342edo: Difference between revisions
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== Theory == | == Theory == | ||
342edo is a very strong 11-limit system. It is, as one would expect, [[consistency|distinctly consistent]] through the [[11-odd-limit]], but goes no higher; nonetheless, it is a [[zeta peak edo]]. A [[comma basis|basis]] for the 11-limit [[comma]]s consists of [[2401/2400]], [[3025/3024]], [[4375/4374]] and [[32805/32768]]. It is the [[optimal patent val]] for 11-limit [[Breedsmic temperaments #Hemitert|hemitert]] temperament, and [[support]]s hemiennealimmal. | 342edo is a very strong 11-limit system. It is, as one would expect, [[consistency|distinctly consistent]] through the [[11-odd-limit]], but goes no higher; nonetheless, it is a [[zeta peak edo]]. A [[comma basis|basis]] for the 11-limit [[comma]]s consists of [[2401/2400]], [[3025/3024]], [[4375/4374]] and [[32805/32768]]. It is the [[optimal patent val]] for 11-limit [[Breedsmic temperaments #Hemitert|hemitert]] temperament, and [[support]]s hemiennealimmal. | ||
If 3.5 cents is taken as the [[just-noticeable difference]], then 342edo may be regarded as the highest EDO whose step size remains individually discernible. However, the [[JND]] is not fixed and depends on the listener and musical context. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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== Approximation to JI == | == Approximation to JI == | ||
=== Zeta peak index === | === Zeta peak index === | ||
{ | {{ZPI | ||
| | | zpi = 2568 | ||
| steps = 341.974850913987 | |||
| step size = 3.50902996753355 | |||
| tempered height = 13.478611 | |||
| pure height = 12.437722 | |||
| integral = 1.890555 | |||
| gap = 20.767404 | |||
| octave = 1200.08824889647 | |||
| consistent = 12 | |||
| distinct = 12 | |||
}} | |||
| 3.50902996753355 | |||
| 13.478611 | |||
| 1.890555 | |||
| 20.767404 | |||
| | |||
| 12 | |||
| 12 | |||
== Regular temperament properties == | == Regular temperament properties == | ||
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| [[Hemienneadecal]] | | [[Hemienneadecal]] | ||
|} | |} | ||
<nowiki />* [[Normal | <nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||
* [[11-odd-limit|Diamond11]]: 43 4 5 6 8 10 14 9 11 9 5 18 15 9 10 9 15 18 5 9 11 9 14 10 8 6 5 4 43 | * [[11-odd-limit|Diamond11]]: 43 4 5 6 8 10 14 9 11 9 5 18 15 9 10 9 15 18 5 9 11 9 14 10 8 6 5 4 43 | ||