318edo: Difference between revisions

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'''318edo''' is the '''318 equal division of the octave''' into equal parts of 3.774 cents each.

== ~~Mappings~~ ==

== Theory ==

~~318 = 2 ×~~ 3 ~~× 53~~, ~~and~~ 318edo is [[~~Contorsion~~|~~contorted~~]] ~~in both~~ the 3-~~limit and the 5-limit, sharing the same mappings with [[53edo]]~~, ~~with inconsistency developing for~~ the ~~3-limit as~~ a ~~result. Besides, it shares its representations~~ of ~~the 11th and 17th [[Overtone series|harmonics]] with [[159edo]]. However, compared to 159edo, the [[patent val]]s differ on the mappings for 7, 13, and 19~~.

At only slightly more than 3.5 cents, the step size of 318edo is really close to being [[unnoticeable comma|unnoticeable]] as is the case with other Mega-EDOs in this vicinity, so the steps themselves run a pretty high risk of blending completely into one another.

== ~~Commas~~ ==

=== Mappings ===

In the 5-limit, it tempers out the same commas as 53edo, including the [[32805/32768|schisma (32805/32768)]], the [[15625/15552|kleisma (15625/15552)]], the [[amity comma|amity comma (1600000/1594323)]], the [[semicomma|semicomma (2109375/2097152)]], the [[vulture comma|vulture comma (10485760000/10460353203)]], etc. In the 7-limit it tempers out the stearnsma (118098/117649) and 589824/588245. In the 11-limit it tempers out the swetisma (540/539), the [[4000/3993|wizardharry (4000/3993)]], the [[9801/9800|kalisma (9801/9800)]] and the [[nexuma|nexus comma (1771561/1769472)]]. In the 13-limit, 1575/1573, 2080/2079, the [[4096/4095|schismina (4096/4095)]], and the [[cantonisma|cantonisma (10985/10976)]] have been found to be tempered out by this EDO. It is of note that 318edo divides the [[4/3]] perfect fourth into two even halves through means other than tempering out the [[676/675|island comma (676/675)]], and that this EDO likewise divides the [[2187/2048|apotome (2187/2048)]] in half by means other than tempering out the [[243/242|rastma (243/242)]].

318 = 2 × 3 × 53, and 318edo is [[Contorsion|contorted]] in both the 3-limit and the 5-limit, sharing the same mappings with [[53edo]], with inconsistency developing for the 3-limit as a result, albeit only late in the [[circle of fifths]]. Besides, it shares its representations of the 11th and 17th [[Overtone series|harmonics]] with [[159edo]]. However, compared to 159edo, the [[patent val]]s differ on the mappings for 7, 13, and 19.

=== Commas ===

In the 5-limit, it tempers out the same commas as 53edo, including the [[32805/32768|schisma (32805/32768)]], the [[15625/15552|kleisma (15625/15552)]], the [[amity comma|amity comma (1600000/1594323)]], the [[semicomma|semicomma (2109375/2097152)]], the [[vulture comma|vulture comma (10485760000/10460353203)]], etc. In the 7-limit it tempers out the stearnsma (118098/117649) and 589824/588245. In the 11-limit it tempers out the swetisma (540/539), the [[4000/3993|wizardharry (4000/3993)]], the [[9801/9800|kalisma (9801/9800)]] and the [[nexuma|nexus comma (1771561/1769472)]]. In the 13-limit, 1575/1573, 2080/2079, the [[4096/4095|schismina (4096/4095)]], and the [[cantonisma|cantonisma (10985/10976)]] have been found to be tempered out by this EDO.

[[Category:Equal divisions of the octave]]

@@ Line 1: / Line 1: @@
 '''318edo''' is the '''318 equal division of the octave''' into equal parts of 3.774 cents each.
-== Mappings ==
+== Theory ==
-= 2 × 3 × 53, and 318edo is [[Contorsion|contorted]] in both the 3-limit and the 5-limit, sharing the same mappings with [[53edo]], with inconsistency developing for the 3-limit as a result. Besides, it shares its representations of the 11th and 17th [[Overtone series|harmonics]] with [[159edo]]. However, compared to 159edo, the [[patent val]]s differ on the mappings for 7, 13, and 19.
+At only slightly more than 3.5 cents, the step size of 318edo is really close to being [[unnoticeable comma|unnoticeable]] as is the case with other Mega-EDOs in this vicinity, so the steps themselves run a pretty high risk of blending completely into one another.
-== Commas ==
+=== Mappings ===
-In the 5-limit, it tempers out the same commas as 53edo, including the [[32805/32768|schisma (32805/32768)]], the [[15625/15552|kleisma (15625/15552)]], the [[amity comma|amity comma (1600000/1594323)]], the [[semicomma|semicomma (2109375/2097152)]], the [[vulture comma|vulture comma (10485760000/10460353203)]], etc. In the 7-limit it tempers out the stearnsma (118098/117649) and 589824/588245. In the 11-limit it tempers out the swetisma (540/539), the [[4000/3993|wizardharry (4000/3993)]], the [[9801/9800|kalisma (9801/9800)]] and the [[nexuma|nexus comma (1771561/1769472)]]. In the 13-limit, 1575/1573, 2080/2079, the [[4096/4095|schismina (4096/4095)]], and the [[cantonisma|cantonisma (10985/10976)]] have been found to be tempered out by this EDO.  It is of note that 318edo divides the [[4/3]] perfect fourth into two even halves through means other than tempering out the [[676/675|island comma (676/675)]], and that this EDO likewise divides the [[2187/2048|apotome (2187/2048)]] in half by means other than tempering out the [[243/242|rastma (243/242)]].
+= 2 × 3 × 53, and 318edo is [[Contorsion|contorted]] in both the 3-limit and the 5-limit, sharing the same mappings with [[53edo]], with inconsistency developing for the 3-limit as a result, albeit only late in the [[circle of fifths]]. Besides, it shares its representations of the 11th and 17th [[Overtone series|harmonics]] with [[159edo]]. However, compared to 159edo, the [[patent val]]s differ on the mappings for 7, 13, and 19.
+=== Commas ===
+In the 5-limit, it tempers out the same commas as 53edo, including the [[32805/32768|schisma (32805/32768)]], the [[15625/15552|kleisma (15625/15552)]], the [[amity comma|amity comma (1600000/1594323)]], the [[semicomma|semicomma (2109375/2097152)]], the [[vulture comma|vulture comma (10485760000/10460353203)]], etc. In the 7-limit it tempers out the stearnsma (118098/117649) and 589824/588245. In the 11-limit it tempers out the swetisma (540/539), the [[4000/3993|wizardharry (4000/3993)]], the [[9801/9800|kalisma (9801/9800)]] and the [[nexuma|nexus comma (1771561/1769472)]]. In the 13-limit, 1575/1573, 2080/2079, the [[4096/4095|schismina (4096/4095)]], and the [[cantonisma|cantonisma (10985/10976)]] have been found to be tempered out by this EDO.
 [[Category:Equal divisions of the octave]]