Pythagorean tuning

Pythagorean tuning is the tuning system in which only 3-limit just intonation intervals are used - that is, intervals generated by stacking perfect fifths of 3/2 and octaves of 2/1 up and down. Pythagorean tuning is a rank-2 system that does not include any tempering, and is thus useful as a basis for notation. When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the chain of fifths. Simple Pythagorean intervals include 3/2, 9/8, 32/27, 81/64, and their octave complements.

Pythagorean tuning and temperaments

Note that Pythagorean tuning often refers to the tuning, not the interpretation, and this is its distinction from the 3-limit - that is, some people consider regular temperaments that are well-tuned in Pythagorean tuning to, themselves, count as Pythagorean.

Schismic and Garibaldi

The most notable example of this is Schismic temperament, which equates the moderately complex Pythagorean interval 8192/6561, the diatonic diminished fourth, to 5/4, which when tuned to just Pythagorean tuning has only 2 cents of error, and its extension Garibaldi, which further equates the double-diminished octave to 7/4, with only 4 cents of error when the former is tuned just.

Monocot

Monocot is the temperament archetype where an octave is the period and a perfect fifth is the generator. Monocot is equivalent to the standard chain of fifths, going ... B♭ - F - C - G - D - A - E - B - F♯ ... , and is strongly associated with the diatonic scale as the MOS form of diatonic is generated by a perfect fifth and octave. Common monocot temperaments include the aforementioned Schismic, as well as Meantone and Archy.

Monocot is the only ploidacot to have an agreed-upon, fully unambiguous scheme for interval and note names.

Generally, "monocot" is broader than "Pythagorean", as Pythagorean implies that the fifth is tuned to a perfect 3/2, while monocot temperaments tune the fifth to a wide range of tunings.

Intervals of Pythagorean diatonic

Minor second

Interval information

Ratio	256/243
Factorization	28 × 3-5
Monzo	[8 -5⟩
Size in cents	90.225¢
Names	Diatonic semitone, diatonic minor second
Special properties	reduced, reduced subharmonic
Harmonic distance (log2 nd)	15.9248
Wilson norm (sopfr(nd))	31
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The 3-limit diatonic semitone, also called the diatonic minor second (m2) and represented by the ratio 256/243, is the smaller of the two seconds (1-step intervals) in the MOS diatonic scale. It is generated by stacking 5 fourths and octave-reducing. In Pythagorean tuning, and thus purely-tuned just intonation, it is approximately 90.2 cents in size, but as an interval in the abstract diatonic scale it may range between 0 and 171 cents, depending on the tuning.

It functions as the small step of the diatonic MOS, and along with the diatonic major second may be used to construct other diatonic intervals. For example, the diatonic minor third is a major second stacked with a minor second. The chromatic semitone is the difference between these two intervals. Note that the chromatic semitone itself is distinct from the diatonic semitone; they are separated by the Pythagorean comma, which separates all enharmonic intervals in Pythagorean tuning, and which, if tempered out, yields 12edo.

As a harmonic interval, the diatonic semitone is usually considered a dissonance, due to its small size and complex ratio.

The diatonic scale contains two minor seconds. In the Ionian mode, minor seconds are found on the third and seventh scale degrees; the others have major seconds. The small number of minor seconds compared to major seconds ensures that thirds that include minor seconds (that is, minor thirds) are roughly evenly distributed with major thirds; in a scale with three small steps and four large steps, for example, six out of the seven thirds are minor.

The diatonic semitone is a product of square superparticulars, denoted S7 * S8². When tempered out, it leads to Blackwood temperament, which tunes the 2.3.7 subgroup to 5edo.

Major second

Interval information

Ratio	9/8
Factorization	2-3 × 32
Monzo	[-3 2⟩
Size in cents	203.91¢
Name	Diatonic major second
Special properties	square superparticular, reduced, reduced harmonic
Harmonic distance (log2 nd)	6.16993
Wilson norm (sopfr(nd))	12
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The diatonic major second (M2), represented by the frequency ratio 9/8, is the larger of the two seconds (1-step intervals) in the MOS form of the diatonic scale. It is generated by stacking 2 fifths and octave-reducing. In Pythagorean tuning (and thus purely-tuned just intonation), it is approximately 203.9 cents in size, but as an interval in the abstract diatonic scale it may range from 171 to 240 cents, depending on the tuning.

It functions as the large step of diatonic, and along with the diatonic semitone (diatonic minor second) may be used to construct other diatonic intervals. For example, the diatonic major third is two major seconds stacked, and the diatonic minor third is a major second stacked with a minor second. The chromatic semitone is the difference between these two intervals.

As a harmonic interval, the diatonic major second is considered a dissonance in most contexts, due to its small size, but can in some contexts (such as arto and tendo theory) be considered a consonance or ambisonance. In 5edo, it is a consonant 8/7 interval much like the chromatic semitone.

The diatonic scale contains five major seconds. In the Ionian mode, major seconds are found on the 1st, 2nd, 4th, 5th, and 6th degrees of the scale; the other two degrees have minor seconds. The large number of major seconds compared to minor seconds ensures that thirds that include minor seconds (that is, minor thirds) are roughly evenly distributed with major thirds; in a scale with three small steps and four large steps, for example, six out of the seven thirds are minor.

Major third

Interval information

Ratio	81/64
Factorization	2-6 × 34
Monzo	[-6 4⟩
Size in cents	407.82¢
Name	Diatonic major third
Special properties	reduced, reduced harmonic
Harmonic distance (log2 nd)	12.3399
Wilson norm (sopfr(nd))	24
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The diatonic major third (M3), represented by the frequency ratio 81/64, is the larger of the two thirds (2-step intervals) in the MOS form of the diatonic scale. It is generated by stacking 4 fifths octave-reduced. In Pythagorean tuning (and thus purely-tuned just intonation), it is approximately 407.8 cents in size, but as an interval in the abstract diatonic scale it may range from 343 to 480 cents, depending on the tuning.

It can be constructed by stacking two diatonic major seconds, and as such may be called the ditone.

As a harmonic interval, the diatonic major third may be considered either a consonance or a dissonance depending on its tuning. Important tuning targets for the diatonic major third are 5/4 (Meantone temperament), 14/11 (Pentacircle temperament), 9/7 (Archytas temperament) and 13/10 (Oceanfront temperament).

The diatonic scale contains three major thirds. In the Ionian mode, major thirds are found on the first, fourth, and fifth degrees of the scale; the other four degrees have minor thirds. This roughly equal distribution leads to diatonic tonality being largely based on the distinction between major and minor thirds and triads.