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365 is the number of days in the common year, and as such this proposes some interesting intereprations.
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== Theory ==
As 365 is the number of days in the common year, and there is a way to implement such a fact into music. An octave stretch of -0.796 cents would compress 365edo to an interesting intepretation: the pure 2/1 would represent 365.24219edo, which is the length of solar days in a tropical year. Therefore, each step is equal to 3.2854936 cents, and there are ''common octaves'' consisting of 365 steps and ''leap octaves'' consisting of 366 steps. The additional step does not amount to a JI mapping on its own, but rather resets the stretched octave to match the pure octave. Given that a standard gamut of music suitable for human hearing uses 8 octaves, this means 2 of them would be leap.
 
=== Using a val with a selected octave stretch ===
In 23-limit, 365eeffgghiii val's octave stretch of -0.79428 cents is very close to the target value, and makes 2/1 correspond to 365.241917 days, or 365 days 5h 48m 21.7s, which is only about 20 seconds short of the tropical year in the present era.


This approach will cover what is essentially "tropical year"-EDO, that is 365.2422edo using the 365eeffgghiii val.
In the 7-limit, the val is the same as patent val for 365edo, and hence it supports the [[hemiwürschmidt]] temperament just like the unstretched pure-octave 365edo does.


Mapping:
=== Mapping used ===
{| class="wikitable"
{| class="wikitable"
!
!2
!3
!5
!7
!11
!13
!17
!19
!23
|-
|-
|Map
! Prime Harmonic
|365
! 2
|579
! 3
|848
! 5
|1025
! 7
|1264
! 11
|1352
! '''13'''
|1493
! '''17'''
|1551
! '''19'''
|1653
! '''23'''
|-
| Map
| 365
| 579
| 848
| 1025
| '''1264'''
| '''1352'''
| '''1493'''
| '''1551'''
| '''1653'''
|-
|-
|Reduced
| Patent val 365edo
|365
| 365
|214
| 579
|118
| 848
|295
| 1025
|169
| '''1263'''
|257
| '''1351'''
|33
| '''1492'''
|91
| '''1550'''
|193
| '''1651'''
|-
| Reduced
| 365
| 214
| 118
| 295
| '''169'''
| '''257'''
| '''33'''
| '''91'''
| '''193'''
|}
|}


== Theory ==
Mappings different from the patent val highlighted in '''bold'''.
365eeffgghiii val shares rank two temperaments with the following vals: {{EDOs|121i, 99ef, 68e, 27egi, 41i, 46p, 58hi, 87, 50, 94}}.
 
=== Using a uniform map ===
It is possible to take interval approximations directly in 365.24219edo.
 
{{harmonics in equal|365.24219}}
 
One might take note that for harmonics 19 and 23, the uniform map features a distinct mapping from the val with the best octave stretching. Overall, the uniform map performs quite poorly in lower limits, though it provides good tunings for the 2.3.5.17.23 subgroup.
 
== Regular temperament properties ==
365eeffgghiii val is used.
 
* 11-limit commas: {896/891, 6144/6125, 6250/6237, 532400/531441}
 
* 23-limit commas: {256/255, 300/299, 352/351, 456/455, 896/891, 1225/1224, 3136/3125, 13608/13585}


==Table of intervals==
==Table of intervals==
   
   
{| class="wikitable"
{| class="wikitable"
|+
!Step
!Note name
!Interval name
!Associated Ratio
<small>based on 365eeffgghiii</small>
|-
|-
|0
! Step
|January 1
! Note name
|Prime, unison
! Interval name
|1/1
! Associated ratio*
|-
|-
|32
| 0
|February 1
| January 1
|Dodecaphonic semitone
| Prime, unison
|
| 1/1
|-
|-
|33
| 32
|February 2
| February 1
|Septendecimal semitone
| Dodecaphonic semitone
|17/16
|  
|-
|-
|59
| 33
|February 28
| February 2
|
| Septendecimal semitone
|
| 17/16
|-
|-
|''59 II''
| 59
|''February 29''
| February 28
|''inserted once every 4 octaves''
|  
|''accumulating octave stretch''
|  
|-
|-
|60
| ''59 II''
|March 1
| ''February 29''
|Dodecaphonic major second
| ''inserted once every 4 octaves''
|
| ''accumulating octave stretch''
|-
|-
|63
| 60
|March 4
| March 1
|Classical major second, meantone
| Dodecaphonic major second
|9/8
|  
|-
|-
|91
| 63
|April 1
| March 4
|Undevicesimal minor third, dodecaphonic minor third
| Classical major second, meantone
|19/16
| 9/8
|-
|-
|118
| 91
|April 28
| April 1
|Classical major third
| Undevicesimal minor third, dodecaphonic minor third
|5/4
| 19/16
|-
|-
|169
| 118
|June 18
| April 28
|Undecimal superfourth
| Classical major third
|11/8
| 5/4
|-
|-
|182
| 169
|July 1
| June 18
|Dodecaphonic tritone
| Undecimal superfourth
|
| 11/8
|-
|-
|193
| 182
|July 12
| July 1
|Vicesimotertial lalala
| Dodecaphonic tritone
|23/16
|  
|-
|-
|214
| 193
|August 2
| July 12
|Perfect fifth
| Vicesimotertial lalala
|3/2
| 23/16
|-
|-
|257
| 214
|September 14
| August 2
|Tridecimal neutral sixth
| Perfect fifth
|13/8
| 3/2
|-
|-
|295
| 257
|October 22
| September 14
|Harmonic seventh
| Tridecimal neutral sixth
|7/4
| 13/8
|-
|-
|365
| 295
|January 1
| October 22
|Octave
| Harmonic seventh
|2/1
| 7/4
|-
| 365
| January 1
| Octave
| 2/1
|}
|}
<nowiki />* Based on 365eeffgghiii val
[[Category:365edo]]
[[Category:Method]]
[[Category:Approaches to tuning systems]]
Retrieved from "https://en.xen.wiki/w/365edo/Eliora%27s_approach"