Pythagorean Tuning - Size of Intervals

Size of Intervals

The table above shows only intervals from D. However, intervals can be formed by starting from each of the above listed 12 notes. Thus, twelve intervals can be defined for each interval type (twelve unisons, twelve semitones, twelve intervals composed of 2 semitones, twelve intervals composed of 3 semitones, etc.).

As explained above, one of the twelve fifths (the wolf fifth) has a different size with respect to the other eleven. For a similar reason, each of the other interval types, except for the unisons and the octaves, has two different sizes in Pythagorean tuning. This is the price paid for seeking just intonation. The tables on the right and below show their frequency ratios and their approximate sizes in cents. Interval names are given in their standard shortened form. For instance, the size of the interval from D to A, which is a perfect fifth (P5), can be found in the seventh column of the row labeled D. Strictly just (or pure) intervals are shown in bold font. Wolf intervals are highlighted in red.

The reason why the interval sizes vary throughout the scale is that the pitches forming the scale are unevenly spaced. Namely, the frequencies defined by construction for the twelve notes determine two different semitones (i.e. intervals between adjacent notes):

1. The minor second (m2), also called diatonic semitone, with size
   
   (e.g. between D and E♭)
2. The augmented unison (A1), also called chromatic semitone, with size
   
   (e.g. between E♭ and E)

Conversely, in an equally tempered chromatic scale, by definition the twelve pitches are equally spaced, all semitones having a size of exactly

As a consequence all intervals of any given type have the same size (e.g., all major thirds have the same size, all fifths have the same size, etc.). The price paid, in this case, is that none of them is justly tuned and perfectly consonant, except, of course, for the unison and the octave.

For a comparison with other tuning systems, see also this table.

By definition, in Pythagorean tuning 11 perfect fifths (P5 in the table) have a size of approximately 701.955 cents (700+ε cents, where ε ≈ 1.955 cents). Since the average size of the 12 fifths must equal exactly 700 cents (as in equal temperament), the other one must have a size of 700−11ε cents, which is about 678.495 cents (the wolf fifth). Notice that, as shown in the table, the latter interval, although enharmonically equivalent to a fifth, is more properly called a diminished sixth (d6). Similarly,

• 9 minor thirds (m3) are ≈ 294.135 cents (300−3ε), 3 augmented seconds (A2) are ≈ 317.595 cents (300+9ε), and their average is 300 cents;
• 8 major thirds (M3) are ≈ 407.820 cents (400+4ε), 4 diminished fourths (d4) are ≈ 384.360 cents (400−8ε), and their average is 400 cents;
• 7 diatonic semitones (m2) are ≈ 90.225 cents (100−5ε), 5 chromatic semitones (A1) are ≈ 113.685 cents (100+7ε), and their average is 100 cents.

In short, similar differences in width are observed for all interval types, except for unisons and octaves, and they are all multiples of ε, the difference between the Pythagorean fifth and the average fifth.

Notice that, as an obvious consequence, each augmented or diminished interval is exactly 12ε (≈ 23.460) cents narrower or wider than its enharmonic equivalent. For instance, the d6 (or wolf fifth) is 12ε cents narrower than each P5, and each A2 is 12ε cents wider than each m3. This interval of size 12ε is known as a Pythagorean comma, exactly equal to the opposite of a diminished second (≈ −23.460 cents). This implies that ε can be also defined as one twelfth of a Pythagorean comma.