27edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|27}} | {{EDO intro|27}} | ||
== Theory == | == Theory == | ||
27edo divides the [[octave]] in 27 equal parts each exactly 44{{frac|4|9}} [[cent]]s in size. However, since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023 cents. More generally, narrowing the steps to between 44.2 and 44.35 cents would be better in theory; [[43edt]], [[70ed6]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, and 2.55 cents, respectively. | 27edo divides the [[octave]] in 27 equal parts each exactly 44{{frac|4|9}} [[cent]]s in size. However, since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023 cents. More generally, narrowing the steps to between 44.2 and 44.35 cents would be better in theory; [[43edt]], [[70ed6]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, and 2.55 cents, respectively. | ||
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== Approximation to JI == | == Approximation to JI == | ||
[[File:27ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 27edo]] | [[File:27ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 27edo]] | ||
=== Interval mappings === | === Interval mappings === | ||
{{15-odd-limit|27}} | {{15-odd-limit|27}} | ||
{{15-odd-limit|27.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 27e val mapping}} | {{15-odd-limit|27.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 27e val mapping}} | ||
=== Zeta peak index === | |||
{| class="wikitable center-all" | |||
|- | |||
! colspan="3" | Tuning | |||
! colspan="3" | Strength | |||
! colspan="2" | Closest edo | |||
! colspan="2" | Integer limit | |||
|- | |||
! ZPI | |||
! Steps per octave | |||
! Step size (cents) | |||
! Height | |||
! Integral | |||
! Gap | |||
! Edo | |||
! Octave (cents) | |||
! Consistent | |||
! Distinct | |||
|- | |||
| [[106zpi]] | |||
| 27.0866140827635 | |||
| 44.3023257293579 | |||
| 6.069233 | |||
| 1.185939 | |||
| 16.215619 | |||
| 27edo | |||
| 1196.16279469266 | |||
| 10 | |||
| 8 | |||
|} | |||
== Regular temperament properties == | == Regular temperament properties == | ||
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| Nugu | | Nugu | ||
| 19th-partial chroma | | 19th-partial chroma | ||
|} | |} | ||