175edo: Difference between revisions
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== Theory == | |||
175et [[tempering out|tempers out]] [[225/224]] and [[1029/1024]], so that it [[support]]s 7-limit [[miracle]], and in fact provides an excellent alternative to [[72edo]] for 7-limit miracle with improved [[5/1|5]] and [[7/1|7]] at the cost of a slightly flatter [[3/1|3]]. In the 11-limit, it tempers out [[243/242]], [[385/384]], [[441/440]] and [[540/539]], and supports 11-limit miracle. In the 13-limit, the 175f val, {{val| 175 277 406 491 605 '''647''' }} tempers out [[351/350]] just as 72 does, and provides a tuning for [[benediction]] temperament very close to the POTE tuning. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|175}} | |||
=== Subsets and supersets === | |||
Since 175 factors into {{factorization|175}}, 175edo has subset edos {{EDOs| 5, 7, 25, and 35 }}. | |||
== Music == | == Music == | ||
* [http://www.archive.org/details/RachmaninoffPlaysBlackjack | ; [[Gene Ward Smith]] | ||
* ''Rachmaninoff Plays Blackjack'' (archived 2010) – [http://www.archive.org/details/RachmaninoffPlaysBlackjack detail] | [http://www.archive.org/download/RachmaninoffPlaysBlackjack/rachman.mp3 play] – [[Blackjack]] in 175edo tuning | |||
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Latest revision as of 14:06, 20 February 2025
| ← 174edo | 175edo | 176edo → |
175 equal divisions of the octave (abbreviated 175edo or 175ed2), also called 175-tone equal temperament (175tet) or 175 equal temperament (175et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 175 equal parts of about 6.86 ¢ each. Each step represents a frequency ratio of 21/175, or the 175th root of 2.
Theory
175et tempers out 225/224 and 1029/1024, so that it supports 7-limit miracle, and in fact provides an excellent alternative to 72edo for 7-limit miracle with improved 5 and 7 at the cost of a slightly flatter 3. In the 11-limit, it tempers out 243/242, 385/384, 441/440 and 540/539, and supports 11-limit miracle. In the 13-limit, the 175f val, ⟨175 277 406 491 605 647] tempers out 351/350 just as 72 does, and provides a tuning for benediction temperament very close to the POTE tuning.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.53 | -2.31 | -1.97 | +1.80 | -2.75 | +2.90 | +2.02 | -2.10 | -2.66 | +2.36 | +2.58 |
| Relative (%) | -36.8 | -33.7 | -28.7 | +26.3 | -40.1 | +42.3 | +29.4 | -30.6 | -38.7 | +34.4 | +37.7 | |
| Steps (reduced) |
277 (102) |
406 (56) |
491 (141) |
555 (30) |
605 (80) |
648 (123) |
684 (159) |
715 (15) |
743 (43) |
769 (69) |
792 (92) | |
Subsets and supersets
Since 175 factors into 52 × 7, 175edo has subset edos 5, 7, 25, and 35.