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74edo: Difference between revisions

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'''74edo''' divides the [[octave]] into 74 equal parts of size 16.216 [[cent]]s each. It is most notable as a [[meantone]] tuning, tempering out [[81/80]] in the [[5-limit]]; 81/80 and [[126/125]] (and hence [[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 [[Meantone_family|13-limit meantone]], aka 13-limit [[huygens]], for which 74edo gives the [[optimal patent val]]; and discarding 144/143 gives a 13-limit 62&74 temperament with half-octave period and two parallel tracks of meantone.
{{Infobox ET}}
{{ED intro}}


74 tunes 11 only 1/30 of a cent sharp, and 13 2.7 cents sharp, making it a distinctly interesting choice for higher-limit meantone.
== Theory ==
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.


[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 Twinkle canon – 74 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin] {{dead link}}
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 ===
{{Harmonics in equal|74}}
 
=== Subsets and supersets ===
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 ==
{{Interval table}}
 
== Notation ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5-szg}}
 
=== 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 ===
==== Evo flavor ====
<imagemap>
File:74-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 685 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 180 106 [[1701/1664]]
rect 180 80 300 106 [[36/35]]
rect 300 80 460 106 [[1053/1024]]
default [[File:74-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:74-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 180 106 [[1701/1664]]
rect 180 80 300 106 [[36/35]]
rect 300 80 460 106 [[1053/1024]]
default [[File:74-EDO_Revo_Sagittal.svg]]
</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.
 
== 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 ==
=== Modern renderings ===
; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=QBqzUWr6gXk ''Maple Leaf Rag''] (1899) – rendered by Francium (2024)
* [https://www.youtube.com/watch?v=oDTF5h9tsSU ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)
 
=== 21st century ===
; [[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:Equal divisions of the octave]]
[[Category:74edo| ]] <!-- main article -->
[[Category:Meantone]]
[[Category:Meantone]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Historical]]
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