Hemimean family: Difference between revisions
m →2.3.5.7.17.19 subgroup: add S-expression-based comma list |
m add way to derive (49/45)/(25/24)^2 using S-expressions starting from the recognition of 3136/3125's ability to be expressed as a ratio of an ultraparticular and a semiparticular |
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The '''hemimean family''' of temperaments are rank-3 temperaments tempering out [[3136/3125]]. | The '''hemimean family''' of temperaments are rank-3 temperaments tempering out [[3136/3125]]. | ||
The hemimean comma, 3136/3125, is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49). | The hemimean comma, 3136/3125, is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49). | ||
If we write this equivalently using S-expressions we obtain ([[128/125|S4/S5]])/([[50/49|S5/S7]]) which can be rearranged to [[16/15|S4]]*[[49/48|S7]]/[[25/24|S5]]<sup>2</sup>. | |||
Then we can optionally replace S4 with a nontrivial equivalent S-expression, S4 = [[36/35|S6]]*[[49/48|S7]]*[[64/63|S8]]; substituting this in and simplifying yields: | |||
S6*S7*S8*S7/S5<sup>2</sup> from which we can obtain an alternative equivalence 3136/3125 = ([[49/45]])/([[25/24]])<sup>2</sup>. | |||
== Hemimean == | == Hemimean == | ||