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 == | == 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) × (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 "[[ | 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 == | == 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, | 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 == | == Etymology == | ||
Latest revision as of 06:42, 15 August 2025
| Interval information |
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 8/7s exceed 20/9. It can also be expressed as the difference between three intervals of S7 = 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).
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 EDOs such as 26, 68, and 99.
Etymology
The comma was named by Xenllium in 2021, from a combination of hemiwürschmidt and gamera, two 7-limit temperaments that share this tempered comma.