User:2^67-1/TempClean sandbox/Pythagorean tuning: Difference between revisions

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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 ==
Line 10: Line 12:
== 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 called [[obikhod]] and 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 obikhod, 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 ==
Line 32: Line 34:
| wa unison
| wa unison
| P1
| P1
| C
| D
|-
|-
| [[2187/2048]]
| [[2187/2048]]
Line 40: Line 42:
| lawa 1sn
| lawa 1sn
| A1
| A1
| C#
| D#
|-
|-
| [[256/243]]
| [[256/243]]
Line 48: Line 50:
| sawa 2nd
| sawa 2nd
| m2
| m2
| Db
| Eb
|-
|-
| [[9/8]]
| [[9/8]]
Line 56: Line 58:
| wa 2nd
| wa 2nd
| M2
| M2
| D
| E
|-
|-
| [[19683/16384]]
| [[19683/16384]]
Line 64: Line 66:
| lawa 2nd
| lawa 2nd
| A2
| A2
| D#
| E#
|-
|-
| [[32/27]]
| [[32/27]]
Line 72: Line 74:
| wa 3rd
| wa 3rd
| m3
| m3
| Eb
| F
|-
|-
| [[81/64]]
| [[81/64]]
Line 80: Line 82:
| lawa 3rd
| lawa 3rd
| M3
| M3
| E
| F#
|-
|-
| [[8192/6561]]
| [[8192/6561]]
Line 88: Line 90:
| sawa 4th
| sawa 4th
| d4
| d4
| Fb
| Gb
|-
|-
| [[4/3]]
| [[4/3]]
Line 96: Line 98:
| wa 4th
| wa 4th
| P4
| P4
| F
| G
|-
|-
| [[729/512]]
| [[729/512]]
Line 104: Line 106:
| lawa 4th
| lawa 4th
| A4
| A4
| F#
| G#
|-
|-
| [[1024/729]]
| [[1024/729]]
Line 112: Line 114:
| sawa 5th
| sawa 5th
| d5
| d5
| Gb
| Ab
|-
|-
| [[3/2]]
| [[3/2]]
Line 120: Line 122:
| wa 5th
| wa 5th
| P5
| P5
| G
| A
|-
|-
| [[6561/4096]]
| [[6561/4096]]
Line 128: Line 130:
| lawa 5th
| lawa 5th
| A5
| A5
| G#
| A#
|-
|-
| [[128/81]]
| [[128/81]]
Line 136: Line 138:
| sawa 6th
| sawa 6th
| m6
| m6
| Ab
| Bb
|-
|-
| [[27/16]]
| [[27/16]]
Line 144: Line 146:
| wa 6th
| wa 6th
| M6
| M6
| A
| B
|-
|-
| [[32768/19683]]
| [[32768/19683]]
Line 152: Line 154:
| sawa 7th
| sawa 7th
| d7
| d7
| Bbb
| Cb
|-
|-
| [[16/9]]
| [[16/9]]
Line 160: Line 162:
| wa 7th
| wa 7th
| m7
| m7
| Bb
| C
|-
|-
| [[243/128]]
| [[243/128]]
Line 168: Line 170:
| lawa 7th
| lawa 7th
| M7
| M7
| B
| C#
|-
|-
| [[4096/2187]]
| [[4096/2187]]
Line 176: Line 178:
| sawa 8ve
| sawa 8ve
| d8
| d8
| Cb
| Db
|-
|-
| [[2/1]]
| [[2/1]]
Line 184: Line 186:
| wa 8ve
| wa 8ve
| P8
| P8
| C
| D
|}
|}

Latest revision as of 13:24, 23 February 2025

English Wikipedia has an article on:

The 3-limit consists of intervals 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]\displaystyle{ 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

EDOs which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the continued fraction for the logarithm of 3 base 2. These are 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 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 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.

Ratio Monzo Size (¢) Color Name Diatonic Category
1/1 [0⟩ 0.000 w1 wa unison P1 D
2187/2048 [-11 7⟩ 113.685 Lw1 lawa 1sn A1 D#
256/243 [8 -5⟩ 90.225 sw2 sawa 2nd m2 Eb
9/8 [-3 2⟩ 203.910 w2 wa 2nd M2 E
19683/16384 [-14 9⟩ 317.595 Lw2 lawa 2nd A2 E#
32/27 [5 -3⟩ 294.135 w3 wa 3rd m3 F
81/64 [-6 4⟩ 407.820 Lw3 lawa 3rd M3 F#
8192/6561 [13 -8⟩ 384.360 sw4 sawa 4th d4 Gb
4/3 [2 -1⟩ 498.045 w4 wa 4th P4 G
729/512 [-9 6⟩ 611.730 Lw4 lawa 4th A4 G#
1024/729 [10 -6⟩ 588.270 sw5 sawa 5th d5 Ab
3/2 [-1 1⟩ 701.955 w5 wa 5th P5 A
6561/4096 [-12 8⟩ 815.640 Lw5 lawa 5th A5 A#
128/81 [7 -4⟩ 792.180 sw6 sawa 6th m6 Bb
27/16 [-4 3⟩ 905.865 w6 wa 6th M6 B
32768/19683 [15 -9⟩ 882.405 sw7 sawa 7th d7 Cb
16/9 [4 -2⟩ 996.090 w7 wa 7th m7 C
243/128 [-7 5⟩ 1109.775 Lw7 lawa 7th M7 C#
4096/2187 [12 -7⟩ 1086.315 sw8 sawa 8ve d8 Db
2/1 [1⟩ 1200.000 w8 wa 8ve P8 D
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