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{{Wikipedia|Pythagorean tuning}}

The '''3-limit''' consists of [[interval]]s that are either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. All 3-limit intervals can be written as <math>2^a \cdot 3^b</math>, where ''a'' and ''b'' can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are [[3/2]], [[4/3]], [[9/8]]. Confining intervals to the 3-limit is known as '''Pythagorean tuning''', and the Pythagorean tuning used in Europe during the Middle Ages is the seed out of which grew the common-practice tradition of Western music, as well as genres derived from it. A 3-limit interval is also known as a Pythagorean interval.

Pythagorean tuning forms the basis of most systems of diatonic interval categories.

== EDO approximation ==
[[EDO]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the [[wikipedia: Continued fraction|continued fraction]] for the logarithm of 3 base 2. These are {{EDOs| 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306 }}, …

Another approach is to find edos which have more accurate 3 than all smaller edos. This results in {{EDOs|1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867 }}, …

== Approaches ==

There are many possible approaches to Pythagorean tuning, and each approach is associated with a different Pythagorean equave. The two most widely-used are [[octave]]-based and [[tritave]]-based Pythagorean.

[[Octave]]-based Pythagorean tuning is essentially how it is used in the common-practice. This gives MOS sizes of 2, 3, 5 ([[2L 3s]] pentic), 7 ([[5L 2s]] diatonic), 12 ([[5L 7s]] chromatic), 17 ([[12L 5s]] enharmonic), 29, 41, and 53.

[[Tritave]]-based Pythagorean tuning is an approach described in [https://arxiv.org/abs/1709.00375 this paper] by M. Schmidmeier. This gives MOS sizes of 2, 3, 5, 8 (3L 5s), 11 (8L 3s), 19 (8L 11s), 27 (19L 8s), 46, and 65. The 11-note scale can be regarded as the diatonic-like scale of tritave-equivalent Pythagorean, and the 19-note scale can be regarded as its respective chromatic-like scale.

== Table of intervals ==
This is a table of 3-limit (or Pythagorean) intervals which can be obtained by stacking nine (or less) fifths or fourths from the tonic and octave-reducing them.
{| class="wikitable center-1 right-3 center-6 center-7"
|-
! [[Ratio]]
! [[Monzo]]
! Size ([[Cent|¢]])
! colspan="2" | [[Kite's color notation|Color Name]]
! colspan="2" | Diatonic Category
|-
| [[1/1]]
| {{Monzo| 0 }}
| 0.000
| w1
| wa unison
| P1
| D
|-
| [[2187/2048]]
| {{Monzo| -11 7 }}
| 113.685
| Lw1
| lawa 1sn
| A1
| D#
|-
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.225
| sw2
| sawa 2nd
| m2
| Eb
|-
| [[9/8]]
| {{Monzo| -3 2 }}
| 203.910
| w2
| wa 2nd
| M2
| E
|-
| [[19683/16384]]
| {{Monzo| -14 9 }}
| 317.595
| Lw2
| lawa 2nd
| A2
| E#
|-
| [[32/27]]
| {{Monzo| 5 -3 }}
| 294.135
| w3
| wa 3rd
| m3
| F
|-
| [[81/64]]
| {{Monzo| -6 4 }}
| 407.820
| Lw3
| lawa 3rd
| M3
| F#
|-
| [[8192/6561]]
| {{Monzo| 13 -8 }}
| 384.360
| sw4
| sawa 4th
| d4
| Gb
|-
| [[4/3]]
| {{Monzo| 2 -1 }}
| 498.045
| w4
| wa 4th
| P4
| G
|-
| [[729/512]]
| {{Monzo| -9 6 }}
| 611.730
| Lw4
| lawa 4th
| A4
| G#
|-
| [[1024/729]]
| {{Monzo| 10 -6 }}
| 588.270
| sw5
| sawa 5th
| d5
| Ab
|-
| [[3/2]]
| {{Monzo| -1 1 }}
| 701.955
| w5
| wa 5th
| P5
| A
|-
| [[6561/4096]]
| {{Monzo| -12 8 }}
| 815.640
| Lw5
| lawa 5th
| A5
| A#
|-
| [[128/81]]
| {{Monzo| 7 -4 }}
| 792.180
| sw6
| sawa 6th
| m6
| Bb
|-
| [[27/16]]
| {{Monzo| -4 3 }}
| 905.865
| w6
| wa 6th
| M6
| B
|-
| [[32768/19683]]
| {{Monzo| 15 -9 }}
| 882.405
| sw7
| sawa 7th
| d7
| Cb
|-
| [[16/9]]
| {{Monzo| 4 -2 }}
| 996.090
| w7
| wa 7th
| m7
| C
|-
| [[243/128]]
| {{Monzo| -7 5 }}
| 1109.775
| Lw7
| lawa 7th
| M7
| C#
|-
| [[4096/2187]]
| {{Monzo| 12 -7 }}
| 1086.315
| sw8
| sawa 8ve
| d8
| Db
|-
| [[2/1]]
| {{Monzo| 1 }}
| 1200.000
| w8
| wa 8ve
| P8
| D
|}