36edo: Difference between revisions
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Thanks to its sevenths, 36edo is an ideal tuning for its size for [[metallic harmony]]. | Thanks to its sevenths, 36edo is an ideal tuning for its size for [[metallic harmony]]. | ||
=== Octave stretch === | |||
36edo has almost 50% relative error on harmonics 5/1 and 11/1. This means that whether one [[Octave stretch|stretches]] or [[Octave shrinking|compresses]] the octave, either way it will improve 36edo’s approximations of [[JI]], but in opposite directions (as long as it is done by the right amount). | |||
What follows is a comparison of stretched- and compressed-octave 36edo tunings. | |||
Of the stretched-octave tunings listed, ''36ed513/256'' performs best in this comparison, approximating all 11-limit primes with less than 36% relative error (<12 cents error). | |||
Of the compressed-octave tunings listed, ''36ed511/256'' performs best in this comparison, approximating all 11-limit primes with less than 36% relative error (<12 cents error). | |||
{| class="wikitable sortable" | |||
! rowspan="2" | Name of tuning !! rowspan="2" | Step size (cents) !! colspan="5" | Error (% step size) !! colspan="5" | Mapping (# steps) | |||
|- | |||
! Prime 2 !! Prime 3 !! Prime 5 !! Prime 7 !! Prime 11 !! Prime 2 !! Prime 3 !! Prime 5 !! Prime 7 !! Prime 11 | |||
|- | |||
! 154zpi | |||
| 33.55 || 23 || 31 || 5 || 41 || 27 || 36 || 57 || 83 || 100 || 124 | |||
|- | |||
! 36ed257/128 | |||
| 33.52 || 20 || 26 || 12 || 50 || 15 || 36 || 57 || 83 || 101 || 124 | |||
|- | |||
! 36ed513/256 | |||
| 33.43 || 10 || 11 || 35 || 23 || 18 || 36 || 57 || 83 || 101 || 124 | |||
|- | |||
! 57edt | |||
| 33.37 || 4 || 0 || 50 || 5 || 40 || 36 || 57 || 83 || 101 || 124 | |||
|- | |||
! 155zpi | |||
| 33.35 || 2 || 3 || 45 || 1 || 48 || 36 || 57 || 84 || 101 || 124 | |||
|- | |||
! 36EDO | |||
| '''33.33'''|| '''0'''|| '''6'''|| '''40'''|| '''8'''|| '''45'''|| '''36'''|| '''57'''|| '''84'''|| '''101'''|| '''125''' | |||
|- | |||
! 36ed511/256 | |||
| 33.24 || 10 || 22 || 18 || 35 || 11 || 36 || 57 || 84 || 101 || 125 | |||
|- | |||
! 156zpi | |||
| 33.15 || 20 || 37 || 5 || 38 || 23 || 36 || 57 || 84 || 102 || 125 | |||
|- | |||
! 36ed255/128 | |||
| 33.145 || 21 || 38 || 6 || 36 || 25 || 36 || 57 || 84 || 102 || 125 | |||
|} | |||
=== Subsets and supersets === | === Subsets and supersets === | ||