These three intervals and their octave equivalents, such as the perfect eleventh and twelfth, are the only absolute consonances of the Pythagorean system.
All other intervals have varying degrees of dissonance, ranging from smooth to rough.
The difference between the tone and the limma is the major semitone or apotome ("part cut off") of 2187/2048.
There is a one-to-one correspondence between interval names (number of scale steps + quality) and frequency ratios.
This contrasts with equal temperament, in which intervals with the same frequency ratio can have different names (e.g., the diminished fifth and the augmented fourth); and with other forms of just intonation, in which intervals with the same name can have different frequency ratios (e.g., 9/8 for the major second from C to D, but 10/9 for the major second from D to E).
Pythagorean perfect fifth on C
Play
ⓘ
: C-G (3/2 ÷ 1/1 = 3/2).
Frequency ratio of the 144 intervals in D-based Pythagorean tuning.
Interval names
are given in their shortened form.
Pure intervals
are shown in
bold
font.
Wolf intervals
are highlighted in red. Numbers larger than 999 are shown as powers of 2 or 3. Other versions of this table are provided
here
and
here
.
Pythagorean perfect fifth on D
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: D-A+ (27/16 ÷ 9/8 = 3/2).
Just perfect fourth
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, one perfect fifth inverted (4/3 ÷ 1/1 = 4/3).
Major tone on C
Play
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: C-D (9/8 ÷ 3/2 = 3/2), two Pythagorean perfect fifths.
Pythagorean small minor seventh (1/1 - 16/9)
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, two perfect fifths inverted.
Pythagorean major sixth on C (1/1 - 27/16)
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, three Pythagorean perfect fifths.
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