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== Theory ==
== Theory ==


<b>26edo</b> divides the [[octave]] into 26 equal parts of 46.154 [[cent]]s each. It tempers out 81/80 in the [[5-limit]], making it a meantone tuning with a very flat fifth. In the [[7-limit]], it tempers out 50/49, 525/512 and 875/864, and supports [[injera]], [[flattone]], [[Jubilismic clan#Lemba|lemba]] and [[Jubilismic clan#Doublewide|doublewide]] temperaments. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13 odd limit]] [[consistent|consistently]]. 26edo has a very good approximation of the harmonic seventh ([[7/4]]).
<b>26edo</b> divides the [[octave]] into 26 equal parts of 46.154 [[Cent|cents]] each. It tempers out 81/80 in the [[5-limit]], making it a meantone tuning with a very flat fifth. In the [[7-limit]], it tempers out 50/49, 525/512 and 875/864, and supports [[injera]], [[flattone]], [[Jubilismic clan#Lemba|lemba]] and [[Jubilismic clan#Doublewide|doublewide]] temperaments. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13 odd limit]] [[consistent|consistently]]. 26edo has a very good approximation of the harmonic seventh ([[7/4]]).


26edo's "minor sixth" (1.6158) is very close to φ ≈ 1.6180 (i. e., the golden ratio).
26edo's "minor sixth" (1.6158) is very close to φ ≈ 1.6180 (i. e., the golden ratio).
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== Intervals ==
== Intervals ==


{| class="wikitable" style="text-align:center;"
{| class="wikitable center-all right-2 left-3"
|-
|-
! Degrees
! Degrees
![[cent]]s
! [[Cents|cents]]
! Approximate Ratios*
! Approximate Ratios*
! Solfege
! Solfege
! Interval <br /> Name
! Interval<br>Name
! Example <br /> in D
! Example<br>in D
|-
|-
| 0
| 0
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|-
|-
| 4
| 4
| 184.615
| 184.62
| [[9/8]], [[10/9]], [[11/10]]
| [[9/8]], [[10/9]], [[11/10]]
| re
| re
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|-
|-
| 9
| 9
| 415.385
| 415.38
| [[9/7]], [[14/11]], [[33/26]]
| [[9/7]], [[14/11]], [[33/26]]
| maa
| maa
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|-
|-
| 13
| 13
|600
|600.00
| [[7/5]], [[10/7]]
| [[7/5]], [[10/7]]
| fi/se
| fi/se
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|-
|-
| 17
| 17
| 784.615
| 784.62
| [[11/7]], [[14/9]]
| [[11/7]], [[14/9]]
| leh
| leh
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|-
|-
| 22
| 22
| 1015.385
| 1015.38
| [[9/5]], [[16/9]], [[20/11]]
| [[9/5]], [[16/9]], [[20/11]]
| te
| te
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|-
|-
| 26
| 26
| 1200
| 1200.00
| 2/1
| 2/1
| do
| do
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== Selected just intervals by error ==
== Selected just intervals by error ==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 26edo (ordered by absolute error).
==== 15-odd-limit interval mappings ====
 
The following table shows how [[15-odd-limit intervals]] are represented in 26edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
=== Best direct mapping, even if inconsistent ===


{| class="wikitable" style="text-align:center;"
{| class="wikitable" style="text-align:center;"
|+Direct mapping (even if inconsistent)
|-
|-
! Interval, complement
! Interval, complement
! Error (abs., in [[cent]]s)
! Error (abs, [[Cent|¢]])
|-
|-
| [[13/12]], [[24/13]]
| [[13/12]], [[24/13]]
| 0.111
| 0.111
|-
|-
| [[8/7]], [[7/4]]
| '''[[8/7]], [[7/4]]'''
| 0.405
| '''0.405'''
|-
|-
| [[14/11]], [[11/7]]
| [[14/11]], [[11/7]]
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| 2.212
| 2.212
|-
|-
| [[11/8]], [[16/11]]
| '''[[11/8]], [[16/11]]'''
| 2.528
| '''2.528'''
|-
|-
| [[13/10]], [[20/13]]
| [[13/10]], [[20/13]]
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| 9.536
| 9.536
|-
|-
| [[4/3]], [[3/2]]
| '''[[4/3]], [[3/2]]'''
| 9.647
| '''9.647'''
|-
|-
| [[16/13]], [[13/8]]
| '''[[16/13]], [[13/8]]'''
| 9.758
| '''9.758'''
|-
|-
| [[7/6]], [[12/7]]
| [[7/6]], [[12/7]]
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| 12.287
| 12.287
|-
|-
| [[15/11]], [[22/15]]
| ''[[15/11]], [[22/15]]''
| 16.895
| ''16.895''
|-
|-
| [[15/13]], [[26/15]]
| [[15/13]], [[26/15]]
| 16.972
| 16.972
|-
|-
| [[5/4]], [[8/5]]
| '''[[5/4]], [[8/5]]'''
| 17.083
| '''17.083'''
|-
|-
| [[7/5]], [[10/7]]
| [[7/5]], [[10/7]]
| 17.488
| 17.488
|-
|-
| [[15/14]], [[28/15]]
| ''[[15/14]], [[28/15]]''
| 19.019
| ''19.019''
|-
|-
| [[9/8]], [[16/9]]
| [[9/8]], [[16/9]]
| 19.295
| 19.295
|-
|-
| [[16/15]], [[15/8]]
| ''[[16/15]], [[15/8]]''
| 19.424
| ''19.424''
|-
|-
| [[11/10]], [[20/11]]
| [[11/10]], [[20/11]]
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| 21.823
| 21.823
|}
|}
=== Patent val mapping ===


{| class="wikitable" style="text-align:center;"
{| class="wikitable" style="text-align:center;"
|+Patent val mapping
|-
|-
! Interval, complement
! Interval, complement
! Error (abs., in [[cent]]s)
! Error (abs, [[Cent|¢]])
|-
|-
| [[13/12]], [[24/13]]
| [[13/12]], [[24/13]]
| 0.111
| 0.111
|-
|-
| [[8/7]], [[7/4]]
| '''[[8/7]], [[7/4]]'''
| 0.405
| '''0.405'''
|-
|-
| [[14/11]], [[11/7]]
| [[14/11]], [[11/7]]
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| 2.212
| 2.212
|-
|-
| [[11/8]], [[16/11]]
| '''[[11/8]], [[16/11]]'''
| 2.528
| '''2.528'''
|-
|-
| [[13/10]], [[20/13]]
| [[13/10]], [[20/13]]
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| 9.536
| 9.536
|-
|-
| [[4/3]], [[3/2]]
| '''[[4/3]], [[3/2]]'''
| 9.647
| '''9.647'''
|-
|-
| [[16/13]], [[13/8]]
| '''[[16/13]], [[13/8]]'''
| 9.758
| '''9.758'''
|-
|-
| [[7/6]], [[12/7]]
| [[7/6]], [[12/7]]
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| 16.972
| 16.972
|-
|-
| [[5/4]], [[8/5]]
| '''[[5/4]], [[8/5]]'''
| 17.083
| '''17.083'''
|-
|-
| [[7/5]], [[10/7]]
| [[7/5]], [[10/7]]
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| 21.823
| 21.823
|-
|-
| [[16/15]], [[15/8]]
| ''[[16/15]], [[15/8]]''
| 26.730
| ''26.730''
|-
|-
| [[15/14]], [[28/15]]
| ''[[15/14]], [[28/15]]''
| 27.135
| ''27.135''
|-
|-
| [[15/11]], [[22/15]]
| ''[[15/11]], [[22/15]]''
| 29.258
| ''29.258''
|}
|}


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The primary triad for orgone temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates [[16/11|16:11]] and 3g approximates [[7/4|7:4]] (and I would call that the definition of Orgone Temperament). That also implies that g approximates the difference between 7:4 and 16:11, which is 77:64, about 320.1 cents.
The primary triad for orgone temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates [[16/11|16:11]] and 3g approximates [[7/4|7:4]] (and I would call that the definition of Orgone Temperament). That also implies that g approximates the difference between 7:4 and 16:11, which is 77:64, about 320.1 cents.


[[37edo|37edo]] is another orgone tuning, and [[89edo|89edo]] is better even than 26. If we take 11 and 26 to be the edges of the Orgone Spectrum, we may fill in the rest of the spectrum thus:
[[37edo]] is another orgone tuning, and [[89edo]] is better even than 26. If we take 11 and 26 to be the edges of the Orgone Spectrum, we may fill in the rest of the spectrum thus:


{| class="wikitable"
{| class="wikitable"
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[[Category:Theory]]
[[Category:Theory]]
[[Category:Twentuning]]
[[Category:Twentuning]]
[[Category:todo:unify precision]]
Retrieved from "https://en.xen.wiki/w/26edo"