72edo: Difference between revisions
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The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33. | The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33. | ||
72et is the only 11-limit regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits [[25/24]] into two equal [[49/48]][[~]][[50/49]]'s, and splits [[28/27]] into two equal [[55/54]]~[[56/55]]'s. It is also an excellent tuning for [[miracle]] temperament, especially the 11-limit version, and the related rank-3 temperament [[prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]]. | |||
The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 72edo could be treated as a 2.3.5.7.11.ϕ.17 temperament. | The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 72edo could be treated as a 2.3.5.7.11.ϕ.17 temperament. | ||
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{{Harmonics in equal|72|columns=9}} | {{Harmonics in equal|72|columns=9}} | ||
{{Harmonics in equal|72|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}} | {{Harmonics in equal|72|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}} | ||
=== Octave stretch === | |||
72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies. | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 72 factors into 2<sup>3</sup> × 3<sup>2</sup>, 72edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36 }}. [[144edo]], which doubles it, provides a possible correction to its approximate harmonic 13. | Since 72 factors into primes as {{nowrap| 2<sup>3</sup> × 3<sup>2</sup> }}, 72edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36 }}. [[144edo]], which doubles it, provides a possible correction to its approximate harmonic 13. | ||
== Intervals == | == Intervals == | ||
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| 20 | | 20 | ||
| 333.3 | | 333.3 | ||
| 17/14, 40/33 | | 17/14, 39/32, 40/33 | ||
| ^^m3, v~3 | | ^^m3, v~3 | ||
| dupminor 3rd, downmid 3rd | | dupminor 3rd, downmid 3rd | ||
| Line 395: | Line 398: | ||
| 32 | | 32 | ||
| 533.3 | | 533.3 | ||
| 15/11, 19/14 | | 15/11, 19/14, ''26/19'' | ||
| ^^4, v~4 | | ^^4, v~4 | ||
| dup 4th, downmid 4th | | dup 4th, downmid 4th | ||
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| 40 | | 40 | ||
| 666.7 | | 666.7 | ||
| 22/15, 28/19 | | ''19/13'', 22/15, 28/19 | ||
| vv5, ^~5 | | vv5, ^~5 | ||
| dud 5th, upmid 5th | | dud 5th, upmid 5th | ||
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| 52 | | 52 | ||
| 866.7 | | 866.7 | ||
| 28/17, 33/20 | | 28/17, 33/20, 64/39 | ||
| ^~6, vvM6 | | ^~6, vvM6 | ||
| upmid 6th, dudmajor 6th | | upmid 6th, dudmajor 6th | ||
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| D | | D | ||
|} | |} | ||
<references group="note" /> | |||
=== Interval quality and chord names in color notation === | === Interval quality and chord names in color notation === | ||
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| C dupmajor or C dup | | C dupmajor or C dup | ||
|} | |} | ||
For a more complete list, see [[Ups and | For a more complete list, see [[Ups and downs notation #Chord names in other EDOs]]. | ||
=== Relationship between primes and rings === | === Relationship between primes and rings === | ||
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{{Sharpness-sharp6-iw|72}} | {{Sharpness-sharp6-iw|72}} | ||
== JI | == Approximation to JI == | ||
[[File:72ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected intervals approximated in 72edo]] | [[File:72ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected intervals approximated in 72edo]] | ||
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[[File:plot72.png|alt=plot72.png|plot72.png]] | [[File:plot72.png|alt=plot72.png|plot72.png]] | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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| Jacobin comma | | Jacobin comma | ||
|} | |} | ||
<references group="note" /> | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
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| 383.3<br>(16.7) | | 383.3<br>(16.7) | ||
| 5/4<br>(81/80) | | 5/4<br>(81/80) | ||
| [[ | | [[Gamelstearn]] | ||
|} | |} | ||
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct | ||
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== Instruments == | == Instruments == | ||
If one can get six 12edo instruments tuned a twelfth-tone apart, it is possible to use these instruments in combination to play the full gamut of 72edo (see Music). | |||
One can also use a skip fretting system: | |||
* [[Skip fretting system 72 2 27]] | |||
Alternatively, an appropriately mapped keyboard of sufficient size is usable for playing 72edo: | Alternatively, an appropriately mapped keyboard of sufficient size is usable for playing 72edo: | ||
* [[Lumatone mapping for 72edo]] | |||
== Music == | == Music == | ||
; [[Bryan Deister]] | |||
* [https://www.youtube.com/shorts/VwVp3RVao_k ''microtonal improvisation in 72edo''] (2025) | |||
; [[Ambient Esoterica]] | ; [[Ambient Esoterica]] | ||
* [https://www.youtube.com/watch?v=seWcDAoQjxY ''Goetic Synchronities''] (2023) | * [https://www.youtube.com/watch?v=seWcDAoQjxY ''Goetic Synchronities''] (2023) | ||
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* [https://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo] | * [https://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo] | ||
* [http://tonalsoft.com/enc/number/72edo.aspx 72-ed2 / 72-edo / 72-ET / 72-tone equal-temperament] on [[Tonalsoft Encyclopedia]] | * [http://tonalsoft.com/enc/number/72edo.aspx 72-ed2 / 72-edo / 72-ET / 72-tone equal-temperament] on [[Tonalsoft Encyclopedia]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||