Editing Pythagorean tuning (section)

== Sizes of intervals ==
[[File:144 intervals in Pythagorean tuning.svg|500px|right|thumb|The 144 intervals in C-based Pythagorean tuning.]]

The tables above only show the frequency ratios of each note with respect to the base note. However, intervals can start from any note and so twelve intervals can be defined for each '''interval type''' – twelve unisons, twelve [[semitone]]s, twelve 2-semitone intervals, 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 interval type except unisons and octaves has two different sizes. The table on the right shows their frequency ratios, with deviations of a [[Pythagorean comma]] coloured.<ref name="Wolf">Wolf intervals are operationally defined herein as intervals composed of 3, 4, 5, 7, 8, or 9 semitones (i.e. major and minor thirds or sixths, perfect fourths or fifths, and their [[enharmonic equivalent]]s) the size of which deviates by more than one [[syntonic comma]] (about 21.5 cents) from the corresponding justly intonated interval. Intervals made up of 1, 2, 6, 10, or 11 semitones (e.g. major and minor seconds or sevenths, tritones, and their [[enharmonic]] equivalents) are considered to be [[consonance and dissonance|dissonant]] even when they are justly tuned, thus they are not marked as wolf intervals even when they deviate from just intonation by more than one syntonic comma.</ref> The deviations arise because the notes determine two different [[semitone]]s:
* The minor second ('''m2'''), also called diatonic semitone, with size <math display="block"> S_1 = {256 \over 243} \approx 90.225 \text{ cents} </math> (e.g. between D and E{{Music|b}})
* The augmented unison ('''A1'''), also called chromatic semitone, with size <math display="block"> S_2 =  {3^7 \over 2^{11}} = {2187 \over 2048} \approx 113.685 \text{ cents} </math> (e.g. between E{{Music|b}} and E)

By contrast, in an [[equal temperament|equally tempered]] chromatic scale, all semitones measure
:<math>S_E=\sqrt[12]2=100.000\text{ cents}</math>
and intervals of any given type have the same size, but none are justly tuned except unisons and octaves.

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''&nbsp;≈&nbsp;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&nbsp;&minus;&nbsp;11''ε'' cents, which is about 678.495 cents (the wolf fifth). 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 third]]s ('''m3''') are ≈&nbsp;294.135 cents (300&nbsp;&minus;&nbsp;3''ε''), 3 [[augmented second]]s ('''A2''') are ≈&nbsp;317.595 cents (300&nbsp;+&nbsp;9''ε''), and their average is 300 cents;
* 8 [[major third]]s ('''M3''') are ≈&nbsp;407.820 cents (400&nbsp;+&nbsp;4''ε''), 4 [[diminished fourth]]s ('''d4''') are ≈&nbsp;384.360 cents (400&nbsp;&minus;&nbsp;8''ε''), and their average is 400 cents;
* 7 diatonic [[semitone]]s ('''m2''') are ≈&nbsp;90.225 cents (100&nbsp;&minus;&nbsp;5''ε''), 5 chromatic semitones ('''A1''') are ≈&nbsp;113.685 cents (100&nbsp;+&nbsp;7''ε''), 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&nbsp;''ε'', the difference between the Pythagorean fifth and the average fifth.

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