20203edo: Difference between revisions
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{{Infobox ET|Consistency=45|Distinct consistency=45}} | |||
{{ | {{ED intro}} | ||
20203edo is a very strong high-limit system, and specializes in the 17- and 19-limit, with | 20203edo is a very strong high-limit system, and specializes in the [[17-limit|17-]] and [[19-limit]], with lower 17- and 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative errors]] than any smaller edo until [[102557edo|102557]] and 128215, respectively. It is also distinctly [[consistent]] through the [[45-odd-limit]], and has a lower [[43-limit]] relative error than any smaller edo except for [[7361edo|7361]], [[14348edo|14348]] and [[17461edo|17461]]. | ||
A 43-limit [[comma basis]] for this temperament is {29792/29791, 32799/32798, 43264/43263, 45696/45695, 47151/47150, 52326/52325, 53361/53360, 69875/69874, 81796/81795, 83521/83520, 87465/87464, 96876/96875, 111112/111111}. In the [[13-limit]] it tempers out [[123201/123200]] and [[1990656/1990625]]; in the [[17-limit]] [[194481/194480]] and [[336141/336140]]; in the [[19-limit]] 89376/89375, 104976/104975, and 165376/165375; in the [[23-limit]] 43264/43263 and 52326/52325 among others. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|20203}} | {{Harmonics in equal|20203|columns=11}} | ||
{{Harmonics in equal|20203|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 20203edo (continued)}} | |||
Latest revision as of 19:02, 2 October 2025
| ← 20202edo | 20203edo | 20204edo → |
20203 equal divisions of the octave (abbreviated 20203edo or 20203ed2), also called 20203-tone equal temperament (20203tet) or 20203 equal temperament (20203et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 20203 equal parts of about 0.0594 ¢ each. Each step represents a frequency ratio of 21/20203, or the 20203rd root of 2.
20203edo is a very strong high-limit system, and specializes in the 17- and 19-limit, with lower 17- and 19-limit relative errors than any smaller edo until 102557 and 128215, respectively. It is also distinctly consistent through the 45-odd-limit, and has a lower 43-limit relative error than any smaller edo except for 7361, 14348 and 17461.
A 43-limit comma basis for this temperament is {29792/29791, 32799/32798, 43264/43263, 45696/45695, 47151/47150, 52326/52325, 53361/53360, 69875/69874, 81796/81795, 83521/83520, 87465/87464, 96876/96875, 111112/111111}. In the 13-limit it tempers out 123201/123200 and 1990656/1990625; in the 17-limit 194481/194480 and 336141/336140; in the 19-limit 89376/89375, 104976/104975, and 165376/165375; in the 23-limit 43264/43263 and 52326/52325 among others.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0002 | +0.0051 | +0.0005 | +0.0061 | +0.0010 | -0.0007 | +0.0072 | +0.0284 | +0.0125 | +0.0221 |
| Relative (%) | +0.0 | +0.3 | +8.7 | +0.9 | +10.3 | +1.6 | -1.2 | +12.0 | +47.8 | +21.0 | +37.2 | |
| Steps (reduced) |
20203 (0) |
32021 (11818) |
46910 (6504) |
56717 (16311) |
69891 (9282) |
74760 (14151) |
82579 (1767) |
85821 (5009) |
91390 (10578) |
98146 (17334) |
100090 (19278) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0246 | +0.0224 | +0.0103 | -0.0213 | -0.0105 | -0.0022 | +0.0186 | -0.0119 | -0.0203 | -0.0015 | -0.0168 |
| Relative (%) | +41.4 | +37.7 | +17.3 | -35.9 | -17.7 | -3.8 | +31.4 | -20.0 | -34.1 | -2.6 | -28.2 | |
| Steps (reduced) |
105247 (4232) |
108239 (7224) |
109627 (8612) |
112219 (11204) |
115721 (14706) |
118847 (17832) |
119819 (18804) |
122553 (1335) |
124243 (3025) |
125053 (3835) |
127355 (6137) | |