Hewuermera comma: Difference between revisions

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'''589824/588245''', the '''hewuermera comma''', is a [[7-limit]] [[small comma]] measuring about 4.6 cents. It is the amount by which a stack of six [[8/7]]s exceed [[20/9]]. It can also be expressed as the difference between three intervals of {{S|7}} = [[49/48]] and S4 = [[16/15]].

'''589824/588245''', the '''hewuermera comma''', is a [[7-limit]] [[small comma]] measuring about 4.6 cents. It is the amount by which a stack of six [[8/7]]s exceed [[20/9]]. It can also be expressed as the difference between three intervals of {{S|7}} = [[49/48]] and S4 = [[16/15]], or as (S6/S7 = [[1728/1715]])/(S7/S8 = [[1029/1024]]).

== Factorizations ==

As a generalized lopsided comma, 589824/588245 has the S-expression of (S46 = [[2116/2115]]) × (S47 = [[2209/2208]])2 × (S48 = [[2304/2303]])3. Equivalent factorizations include (S48 = 2304/2303) × (S(47/2) = [[2209/2205]]) and (S46×S47×S48 = [[736/735]]) × (S47×S48 = [[1128/1127]]) × (S48 = 2304/2303).

As a generalized lopsided comma, 589824/588245 has the S-expression of (S46 = [[2116/2115]]) × (S47 = [[2209/2208]])2 × (S48 = [[2304/2303]])3. Equivalent factorizations include (S48 = 2304/2303)2 × (S(47/2) = [[2209/2205]]) and (S46×S47×S48 = [[736/735]]) × (S47×S48 = [[1128/1127]]) × (S48 = 2304/2303).

The S-factorization indicates that 23 and 47 are particularly natural primes to incorporate into hewuermera temperaments (by tempering out S46, S47, and S48); in particular, three 8/7s reach a "[[1/2-comma meantone]] fifth" identified with [[70/47]][[~]][[94/63]].

== Temperaments ==

Tempering it out alone in the 7-limit leads to the '''hewuermera''' [[rank-3 temperament]], and in combination with another comma, to the [[hewuermera temperaments]], the most notable of which are [[hemiwürschmidt]] and [[mothra]]. For highest accuracy, this comma demands that 8/7 be tuned slightly flat, as it is in [[EDO]]s such as {{EDOs|26, 99, and ~~161~~}}.

Tempering it out alone in the 7-limit leads to the '''hewuermera''' [[rank-3 temperament]], and in combination with another comma, to the [[hewuermera temperaments]], the most notable of which are [[hemiwürschmidt]] and [[mothra]]. For highest accuracy, this comma demands that 8/7 be tuned slightly flat, as it is in [[EDO]]s such as {{EDOs|26, 68, and 99}}.

== Etymology ==

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-'''589824/588245''', the '''hewuermera comma''', is a [[7-limit]] [[small comma]] measuring about 4.6 cents. It is the amount by which a stack of six [[8/7]]s exceed [[20/9]]. It can also be expressed as the difference between three intervals of {{S|7}} = [[49/48]] and S4 = [[16/15]].
+'''589824/588245''', the '''hewuermera comma''', is a [[7-limit]] [[small comma]] measuring about 4.6 cents. It is the amount by which a stack of six [[8/7]]s exceed [[20/9]]. It can also be expressed as the difference between three intervals of {{S|7}} = [[49/48]] and S4 = [[16/15]], or as (S6/S7 = [[1728/1715]])/(S7/S8 = [[1029/1024]]).
 == Factorizations ==
-As a generalized lopsided comma, 589824/588245 has the S-expression of (S46 = [[2116/2115]]) × (S47 = [[2209/2208]])<sup>2</sup> × (S48 = [[2304/2303]])<sup>3</sup>. Equivalent factorizations include (S48 = 2304/2303) × (S(47/2) = [[2209/2205]]) and (S46×S47×S48 = [[736/735]]) × (S47×S48 = [[1128/1127]]) × (S48 = 2304/2303).
+As a generalized lopsided comma, 589824/588245 has the S-expression of (S46 = [[2116/2115]]) × (S47 = [[2209/2208]])<sup>2</sup> × (S48 = [[2304/2303]])<sup>3</sup>. Equivalent factorizations include (S48 = 2304/2303)<sup>2</sup> × (S(47/2) = [[2209/2205]]) and (S46×S47×S48 = [[736/735]]) × (S47×S48 = [[1128/1127]]) × (S48 = 2304/2303).
 The S-factorization indicates that 23 and 47 are particularly natural primes to incorporate into hewuermera temperaments (by tempering out S46, S47, and S48); in particular, three 8/7s reach a "[[1/2-comma meantone]] fifth" identified with [[70/47]][[~]][[94/63]].
 == Temperaments ==
-Tempering it out alone in the 7-limit leads to the '''hewuermera''' [[rank-3 temperament]], and in combination with another comma, to the [[hewuermera temperaments]], the most notable of which are [[hemiwürschmidt]] and [[mothra]]. For highest accuracy, this comma demands that 8/7 be tuned slightly flat, as it is in [[EDO]]s such as {{EDOs|26, 99, and 161}}.
+Tempering it out alone in the 7-limit leads to the '''hewuermera''' [[rank-3 temperament]], and in combination with another comma, to the [[hewuermera temperaments]], the most notable of which are [[hemiwürschmidt]] and [[mothra]]. For highest accuracy, this comma demands that 8/7 be tuned slightly flat, as it is in [[EDO]]s such as {{EDOs|26, 68, and 99}}.
 == Etymology ==