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{{Main|365edo}}
{{breadcrumb}}
== Theory ==
== 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. In 23-limit, 365eeffgghiii val's octave stretch of -0.79428 cents is very close, 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.
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.


This approach will cover what is essentially "tropical year"-EDO, that is 365.2422edo using the 365eeffgghiii val. 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.   


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.
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.
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=== Mapping used ===
=== Mapping used ===
{| class="wikitable"
{| class="wikitable"
!Prime Harmonic
!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'''
|-
|-
|Patent val 365edo
| Map
|365
| 365
|579
| 579
|848
| 848
|1025
| 1025
|'''1263'''
| '''1264'''
|'''1351'''
| '''1352'''
|'''1492'''
| '''1493'''
|'''1550'''
| '''1551'''
|'''1651'''
| '''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'''
|}
|}


Mappings different from the patent val highlighted in '''bold'''.
Mappings different from the patent val highlighted in '''bold'''.
=== 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 ==
== Regular temperament properties ==
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{| 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*
|-
| 0
| January 1
| Prime, unison
| 1/1
|-
|-
|32
| 32
|February 1
| February 1
|Dodecaphonic semitone
| Dodecaphonic semitone
|
|  
|-
|-
|33
| 33
|February 2
| February 2
|Septendecimal semitone
| Septendecimal semitone
|17/16
| 17/16
|-
|-
|59
| 59
|February 28
| February 28
|
|  
|
|  
|-
|-
|''59 II''
| ''59 II''
|''February 29''
| ''February 29''
|''inserted once every 4 octaves''
| ''inserted once every 4 octaves''
|''accumulating octave stretch''
| ''accumulating octave stretch''
|-
|-
|60
| 60
|March 1
| March 1
|Dodecaphonic major second
| Dodecaphonic major second
|
|  
|-
|-
|63
| 63
|March 4
| March 4
|Classical major second, meantone
| Classical major second, meantone
|9/8
| 9/8
|-
|-
|91
| 91
|April 1
| April 1
|Undevicesimal minor third, dodecaphonic minor third
| Undevicesimal minor third, dodecaphonic minor third
|19/16
| 19/16
|-
|-
|118
| 118
|April 28
| April 28
|Classical major third
| Classical major third
|5/4
| 5/4
|-
|-
|169
| 169
|June 18
| June 18
|Undecimal superfourth
| Undecimal superfourth
|11/8
| 11/8
|-
|-
|182
| 182
|July 1
| July 1
|Dodecaphonic tritone
| Dodecaphonic tritone
|
|  
|-
|-
|193
| 193
|July 12
| July 12
|Vicesimotertial lalala
| Vicesimotertial lalala
|23/16
| 23/16
|-
|-
|214
| 214
|August 2
| August 2
|Perfect fifth
| Perfect fifth
|3/2
| 3/2
|-
|-
|257
| 257
|September 14
| September 14
|Tridecimal neutral sixth
| Tridecimal neutral sixth
|13/8
| 13/8
|-
|-
|295
| 295
|October 22
| October 22
|Harmonic seventh
| Harmonic seventh
|7/4
| 7/4
|-
|-
|365
| 365
|January 1
| January 1
|Octave
| Octave
|2/1
| 2/1
|}
|}
<nowiki />* Based on 365eeffgghiii val


[[Category:365edo]]
[[Category:365edo]]
[[Category:Method]]
[[Category:Method]]
[[Category:Approaches to tuning systems]]
Retrieved from "https://en.xen.wiki/w/365edo/Eliora%27s_approach"