74edo: Difference between revisions

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 edo is most notable as a [[meantone]] tuning, [[tempering out]] [[81/80]] in the [[5-limit]]; [[126/125]] and [[225/224]] in the [[7-limit]]; [[99/98]], [[176/175]] and [[441/440]] in the [[11-limit]]; and [[144/143]] and [[847/845]] in the [[13-limit]]. Discarding 847/845 from that gives the 13-limit meantone extension [[grosstone]], for which 74edo gives the [[optimal patent val]]; and discarding 144/143 gives [[semimeantone]], a 13-limit 62 &amp; 74 temperament with half-octave period and two parallel tracks of meantone.
-edo tunes [[harmonic]] [[11/1|11]] only 1/30 of a cent sharp, and [[13/1|13]] 2.7 cents sharp, making it a distinctly interesting choice for higher-limit meantone.
+edo tunes [[harmonic]] [[11/1|11]] only 1/30 of a cent sharp, and [[13/1|13]] only 2.7 cents sharp, making it a distinctly interesting choice for higher-limit meantone.
 === Odd harmonics ===
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
-Since 74 factors into {{factorization|74}}, 74edo contains [[2edo]] and [[37edo]] as its subsets.
+Since 74 factors into {{nowrap| 2 × 37 }}, 74edo contains [[2edo]] and [[37edo]] as its subsets; of these, 37edo has the same highly accurate prime harmonics in the no-3s [[13-limit]].
 == Intervals ==
@@ Line 17: / Line 17: @@
 == Notation ==
-===Ups and downs notation===
+=== Stein–Zimmermann–Gould notation ===
-edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
+[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
+{{Sharpness-sharp5-szg}}
+=== Kite's ups and downs notation ===
+edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
 {{Sharpness-sharp5a}}
-Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. It uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
-{{Sharpness-sharp5}}
 === Sagittal notation ===
 ==== Evo flavor ====
@@ Line 47: / Line 50: @@
 </imagemap>
-In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
+In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation #Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.
+== Regular temperament properties ==
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[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| -117 74 }}
+| {{Mapping| 74 117 }}
+| +1.469
+| 1.47
+| 9.06
+|-
+| 2.3.5
+| 81/80, {{monzo| 41 2 -19 }}
+| {{Mapping| 74 117 172 }}
+| +0.564
+| 1.75
+| 10.80
+|-
+| 2.3.5.7
+| 81/80, 126/125, 4194304/4117715
+| {{Mapping| 74 117 172 208 }}
+| +0.053
+| 1.76
+| 10.83
+|-
+| 2.3.5.7.11
+| 81/80, 99/98, 126/125, 65536/65219
+| {{Mapping| 74 117 172 208 256 }}
+| +0.041
+| 1.57
+| 9.69
+|-
+| 2.3.5.7.11.13
+| 81/80, 99/98, 126/125, 144/143, 847/845
+| {{Mapping| 74 117 172 208 256 274 }}
+| −0.089
+| 1.46
+| 9.02
+|}
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br>per 8ve
+! Generator*
+! Cents*
+! Associated<br>ratio*
+! Temperaments
+|-
+| 1
+| 7\74
+| 113.5
+| 16/15
+| [[Misneb]]
+|-
+| 1
+| 31\74
+| 502.7
+| 4/3
+| [[Meantone]]
+|-
+| 1
+| 33\74
+| 535.1
+| 15/11
+| [[Maquila]]
+|-
+| 2
+| 10\74
+| 162.2
+| 11/10
+| [[Gwazy]]
+|-
+| 2
+| 20\74<br>(17\74)
+| 324.3<br>(275.7)
+| 77/64<br>(7/6)
+| [[Orphic]]
+|-
+| 2
+| 31\74<br>(6\74)
+| 502.7<br>(97.3)
+| 4/3<br>(55/52)
+| [[Semimeantone]]
+|-
+| 37
+| 31\74<br>(1\74)
+| 502.7<br>(16.2)
+| 4/3<br>(121/120)
+| [[Rubidium]]
+|}
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
+== Instruments ==
+* [[Lumatone mapping for 74edo]]
 == Music ==
@@ Line 58: / Line 166: @@
 ; [[Claudi Meneghin]]
 * ''Twinkle canon'' (2012) – [https://web.archive.org/web/20171009205013/http://soonlabel.com/xenharmonic/archives/573 detail] | [https://web.archive.org/web/20201127015514/http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 play]
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/ylOGUb395Gg ''microtonal improvisation in 74edo''] (2025)
+* [https://www.youtube.com/shorts/dB9_3TC3yPo ''Resonance - Home (microtonal cover in 74edo)''] (2026)
 [[Category:Meantone]]
 [[Category:Listen]]
 [[Category:Historical]]