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'''55EDT''' is the [[Edt|equal division of the third harmonic]] into 55 parts of 34.5810 [[cent|cents]] each, corresponding to 34.7011 [[edo]]. It is related to the regular temperament which tempers out 420175/419904 and 205891132094649/204800000000000 in the 7-limit, which is supported by 243, 347, and 590 EDOs among others. | {{Infobox ET}} | ||
'''55EDT''' is the [[Edt|equal division of the third harmonic]] into 55 parts of 34.5810 [[cent|cents]] each, corresponding to 34.7011 [[edo]]. It is related to the regular temperament which tempers out 420175/419904 and 205891132094649/204800000000000 in the 7-limit, which is supported by [[243edo|243]], [[347edo|347]], and [[590edo|590]] EDOs among others. | |||
=Related regular temperaments= | == Intervals == | ||
==243&347 temperament== | {{Interval table}} | ||
===7-limit=== | |||
== Harmonics == | |||
{{Harmonics in equal | |||
| steps = 55 | |||
| num = 3 | |||
| denom = 1 | |||
| intervals = integer | |||
}} | |||
{{Harmonics in equal | |||
| steps = 55 | |||
| num = 3 | |||
| denom = 1 | |||
| start = 12 | |||
| collapsed = 1 | |||
| intervals = integer | |||
}} | |||
==Related regular temperaments== | |||
===243&347 temperament=== | |||
====7-limit==== | |||
Commas: 420175/419904, |-22 30 -11> | Commas: 420175/419904, |-22 30 -11> | ||
| Line 10: | Line 30: | ||
Map: [<1 0 -2 2|, <0 55 150 28|] | Map: [<1 0 -2 2|, <0 55 150 28|] | ||
EDOs: 243, 347, 590 | EDOs: {{EDOs|243, 347, 590}} | ||
===11-limit=== | ====11-limit==== | ||
Commas: 137781/137500, 352947/352000, 16808715/16777216 | Commas: 137781/137500, 352947/352000, 16808715/16777216 | ||
| Line 19: | Line 39: | ||
Map: [<1 0 -2 2 10|, <0 55 150 28 -227|] | Map: [<1 0 -2 2 10|, <0 55 150 28 -227|] | ||
EDOs: 243, 347, 590 | EDOs: {{EDOs|243, 347, 590}} | ||
===13-limit=== | ====13-limit==== | ||
Commas: 4459/4455, 15379/15360, 67392/67375, 83349/83200 | Commas: 4459/4455, 15379/15360, 67392/67375, 83349/83200 | ||
| Line 28: | Line 48: | ||
Map: [<1 0 -2 2 10 2|, <0 55 150 28 -227 59|] | Map: [<1 0 -2 2 10 2|, <0 55 150 28 -227 59|] | ||
EDOs: 243, 347, 590 | EDOs: {{EDOs|243, 347, 590}} | ||
Latest revision as of 19:23, 1 August 2025
| ← 54edt | 55edt | 56edt → |
55EDT is the equal division of the third harmonic into 55 parts of 34.5810 cents each, corresponding to 34.7011 edo. It is related to the regular temperament which tempers out 420175/419904 and 205891132094649/204800000000000 in the 7-limit, which is supported by 243, 347, and 590 EDOs among others.
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 34.6 | 23.6 | |
| 2 | 69.2 | 47.3 | 25/24, 27/26 |
| 3 | 103.7 | 70.9 | 18/17, 33/31 |
| 4 | 138.3 | 94.5 | |
| 5 | 172.9 | 118.2 | 21/19, 31/28 |
| 6 | 207.5 | 141.8 | 26/23 |
| 7 | 242.1 | 165.5 | 31/27 |
| 8 | 276.6 | 189.1 | 27/23 |
| 9 | 311.2 | 212.7 | 6/5 |
| 10 | 345.8 | 236.4 | 11/9 |
| 11 | 380.4 | 260 | |
| 12 | 415 | 283.6 | 14/11, 33/26 |
| 13 | 449.6 | 307.3 | 22/17 |
| 14 | 484.1 | 330.9 | |
| 15 | 518.7 | 354.5 | 23/17, 31/23 |
| 16 | 553.3 | 378.2 | |
| 17 | 587.9 | 401.8 | |
| 18 | 622.5 | 425.5 | 33/23 |
| 19 | 657 | 449.1 | 19/13 |
| 20 | 691.6 | 472.7 | |
| 21 | 726.2 | 496.4 | |
| 22 | 760.8 | 520 | 14/9 |
| 23 | 795.4 | 543.6 | |
| 24 | 829.9 | 567.3 | 21/13, 29/18 |
| 25 | 864.5 | 590.9 | 28/17 |
| 26 | 899.1 | 614.5 | |
| 27 | 933.7 | 638.2 | |
| 28 | 968.3 | 661.8 | |
| 29 | 1002.8 | 685.5 | |
| 30 | 1037.4 | 709.1 | 31/17 |
| 31 | 1072 | 732.7 | 13/7 |
| 32 | 1106.6 | 756.4 | |
| 33 | 1141.2 | 780 | 27/14, 29/15 |
| 34 | 1175.8 | 803.6 | |
| 35 | 1210.3 | 827.3 | |
| 36 | 1244.9 | 850.9 | |
| 37 | 1279.5 | 874.5 | 23/11 |
| 38 | 1314.1 | 898.2 | |
| 39 | 1348.7 | 921.8 | |
| 40 | 1383.2 | 945.5 | |
| 41 | 1417.8 | 969.1 | |
| 42 | 1452.4 | 992.7 | |
| 43 | 1487 | 1016.4 | 26/11, 33/14 |
| 44 | 1521.6 | 1040 | |
| 45 | 1556.1 | 1063.6 | 27/11 |
| 46 | 1590.7 | 1087.3 | 5/2 |
| 47 | 1625.3 | 1110.9 | 23/9 |
| 48 | 1659.9 | 1134.5 | |
| 49 | 1694.5 | 1158.2 | |
| 50 | 1729.1 | 1181.8 | 19/7 |
| 51 | 1763.6 | 1205.5 | |
| 52 | 1798.2 | 1229.1 | 17/6, 31/11 |
| 53 | 1832.8 | 1252.7 | 26/9 |
| 54 | 1867.4 | 1276.4 | |
| 55 | 1902 | 1300 | 3/1 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.3 | +0.0 | -13.9 | +14.7 | +10.3 | -14.5 | -3.6 | +0.0 | -9.5 | -1.6 | -13.9 |
| Relative (%) | +29.9 | +0.0 | -40.2 | +42.6 | +29.9 | -41.8 | -10.3 | +0.0 | -27.5 | -4.6 | -40.2 | |
| Steps (reduced) |
35 (35) |
55 (0) |
69 (14) |
81 (26) |
90 (35) |
97 (42) |
104 (49) |
110 (0) |
115 (5) |
120 (10) |
124 (14) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -14.2 | -4.1 | +14.7 | +6.8 | +5.5 | +10.3 | -14.1 | +0.8 | -14.5 | +8.7 | +0.9 |
| Relative (%) | -40.9 | -12.0 | +42.6 | +19.5 | +16.0 | +29.9 | -40.8 | +2.4 | -41.8 | +25.3 | +2.7 | |
| Steps (reduced) |
128 (18) |
132 (22) |
136 (26) |
139 (29) |
142 (32) |
145 (35) |
147 (37) |
150 (40) |
152 (42) |
155 (45) |
157 (47) | |
Related regular temperaments
243&347 temperament
7-limit
Commas: 420175/419904, |-22 30 -11>
POTE generator: ~49/48 = 34.5742
Map: [<1 0 -2 2|, <0 55 150 28|]
11-limit
Commas: 137781/137500, 352947/352000, 16808715/16777216
POTE generator: ~49/48 = 34.5761
Map: [<1 0 -2 2 10|, <0 55 150 28 -227|]
13-limit
Commas: 4459/4455, 15379/15360, 67392/67375, 83349/83200
POTE generator: ~49/48 = 34.5762
Map: [<1 0 -2 2 10 2|, <0 55 150 28 -227 59|]