User:2^67-1/TempClean sandbox/Pythagorean tuning - Revision history

2^67-1: /* Table of intervals */

2025-02-23T13:24:38Z

Table of intervals

← Older revision		Revision as of 13:24, 23 February 2025
Line 34:		Line 34:
	\| wa unison		\| wa unison
	\| P1		\| P1
	\| C		\| D
	\|-		\|-
	\| [[2187/2048]]		\| [[2187/2048]]
Line 42:		Line 42:
	\| lawa 1sn		\| lawa 1sn
	\| A1		\| A1
	\| C#		\| D#
	\|-		\|-
	\| [[256/243]]		\| [[256/243]]
Line 50:		Line 50:
	\| sawa 2nd		\| sawa 2nd
	\| m2		\| m2
	\| Db		\| Eb
	\|-		\|-
	\| [[9/8]]		\| [[9/8]]
Line 58:		Line 58:
	\| wa 2nd		\| wa 2nd
	\| M2		\| M2
	\| D		\| E
	\|-		\|-
	\| [[19683/16384]]		\| [[19683/16384]]
Line 66:		Line 66:
	\| lawa 2nd		\| lawa 2nd
	\| A2		\| A2
	\| D#		\| E#
	\|-		\|-
	\| [[32/27]]		\| [[32/27]]
Line 74:		Line 74:
	\| wa 3rd		\| wa 3rd
	\| m3		\| m3
	\| Eb		\| F
	\|-		\|-
	\| [[81/64]]		\| [[81/64]]
Line 82:		Line 82:
	\| lawa 3rd		\| lawa 3rd
	\| M3		\| M3
	\| E		\| F#
	\|-		\|-
	\| [[8192/6561]]		\| [[8192/6561]]
Line 90:		Line 90:
	\| sawa 4th		\| sawa 4th
	\| d4		\| d4
	\| Fb		\| Gb
	\|-		\|-
	\| [[4/3]]		\| [[4/3]]
Line 98:		Line 98:
	\| wa 4th		\| wa 4th
	\| P4		\| P4
	\| F		\| G
	\|-		\|-
	\| [[729/512]]		\| [[729/512]]
Line 106:		Line 106:
	\| lawa 4th		\| lawa 4th
	\| A4		\| A4
	\| F#		\| G#
	\|-		\|-
	\| [[1024/729]]		\| [[1024/729]]
Line 114:		Line 114:
	\| sawa 5th		\| sawa 5th
	\| d5		\| d5
	\| Gb		\| Ab
	\|-		\|-
	\| [[3/2]]		\| [[3/2]]
Line 122:		Line 122:
	\| wa 5th		\| wa 5th
	\| P5		\| P5
	\| G		\| A
	\|-		\|-
	\| [[6561/4096]]		\| [[6561/4096]]
Line 130:		Line 130:
	\| lawa 5th		\| lawa 5th
	\| A5		\| A5
	\| G#		\| A#
	\|-		\|-
	\| [[128/81]]		\| [[128/81]]
Line 138:		Line 138:
	\| sawa 6th		\| sawa 6th
	\| m6		\| m6
	\| Ab		\| Bb
	\|-		\|-
	\| [[27/16]]		\| [[27/16]]
Line 146:		Line 146:
	\| wa 6th		\| wa 6th
	\| M6		\| M6
	\| A		\| B
	\|-		\|-
	\| [[32768/19683]]		\| [[32768/19683]]
Line 154:		Line 154:
	\| sawa 7th		\| sawa 7th
	\| d7		\| d7
	\| ~~Bbb~~		\| Cb
	\|-		\|-
	\| [[16/9]]		\| [[16/9]]
Line 162:		Line 162:
	\| wa 7th		\| wa 7th
	\| m7		\| m7
	\| Bb		\| C
	\|-		\|-
	\| [[243/128]]		\| [[243/128]]
Line 170:		Line 170:
	\| lawa 7th		\| lawa 7th
	\| M7		\| M7
	\| B		\| C#
	\|-		\|-
	\| [[4096/2187]]		\| [[4096/2187]]
Line 178:		Line 178:
	\| sawa 8ve		\| sawa 8ve
	\| d8		\| d8
	\| Cb		\| Db
	\|-		\|-
	\| [[2/1]]		\| [[2/1]]
Line 186:		Line 186:
	\| wa 8ve		\| wa 8ve
	\| P8		\| P8
	\| C		\| D
	\|}		\|}

2^67-1 at 13:21, 23 February 2025

2025-02-23T13:21:21Z

← Older revision		Revision as of 13:21, 23 February 2025
Line 2:		Line 2:

	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 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 ==

2^67-1: Removed ambiguity

2024-08-26T04:10:23Z

Removed ambiguity

← Older revision		Revision as of 04:10, 26 August 2024
Line 14:		Line 14:
	[[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, 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.

	== Table of intervals ==		== Table of intervals ==

2^67-1: Created page with "{{Wikipedia|Pythagorean tuning}} The '''3-limit''' consists of intervals that are either an integer whose only prime factors are 2 and 3, the reciprocal of such an intege..."

2024-08-26T01:34:49Z

Created page with "{{Wikipedia|Pythagorean tuning}} The '''3-limit''' consists of intervals that are either an integer whose only prime factors are 2 and 3, the reciprocal of such an intege..."

New page

{{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.

== 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 two possible approaches to Pythagorean tuning, namely [[octave]]-based and [[tritave]]-based.

[[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.

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