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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.
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 approximation ==
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== Approaches ==
== Approaches ==


There are two possible approaches to Pythagorean tuning, namely [[octave]]-based and [[tritave]]-based.
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.
[[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, 11, 19, 25, 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.
[[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 ==
== Table of intervals ==
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| wa unison
| wa unison
| P1
| P1
| C
| D
|-
|-
| [[2187/2048]]
| [[2187/2048]]
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| lawa 1sn
| lawa 1sn
| A1
| A1
| C#
| D#
|-
|-
| [[256/243]]
| [[256/243]]
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| sawa 2nd
| sawa 2nd
| m2
| m2
| Db
| Eb
|-
|-
| [[9/8]]
| [[9/8]]
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| wa 2nd
| wa 2nd
| M2
| M2
| D
| E
|-
|-
| [[19683/16384]]
| [[19683/16384]]
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| lawa 2nd
| lawa 2nd
| A2
| A2
| D#
| E#
|-
|-
| [[32/27]]
| [[32/27]]
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| wa 3rd
| wa 3rd
| m3
| m3
| Eb
| F
|-
|-
| [[81/64]]
| [[81/64]]
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| lawa 3rd
| lawa 3rd
| M3
| M3
| E
| F#
|-
|-
| [[8192/6561]]
| [[8192/6561]]
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| sawa 4th
| sawa 4th
| d4
| d4
| Fb
| Gb
|-
|-
| [[4/3]]
| [[4/3]]
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| wa 4th
| wa 4th
| P4
| P4
| F
| G
|-
|-
| [[729/512]]
| [[729/512]]
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| lawa 4th
| lawa 4th
| A4
| A4
| F#
| G#
|-
|-
| [[1024/729]]
| [[1024/729]]
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| sawa 5th
| sawa 5th
| d5
| d5
| Gb
| Ab
|-
|-
| [[3/2]]
| [[3/2]]
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| wa 5th
| wa 5th
| P5
| P5
| G
| A
|-
|-
| [[6561/4096]]
| [[6561/4096]]
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| lawa 5th
| lawa 5th
| A5
| A5
| G#
| A#
|-
|-
| [[128/81]]
| [[128/81]]
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| sawa 6th
| sawa 6th
| m6
| m6
| Ab
| Bb
|-
|-
| [[27/16]]
| [[27/16]]
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| wa 6th
| wa 6th
| M6
| M6
| A
| B
|-
|-
| [[32768/19683]]
| [[32768/19683]]
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| sawa 7th
| sawa 7th
| d7
| d7
| Bbb
| Cb
|-
|-
| [[16/9]]
| [[16/9]]
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| wa 7th
| wa 7th
| m7
| m7
| Bb
| C
|-
|-
| [[243/128]]
| [[243/128]]
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| lawa 7th
| lawa 7th
| M7
| M7
| B
| C#
|-
|-
| [[4096/2187]]
| [[4096/2187]]
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| sawa 8ve
| sawa 8ve
| d8
| d8
| Cb
| Db
|-
|-
| [[2/1]]
| [[2/1]]
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| wa 8ve
| wa 8ve
| P8
| P8
| C
| D
|}
|}
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