31edo/Individual degrees - Revision history

FloraC at 08:07, 1 January 2025

2025-01-01T08:07:28Z

← Older revision		Revision as of 08:07, 1 January 2025
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	A single step of 31-edo is about 38.71¢. Intervals around this size are called ''dieses'' (singular '''diesis'''). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G♯, to B♯, but {{nowrap\|B♯ ≠ C}}), or four minor thirds (C to E♭ to G♭ to B𝄫 to {{nowrap\|D𝄫 ≠ C}}). In the [[11-limit]], the diesis stands in for just ratios {{nowrap\|56:55 (31.19)\|55:54 (31.77¢)\|49:48 (39.70¢)\|45:44 (38.91¢)\|36:35 (48.77¢)\|33:32 (53.27¢)}}, and others. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. Demonstrated in [[SpiralProgressions]].		A single step of 31-edo is about 38.71¢. Intervals around this size are called ''dieses'' (singular '''diesis'''). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G♯, to B♯, but {{nowrap\|B♯ ≠ C}}), or four minor thirds (C to E♭ to G♭ to B𝄫 to {{nowrap\|D𝄫 ≠ C}}). In the [[11-limit]], the diesis stands in for just ratios {{nowrap\|56:55 (31.19)\|55:54 (31.77¢)\|49:48 (39.70¢)\|45:44 (38.91¢)\|36:35 (48.77¢)\|33:32 (53.27¢)}}, and others. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. Demonstrated in [[SpiralProgressions]].

	== 2\31 – minor semitone, chromatic semitone, small minor 2nd or downminor 2nd ==		== 2\31 – minor semitone, chromatic semitone, small minor 2nd, or downminor 2nd ==
	The difference between a major and minor third. The more "expressive" of the "half steps," and the larger of 31's two "microtones". In meantone, it is the ''chromatic semitone'', the interval that distinguishes major and minor intervals of the same generic interval class (e.g. thirds). 2\31 stands in for just ratios {{nowrap\|28:27 (62.96¢)\|25:24 (70.67¢)\|22:21 (80.54¢)\|21:20 (84.45¢)}}, and others. Generates [[Valentine\|valentine temperament]]—aka [[Armodue theory#Semi-equalized Armodue\|semi-equalized Armodue]].		The difference between a major and minor third. The more "expressive" of the "half steps," and the larger of 31's two "microtones". In meantone, it is the ''chromatic semitone'', the interval that distinguishes major and minor intervals of the same generic interval class (e.g. thirds). 2\31 stands in for just ratios {{nowrap\|28:27 (62.96¢)\|25:24 (70.67¢)\|22:21 (80.54¢)\|21:20 (84.45¢)}}, and others. Generates [[Valentine\|valentine temperament]]—aka [[Armodue theory#Semi-equalized Armodue\|semi-equalized Armodue]].

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2025-01-01T08:06:57Z

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ArrowHead294 at 14:41, 31 December 2024

2024-12-31T14:41:18Z

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ArrowHead294 at 15:13, 26 November 2024

2024-11-26T15:13:48Z

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Inthar: /* 5\31 – whole tone or major 2nd */

2023-06-06T19:12:37Z

5\31 – whole tone or major 2nd

← Older revision		Revision as of 19:12, 6 June 2023
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	== 5\31 – whole tone or major 2nd ==		== 5\31 – whole tone or major 2nd ==
	A rather smallish whole tone. Sometimes called melodically dull. As it falls between (and functions as) just whole tones 9:8 and 10:9, 5\31 is considered a "meantone". Two meantones make a near-just major third. Perhaps it is worth noting that its relative narrowness (to JI 9/8) makes it easier to distinguish from the 8/7 approximation. And although it is over 10¢ flat of 9/8, 5\31 can function as a somewhat "active" (as opposed to perfectly stable) harmonic ninth, and it can be effective in combination with the also-narrow 11th harmonic. Indeed, the 11/9 approximation is excellent. Try, for instance 31's version of a 4:6:9:11 chord (steps 0-18-36-45). Generates [[Hemithirds\|hemithirds temperament]] and [[Hemiwürschmidt\|hemiwürschmidt temperament]].		A rather smallish whole tone, 0.99¢ sharp of [[19/17]] (192.56¢). Sometimes called melodically dull. As it falls between (and functions as) just whole tones 9:8 and 10:9, 5\31 is considered a "meantone". Two meantones make a near-just major third. Perhaps it is worth noting that its relative narrowness (to JI 9/8) makes it easier to distinguish from the 8/7 approximation. And although it is over 10¢ flat of 9/8, 5\31 can function as a somewhat "active" (as opposed to perfectly stable) harmonic ninth, and it can be effective in combination with the also-narrow 11th harmonic. Indeed, the 11/9 approximation is excellent. Try, for instance 31's version of a 4:6:9:11 chord (steps 0-18-36-45). Generates [[Hemithirds\|hemithirds temperament]] and [[Hemiwürschmidt\|hemiwürschmidt temperament]].

	MOS Scales generated by 5\31		MOS Scales generated by 5\31

Inthar: /* 14\31 – superfourth or up 4th */

2023-06-06T19:11:00Z

14\31 – superfourth or up 4th

← Older revision		Revision as of 19:11, 6 June 2023
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	== 14\31 – superfourth or up 4th ==		== 14\31 – superfourth or up 4th ==
	Exactly twice a subminor third, and 1.1c flat of [[26/19]] (543.0¢). When viewing 31edo as a 11-limit temperament, the interval functions as both the 11:8 (551.32¢) and 15:11 (536.95¢) undecimal superfourths (121/120 is tempered out). Thus it makes possible a symmetrical tempered version of an 8:11:15 triad. As 11/8, 14\31 is about 9¢ flat; however, it fits nicely with the also-flat 9/8, allowing a near-just 11/9. Nonetheless, most 11-limit chords in 31edo have a somewhat unstable quality which distinguishes them from their just counterparts. Generates [[casablanca]] and [[joan]] temperaments.		Exactly twice a subminor third, and 1.08c flat of [[26/19]] (543.02¢). When viewing 31edo as a 11-limit temperament, the interval functions as both the 11:8 (551.32¢) and 15:11 (536.95¢) undecimal superfourths (121/120 is tempered out). Thus it makes possible a symmetrical tempered version of an 8:11:15 triad. As 11/8, 14\31 is about 9¢ flat; however, it fits nicely with the also-flat 9/8, allowing a near-just 11/9. Nonetheless, most 11-limit chords in 31edo have a somewhat unstable quality which distinguishes them from their just counterparts. Generates [[casablanca]] and [[joan]] temperaments.

	MOS Scales generated by 14\31		MOS Scales generated by 14\31

Inthar: /* 12\31 – narrow 4th, subfourth or down 4th */

2023-06-06T19:09:05Z

12\31 – narrow 4th, subfourth or down 4th

← Older revision		Revision as of 19:09, 6 June 2023
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	== 12\31 – narrow 4th, subfourth or down 4th ==		== 12\31 – narrow 4th, subfourth or down 4th ==
	Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 21:16 (470.78¢). Although 31edo does not offer reasonable approximations of the 17th or 13th harmonics, 12\31 is less than 0.09¢ flat of the [[17-limit]] interval [[17/13]] (464.43¢); combining this with 17\31's 1.1-cent-sharp approximation to [[19/13]] yields a good 13:17:19, which helps make this identity clear. This interval and its inversion 19\31 (735.48¢, a superfifth) are notable for being the only intervals in the 31edo octave larger than the 3\31 diatonic semitone (and smaller than its inversion, 28\31) that are not 11-odd-limit consonances, and the only intervals larger than 2\31 and smaller than 29\31 that are not 15-odd-limit consonances. Generates [[A-Team]] and [[semisept]] temperaments.		Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 21:16 (470.78¢). Although 31edo does not offer reasonable approximations of the 17th or 13th harmonics, 12\31 is less than 0.09¢ flat of the [[17-limit]] interval [[17/13]] (464.43¢); combining this with 17\31's 1.08-cent-sharp approximation to [[19/13]] yields a good 13:17:19, which helps make this identity clear. This interval and its inversion 19\31 (735.48¢, a superfifth) are notable for being the only intervals in the 31edo octave larger than the 3\31 diatonic semitone (and smaller than its inversion, 28\31) that are not 11-odd-limit consonances, and the only intervals larger than 2\31 and smaller than 29\31 that are not 15-odd-limit consonances. Generates [[A-Team]] and [[semisept]] temperaments.

	MOS Scales generated by 12\31		MOS Scales generated by 12\31

Inthar: /* 12\31 – narrow 4th, subfourth or down 4th */

2023-06-06T19:08:30Z

12\31 – narrow 4th, subfourth or down 4th

← Older revision		Revision as of 19:08, 6 June 2023
Line 2,217:		Line 2,217:

	== 12\31 – narrow 4th, subfourth or down 4th ==		== 12\31 – narrow 4th, subfourth or down 4th ==
	Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 21:16 (470.78¢). Although 31edo does not offer reasonable approximations of the 17th or 13th harmonics, 12\31 is less than 0.1¢ flat of [[17-limit]] interval [[17/13]] (464.43¢); combining this with 17\31's 1.1-cent-sharp approximation to [[19/13]] yields a good 13:17:19, which helps make this identity clear. This interval and its inversion 19\31 (735.48¢, a superfifth) are notable for being the only intervals in the 31edo octave larger than the 3\31 diatonic semitone (and smaller than its inversion, 28\31) that are not 11-odd-limit consonances, and the only intervals larger than 2\31 and smaller than 29\31 that are not 15-odd-limit consonances. Generates [[A-Team]] and [[semisept]] temperaments.		Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 21:16 (470.78¢). Although 31edo does not offer reasonable approximations of the 17th or 13th harmonics, 12\31 is less than 0.09¢ flat of the [[17-limit]] interval [[17/13]] (464.43¢); combining this with 17\31's 1.1-cent-sharp approximation to [[19/13]] yields a good 13:17:19, which helps make this identity clear. This interval and its inversion 19\31 (735.48¢, a superfifth) are notable for being the only intervals in the 31edo octave larger than the 3\31 diatonic semitone (and smaller than its inversion, 28\31) that are not 11-odd-limit consonances, and the only intervals larger than 2\31 and smaller than 29\31 that are not 15-odd-limit consonances. Generates [[A-Team]] and [[semisept]] temperaments.

	MOS Scales generated by 12\31		MOS Scales generated by 12\31