Pythagorean tuning - Revision history

Vector at 05:20, 28 March 2026

2026-03-28T05:20:05Z

← Older revision		Revision as of 05:20, 28 March 2026
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	==== Schismic and Garibaldi ====		==== Schismic and Garibaldi ====
			''Main article: [[Schismic]]''

	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. There is also no reason, if you are using Schismic, to not further equate 19/16 to 32/27, at only 3 cents of error.		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. There is also no reason, if you are using Schismic, to not further equate 19/16 to 32/27, at only 3 cents of error.

Inthar: /* Schismic and Garibaldi */

2026-03-28T04:14:28Z

Schismic and Garibaldi

← Older revision		Revision as of 04:14, 28 March 2026
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	==== Schismic and Garibaldi ====		==== 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. There is also no reason, if you are using Schismic, to not further equate 19/16 to 32/27, at only 3 cents of error.		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. There is also no reason, if you are using Schismic, to not further equate 19/16 to 32/27, at only 3 cents of error.

			Garibaldi in particular has the advantage of equating both 81/80 and 64/63 to the same interval, in particular the Pythagorean comma. Thus Garibaldi supports Hemifamitt temperament.

	== Monocot ==		== Monocot ==

Vector at 05:39, 27 March 2026

2026-03-27T05:39:44Z

← Older revision		Revision as of 05:39, 27 March 2026
Line 5:		Line 5:

	==== Schismic and Garibaldi ====		==== 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. There is also no reason, if you are using Schismic, to not further equate 19/16 to 32/27.		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. There is also no reason, if you are using Schismic, to not further equate 19/16 to 32/27, at only 3 cents of error.

	== Monocot ==		== Monocot ==

Vector at 05:39, 27 March 2026

2026-03-27T05:39:07Z

← Older revision		Revision as of 05:39, 27 March 2026
Line 5:		Line 5:

	==== Schismic and Garibaldi ====		==== 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.		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. There is also no reason, if you are using Schismic, to not further equate 19/16 to 32/27.

	== Monocot ==		== Monocot ==

Inthar: /* Minor second */

2026-03-16T19:22:54Z

Minor second

← Older revision		Revision as of 19:22, 16 March 2026
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	== Intervals of Pythagorean diatonic ==		== Intervals of Pythagorean diatonic ==
	=== Minor second ===		=== Minor second ===
	{{Infobox interval\|256/243~~\|names=Chromatic semitone, augmented unison\|name=Chromatic semitone, augmented unison~~\|Name=~~Diatonic~~ semitone, diatonic minor second}}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\|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.		{{Infobox interval\|256/243\|Name=diatonic semitone, diatonic minor second, limma}}The 3-limit '''diatonic semitone''', also called the '''diatonic minor second (m2)''' or the '''limma''', 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\|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 [[12edo#Compton temperament\|Pythagorean comma]], which separates all [[enharmonic]] intervals in Pythagorean tuning, and which, if tempered out, yields 12edo.		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 [[12edo#Compton temperament\|Pythagorean comma]], which separates all [[enharmonic]] intervals in Pythagorean tuning, and which, if tempered out, yields 12edo.

Inthar at 05:16, 12 March 2026

2026-03-12T05:16:19Z

← Older revision		Revision as of 05:16, 12 March 2026
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	{{Interval regions}}		{{Interval regions}}
			{{Cat\|JI groups}}

Vector at 10:24, 6 March 2026

2026-03-06T10:24:32Z

← Older revision		Revision as of 10:24, 6 March 2026
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	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.		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.

			{{Interval regions}}

Inthar: /* Monocot */

2026-02-26T10:52:12Z

Monocot

← Older revision		Revision as of 10:52, 26 February 2026
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	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 [[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.		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.		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.

Inthar: /* Minor second */

2026-02-26T10:50:47Z

Minor second

← Older revision		Revision as of 10:50, 26 February 2026
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	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 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 [[Interseptimal diesis\|S7]] * [[Archy\|S8]]<sup>2</sup>. When tempered out, it leads to ~~blackwood~~ temperament, which tunes the [[2.3.7 subgroup]] to [[Equipentatonic#5edo\|5edo]].		The diatonic semitone is a product of square superparticulars, denoted [[Interseptimal diesis\|S7]] * [[Archy\|S8]]<sup>2</sup>. When tempered out, it leads to Blackwood temperament, which tunes the [[2.3.7 subgroup]] to [[Equipentatonic#5edo\|5edo]].

	=== Major second ===		=== Major second ===
	{{Infobox interval\|9/8\|names=Chromatic semitone, augmented unison\|name=Chromatic semitone, augmented unison\|Name=Diatonic major second}}		{{Infobox interval\|9/8\|names=Chromatic semitone, augmented unison\|name=Chromatic semitone, augmented unison\|Name=Diatonic major second}}

Inthar: /* Schismic and Garibaldi */

2026-02-26T10:49:52Z

Schismic and Garibaldi

← Older revision		Revision as of 10:49, 26 February 2026
Line 5:		Line 5:

	==== Schismic and Garibaldi ====		==== 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.		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 ==