7ed5/4: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
'''7ED5/4''' is the [[Equal-step tuning|equal division]] of the [[5/4|just major third]] into seven parts of 55.1877 [[cent|cents]] each, corresponding to 21.7440 [[EDO|edo]] (very nearly [[61ed7]]). It is related to the [[Hemifamity temperaments|alphaquarter temperament]] and the 5-limit temperament which tempers out |234 -7 -96> (0.198463 cents, 5-limit 1783&7980 comma). | '''7ED5/4''' is the [[Equal-step tuning|equal division]] of the [[5/4|just major third]] into seven parts of 55.1877 [[cent|cents]] each, corresponding to 21.7440 [[EDO|edo]] (very nearly [[61ed7]]). It is related to the [[Hemifamity temperaments|alphaquarter temperament]] and the 5-limit temperament which tempers out |234 -7 -96> (0.198463 cents, 5-limit 1783&7980 comma). | ||
== Harmonics == | |||
{{Harmonics in equal | |||
| steps = 7 | |||
| num = 5 | |||
| denom = 4 | |||
}} | |||
{{Harmonics in equal | |||
| steps = 7 | |||
| num = 5 | |||
| denom = 4 | |||
| start = 12 | |||
| collapsed = 1 | |||
}} | |||
==Intervals== | ==Intervals== | ||
| Line 397: | Line 411: | ||
[[Category:Major third]] | [[Category:Major third]] | ||
[[Category:Equal-step tuning]] | [[Category:Equal-step tuning]] | ||
Latest revision as of 19:23, 1 August 2025
| ← 6ed5/4 | 7ed5/4 | 8ed5/4 → |
(semiconvergent)
7ED5/4 is the equal division of the just major third into seven parts of 55.1877 cents each, corresponding to 21.7440 edo (very nearly 61ed7). It is related to the alphaquarter temperament and the 5-limit temperament which tempers out |234 -7 -96> (0.198463 cents, 5-limit 1783&7980 comma).
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +14.1 | -25.6 | -26.9 | -26.9 | -11.4 | -2.4 | -12.8 | +4.0 | -12.8 | -12.2 | +2.7 |
| Relative (%) | +25.6 | -46.3 | -48.8 | -48.8 | -20.7 | -4.3 | -23.2 | +7.3 | -23.2 | -22.2 | +4.9 | |
| Steps (reduced) |
22 (1) |
34 (6) |
43 (1) |
50 (1) |
56 (0) |
61 (5) |
65 (2) |
69 (6) |
72 (2) |
75 (5) |
78 (1) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -25.5 | +11.8 | +2.7 | +1.3 | +6.7 | +18.2 | -20.2 | +1.3 | +27.2 | +1.9 | -19.9 |
| Relative (%) | -46.2 | +21.3 | +4.9 | +2.4 | +12.2 | +32.9 | -36.7 | +2.4 | +49.4 | +3.4 | -36.0 | |
| Steps (reduced) |
80 (3) |
83 (6) |
85 (1) |
87 (3) |
89 (5) |
91 (0) |
92 (1) |
94 (3) |
96 (5) |
97 (6) |
98 (0) | |
Intervals
| degree | cents value | ratio |
|---|---|---|
| 0 | 0.0000 | 1/1 |
| 1 | 55.1877 | (5/4)1/7 |
| 2 | 110.3753 | (5/4)2/7 |
| 3 | 165.5630 | (5/4)3/7 |
| 4 | 220.7507 | (5/4)4/7 |
| 5 | 275.9384 | (5/4)5/7 |
| 6 | 331.1260 | (5/4)6/7 |
| 7 | 386.3137 | 5/4 |
| 8 | 441.5014 | (5/4)8/7 |
| 9 | 496.6891 | (5/4)9/7 |
| 10 | 551.8767 | (5/4)10/7 |
| 11 | 607.0644 | (5/4)11/7 |
| 12 | 662.2521 | (5/4)12/7 |
| 13 | 717.4398 | (5/4)13/7 |
| 14 | 772.6274 | (5/4)2 = 25/16 |
| 15 | 827.8151 | (5/4)15/7 |
| 16 | 883.0028 | (5/4)16/7 |
| 17 | 938.1904 | (5/4)17/7 |
| 18 | 993.3781 | (5/4)18/7 |
| 19 | 1048.5658 | (5/4)19/7 |
| 20 | 1103.7535 | (5/4)20/7 |
| 21 | 1158.9411 | (5/4)3 = 125/64 |
| 22 | 1214.1288 | (5/4)22/7 |
| 23 | 1269.3165 | (5/4)23/7 |
| 24 | 1324.5042 | (5/4)24/7 |
| 25 | 1379.6918 | (5/4)25/7 |
| 26 | 1434.8795 | (5/4)26/7 |
| 27 | 1490.0672 | (5/4)27/7 |
| 28 | 1545.2549 | (5/4)4 = 625/256 |
| 29 | 1600.4425 | (5/4)29/7 |
| 30 | 1655.6302 | (5/4)30/7 |
| 31 | 1710.8179 | (5/4)31/7 |
| 32 | 1766.0055 | (5/4)32/7 |
| 33 | 1821.1932 | (5/4)33/7 |
| 34 | 1876.3809 | (5/4)34/7 |
| 35 | 1931.5686 | (5/4)5 = 3125/1024 |
| 36 | 1986.7562 | (5/4)36/7 |
| 37 | 2041.9439 | (5/4)37/7 |
| 38 | 2097.1316 | (5/4)38/7 |
| 39 | 2152.3193 | (5/4)39/7 |
| 40 | 2207.5069 | (5/4)40/7 |
| 41 | 2262.6946 | (5/4)41/7 |
| 42 | 2317.8823 | (5/4)6 = 15625/4096 |
| 43 | 2373.0700 | (5/4)43/7 |
| 44 | 2428.2576 | (5/4)44/7 |
| 45 | 2483.4453 | (5/4)45/7 |
| 46 | 2538.6330 | (5/4)46/7 |
| 47 | 2593.8207 | (5/4)47/7 |
| 48 | 2649.0083 | (5/4)48/7 |
| 49 | 2704.1960 | (5/4)7 = 78125/16384 |
| 50 | 2759.3837 | (5/4)50/7 |
| 51 | 2814.5713 | (5/4)51/7 |
| 52 | 2869.7590 | (5/4)52/7 |
| 53 | 2924.9467 | (5/4)53/7 |
| 54 | 2980.1344 | (5/4)54/7 |
| 55 | 3035.3220 | (5/4)55/7 |
| 56 | 3090.5097 | (5/4)8 = 390625/65536 |
| 57 | 3145.6974 | (5/4)57/7 |
| 58 | 3200.8851 | (5/4)58/7 |
| 59 | 3256.0727 | (5/4)59/7 |
| 60 | 3311.2604 | (5/4)60/7 |
| 61 | 3366.4481 | (5/4)61/7 |
| 62 | 3421.6358 | (5/4)62/7 |
| 63 | 3476.8234 | (5/4)9 = 1953125/262144 |
| 64 | 3532.0111 | (5/4)64/7 |
| 65 | 3587.1988 | (5/4)65/7 |
| 66 | 3642.3864 | (5/4)66/7 |
| 67 | 3697.5741 | (5/4)67/7 |
| 68 | 3752.7618 | (5/4)68/7 |
| 69 | 3807.9495 | (5/4)69/7 |
| 70 | 3863.1371 | (5/4)10 = 9765625/1048576 |
| 71 | 3918.3248 | (5/4)71/7 |
| 72 | 3973.5125 | (5/4)72/7 |
| 73 | 4028.7002 | (5/4)73/7 |
| 74 | 4083.8878 | (5/4)74/7 |
| 75 | 4139.0755 | (5/4)75/7 |
| 76 | 4194.2632 | (5/4)76/7 |
| 77 | 4249.4509 | (5/4)11 = 48828125/4194304 |
| 78 | 4304.6385 | (5/4)78/7 |
| 79 | 4359.8262 | (5/4)79/7 |
| 80 | 4415.0139 | (5/4)80/7 |
| 81 | 4470.2015 | (5/4)81/7 |
| 82 | 4525.3892 | (5/4)82/7 |
| 83 | 4580.5769 | (5/4)83/7 |
| 84 | 4635.7646 | (5/4)12 = 244140625/16777216 |
| 85 | 4690.9522 | (5/4)85/7 |
| 86 | 4746.1399 | (5/4)86/7 |
| 87 | 4801.3276 | (5/4)87/7 |
| 88 | 4856.5153 | (5/4)88/7 |
| 89 | 4911.7029 | (5/4)89/7 |
| 90 | 4966.8906 | (5/4)90/7 |
| 91 | 5022.0783 | (5/4)13 = 1220703125/67108864 |
7ED5/4 as a generator
Alphaquarter
7ED5/4 leads the alphaquarter temperament using its three steps for 11/10, its nine steps for 4/3, and its 61 steps for 7/1. Alphaquarter tempers out 3025/3024, 4000/3993, and 5120/5103 in the 11-limit, supported by 87edo, 152edo, 239edo, and 391edo among others.
1783&7980 temperament
7ED5/4 leads 1783&7980 temperament using its 96 steps for 64/3 (four octaves plus just perefect fourth).
Comma: |234 -7 -96>
POTE generator: 55.188
Mapping: [<1 6 2|, <0 -96 7|]
EDOs: 1783, 4414, 6197, 7980, 9763, 11546, 14177
Badness: 0.1157