI in Amplified Futures?

JI tuning produces harmonic locking — when multiple voices are in JI relationships, their overtone series align and reinforce each other acoustically. This creates a richer, more resonant sound than equal temperament at the cost of being tied to a specific root.

In the context of massed oscillators (StringMassCore 16 voices, DroneClone 8 voices), this locking produces the characteristic "wall" resonance. With equal temperament the same mass of voices sounds more "electronic" and less cohesive.

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Ptolemaic Just Intonation (12-note scale)

StringMassCore's JUST mode uses a 12-note chromatic scale in Ptolemaic (5-limit) just intonation. These ratios use only the prime factors 2, 3, and 5 (no 7 — that's the septimal system).

Degree	Ratio	Cents	Note from C	12-TET note	Deviation
1	1:1	0	C	C	0¢
b2	16:15	112	Db	C#	+12¢
2	9:8	204	D	D	+4¢
b3	6:5	316	Eb	Eb	+16¢
3	5:4	386	E	E	−14¢
4	4:3	498	F	F	−2¢
b5	45:32	590	F#	F#	−10¢
5	3:2	702	G	G	+2¢
b6	8:5	814	Ab	Ab	+14¢
6	5:3	884	A	A	−16¢
b7	16:9	996	Bb	Bb	−4¢
7	15:8	1088	B	B	−12¢
8	2:1	1200	C	C	0¢

The most striking deviations from 12-TET are the major 3rd (E, −14¢), minor 3rd (Eb, +16¢), and major 6th (A, −16¢). These make JI chords sound "warmer" and more "in-tune" acoustically but slightly "off" compared to equal temperament expectations.

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Odd Harmonic Series — StringMassCore HARM mode

StringMassCore's HARM mode distributes 16 voices across 8 sections based on the odd harmonics (1, 3, 5, 7, 9, 11, 13, 15) folded into one octave.

Section	Harmonic	Octave-folded ratio	Cents above root	Note from C	Musical interval
1	1st	1:1	0	C	Unison
2	3rd	3:2	702	G	Perfect 5th
3	5th	5:4	386	E	Major 3rd
4	7th	7:4	969	Bb♭	Septimal minor 7th
5	9th	9:8	204	D	Major 2nd
6	11th	11:8	551	F+	11th harmonic (neutral 4th)
7	13th	13:8	841	Ab+	13th harmonic (neutral 6th)
8	15th	15:8	1088	B	Major 7th

"Octave-folded" means dividing the harmonic down by 2 until it falls within one octave (1:1 to 2:1). For example, the 3rd harmonic (3:1 = two octaves + perfect 5th) becomes 3:2 (within one octave).

The chord implied by these 8 sections on root C is: C – D – E – G – Bb♭ – F+ – Ab+ – B

This is not a standard Western chord — it contains:

• A perfect major triad (C, E, G)
• A septimal 7th (Bb♭, 31¢ flat of 12-TET Bb)
• The 11th harmonic F+ (49¢ above F, no 12-TET equivalent)
• The 13th harmonic Ab+ (41¢ above Ab, no 12-TET equivalent)
• A nearly-pure major 7th (B)

It is the natural acoustic chord — the chord that resonates above any bass note in nature. It is the chord of the harmonic series itself.

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Common Chords in V/OCT

Practical reference for patching multi-channel V/OCT sources. Root = 0V (C4) in all examples.

Standard triads (12-TET)

Chord	Intervals	V/OCT values	Notes
Major	R, M3, P5	0, 0.333, 0.583	C–E–G
Minor	R, m3, P5	0, 0.250, 0.583	C–Eb–G
Diminished	R, m3, d5	0, 0.250, 0.500	C–Eb–Gb
Augmented	R, M3, A5	0, 0.333, 0.667	C–E–G#
Sus2	R, M2, P5	0, 0.167, 0.583	C–D–G
Sus4	R, P4, P5	0, 0.417, 0.583	C–F–G
Power	R, P5	0, 0.583	C–G

Extended (12-TET)

Chord	Intervals	V/OCT values	Notes
Major 7	R, M3, P5, M7	0, 0.333, 0.583, 0.917	C–E–G–B
Dominant 7	R, M3, P5, m7	0, 0.333, 0.583, 0.833	C–E–G–Bb
Minor 7	R, m3, P5, m7	0, 0.250, 0.583, 0.833	C–Eb–G–Bb
Major add9	R, M2, M3, P5	0, 0.167, 0.333, 0.583	C–D–E–G

Just intonation chords

These use pure JI ratios, not 12-TET semitones. The differences are subtle but audible — JI chords have zero beating.

Chord	Ratios	V/OCT values	Note
JI Major	1:1, 5:4, 3:2	0, 0.322, 0.585	M3 is 14¢ flatter than 12-TET
JI Minor	1:1, 6:5, 3:2	0, 0.263, 0.585	m3 is 16¢ sharper than 12-TET
Harmonic 7th	1:1, 5:4, 3:2, 7:4	0, 0.322, 0.585, 0.807	The "barbershop 7th" — 31¢ flat Bb

Microtonal clusters from HarmonicPressure

Using HarmonicPressure in JUST mode, root C4:

PARTIAL	COUNT	Chord name	V/OCT cluster
1	4	Low harmonic tetrad	C4, C5, G5, C6
1	6	Major-ish hexad	C4, C5, G5, C6, E6♭, G6
1	8	Full octave cluster	C4–C5–G5–C6–E6♭–G6–Bb6♭♭–C7
3	4	5th-anchored cluster	G5, C6, E6♭, G6
5	4	3rd-anchored cluster	E6♭, G6, Bb6♭♭, C7
7	4	Septimal cluster	Bb6♭♭, C7, D7, E7♭

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Quick Cheat Sheet

V/OCT → Interval (one octave from C)

0.000  = C    (unison)
0.083  = C#
0.167  = D
0.250  = Eb   ← minor 3rd
0.333  = E    ← major 3rd (12-TET), JI major 3rd is 0.322
0.417  = F
0.500  = F#   ← tritone
0.583  = G    ← perfect 5th
0.667  = Ab
0.750  = A
0.833  = Bb   ← minor 7th
0.917  = B
1.000  = C    (octave)

Partial → note from C (quick lookup)

P1  = C  (fundamental)
P2  = C  +1 octave
P3  = G  +2¢ (octave + P5)
P4  = C  +2 octaves
P5  = E  −14¢ (M3 + 2 oct)
P6  = G  +2¢ (P5 + 2 oct)
P7  = Bb −31¢ (flat! septimal m7)
P8  = C  +3 octaves
P9  = D  +4¢
P10 = E  −14¢
P11 = F+ +49¢ (no 12-TET equiv.)
P12 = G  +2¢
P13 = Ab +41¢ (no 12-TET equiv.)
P14 = Bb −31¢
P15 = B  −12¢
P16 = C  +4 octaves

Key frequencies

A2  = 110.00 Hz  (DroneCore low)
E3  = 164.81 Hz
A3  = 220.00 Hz
C4  = 261.63 Hz  (middle C, 0V)
A4  = 440.00 Hz  (concert A)
C5  = 523.25 Hz
A5  = 880.00 Hz
C6  = 1046.50 Hz

JI ratios vs 12-TET (in cents)

P5:  JI 702¢  vs 12-TET 700¢  → +2¢  (barely noticeable)
M3:  JI 386¢  vs 12-TET 400¢  → −14¢ (audible, warmer)
m3:  JI 316¢  vs 12-TET 300¢  → +16¢ (audible)
m7:  JI 969¢  vs 12-TET 1000¢ → −31¢ (dramatic, blues feel)
M7:  JI 1088¢ vs 12-TET 1100¢ → −12¢ (noticeable)

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Further reading

• The harmonic series: any standard acoustics textbook. The Science of Sound (Rossing/Moore) is thorough.
• Just intonation: Kyle Gann's Just Intonation Explained — accessible online introduction.
• Septimal harmony: Ben Johnston's microtonal notation system explores septimal intervals in a composerly context.
• Ptolemaic tuning: Harry Partch's Genesis of a Music — foundational text on JI practice.

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See also: Harmonic-Pressure · String-Mass-Core · DroneClone · Home