Hemimean family: Difference between revisions
Rework intro to highlight the most important facts; improve readability |
→Hemimean orion: correct location of S-expression-based comma list to avoid confusion w.r.t subgroup |
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=== Hemimean orion === | === Hemimean orion === | ||
As the second generator of hemimean, [[28/25]], is close to [[19/17]], and as the latter is the [[mediant]] of [[10/9]] and [[9/8]], it is natural to extend hemimean to the 2.3.5.7.17.19 subgroup by tempering out ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[semiparticular]] (5/4)/(19/17)<sup>2</sup> = [[1445/1444]]. Notice 3136/3125 = (476/475)([[2128/2125]]) and that 2128/2125 = ([[1216/1215]])([[1701/1700]]), so it makes sense to temper out 1216/1215 and/or 1701/1700 as well. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error. | As the second generator of hemimean, [[28/25]], is close to [[19/17]], and as the latter is the [[mediant]] of [[10/9]] and [[9/8]], it is natural to extend hemimean to the 2.3.5.7.17.19 subgroup by tempering out ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[semiparticular]] (5/4)/(19/17)<sup>2</sup> = [[1445/1444]]. Notice 3136/3125 = (476/475)([[2128/2125]]) and that 2128/2125 = ([[1216/1215]])([[1701/1700]]), so it makes sense to temper out 1216/1215 and/or 1701/1700 as well. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error. | ||
Subgroup: 2.3.5.7.17 | Subgroup: 2.3.5.7.17 | ||
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==== 2.3.5.7.17.19 subgroup ==== | ==== 2.3.5.7.17.19 subgroup ==== | ||
Subgroup: 2.3.5.7.17.19 | Subgroup: 2.3.5.7.17.19 | ||
The S-expression-based comma list is { [[1216/1215|S16/S18]], [[1445/1444|S17/S19]], [[1701/1700|S18/S20]](, ([[136/135|S16*S17]])/([[190/189|S19*S20]]) = [[476/475|S16/S18 * S17/S19 * S18/S20]]) }. | |||
Comma list: 476/475, 1216/1215, 1445/1444 | Comma list: 476/475, 1216/1215, 1445/1444 | ||