342edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== 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 === | ||
| Line 15: | Line 17: | ||
== 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 == | ||
| Line 51: | Line 36: | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve stretch (¢) | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 84: | Line 69: | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br>per 8ve | ! Periods<br />per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br>ratio* | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 109: | Line 94: | ||
|- | |- | ||
| 2 | | 2 | ||
| 124\342<br>(47\342) | | 124\342<br />(47\342) | ||
| 435.09<br>(164.91) | | 435.09<br />(164.91) | ||
| 9/7<br>(11/10) | | 9/7<br />(11/10) | ||
| [[Semisupermajor]] | | [[Semisupermajor]] | ||
|- | |- | ||
| 2 | | 2 | ||
| 142\342<br>(29\342) | | 142\342<br />(29\342) | ||
| 498.25<br>(101.75) | | 498.25<br />(101.75) | ||
| 4/3<br>(35/33) | | 4/3<br />(35/33) | ||
| [[Bipont]] | | [[Bipont]] | ||
|- | |- | ||
| 3 | | 3 | ||
| 71\342<br>(43\342) | | 71\342<br />(43\342) | ||
| 249.12<br>(150.88) | | 249.12<br />(150.88) | ||
| 15/13<br>(12/11) | | 15/13<br />(12/11) | ||
| [[Hemiterm]] | | [[Hemiterm]] | ||
|- | |- | ||
| 6 | | 6 | ||
| 97\342<br>(17\342) | | 97\342<br />(17\342) | ||
| 340.35<br>(59.65) | | 340.35<br />(59.65) | ||
| 162/133<br>(88/85) | | 162/133<br />(88/85) | ||
| [[Semiseptichrome]] | | [[Semiseptichrome]] | ||
|- | |- | ||
| 6 | | 6 | ||
| 142\342<br>(28\342) | | 142\342<br />(28\342) | ||
| 498.25<br>(98.25) | | 498.25<br />(98.25) | ||
| 4/3<br>(18/17) | | 4/3<br />(18/17) | ||
| [[Semiterm]] | | [[Semiterm]] | ||
|- | |- | ||
| 9 | | 9 | ||
| 63\342<br>(13\342) | | 63\342<br />(13\342) | ||
| 221.05<br>(45.61) | | 221.05<br />(45.61) | ||
| 25/22<br>(77/75) | | 25/22<br />(77/75) | ||
| [[Quadraennealimmal]] | | [[Quadraennealimmal]] | ||
|- | |- | ||
| 18 | | 18 | ||
| 71\342<br>(5\342) | | 71\342<br />(5\342) | ||
| 249.12<br>(17.54) | | 249.12<br />(17.54) | ||
| 15/13<br>(99/98) | | 15/13<br />(99/98) | ||
| [[Hemiennealimmal]] | | [[Hemiennealimmal]] | ||
|- | |- | ||
| 38 | | 38 | ||
| 142\342<br>(2\342) | | 142\342<br />(2\342) | ||
| 498.25<br>(7.02) | | 498.25<br />(7.02) | ||
| 4/3<br>(225/224) | | 4/3<br />(225/224) | ||
| [[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 == | |||
* [[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 | |||