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| In this table, dup is equivalent to quidsharp, trup is equivalent to quudsharp, trudsharp is equivalent to quup, dudsharp is equivalent to quip, etc. | | In this table, dup is equivalent to quidsharp, trup is equivalent to quudsharp, trudsharp is equivalent to quup, dudsharp is equivalent to quip, etc. |
| {{sharpness-sharp7a}} | | {{Ups and downs sharpness}} |
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| Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used: | | Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used: |
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| == Octave stretch or compression == | | == Octave stretch or compression == |
| 42edo’s inaccurate 3rd and 5th harmonics can be greatly improved through [[stretched and compressed tuning|stretching or compressing]] octaves. Both approaches work about equally well but in opposite directions, giving two quite different flavors of tuning to play with. | | 42edo’s inaccurate 3rd and 5th harmonics can be improved through [[stretched and compressed tuning|stretching or compressing]] octaves. Both approaches work about equally well but in opposite directions, giving two quite different flavors of tuning to play with. |
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| What follows is a comparison of stretched- and compressed-octave 42edo tunings.
| | * Good stretched options: [[ed6|108ed6]], [[ed5|97ed5]], [[zpi|189zpi]], [[ed12|150ed12]] |
| | | * Good compressed options: [[ed7|118ed7]], [[ed12|151ed12]], [[ed6|109ed6]], [[zpi|191zpi]] |
| ; [[ed6|108ed6]]
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| * Octave size: 1206.3{{c}}
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| Stretching the octave of 42edo by around 6{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. This approximates all harmonics up to 16 within 13.3{{c}}. The tuning 108ed6 does this. So does the tuning [[ed5|97ed5]] whose octave differs by only 0.1{{c}}.
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| {{Harmonics in equal|108|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 108ed6}}
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| {{Harmonics in equal|108|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 108ed6 (continued)}}
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| ; [[zpi|189zpi]]
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| * Step size: 28.689{{c}}, octave size: 1204.9{{c}}
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| Stretching the octave of 42edo by around 5{{c}} results in improved primes 3, 5 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.9{{c}}. The tuning 189zpi does this.
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| {{Harmonics in cet|28.689|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 189zpi}}
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| {{Harmonics in cet|28.689|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 189zpi (continued)}}
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| ; [[ed12|150ed12]]
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| * Octave size: 1204.5{{c}} | |
| Stretcing the octave of 42edo by around 4.5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 13.6{{c}}. The tuning 150ed12 does this.
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| {{Harmonics in equal|150|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 150ed12}}
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| {{Harmonics in equal|150|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 150ed12 (continued)}}
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| ; [[equal tuning|145ed11]]
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| * Octave size: 1202.5{{c}}
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| Stretching the octave of 42edo by around 2.5{{c}} results in improved primes 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 11.9{{c}}. The tuning 145ed11 does this.
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| {{Harmonics in equal|145|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 145ed11}}
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| {{Harmonics in equal|145|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 145ed11 (continued)}}
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| ; 42edo
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| * Step size: 28.571{{c}}, octave size: 1200.0{{c}}
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| Pure-octaves 42edo approximates all harmonics up to 16 within 13.7{{c}}. The tuning [[zpi|190zpi]] is almost exactly the same as pure-octaves 42edo, its octave differing by less than 0.05{{c}}.
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| {{Harmonics in equal|42|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42edo}}
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| {{Harmonics in equal|42|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42edo (continued)}}
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| ; [[ed7|118ed7]]
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| * Step size: Octave size: 1199.1{{c}}
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| Compressing the octave of 42edo by around 1{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 13.2{{c}}. The tuning 118ed7 does this.
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| {{Harmonics in equal|118|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 118ed7}}
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| {{Harmonics in equal|118|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 118ed7 (continued)}}
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| ; [[WE|42et, 13-limit WE tuning]]
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| * Step size: 28.534{{c}}, octave size: 1198.4{{c}}
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| Compressing the octave of 42edo by around 1.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
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| Of the tunings discussed in this section, 13-limit WE and TE are the only ones to approximate all harmonics up to 10 within 10 cents, making them a good all-round choice.
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| {{Harmonics in cet|28.534|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42et, 13-limit WE tuning}}
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| {{Harmonics in cet|28.534|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42et, 13-limit WE tuning (continued)}}
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| ; [[ed12|151ed12]]
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| * Step size: Octave size: 1196.6{{c}}
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| Compressing the octave of 42edo by around 3.5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 13.7{{c}}. The tuning 151ed12 does this. So do the 7-limit [[WE]] and [[TE]] tunings of 42et, whose octaves are within 0.3{{c}} of 151ed12.
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| {{Harmonics in equal|151|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 151ed12}}
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| {{Harmonics in equal|151|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 151ed12 (continued)}}
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| ; [[ed6|109ed6]]
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| * Octave size: 1195.2{{c}}
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| Compressing the octave of 42edo by around 5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 14.2{{c}}. The tuning 109ed6 does this.
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| {{Harmonics in equal|109|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 109ed6}}
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| {{Harmonics in equal|109|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 109ed6 (continued)}}
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| ; [[zpi|191zpi]]
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| * Step size: 28.444{{c}}, octave size: 1194.6{{c}}
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| Compressing the octave of 42edo by around 5.5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 12.4{{c}}. The tuning 191zpi does this.
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| {{Harmonics in cet|28.444|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 191zpi}}
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| {{Harmonics in cet|28.444|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 191zpi (continued)}}
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| ; [[67edt]]
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| * Step size: 28.387{{c}}, octave size: 1192.3{{c}}
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| Compressing the octave of 42edo by around 7.5{{c}} results in improved primes 3, 5 and 11, but worse primes 2 and 7. This approximates all harmonics up to 16 within 12.9{{c}}. The tuning 67edt does this.
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| {{Harmonics in equal|67|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 67edt}}
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| {{Harmonics in equal|67|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 67edt (continued)}}
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| == Scales == | | == Scales == |
| | ; [[MOS scale]]s |
| {{main|List of MOS scales in 42edo}} | | {{main|List of MOS scales in 42edo}} |
| ; [[MOS scale]]s
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| * Eugene/Tritikleismic[9]: '''3 8 3 3 8 3 3 8 3''' | | * Eugene/Tritikleismic[9]: '''3 8 3 3 8 3 3 8 3''' |
| * Eugene/Tritikleismic[15]: '''3 3 2 3 3 3 3 2 3 3 3 3 2 3 3''' | | * Eugene/Tritikleismic[15]: '''3 3 2 3 3 3 3 2 3 3 3 3 2 3 3''' |
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| * 7-tone pelog: 4 5 9 6 3 10 5 | | * 7-tone pelog: 4 5 9 6 3 10 5 |
| * 5-tone slendro: 8 9 8 9 8 | | * 5-tone slendro: 8 9 8 9 8 |
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| | ; Other scales |
| | * 12-tone 6&7edo scale: 6 1 5 2 4 3 3 4 2 5 1 6 |
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| == Instruments == | | == Instruments == |
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| * [https://www.youtube.com/watch?v=PJw8gZyNPjg ''improv 42edo''] (2023) | | * [https://www.youtube.com/watch?v=PJw8gZyNPjg ''improv 42edo''] (2023) |
| * [https://www.youtube.com/watch?v=ljaSpsQP2qc ''Improvisation in 42edo''] (2023), transcribed by [[Stephen Weigel]] (2024) | | * [https://www.youtube.com/watch?v=ljaSpsQP2qc ''Improvisation in 42edo''] (2023), transcribed by [[Stephen Weigel]] (2024) |
| | * [https://www.youtube.com/watch?v=cL6CY3U9mHM ''42edo groove''] (2025) |
| | * ''A Hunger Awakes - 42edo'' (2026) |
| | ** [https://www.youtube.com/shorts/B90JT_SxSSE <nowiki>[short]</nowiki>] (Lumatone view) |
| | ** [https://www.youtube.com/watch?v=VwHqWffglj4 <nowiki>[full version]</nowiki>] (music video with stop-motion by [[Jelly Eyes]]) |
| | * ''Waltz in 42edo'' (2026) |
| | ** [https://www.youtube.com/shorts/D_YgzRJFg8I <nowiki>[short]</nowiki>] (Lumatone view) |
| | ** [https://www.youtube.com/watch?v=QyglWQ_0bIk <nowiki>[full version]</nowiki>] |
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| ; [[James Kukula]] | | ; [[James Kukula]] |
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| * [https://www.youtube.com/watch?v=ORy7nv6SnN8 ''Glory of Them''] (2024) | | * [https://www.youtube.com/watch?v=ORy7nv6SnN8 ''Glory of Them''] (2024) |
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| [[Category:Augene]] | | ; [[Stephen Weigel]] |
| | * [https://www.youtube.com/watch?v=tLmaQK10aYM ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!''] (2025; mostly in 42edo, but also some in 40edo) |
| | ** [https://www.youtube.com/watch?v=NE77rwCsGHw live performance of the above in Munich, Germany] (2026) |
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| | [[Category:Augmented]] |
| {{Todo|review|add rank 2 temperaments table}} | | {{Todo|review|add rank 2 temperaments table}} |