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27edo: Difference between revisions

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| Prime factorization = 3<sup>3</sup>
| Prime factorization = 3<sup>3</sup>
| Subgroup = 2.3.5.7.13.19
| Subgroup = 2.3.5.7.13.19
| Step size = 44.444
| Step size = 44.444¢
| Fifth type = [[superpyth]] 16\27 711.111¢
| Fifth type = [[superpyth]] 16\27 = 711.111¢
| Major 2nd = 5\27 = 222¢
| Minor 2nd = 1\27 = 44¢
| Augmented 1sn = 4\27 = 178¢
| Important MOS = [[superpyth]] diatonic 5L2s 5551551 (16\27, 1\1)<br/> [[augmented]] ([[augene]]) 3L6s 522522522  (2\27, 1\3)<br/> [[beatles]] 3L4s 5353533 (8\27, 1\1)<br/> [[beatles]] 7L3s 3332332332 (9\27, 1\1)<br/> [[sensi]] 3L5s 43343343 (10\27, 1\1)<br/>[[tetracot]] 6L1s 4444443 (4\27, 1\1)<br/>[[octacot]] 13L1s 22222222222221 (2\27, 1\1)
| Important MOS = [[superpyth]] diatonic 5L2s 5551551 (16\27, 1\1)<br/> [[augmented]] ([[augene]]) 3L6s 522522522  (2\27, 1\3)<br/> [[beatles]] 3L4s 5353533 (8\27, 1\1)<br/> [[beatles]] 7L3s 3332332332 (9\27, 1\1)<br/> [[sensi]] 3L5s 43343343 (10\27, 1\1)<br/>[[tetracot]] 6L1s 4444443 (4\27, 1\1)<br/>[[octacot]] 13L1s 22222222222221 (2\27, 1\1)
}}
}}
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== Theory ==
== Theory ==
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
!prime 17
! prime 19
|-
! rowspan="2" |Error
! absolute (¢)
| 0
|  +9.16
|  +13.7
|  +9.0
|  -18.0
|  +3.9
| -16.1
|  +13.6
|-
! [[Relative error|relative]] (%)
| 0
|  +21
|  +31
|  +20
|  -40.5
|  +9
| -36
|  +31
|-
! colspan="2" |[[nearest edomapping]]
|27
|16
|9
|22
|12
|19
|2
|7
|-
! colspan="2" |[[fifthspan]]
|0
| +1
| +9
| -2
| -6
| +13
| -10
| -8
|}


If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent|cents]] in size. However, 27 is a prime candidate for [[Octave shrinking|octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the [[5/4|third]], [[3/2|fifth]] and [[7/4]] sharply.
If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent|cents]] in size. However, 27 is a prime candidate for [[octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the [[5/4|third]], [[3/2|fifth]] and [[7/4]] sharply.  


Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as [[12edo]], sharp 13 2/3 cents. The result is that [[6/5]], [[7/5]] and especially [[7/6]] are all tuned more accurately than this. It can be considered the superpythagorean counterpart of [[19edo]], as it's 5th is audibly indistinguishable from 1/3 [[septimal comma]] superpyth in the same way that 19edo is audibly indistinguishable from [[1/3 syntonic comma meantone]], resulting in three of them reaching a near perfect minor third/major 6th in both.
Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as [[12edo]], sharp 13 2/3 cents. The result is that [[6/5]], [[7/5]] and especially [[7/6]] are all tuned more accurately than this. It can be considered the superpythagorean counterpart of [[19edo]], as it's 5th is audibly indistinguishable from 1/3 [[septimal comma]] superpyth in the same way that 19edo is audibly indistinguishable from [[1/3 syntonic comma meantone]], resulting in three of them reaching a near perfect minor third/major 6th in both.
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| ^1, m2
| ^1, m2
| up-unison, minor 2nd
| up-unison, minor 2nd
| Eb
| ^D, Eb
| di
| di
|-
|-
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| 88.89
| 88.89
| [[16/15]], [[21/20]], [[25/24]], [[19/18]], [[20/19]]
| [[16/15]], [[21/20]], [[25/24]], [[19/18]], [[20/19]]
| ^m2
| ^^1, ^m2
| upminor 2nd
| double-up 1sn, upminor 2nd
| ^Eb
| ^^D, ^Eb
| ra
| ra
|-
|-
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| 133.33
| 133.33
| [[15/14]], [[14/13]], [[13/12]]
| [[15/14]], [[14/13]], [[13/12]]
| ~2
| vA1, ~2
| mid 2nd
| downaug 1sn, mid 2nd
| vD#
| vD#, vvE
| ru
| ru
|-
|-
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| 177.78
| 177.78
| [[10/9]]
| [[10/9]]
| vM2
| A1, vM2
| downmajor 2nd
| aug 1sn, downmajor 2nd
| D#
| D#, vE
| reh
| reh
|-
|-
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| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
| Gb
| ^F
| me
| me
|-
|-
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| ~3
| ~3
| mid 3rd
| mid 3rd
|^Gb
|^^F
| mu
| mu
|-
|-
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| ^4
| ^4
| up 4th
| up 4th
| Ab
| ^G
| fih
| fih
|-
|-
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| v5
| v5
| down fifth
| down fifth
| G#
| vA
| sih
| sih
|-
|-
Line 190: Line 245:
| ~6
| ~6
| mid 6th
| mid 6th
| vA#
| vvB
| lu
| lu
|-
|-
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| vM6
| vM6
| downmajor 6th
| downmajor 6th
| A#
| vB
| la
| la
|-
|-
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| ^m7
| ^m7
| upminor 7th
| upminor 7th
| Db
| ^C
| te
| te
|-
|-
Line 230: Line 285:
| ~7
| ~7
| mid 7th
| mid 7th
| ^Db
| ^^C
| tu
| tu
|-
|-
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=== Selected just intervals by error ===
=== Selected just intervals by error ===
{| class="wikitable center-all"
The following table shows how [[15-odd-limit intervals]] are represented in 27edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 19
|-
! rowspan="2" |Error
! absolute (¢)
| 0.00
| +9.16
| +13.69
| +8.95
| -17.98
| +3.92
| +13.60
|-
! relative (%)
| 0.0
| +20.6
| +30.8
| +20.1
| -40.5
| +8.8
| +30.6
|}
 
The following table shows how [[15-odd-limit intervals]] are represented in 27edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.  
 
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ Direct mapping (even if inconsistent)
|+ Direct mapping (even if inconsistent)
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