109edo: Difference between revisions

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~~==Theory==~~

109edo [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is a strong 2.5.7.11.19.23.31 subgroup tuning, with errors of less than 10% on all harmonics ~~and an exceptionally small error of 0~~.~~16% on the 7th harmonic~~.

== Theory ==

109edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log<sub>2</sub>7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2254, 2300/2299, 6860/6859, 10241/10240.

=== Prime harmonics ===

=== Subsets and supersets ===

109edo is the 29th [[prime ~~EDO~~]].

109edo is the 29th [[prime edo]], following [[107edo]] and before [[113edo]]. [[436edo]], which slices each step of 109edo in four, provides correction for the approximation to harmonic 3.

=== Nonoctave temperaments ===

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[[~~Category~~:~~Equal divisions of the octave~~|~~###~~]]<!-~~- 3-digit number -->~~

== Music ==

[[~~Category:Prime EDO~~]]

; [[Francium]]

* "Teenagerges" from ''TOTMC September to December 2024'' (2024) – [https://open.spotify.com/track/4oQglJSEyp6CsL5RNWuiBy Spotify] | [https://francium223.bandcamp.com/track/teenagerges Bandcamp] | [https://www.youtube.com/watch?v=v_J71U392_k YouTube] – in Tetracot[13], 109edo tuning

* "Catbabel" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/0T7nW3ziEFvjV8c7v1EaMB Spotify] | [https://francium223.bandcamp.com/track/catbabel Bandcamp] | [https://www.youtube.com/watch?v=gtnTdPqiTDQ YouTube]

== See also ==

* [[109-7-comma]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|109}}
+{{ED intro}}
-==Theory==
-edo [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.
-edo is a strong 2.5.7.11.19.23.31 subgroup tuning, with errors of less than 10% on all harmonics and an exceptionally small error of 0.16% on the 7th harmonic.
+== Theory ==
+edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.
+edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log<sub>2</sub>7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2254, 2300/2299, 6860/6859, 10241/10240.
 === Prime harmonics ===
-{{harmonics in equal|109}}
+{{Harmonics in equal|109|columns=16}}
 === Subsets and supersets ===
-edo is the 29th [[prime EDO]].
+edo is the 29th [[prime edo]], following [[107edo]] and before [[113edo]]. [[436edo]], which slices each step of 109edo in four, provides correction for the approximation to harmonic 3.
 === Nonoctave temperaments ===
@@ Line 17: / Line 19: @@
 {{Interval table}}
-[[Category:Equal divisions of the octave|###]]<!-- 3-digit number -->
+== Music ==
-[[Category:Prime EDO]]
+; [[Francium]]
+* "Teenagerges" from ''TOTMC September to December 2024'' (2024) – [https://open.spotify.com/track/4oQglJSEyp6CsL5RNWuiBy Spotify] | [https://francium223.bandcamp.com/track/teenagerges Bandcamp] | [https://www.youtube.com/watch?v=v_J71U392_k YouTube] – in Tetracot[13], 109edo tuning
+* "Catbabel" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/0T7nW3ziEFvjV8c7v1EaMB Spotify] | [https://francium223.bandcamp.com/track/catbabel Bandcamp] | [https://www.youtube.com/watch?v=gtnTdPqiTDQ YouTube]
+== See also ==
+* [[109-7-comma]]