293edo: Difference between revisions
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'''293EDO''' is the [[EDO|equal division of the octave]] into 293 parts of 4.0956 [[cent]]s each. | '''293EDO''' is the [[EDO|equal division of the octave]] into 293 parts of 4.0956 [[cent]]s each. | ||
293EDO is the 62nd [[prime EDO]]. | 293EDO is the 62nd [[prime EDO]]. | ||
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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, 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. | ||
{| class="wikitable" | |||
{| class="wikitable right-3" | |||
|+Selected intervals | |+Selected intervals | ||
|- | |- | ||
! Degree | |||
! Name | |||
! Cents | |||
! Approximate ratios | |||
|- | |- | ||
|5 | | 0 | ||
|Minor leap week interval | | Unison, prime | ||
| | | 0.0000 | ||
|85/84 | | 1/1 exact | ||
|- | |||
| 5 | |||
| Minor leap week interval | |||
| 20.4778 | |||
| 85/84 | |||
|- | |- | ||
|6 | | 6 | ||
|Major leap week interval | | Major leap week interval | ||
| | | 24.5734 | ||
|71/70 | | 71/70 | ||
|- | |- | ||
|11 | | 11 | ||
|Bundle of 2 | | Bundle of 2 | ||
| | | 45.0512 | ||
| | | | ||
|- | |- | ||
|17 | | 17 | ||
|Bundle of 3 | | Bundle of 3 | ||
| | | 69.6246 | ||
| | | | ||
|- | |- | ||
|18 | | 18 | ||
|Vicesimotertial quarter-tone | | Vicesimotertial quarter-tone | ||
| | | 73.7201 | ||
|[[24/23]] | | [[24/23]] | ||
|- | |- | ||
|45 | | 45 | ||
|Minor subcycle | | Minor subcycle | ||
| | | 184.3003 | ||
| | | | ||
|- | |- | ||
|47 | | 47 | ||
|Undevicesimal meantone | | Undevicesimal meantone | ||
| | | 192.4915 | ||
|[[19/17]] | | [[19/17]] | ||
|- | |- | ||
|77 | | 77 | ||
|Minor third | | Minor third | ||
| | | 315.3584 | ||
|[[6/5]] | | [[6/5]] | ||
|- | |- | ||
|79 | | 79 | ||
|Major subcycle | | Major subcycle | ||
| | | 323.5495 | ||
| | | | ||
|- | |- | ||
|115 | | 115 | ||
|21st harmonic | | 21st harmonic | ||
| | | 470.9898 | ||
|[[21/16]] | | [[21/16]] | ||
|- | |- | ||
|116 | | 116 | ||
| | | | ||
| | | 475.0853 | ||
|[[25/19]] | | [[25/19]] | ||
|- | |- | ||
|163 | | 163 | ||
| | | | ||
| | | 667.5768 | ||
|[[25/17]] | | [[25/17]] | ||
|- | |- | ||
|191 | | 191 | ||
| | | | ||
|[[11/7]] | | 782.2526 | ||
| [[11/7]] | |||
|- | |- | ||
|293 | | 293 | ||
|Perfect octave | | Perfect octave | ||
| | | 1200.0000 | ||
|2/1 exact | | 2/1 exact | ||
|} | |} | ||
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== Links == | == Links == | ||
[https://individual.utoronto.ca/kalendis/leap/52-293-sym454-leap-years.htm 52/293 Symmetry454 Leap Years] | * [https://individual.utoronto.ca/kalendis/leap/52-293-sym454-leap-years.htm 52/293 Symmetry454 Leap Years] | ||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
[[Category:Prime EDO]] | [[Category:Prime EDO]] | ||