107edo: Difference between revisions

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'''107edo''' is the [[EDO|equal division of the octave]] into 107 parts of 11.214953271 cents each. It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 7-limit: ~~<~~107 170 248 300| (patent val), ~~<~~107 169 248 300| (107b), ~~<~~107 170 249 300| (107c), and ~~<~~107 170 249 301| (107cd). Using the patent val, it tempers out the [[Magic family|small diesis]], 3125/3072 and 33554432000/31381059609 in the 5-limit; 1029/1024, 2240/2187, and 3125/3087 in the 7-limit; 100/99, 1232/1215, and 1331/1323 in the 11-limit. Using the 107b val, it tempers out the [[syntonic comma]], 81/80 and |-61 -1 27~~&gt~~; in the 5-limit; 2401/2400, 2430/2401, and 234375/229376 in the 7-limit; 385/384, 1350/1331, 1375/1372, and 1944/1925 in the 11-limit. Using the 107c val, it tempers out the immunity comma, 1638400/1594323 and the valentine comma, 1990656/1953125 in the 5-limit; 126/125, 1029/1024, and 307328/295245 in the 7-limit; 121/120, 176/175, 441/440, and 184877/177147 in the 11-limit. Using the 107cd val, it tempers out 1728/1715, 4000/3969, and 28672/28125 in the 7-limit; 121/120, 896/891, 1375/1372, and 3168/3125 in the 11-limit.

'''107edo''' is the [[EDO|equal division of the octave]] into 107 parts of 11.214953271 cents each. It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 7-limit: {{val|107 170 248 300}} (patent val), {{val|107 169 248 300}} (107b), {{val|107 170 249 300}} (107c), and {{val|107 170 249 301}} (107cd). Using the patent val, it tempers out the [[Magic family|small diesis]], [[3125/3072]] and 33554432000/31381059609 in the 5-limit; [[1029/1024]], 2240/2187, and 3125/3087 in the 7-limit; [[100/99]], 1232/1215, and 1331/1323 in the 11-limit. Using the 107b val, it tempers out the [[syntonic comma]], [[81/80]] and {{monzo|-61 -1 27}}; in the 5-limit; [[2401/2400]], [[2430/2401]], and 234375/229376 in the 7-limit; [[385/384]], 1350/1331, 1375/1372, and 1944/1925 in the 11-limit. Using the 107c val, it tempers out the immunity comma, 1638400/1594323 and the valentine comma, 1990656/1953125 in the 5-limit; [[126/125]], [[1029/1024]], and 307328/295245 in the 7-limit; [[121/120]], [[176/175]], [[441/440]], and 184877/177147 in the 11-limit. Using the 107cd val, it tempers out [[1728/1715]], 4000/3969, and 28672/28125 in the 7-limit; 121/120, [[896/891]], 1375/1372, and 3168/3125 in the 11-limit.

It is the 28th [[prime edo]].

==Theory==

Since 107edo has a step of 11.214953271 cents, it also allows one to use its MOS scales as circulating temperaments.

{| class="wikitable"

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|85

|22L 63s

|}

==Regular temperament properties==

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

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

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

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

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

! colspan="2" |Tuning Error

|-

![[TE error|Absolute]] (¢)

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

|-

|2.3

|{{monzo|170 -107}}

|{{val|107 170}}

| -1.4471

| 1.4453

| 12.89

|-

|2.3.5

|3125/3072, {{monzo|18 -23 8}}

|{{val|107 170 248}}

| -0.2497

| 2.0685

| 18.44

|-

|2.3.5.7

|2240/2187, 1029/1024, 3125/3072

|{{val|107 170 248 300}}

| +0.1987

| 1.9529

| 17.41

|-

|2.3.5.7.11

|100/99, 1232/1215, 1375/1372, 1029/1024

|{{val|107 170 248 300 370}}

| +0.2622

| 1.7513

| 15.62

|-

|2.3.5.7.11.13

|100/99, 196/195, 275/273, 1232/1215, 1029/1024

|{{val|107 170 248 300 370 396}}

| +0.1917

| 1.6065

| 14.32

|-

|2.3.5.7.11.13.17

|100/99, 196/195, 136/135, 275/273, 1232/1215, 1547/1536

|{{val|107 170 248 300 370 396}}

| +0.3048

| 1.5129

| 13.49

|}

[[Category:Equal divisions of the octave|###]]

[[Category:Prime EDO]]

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 {{Infobox ET}}
-'''107edo''' is the [[EDO|equal division of the octave]] into 107 parts of 11.214953271 cents each. It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 7-limit: &lt;107 170 248 300| (patent val), &lt;107 169 248 300| (107b), &lt;107 170 249 300| (107c), and &lt;107 170 249 301| (107cd). Using the patent val, it tempers out the [[Magic family|small diesis]], 3125/3072 and 33554432000/31381059609 in the 5-limit; 1029/1024, 2240/2187, and 3125/3087 in the 7-limit; 100/99, 1232/1215, and 1331/1323 in the 11-limit. Using the 107b val, it tempers out the [[syntonic comma]], 81/80 and |-61 -1 27&gt; in the 5-limit; 2401/2400, 2430/2401, and 234375/229376 in the 7-limit; 385/384, 1350/1331, 1375/1372, and 1944/1925 in the 11-limit. Using the 107c val, it tempers out the immunity comma, 1638400/1594323 and the valentine comma, 1990656/1953125 in the 5-limit; 126/125, 1029/1024, and 307328/295245 in the 7-limit; 121/120, 176/175, 441/440, and 184877/177147 in the 11-limit. Using the 107cd val, it tempers out 1728/1715, 4000/3969, and 28672/28125 in the 7-limit; 121/120, 896/891, 1375/1372, and 3168/3125 in the 11-limit.
+'''107edo''' is the [[EDO|equal division of the octave]] into 107 parts of 11.214953271 cents each. It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 7-limit: {{val|107 170 248 300}} (patent val), {{val|107 169 248 300}} (107b), {{val|107 170 249 300}} (107c), and {{val|107 170 249 301}} (107cd). Using the patent val, it tempers out the [[Magic family|small diesis]], [[3125/3072]] and 33554432000/31381059609 in the 5-limit; [[1029/1024]], 2240/2187, and 3125/3087 in the 7-limit; [[100/99]], 1232/1215, and 1331/1323 in the 11-limit. Using the 107b val, it tempers out the [[syntonic comma]], [[81/80]] and {{monzo|-61 -1 27}}; in the 5-limit; [[2401/2400]], [[2430/2401]], and 234375/229376 in the 7-limit; [[385/384]], 1350/1331, 1375/1372, and 1944/1925 in the 11-limit. Using the 107c val, it tempers out the immunity comma, 1638400/1594323 and the valentine comma, 1990656/1953125 in the 5-limit; [[126/125]], [[1029/1024]], and 307328/295245 in the 7-limit; [[121/120]], [[176/175]], [[441/440]], and 184877/177147 in the 11-limit. Using the 107cd val, it tempers out [[1728/1715]], 4000/3969, and 28672/28125 in the 7-limit; 121/120, [[896/891]], 1375/1372, and 3168/3125 in the 11-limit.
 It is the 28th [[prime edo]].
+==Theory==
 Since 107edo has a step of 11.214953271 cents, it also allows one to use its MOS scales as circulating temperaments.
 {| class="wikitable"
@@ Line 268: / Line 269: @@
 |85
 |22L 63s
+|}
+{{Harmonics in equal|107}}
+==Regular temperament properties==
+{| class="wikitable center-4 center-5 center-6"
+! rowspan="2" |[[Subgroup]]
+! rowspan="2" |[[Comma list|Comma List]]
+! rowspan="2" |[[Mapping]]
+! rowspan="2" |Optimal<br>8ve Stretch (¢)
+! colspan="2" |Tuning Error
+|-
+![[TE error|Absolute]] (¢)
+![[TE simple badness|Relative]] (%)
+|-
+|2.3
+|{{monzo|170 -107}}
+|{{val|107 170}}
+| -1.4471
+| 1.4453
+| 12.89
+|-
+|2.3.5
+|3125/3072, {{monzo|18 -23 8}}
+|{{val|107 170 248}}
+| -0.2497
+| 2.0685
+| 18.44
+|-
+|2.3.5.7
+|2240/2187, 1029/1024, 3125/3072
+|{{val|107 170 248 300}}
+| +0.1987
+| 1.9529
+| 17.41
+|-
+|2.3.5.7.11
+|100/99, 1232/1215, 1375/1372, 1029/1024
+|{{val|107 170 248 300 370}}
+| +0.2622
+| 1.7513
+| 15.62
+|-
+|2.3.5.7.11.13
+|100/99, 196/195, 275/273, 1232/1215, 1029/1024
+|{{val|107 170 248 300 370 396}}
+| +0.1917
+| 1.6065
+| 14.32
+|-
+|2.3.5.7.11.13.17
+|100/99, 196/195, 136/135, 275/273, 1232/1215, 1547/1536
+|{{val|107 170 248 300 370 396}}
+| +0.3048
+| 1.5129
+| 13.49
 |}
 [[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Prime EDO]]