365edo/Eliora's approach: Difference between revisions

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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. 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.
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.
Line 10: Line 10:
=== 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'''
|}
|}


Line 74: Line 75:
   
   
{| 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]]

Revision as of 20:32, 25 November 2024

< 365edo

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.

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 used

Prime Harmonic 2 3 5 7 11 13 17 19 23
Map 365 579 848 1025 1264 1352 1493 1551 1653
Patent val 365edo 365 579 848 1025 1263 1351 1492 1550 1651
Reduced 365 214 118 295 169 257 33 91 193

Mappings different from the patent val highlighted in bold.

Using a uniform map

It is possible to take interval approximations directly in 365.24219edo.


Approximation of prime harmonics in 365.24219edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -0.80 +0.34 -0.22 -1.20 +1.54 +1.46 +0.28 +1.57 -0.64 -1.12 -1.58
Relative (%) -24.2 +10.5 -6.6 -36.4 +47.0 +44.3 +8.6 +47.8 -19.6 -34.0 -48.2
Steps
(reduced) 		365
(365) 		579
(213.75781) 		848
(117.51562) 		1025
(294.51562) 		1264
(168.27343) 		1352
(256.27343) 		1493
(32.03124) 		1552
(91.03124) 		1652
(191.03124) 		1774
(313.03124) 		1809
(348.03124)

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

Step Note name Interval name Associated ratio*
0 January 1 Prime, unison 1/1
32 February 1 Dodecaphonic semitone
33 February 2 Septendecimal semitone 17/16
59 February 28
59 II February 29 inserted once every 4 octaves accumulating octave stretch
60 March 1 Dodecaphonic major second
63 March 4 Classical major second, meantone 9/8
91 April 1 Undevicesimal minor third, dodecaphonic minor third 19/16
118 April 28 Classical major third 5/4
169 June 18 Undecimal superfourth 11/8
182 July 1 Dodecaphonic tritone
193 July 12 Vicesimotertial lalala 23/16
214 August 2 Perfect fifth 3/2
257 September 14 Tridecimal neutral sixth 13/8
295 October 22 Harmonic seventh 7/4
365 January 1 Octave 2/1

* Based on 365eeffgghiii val

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