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→Music: Add Bryan Deister's ''74edo improv'' (2026); convert 1 microtonal cover by Bryan Deister to Modern Renderings format
 
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74edo 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 & 74 temperament with half-octave period and two parallel tracks of meantone.
74edo 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 & 74 temperament with half-octave period and two parallel tracks of meantone.


74edo 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.
74edo 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 ===
=== Odd harmonics ===
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=== Subsets and supersets ===
=== 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 ==
== Intervals ==
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== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould notation ===
[[Ups and downs notation]] uses free-standing arrows, but the arrows can be connected to conventional accidentals using [[Helmholtz–Ellis notation|Helmholtz–Ellis]] accidentals:
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5}}
{{Sharpness-sharp5-szg}}
Here, a sharp raises by five steps, and a flat lowers by five steps, so single and double arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with triple arrows.
 
=== Kite's ups and downs notation ===
74edo 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}}


=== Sagittal notation ===
=== Sagittal notation ===
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</imagemap>
</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 ==
== Music ==
=== Modern renderings ===
=== Modern renderings ===
; Home (Randy Goffe)
* [https://www.youtube.com/shorts/dB9_3TC3yPo ''Resonance''] (2014) – microtonal cover in 74edo by [[Bryan Deister]] (2026)
; {{W|Scott Joplin}}
; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=QBqzUWr6gXk ''Maple Leaf Rag''] (1899) – rendered by Francium (2024)
* [https://www.youtube.com/watch?v=QBqzUWr6gXk ''Maple Leaf Rag''] (1899) – rendered by Francium (2024)
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; [[Claudi Meneghin]]
; [[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]
* ''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/vi-KljzPkGs ''74edo improv''] (2026)


[[Category:Meantone]]
[[Category:Meantone]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Historical]]
[[Category:Historical]]
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