293edo: Difference between revisions
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== Theory == | == Theory == | ||
{{primes in edo|293|columns=14}} | {{primes in edo|293|columns=14}} | ||
293 edo does not approximate prime harmonics well all the way into the 41st. The first harmonic that it approximates within 1 standard deviation of one step is 43rd, which is 10 cents flat compared to the just intonated interval. | 293 edo does not approximate prime harmonics well all the way into the 41st, unless 30-relative cent errors are considered "well", in which case it equally represents all of them. The first harmonic that it approximates within 1 standard deviation of one step is 43rd, which is 10 cents flat compared to the just intonated interval. | ||
When it comes to the intervals that are not octave-reduced prime harmonics, some which are well-approximated are [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], and [[25/19]]. [[21/16]], which is a composite octave-reduced harmonic, is also well represented. These numbers are related to poor approximation of prime harmonics by cancelling out of the errors. For example, 19th and 17th harmoincs have +36 and +37 error respectively, which cancels out to | When it comes to the intervals that are not octave-reduced prime harmonics, some which are well-approximated are [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], and [[25/19]]. [[21/16]], which is a composite octave-reduced harmonic, is also well represented. These numbers are related to poor approximation of prime harmonics by cancelling out of the errors. For example, 19th and 17th harmoincs have +36 and +37 error respectively, which together cancels out to 1. | ||
One step of 293edo is at the edge of human pitch perception of 3.5 cents. When combined with low harmonicity, this opens 293edo to a wide range of interpretations. | One step of 293edo is at the edge of human pitch perception of 3.5 cents. When combined with low harmonicity, this opens 293edo to a wide range of interpretations. | ||
{| class="wikitable" | |||
|+Selected intervals | |||
!Degree | |||
!Name | |||
!Cents | |||
!Approximate ratios | |||
|- | |||
|0 | |||
|Unison, prime | |||
|0.0000 | |||
|1/1 exact | |||
|- | |||
|5 | |||
|Minor leap week interval | |||
| | |||
|85/84 | |||
|- | |||
|6 | |||
|Major leap week interval | |||
| | |||
|71/70 | |||
|- | |||
|11 | |||
|Bundle of 2 | |||
| | |||
| | |||
|- | |||
|17 | |||
|Bundle of 3 | |||
| | |||
| | |||
|- | |||
|18 | |||
|Vicesimotertial quarter-tone | |||
| | |||
|[[24/23]] | |||
|- | |||
|45 | |||
|Minor subcycle | |||
| | |||
| | |||
|- | |||
|47 | |||
|Undevicesimal meantone | |||
| | |||
|[[19/17]] | |||
|- | |||
|77 | |||
|Minor third | |||
| | |||
|[[6/5]] | |||
|- | |||
|79 | |||
|Major subcycle | |||
| | |||
| | |||
|- | |||
|115 | |||
|21st harmonic | |||
| | |||
|[[21/16]] | |||
|- | |||
|116 | |||
| | |||
| | |||
|[[25/19]] | |||
|- | |||
|163 | |||
| | |||
| | |||
|[[25/17]] | |||
|- | |||
|191 | |||
| | |||
| | |||
|[[11/7]] | |||
|- | |||
|293 | |||
|Perfect octave | |||
| | |||
|2/1 exact | |||
|} | |||
== Tempered commas == | == Tempered commas == | ||
293edo tempers out 1224440064/1220703125 (parakleisma) and 1121008359375/1099511627776 in the 5-limit. Using the patent val, it tempers out 225/224, 2500000/2470629, and 344373768/341796875 in the 7-limit; 6250/6237, 8019/8000, 14700/14641, and 16896/16807 in the 11-limit; 351/350, 625/624, 1625/1617, and 13122/13013 in the 13-limit; 715/714, 850/847, 1089/1088, 1377/1375, 2058/2057, and 2880/2873 in the 17-limit. | 293edo tempers out 1224440064/1220703125 (parakleisma) and 1121008359375/1099511627776 in the 5-limit. Using the patent val, it tempers out 225/224, 2500000/2470629, and 344373768/341796875 in the 7-limit; 6250/6237, 8019/8000, 14700/14641, and 16896/16807 in the 11-limit; 351/350, 625/624, 1625/1617, and 13122/13013 in the 13-limit; 715/714, 850/847, 1089/1088, 1377/1375, 2058/2057, and 2880/2873 in the 17-limit. | ||