31edo: Difference between revisions
→As a tuning of other temperaments: expand section |
m Text replacement - "strict zeta edo" to "strict zeta edo" |
||
| (6 intermediate revisions by 3 users not shown) | |||
| Line 17: | Line 17: | ||
Prime 11 is somewhat less accurate, making intervals like [[11/8]] off by about 9 cents. However, intervals like [[11/9]] and [[11/6]] are approximated quite well because the errors cancel out. This makes 31edo a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s. | Prime 11 is somewhat less accurate, making intervals like [[11/8]] off by about 9 cents. However, intervals like [[11/9]] and [[11/6]] are approximated quite well because the errors cancel out. This makes 31edo a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s. | ||
Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the [[7-odd-limit|7-]], [[9-odd-limit|9-]], and [[11-odd-limit]], which it is consistent through. It is also a [[ | Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the [[7-odd-limit|7-]], [[9-odd-limit|9-]], and [[11-odd-limit]], which it is consistent through. It is also a [[strict zeta edo]], meaning that it is a zeta peak, zeta peak integer, zeta integral, and zeta gap edo all at once. | ||
One step of 31edo, measuring about 38.7{{c}}, is called a [[diesis]] because it stands in for several intervals called ''dieses'' (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. 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. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently. | One step of 31edo, measuring about 38.7{{c}}, is called a [[diesis]] because it stands in for several intervals called ''dieses'' (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. 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. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently. | ||
In terms of interval categories, because 31edo is a meantone system, the major and minor seconds, thirds, sixth, and sevenths on the chain of fifths are equated to [[5-limit]] intervals, those being [[16/15]], [[10/9]], [[6/5]], [[5/4]], and their [[octave complement]]s. 31edo maps the chromatic semitone to two steps, meaning there are "[[neutral (interval quality)|neutral]]" intervals between minor and major ones, which are not found in [[12edo]]. They can be represented by [[11-limit]] intervals, with [[11/10]]~[[12/11]] being a neutral second, and [[11/9]]~[[27/22]] a neutral third. One step in the other direction from the classical intervals are the subminor and supermajor intervals, which can be seen as intervals of prime [[7/1|7]]. The subminor second is [[21/20]]~[[28/27]], the supermajor second [[8/7]], the subminor third [[7/6]], and the supermajor third [[9/7]]~[[14/11]]. 31edo thus has five varieties of seconds and thirds, which is much more than the two varieties in 12edo. | In terms of interval categories, because 31edo is a meantone system, the major and minor seconds, thirds, sixth, and sevenths on the chain of fifths are equated to [[5-limit]] intervals, those being [[16/15]], [[10/9]], [[6/5]], [[5/4]], and their [[octave complement]]s. 31edo maps the chromatic semitone to two steps, meaning there are "[[neutral (interval quality)|neutral]]" intervals between minor and major ones, which are not found in [[12edo]]. They can be represented by [[11-limit]] intervals, with [[11/10]]~[[12/11]] being a neutral second, and [[11/9]]~[[27/22]] a neutral third. One step in the other direction from the classical intervals are the subminor and supermajor intervals, which can be seen as intervals of prime [[7/1|7]]. The subminor second is [[21/20]]~[[28/27]], the supermajor second [[8/7]], the subminor third [[7/6]], and the supermajor third [[9/7]]~[[14/11]]. 31edo thus has five varieties of seconds and thirds each, which is much more than the two varieties available in 12edo. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 69: | Line 69: | ||
| 38.7 | | 38.7 | ||
| Super-unison | | Super-unison | ||
| [[36/35]], [[45/44]], [[49/48]], [[50/49]], [[64/63]] | | [[36/35]], [[45/44]], [[49/48]], [[50/49]], [[64/63]], [[128/125]] | ||
| {{UDnote|step=1}} | | {{UDnote|step=1}} | ||
|- | |- | ||
| Line 87: | Line 87: | ||
| 154.8 | | 154.8 | ||
| Neutral second | | Neutral second | ||
| [[11/10]], [[12/11]], [[13/12]] | | [[11/10]], [[12/11]], [[13/12]], [[35/32]] | ||
| {{UDnote|step=4}} | | {{UDnote|step=4}} | ||
|- | |- | ||
| Line 147: | Line 147: | ||
| 541.9 | | 541.9 | ||
| Superfourth | | Superfourth | ||
| [[11/8]], [[15/11]], [[26/19]], ''[[18/13]]'' | | [[11/8]], [[15/11]], [[26/19]], ''[[18/13]]'', [[48/35]] | ||
| {{UDnote|step=14}} | | {{UDnote|step=14}} | ||
|- | |- | ||
| Line 165: | Line 165: | ||
| 658.1 | | 658.1 | ||
| Subfifth | | Subfifth | ||
| [[16/11]], [[19/13]], [[22/15]], ''[[13/9]]'' | | [[16/11]], [[19/13]], [[22/15]], ''[[13/9]]'', [[35/24]] | ||
| {{UDnote|step=17}} | | {{UDnote|step=17}} | ||
|- | |- | ||
| Line 225: | Line 225: | ||
| 1045.2 | | 1045.2 | ||
| Neutral seventh | | Neutral seventh | ||
| [[11/6]], [[20/11]], [[24/13]] | | [[11/6]], [[20/11]], [[24/13]], [[64/35]] | ||
| {{UDnote|step=27}} | | {{UDnote|step=27}} | ||
|- | |- | ||
| Line 243: | Line 243: | ||
| 1161.3 | | 1161.3 | ||
| Sub-octave | | Sub-octave | ||
| [[35/18]], [[49/25]], [[63/32]], [[88/45]], [[96/49]] | | [[35/18]], [[49/25]], [[63/32]], [[88/45]], [[96/49]], [[125/64]] | ||
| {{UDnote|step=30}} | | {{UDnote|step=30}} | ||
|- | |- | ||
| Line 1,518: | Line 1,518: | ||
=== Other Instruments === | === Other Instruments === | ||
[[File:31edo array kalimba.jpg|none|thumb|640x640px|31edo array kalimba built by Tristan Bay; 3 octaves, 94 keys, and laid out in circle-of-fourths meantone tuning]] | [[File:31edo array kalimba.jpg|none|thumb|640x640px|31edo array kalimba built by [[Tristan Bay]]; 3 octaves, 94 keys, and laid out in circle-of-fourths meantone tuning]] | ||
=== Lumatone === | === Lumatone === | ||
| Line 1,525: | Line 1,525: | ||
=== Skip fretting === | === Skip fretting === | ||
'''[[Skip fretting system 31 2 9]]''' is a [[skip fretting]] system for 31edo. | '''[[Skip fretting system 31 2 9]]''' is a [[skip fretting]] system for 31edo. | ||
'''[[Skip fretting system 31 3 7]]''' is another skip fretting system for 31edo. | '''[[Skip fretting system 31 3 7]]''' is another skip fretting system for 31edo. | ||
'''Skip fretting system 31 2 5''' is another skip fretting system for 31edo. All examples on this page are for 7-string [[guitar]]. | '''Skip fretting system 31 2 5''' is another skip fretting system for 31edo. All examples on this page are for 7-string [[guitar]]. | ||