293edo: Difference between revisions
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{{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, 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. | 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 together cancels out to 1. | 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. | ||
Likewise, 293edo also can be interpreted as a dual-interval tuning, with ''two notes'' instead of one assigned to a particular interval. | |||
{| class="wikitable mw-collapsible right-3" | {| class="wikitable mw-collapsible right-3" | ||
| Line 24: | Line 26: | ||
| 0.0000 | | 0.0000 | ||
| 1/1 exact | | 1/1 exact | ||
|- | |||
|1 | |||
|Limit-tone | |||
|4.0596 | |||
|423/422 | |||
|- | |- | ||
| 5 | | 5 | ||
| Line 48: | Line 55: | ||
| Vicesimotertial quarter-tone | | Vicesimotertial quarter-tone | ||
| 73.7201 | | 73.7201 | ||
| [[24/23]] | |[[24/23]] | ||
|- | |- | ||
| 45 | | 45 | ||
| Line 58: | Line 65: | ||
| Undevicesimal meantone | | Undevicesimal meantone | ||
| 192.4915 | | 192.4915 | ||
| [[19/17]] | |[[19/17]] | ||
|- | |||
|56 | |||
|Minor septimal second | |||
|229.3515 | |||
|8/7, 214/187 | |||
|- | |||
|57 | |||
|Major septimal second | |||
|233.4471 | |||
|8/7, 266/233 | |||
|- | |||
|62 | |||
|Leap week accumulator | |||
|253.9249 | |||
|755/652 | |||
|- | |- | ||
| 77 | | 77 | ||
| Minor third | | Minor third | ||
| 315.3584 | | 315.3584 | ||
| [[6/5]] | |[[6/5]] | ||
|- | |- | ||
| 79 | | 79 | ||
| Line 73: | Line 95: | ||
| 21st harmonic | | 21st harmonic | ||
| 470.9898 | | 470.9898 | ||
| [[21/16]] | |[[21/16]] | ||
|- | |- | ||
| 116 | | 116 | ||
| | |25 over 19 | ||
| 475.0853 | | 475.0853 | ||
| [[25/19]] | |[[25/19]] | ||
|- | |||
|125 | |||
|43rd harmonic | |||
|511.9454 | |||
|[[43/32]] | |||
|- | |- | ||
| 163 | | 163 | ||
| | |Vengeance subfifth | ||
| 667.5768 | | 667.5768 | ||
| [[25/17]] | |[[25/17]] | ||
|- | |||
|171 | |||
|Perfect fifth | |||
|700.3413 | |||
|[[3/2]] | |||
|- | |||
|172 | |||
|"Major" fifth | |||
|704.4369 | |||
|347/231 | |||
|- | |- | ||
| 191 | | 191 | ||
| | |Undecimal minor sixth | ||
| 782.2526 | | 782.2526 | ||
| [[11/7]] | |[[11/7]] | ||
|- | |||
|236 | |||
|Minor harmonic seventh | |||
|966.5529 | |||
|[[7/4]], 187/107 | |||
|- | |||
|237 | |||
|Major harmonic seventh | |||
|970.6485 | |||
|[[7/4]], 233/133 | |||
|- | |||
|260 | |||
|Leap day accumulator | |||
|1064.8464 | |||
|468/253 | |||
|- | |- | ||
| 293 | | 293 | ||