Working with Tones¶

A Tone represents a single musical note, optionally with an octave number in scientific pitch notation (e.g. C4 = middle C).

What is a Tone?¶

A musical tone is a sound with a definite pitch — a periodic vibration at a specific frequency. In the Western 12-tone system, the octave (a 2:1 frequency ratio) is divided into 12 equal steps called semitones or half steps. Two semitones make a whole step (whole tone).

The 12 chromatic tones are:

C  C#/Db  D  D#/Eb  E  F  F#/Gb  G  G#/Ab  A  A#/Bb  B

Notes with two names (like C# and Db) are enharmonic equivalents — different names for the same pitch. Whether you call it C# or Db depends on the musical context (key signature, harmonic function).

Scientific Pitch Notation¶

Each tone can be assigned an octave number. The standard is scientific pitch notation, where the octave number increments at C:

... B3  C4  C#4  D4 ... A4  B4  C5  C#5 ...
         ^                        ^
     middle C              one octave up

Key reference points:

A4 = 440 Hz — the international tuning standard (ISO 16)
C4 = 261.63 Hz — middle C on the piano
A0 = 27.5 Hz — the lowest A on a standard piano
C8 = 4186 Hz — the highest C on a standard piano

Creating Tones¶

from pytheory import Tone

# From a string (most common)
c4 = Tone.from_string("C4")
cs4 = Tone.from_string("C#4")

# Direct construction
d = Tone(name="D", octave=3)

# With a specific system
a4 = Tone.from_string("A4", system="western")

Properties¶

>>> c4 = Tone.from_string("C4")
>>> c4.name
'C'
>>> c4.octave
4
>>> c4.full_name
'C4'
>>> str(c4)
'C4'

Pitch and Frequency¶

Every tone vibrates at a specific frequency measured in Hertz (Hz — cycles per second). The relationship between pitch and frequency is logarithmic: each octave doubles the frequency, and each semitone multiplies by the 12th root of 2 (~1.05946).

>>> a4 = Tone.from_string("A4", system="western")
>>> a4.frequency
440.0

>>> Tone.from_string("A3", system="western").frequency
220.0    # One octave down = half the frequency

>>> Tone.from_string("C4", system="western").frequency
261.63   # Middle C

Temperament¶

Temperament is the system used to tune the intervals between notes. Different temperaments produce slightly different frequencies for the same note name:

Equal temperament (default): Every semitone has an identical frequency ratio of 2^(1/12). This is the modern standard — it allows free modulation between all keys but no interval is acoustically “pure” except the octave.
Pythagorean temperament: Built entirely from pure perfect fifths (3:2 ratio). Produces beatless fifths but introduces the “Pythagorean comma” — a small discrepancy when 12 fifths don’t quite equal 7 octaves. Used in medieval European music.
Quarter-comma meantone: Tunes major thirds to the pure ratio of 5:4, distributing the resulting error across the fifths. Dominant in Renaissance and Baroque music (15th–18th century). Sounds beautiful in closely related keys but “wolf intervals” make distant keys unusable.

>>> a4.pitch(temperament="equal")
440.0
>>> a4.pitch(temperament="pythagorean")
440.0    # A4 is always 440 (it's the reference)

>>> c5 = Tone.from_string("C5", system="western")
>>> c5.pitch(temperament="equal")
523.25
>>> c5.pitch(temperament="pythagorean")
521.48   # Slightly different!

Symbolic Pitch¶

Pass symbolic=True to get exact pitch ratios as SymPy expressions instead of floating-point approximations. This is useful for mathematical analysis, proving tuning relationships, or comparing temperaments with exact arithmetic.

>>> a4 = Tone.from_string("A4", system="western")

# Equal temperament: irrational ratios (roots of 2)
>>> a4.pitch(symbolic=True)
440
>>> Tone.from_string("C5", system="western").pitch(symbolic=True)
440*2**(1/4)

# Pythagorean: pure rational ratios (powers of 3/2)
>>> Tone.from_string("G4", system="western").pitch(
...     temperament="pythagorean", symbolic=True)
660

# Compare the major third across temperaments
>>> e4 = Tone.from_string("E4", system="western")
>>> e4.pitch(temperament="equal", symbolic=True)
440*2**(1/3)
>>> e4.pitch(temperament="pythagorean", symbolic=True)
12160/27
>>> e4.pitch(temperament="meantone", symbolic=True)
550

# Symbolic expressions can be evaluated to any precision
>>> e4.pitch(symbolic=True).evalf(50)
329.62755691286991583007431157433859631791591649985

The symbolic output reveals why temperaments differ: equal temperament uses irrational numbers (roots of 2), Pythagorean uses powers of 3/2 (rational but accumulating error), and meantone tunes thirds to the pure 5/4 ratio (sacrificing fifths).

Intervals and Arithmetic¶

An interval is the distance between two pitches, measured in semitones. Intervals have both a quantity (number of scale steps) and a quality (perfect, major, minor, augmented, diminished).

Common intervals:

Semitones   Name              Sound
─────────   ────              ─────
         Unison            Same note
         Minor 2nd         Tense, dissonant (Jaws theme)
         Major 2nd         A whole step (Do-Re)
         Minor 3rd         Sad, dark (Greensleeves)
         Major 3rd         Happy, bright (Kumbaya)
         Perfect 4th       Open, hollow (Here Comes the Bride)
         Tritone           Unstable, tense (The Simpsons)
         Perfect 5th       Strong, stable (Star Wars)
         Minor 6th         Bittersweet
         Major 6th         Warm (My Bonnie)
        Minor 7th         Bluesy (Star Trek TOS)
        Major 7th         Dreamy, yearning
        Octave            Same note, higher

Tones support + and - operators for semitone math:

>>> c4 = Tone.from_string("C4", system="western")
>>> c4 + 4        # Major third up
<Tone E4>
>>> c4 + 7        # Perfect fifth up
<Tone G4>
>>> c4 + 12       # Octave up
<Tone C5>

Subtracting two tones gives the semitone distance:

>>> g4 = Tone.from_string("G4", system="western")
>>> g4 - c4       # Perfect fifth = 7 semitones
7

>>> c5 = Tone.from_string("C5", system="western")
>>> c5 - c4       # Octave = 12 semitones
12

Comparison and Sorting¶

Tones can be compared and sorted by pitch frequency:

>>> c4 < g4
True
>>> sorted([g4, c4, e4])
[<Tone C4>, <Tone E4>, <Tone G4>]

Equality checks note name and octave:

>>> c4 == "C"      # Compare with string (name only)
True
>>> c4 == Tone(name="C", octave=4)
True

The Overtone Series¶

Every tone you hear is actually a composite of many frequencies. When a string vibrates, it doesn’t just vibrate as a whole — it also vibrates in halves, thirds, quarters, and so on, producing the harmonic series:

>>> a4 = Tone.from_string("A4", system="western")
>>> a4.overtones(8)
[440.0, 880.0, 1320.0, 1760.0, 2200.0, 2640.0, 3080.0, 3520.0]

These harmonics correspond to musical intervals:

Harmonic  Frequency  Interval from fundamental
1st       440 Hz     Unison (A4)
2nd       880 Hz     Octave (A5)
3rd       1320 Hz    Octave + perfect 5th (E6)
4th       1760 Hz    Two octaves (A6)
5th       2200 Hz    Two octaves + major 3rd (C#7)
6th       2640 Hz    Two octaves + perfect 5th (E7)
7th       3080 Hz    Two octaves + minor 7th (≈G7, slightly flat)
8th       3520 Hz    Three octaves (A7)

The overtone series is why a perfect fifth sounds consonant — the 3rd harmonic of the lower note matches the 2nd harmonic of the upper note. It’s also why the major triad (root, major 3rd, perfect 5th) feels “natural” — these intervals appear in the first 6 harmonics.

Different instruments emphasize different harmonics, which is why a violin and a flute playing the same note sound different. This quality is called timbre.

Enharmonic Equivalents¶

In equal temperament, C# and Db are the same pitch (they have the same frequency). They’re called enharmonic equivalents. Which name you use depends on context:

In the key of D major (2 sharps), you write C#
In the key of Gb major (6 flats), you write Db

The rule: each letter name should appear exactly once in a scale. The D major scale is D E F# G A B C# — not D E Gb G A B Db, even though F#=Gb and C#=Db.

PyTheory uses sharps by default (following the tone list ordering), but tones carry their enharmonic equivalents:

>>> Tone.from_tuple(("C#", "Db")).names()
['C#', 'Db']

The Circle of Fifths¶

The circle of fifths is the most important diagram in Western music theory. Starting from any note and ascending by perfect fifths (7 semitones), you pass through all 12 chromatic tones before returning to the starting note:

>>> t = Tone.from_string("C4", system="western")
>>> for i in range(12):
...     print(t.name, end=" ")
...     t = t + 7
C G D A E B F# C# G# D# A# F

Each step clockwise adds one sharp to the key signature; each step counter-clockwise (ascending by fourths = 5 semitones) adds one flat.