3.5.7 subgroup: Difference between revisions
CompactStar (talk | contribs) removing this section that I wrote years ago, I will rewrite this later, and besides Mintaka is more important to discuss in that than 3.5.11.13 |
CompactStar (talk | contribs) |
||
| Line 42: | Line 42: | ||
Among the edts tempering out 245/243, 13edt stands out. An apt analogy can be drawn with edos supporting meantone: 4edt and 9edt are to BP what 5edo and 7edo are to meantone. However, in contrast to meantone, the simplest edt supporting the BP enneatonic scale – 13edt, the traditional tempered BP scale – is the most accurate and lowest in tuning error until 56edt. However, there are many edts supporting BP temperament which also support extensions to higher-limit temperaments; in particular 2, 3, and 4 times 13 in the form of {{EDTs| 26, 39, and 52 as well as 56EDT.}} For tempering out 16875/16807, 13edt again stands out, though much better accuracy can be found in more complex divisions such as 114edt or 127edt. All of this explains the focus on 13edt to the exclusion of other edts among practitioners of the art of non-octave composition, but it must be noted that the analysis is only valid if consideration is confined to 3.5.7, which is exactly analogous to confining it to the 5-limit with edos. There's a whole other world out there which has not been much explored. | Among the edts tempering out 245/243, 13edt stands out. An apt analogy can be drawn with edos supporting meantone: 4edt and 9edt are to BP what 5edo and 7edo are to meantone. However, in contrast to meantone, the simplest edt supporting the BP enneatonic scale – 13edt, the traditional tempered BP scale – is the most accurate and lowest in tuning error until 56edt. However, there are many edts supporting BP temperament which also support extensions to higher-limit temperaments; in particular 2, 3, and 4 times 13 in the form of {{EDTs| 26, 39, and 52 as well as 56EDT.}} For tempering out 16875/16807, 13edt again stands out, though much better accuracy can be found in more complex divisions such as 114edt or 127edt. All of this explains the focus on 13edt to the exclusion of other edts among practitioners of the art of non-octave composition, but it must be noted that the analysis is only valid if consideration is confined to 3.5.7, which is exactly analogous to confining it to the 5-limit with edos. There's a whole other world out there which has not been much explored. | ||
The next simplest no-twos subgroups after 3.5.7 are probably the [[3.5.11 subgroup]] and [[3.7.11 subgroup]], which could be considered as analogous to 2.3.7 and. 2.5.7. 13edt actually does poorly on both of these due to a lack of approximation of the 11th harmonic. | |||
[[Category:Just intonation subgroups|#]] | [[Category:Just intonation subgroups|#]] | ||
[[Category:Rank-3 temperaments|#]] | [[Category:Rank-3 temperaments|#]] | ||
[[Category:7-limit|#]] | [[Category:7-limit|#]] | ||