22edo: Difference between revisions
m "Assumed subgroup" lol what |
Standardize the layout somewhat |
||
| Line 8: | Line 8: | ||
{{Wikipedia|22 equal temperament}} | {{Wikipedia|22 equal temperament}} | ||
{{ED intro}} Because it distinguishes [[10/9]] and [[9/8]], it is not a [[meantone]] system. | {{ED intro}} Because it distinguishes [[10/9]] and [[9/8]], it is not a [[meantone]] system. | ||
== History == | |||
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}. Inspired by the supposed division of the octave into 22 unequal parts in the [[Indian music|music theory of India]], Bosanquet noted that such an equal division was capable of representing 5-limit music with tolerable accuracy. In this he was followed in the twentieth century by theorist José Würschmidt, who noted it as a possible next step after [[19edo]], and {{w|James Murray Barbour|J. Murray Barbour}} in his classic survey of tuning history, ''Tuning and Temperament''. | |||
== Theory == | == Theory == | ||
| Line 22: | Line 25: | ||
22edo is also the third-smallest edo (after [[10edo]] and [[15edo]]) that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]]. | 22edo is also the third-smallest edo (after [[10edo]] and [[15edo]]) that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]]. | ||
=== | === Prime harmonics === | ||
{{Harmonics in equal|22|columns=11}} | |||
=== As a tuning of other temperaments === | |||
==== Observance of 81/80 ==== | ==== Observance of 81/80 ==== | ||
22edo, unlike 12 and 19, is not a system of [[meantone]] temperament, and as such it distinguishes a number of [[3-limit]] and [[5-limit]] intervals that meantone tunings (most notably [[12edo]], [[19edo]], 31edo, and 43edo) do not distinguish, such as the two whole tones of 9/8 and 10/9. Indeed, these distinctions are significantly exaggerated in 22edo in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]], and [[53edo]], allowing many opportunities for alternate interpretations of their harmony. As a result of the observance of 81/80, the standard 5-limit diatonic scale does not collapse to [[5L 2s|mosdiatonic]] as in meantone systems. Instead, it is a ternary scale | 22edo, unlike 12 and 19, is not a system of [[meantone]] temperament, and as such it distinguishes a number of [[3-limit]] and [[5-limit]] intervals that meantone tunings (most notably [[12edo]], [[19edo]], 31edo, and 43edo) do not distinguish, such as the two whole tones of 9/8 and 10/9. Indeed, these distinctions are significantly exaggerated in 22edo in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]], and [[53edo]], allowing many opportunities for alternate interpretations of their harmony. As a result of the observance of 81/80, the standard 5-limit diatonic scale does not collapse to [[5L 2s|mosdiatonic]] as in meantone systems. Instead, it is a ternary scale | ||
| Line 54: | Line 60: | ||
=== Higher-limit interpretations === | === Higher-limit interpretations === | ||
22edo can also be treated as adding harmonics 3 and 5 to [[11edo]]'s 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. Let us also mind its approximation of the 31st harmonic is within half a cent, which is very accurate. It also approximates some intervals involving the 29th harmonic well, especially 29/24, which is also matched within half a cent. This leaves us with 2.3.5.7.11.17.29.31. | 22edo can also be treated as adding harmonics 3 and 5 to [[11edo]]'s 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. Let us also mind its approximation of the 31st harmonic is within half a cent, which is very accurate. It also approximates some intervals involving the 29th harmonic well, especially 29/24, which is also matched within half a cent. This leaves us with 2.3.5.7.11.17.29.31. | ||
== Intervals == | == Intervals == | ||