03Frequency becomes a musical coordinate

Pitch & tuning

Pitch is the way hearing organizes repeating sound from low to high. Tuning chooses the frequency positions; a tonal center supplies a reference; a pitch collection selects available relationships; register places each pitch in an octave.

frequency
physical rate→tuning
available positions→center + collection
musical context→pitch + register
performed note

Equal steps are frequency ratios

In twelve-tone equal temperament, moving one semitone multiplies frequency by 2^(1/12). Moving twelve semitones doubles it, which listeners usually hear as the same pitch class in another octave.

Equal pitch steps ≠ equal Hz stepsf(n) = 220 × 2^(n/12)
0220.00 Hz1.000×
1233.08 Hz1.059×
2246.94 Hz1.122×
3261.63 Hz1.189×
4277.18 Hz1.260×
5293.66 Hz1.335×
6311.13 Hz1.414×
7329.63 Hz1.498×
8349.23 Hz1.587×
9369.99 Hz1.682×
10392.00 Hz1.782×
11415.30 Hz1.888×
12440.00 Hz2.000×
equal visual width = equal perceived pitch stepfrequency doubles at +12
f(n) = f₀ × 2^(n/12)n is semitone distance. The formula describes 12-TET tuning—not every tuning system and not the way a tradition uses its pitches.

Keep four decisions separate

DecisionQuestionExample
TuningWhich frequency positions exist?12-TET, just intonation, a specific instrument tuning
Tonal centerWhich pitch currently feels like reference zero?C = 0
CollectionWhich relative positions are foregrounded?0 2 4 5 7 9 11
RegisterWhich octave contains the event?C3, C4, C5
Contextual example

Transpose one shape

12-TET · center C · major collection
major frame relative to the center0 → 4 → 7 → 12
0root
43
75
12octave
Hear
The interval identity survives while the absolute frequencies move.
Use
Reuse one bass, chord or melodic shape at another tonal center.
Why it fits
Adding the same root offset preserves every internal semitone distance.
Contextual example

Change one degree, keep the center

12-TET · center D · minor → Dorian
minor0 2 3 5 7 8 10
0root1·223♭34·546·758♭69·10♭711·
Dorian0 2 3 5 7 9 10
0root1·223♭34·546·758·9610♭711·
Hear
The natural sixth brightens the collection without moving the tonal center.
Use
Introduce a contrasting color while a D pedal or bass anchor continues.
Why it fits
Six pitch classes remain common; only the minor sixth moves up one semitone.
Detailed lookupSemitone offsets and compact notationPitch collections and cultural caveats
Keep exploring

Terms, relations and sources

Key terms

frequencytuningtonal centercollection

Related chapters

Tunings & traditionsHarmony & motion

Detailed lookup

Relative pitch notationPitch collectionsNotation legend

Evidence

Editorial sources

Models state their boundary; tradition-specific claims require specialist sources.