317edo: Difference between revisions

Line 3:

== Theory ==

~~317et~~ is only consistent to the [[5-odd-limit]] and ~~the~~ [[harmonic]] 3 is about halfway its steps. ~~Using~~ the ~~patent val~~, it ~~tempers out 589824/588245~~, [[~~16875/16807~~]], [[~~65625~~/~~65536~~]] ~~and 49009212/48828125 in the 7-limit; 1835008/1830125, 14700/14641, 2097152/2096325~~, [[~~4000~~/~~3993~~]], ~~2734375/2725888, 1953125/1951488, 172032/171875,~~ [[~~441~~/~~440~~]]~~, 5767168/5764801, 825000/823543, 537109375/536870912, 134775333/134217728, 160083/160000, 16808715/16777216, 539055/537824 and 3294225/3294172 in the 11-limit~~.

317edo is only [[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway its steps. To start with, it can be considered in the 2.9.5.7 [[subgroup]], in which it is strong, [[tempering out]] [[420175/419904]], [[703125/702464]], and [[33554432/33480783]].

===Odd harmonics===

Using the [[patent val]] nonetheless, the equal temperament tempers out 78732/78125 ([[sensipent comma]] in the 5-limit; [[16875/16807]], [[65625/65536]], 589824/588245, and 49009212/48828125 in the 7-limit. In the 11-limit, the 317e [[val]] tempers out [[540/539]], 1375/1372, and [[3025/3024]], whereas the patent val tempers out [[441/440]], [[4000/3993]], and 14700/14641.

=== Odd harmonics ===

===Subsets and supersets===

=== Subsets and supersets ===

317edo is the 66th [[prime edo]]. [[634edo]], which doubles it, gives a good correction to the harmonic 3.

==Regular temperament properties==

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" |[[Subgroup]]

! rowspan="2" | [[Subgroup]]

! rowspan="2" |[[Comma list|Comma List]]

! rowspan="2" | [[Comma list|Comma List]]

! rowspan="2" |[[Mapping]]

! rowspan="2" | [[Mapping]]

! rowspan="2" |Optimal<br>8ve Stretch (¢)

! rowspan="2" | Optimal<br>8ve Stretch (¢)

! colspan="2" |Tuning Error

! colspan="2" | Tuning Error

|-

![[TE error|Absolute]] (¢)

! [[TE error|Absolute]] (¢)

![[TE simple badness|Relative]] (%)

! [[TE simple badness|Relative]] (%)

|-

|2.9

| 2.9

|{{monzo|1005 -317}}

| {{monzo| 1005 -317 }}

|{{mapping|317 1005}}

| {{mapping| 317 1005 }}

| -0.0799

| 0.0799

| 2.11

|-

|2.9.5

| 2.9.5

|{{monzo|-53 5 16}}, {{monzo|33 -17 9}}

| {{monzo| -53 5 16 }}, {{monzo| 33 -17 9 }}

|{{mapping|317 1005 736}}

| {{mapping| 317 1005 736 }}

| -0.0254

| 0.1009

| 2.67

|-

|2.9.5.7

| 2.9.5.7

|420175/419904, 703125/702464, 33554432/33480783

| 420175/419904, 703125/702464, 33554432/33480783

|{{mapping|317 1005 736 890}}

| {{mapping| 317 1005 736 890 }}

| -0.0422

| 0.0921

| 2.43

|-

|2.9.5.7.11

| 2.9.5.7.11

|6250/6237, 12005/11979, 46656/46585, 151263/151250

| 6250/6237, 12005/11979, 46656/46585, 151263/151250

|{{mapping|317 1005 736 890 1097}}

| {{mapping| 317 1005 736 890 1097 }}

| -0.1126

| 0.1631

| 4.31

|-

|2.9.5.7.11.13

| 2.9.5.7.11.13

|1575/1573, 4459/4455, 6250/6237, 67392/67375, 190125/189728

| 1575/1573, 4459/4455, 6250/6237, 67392/67375, 190125/189728

|{{mapping|317 1005 736 890 1097 1173}}

| {{mapping| 317 1005 736 890 1097 1173 }}

| -0.0871

| 0.1594

| 4.21

|-

|2.9.5.7.11.13.17

| 2.9.5.7.11.13.17

|936/935, 1225/1224, 1575/1573, 12376/12375, 17920/17901, 34000/33957

| 936/935, 1225/1224, 1575/1573, 12376/12375, 17920/17901, 34000/33957

|{{mapping|317 1005 736 890 1097 1173 1296}}

| {{mapping| 317 1005 736 890 1097 1173 1296 }}

| -0.1109

| 0.1587

Line 67:

Line 69:

== Music ==

; [[Francium]]

* "Narrator Of Myths" from ''Mysteries'' (2023) [https://open.spotify.com/track/6AYSSw1kVKDDzWVYMyxafv Spotify] | [https://francium223.bandcamp.com/track/narrator-of-myths Bandcamp] | [https://www.youtube.com/watch?v=fNmEjgVonHk YouTube]

* "Narrator Of Myths" from ''Mysteries'' (2023) – [https://open.spotify.com/track/6AYSSw1kVKDDzWVYMyxafv Spotify] | [https://francium223.bandcamp.com/track/narrator-of-myths Bandcamp] | [https://www.youtube.com/watch?v=fNmEjgVonHk YouTube]

[[Category:Listen]]

@@ Line 3: / Line 3: @@
 == Theory ==
-et is only consistent to the [[5-odd-limit]] and the [[harmonic]] 3 is about halfway its steps. Using the patent val, it tempers out 589824/588245, [[16875/16807]], [[65625/65536]] and 49009212/48828125 in the 7-limit; 1835008/1830125, 14700/14641, 2097152/2096325, [[4000/3993]], 2734375/2725888, 1953125/1951488, 172032/171875, [[441/440]], 5767168/5764801, 825000/823543, 537109375/536870912, 134775333/134217728, 160083/160000, 16808715/16777216, 539055/537824 and 3294225/3294172 in the 11-limit.
+edo is only [[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway its steps. To start with, it can be considered in the 2.9.5.7 [[subgroup]], in which it is strong, [[tempering out]] [[420175/419904]], [[703125/702464]], and [[33554432/33480783]].
-===Odd harmonics===
+Using the [[patent val]] nonetheless, the equal temperament tempers out 78732/78125 ([[sensipent comma]] in the 5-limit; [[16875/16807]], [[65625/65536]], 589824/588245, and 49009212/48828125 in the 7-limit. In the 11-limit, the 317e [[val]] tempers out [[540/539]], 1375/1372, and [[3025/3024]], whereas the patent val tempers out [[441/440]], [[4000/3993]], and 14700/14641.
+=== Odd harmonics ===
 {{Harmonics in equal|317}}
-===Subsets and supersets===
+=== Subsets and supersets ===
 edo is the 66th [[prime edo]]. [[634edo]], which doubles it, gives a good correction to the harmonic 3.
-==Regular temperament properties==
+== Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |[[Subgroup]]
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" |[[Comma list|Comma List]]
+! rowspan="2" | [[Comma list|Comma List]]
-! rowspan="2" |[[Mapping]]
+! rowspan="2" | [[Mapping]]
-! rowspan="2" |Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
-! colspan="2" |Tuning Error
+! colspan="2" | Tuning Error
 |-
-![[TE error|Absolute]] (¢)
+! [[TE error|Absolute]] (¢)
-![[TE simple badness|Relative]] (%)
+! [[TE simple badness|Relative]] (%)
 |-
-|2.9
+| 2.9
-|{{monzo|1005 -317}}
+| {{monzo| 1005 -317 }}
-|{{mapping|317 1005}}
+| {{mapping| 317 1005 }}
 | -0.0799
 | 0.0799
 | 2.11
 |-
-|2.9.5
+| 2.9.5
-|{{monzo|-53 5 16}}, {{monzo|33 -17 9}}
+| {{monzo| -53 5 16 }}, {{monzo| 33 -17 9 }}
-|{{mapping|317 1005 736}}
+| {{mapping| 317 1005 736 }}
 | -0.0254
 | 0.1009
 | 2.67
 |-
-|2.9.5.7
+| 2.9.5.7
-|420175/419904, 703125/702464, 33554432/33480783
+| 420175/419904, 703125/702464, 33554432/33480783
-|{{mapping|317 1005 736 890}}
+| {{mapping| 317 1005 736 890 }}
 | -0.0422
 | 0.0921
 | 2.43
 |-
-|2.9.5.7.11
+| 2.9.5.7.11
-|6250/6237, 12005/11979, 46656/46585, 151263/151250
+| 6250/6237, 12005/11979, 46656/46585, 151263/151250
-|{{mapping|317 1005 736 890 1097}}
+| {{mapping| 317 1005 736 890 1097 }}
 | -0.1126
 | 0.1631
 | 4.31
 |-
-|2.9.5.7.11.13
+| 2.9.5.7.11.13
-|1575/1573, 4459/4455, 6250/6237, 67392/67375, 190125/189728
+| 1575/1573, 4459/4455, 6250/6237, 67392/67375, 190125/189728
-|{{mapping|317 1005 736 890 1097 1173}}
+| {{mapping| 317 1005 736 890 1097 1173 }}
 | -0.0871
 | 0.1594
 | 4.21
 |-
-|2.9.5.7.11.13.17
+| 2.9.5.7.11.13.17
-|936/935, 1225/1224, 1575/1573, 12376/12375, 17920/17901, 34000/33957
+| 936/935, 1225/1224, 1575/1573, 12376/12375, 17920/17901, 34000/33957
-|{{mapping|317 1005 736 890 1097 1173 1296}}
+| {{mapping| 317 1005 736 890 1097 1173 1296 }}
 | -0.1109
 | 0.1587
@@ Line 67: / Line 69: @@
 == Music ==
 ; [[Francium]]
-* "Narrator Of Myths" from ''Mysteries'' (2023) [https://open.spotify.com/track/6AYSSw1kVKDDzWVYMyxafv Spotify] | [https://francium223.bandcamp.com/track/narrator-of-myths Bandcamp] | [https://www.youtube.com/watch?v=fNmEjgVonHk YouTube]
+* "Narrator Of Myths" from ''Mysteries'' (2023) – [https://open.spotify.com/track/6AYSSw1kVKDDzWVYMyxafv Spotify] | [https://francium223.bandcamp.com/track/narrator-of-myths Bandcamp] | [https://www.youtube.com/watch?v=fNmEjgVonHk YouTube]
 [[Category:Listen]]