42edo: Difference between revisions
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=== Odd harmonics === | === Odd harmonics === | ||
While not an accurate tuning on the full [[7-limit]], 42edo does an excellent job on the 2.9.15.7.33.39 [[k*N subgroups|2*42 subgroup]], having the same tuning on it as does [[84edo]]. On this subgroup 42 has the same [[comma]]s as 84. | While not an accurate tuning on the full [[7-limit]], 42edo does an excellent job on the 2.9.15.7.33.39 [[k*N subgroups|2*42 subgroup]], having the same tuning on it as does [[84edo]]. On this subgroup 42 has the same [[comma]]s as 84. | ||
{{Harmonics in equal|42}} | {{Harmonics in equal|42}} | ||
=== Octave stretch === | === Octave stretch === | ||
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The following table compares three stretched tunings of 42edo: | The following table compares three stretched tunings of 42edo: | ||
{| class="wikitable" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|- | |- | ||
! Tuning | ! Tuning | ||
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The following table compares three compressed tunings of 42edo: | The following table compares three compressed tunings of 42edo: | ||
{| class="wikitable" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|- | |- | ||
! Tuning | ! Tuning | ||
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=== Sagittal notation === | === Sagittal notation === | ||
==== Best fifth notation ==== | ==== Best fifth notation ==== | ||
This notation uses the same sagittal sequence as [[35edo#Second-best fifth notation|35b]]. | This notation uses the same sagittal sequence as [[35edo #Second-best fifth notation|35b]]. | ||
===== Evo flavor ===== | ===== Evo flavor ===== | ||
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==== Second-best fifth notation ==== | ==== Second-best fifth notation ==== | ||
This notation uses the same sagittal sequence as [[47edo#Sagittal notation| | This notation uses the same sagittal sequence as [[47edo#Sagittal notation|47edo]], and is a superset of the notations for edos [[21edo #Sagittal notation|21]], [[14edo #Sagittal notation|14]], and [[7edo #Sagittal notation|7]]. | ||
<imagemap> | <imagemap> | ||
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default [[File:42b_Sagittal.svg]] | default [[File:42b_Sagittal.svg]] | ||
</imagemap> | </imagemap> | ||
== JI approximation == | |||
{{Q-odd-limit intervals}} | |||
=== Zeta peak index === | |||
{{ZPI | |||
| zpi = 190 | |||
| steps = 41.9988672418690 | |||
| step size = 28.5721991759747 | |||
| tempered height = 2.689436 | |||
| pure height = 2.688894 | |||
| integral = 0.303886 | |||
| gap = 9.541545 | |||
| octave = 1200.03236539094 | |||
| consistent = 8 | |||
| distinct = 8 | |||
}} | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{Monzo| 67 -42 }} | |||
| {{Mapping| 42 67 }} | |||
| −3.89 | |||
| 3.88 | |||
| 13.57 | |||
|- | |||
| 2.3.5 | |||
| 128/125, 5000000/4782969 | |||
| {{Mapping| 42 67 98 }} | |||
| −4.55 | |||
| 3.30 | |||
| 11.55 | |||
|- | |||
| 2.3.5.7 | |||
| 64/63, 126/125, 6860/6561 | |||
| {{Mapping| 22 35 51 62 }} | |||
| −3.65 | |||
| 3.26 | |||
| 11.42 | |||
|} | |||
== Scales == | == Scales == | ||
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[[Category:Augene]] | [[Category:Augene]] | ||
{{Todo|review|add rank 2 temperaments table}} | |||