User:FloraC/Sequence of equal temperaments by error/Supplement: Difference between revisions
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The '''tier''' of an equal temperament is defined as the number of GPVs before it that is tuned more efficient than it. Efficiency can be determined in various ways, but the most common way is the TE relative error. | The '''tier''' of an equal temperament is defined as the number of GPVs before it that is tuned more efficient than it. Efficiency can be determined in various ways, but the most common way is the TE relative error. | ||
GPVs starting with 0 are not accounted as they would meddle with the results. For readability reasons, results are shown | GPVs starting with 0 are not accounted as they would meddle with the results. For readability reasons, results smaller than 5 are not shown. | ||
; 3-limit: | ; 3-limit: | ||
| Line 31: | Line 31: | ||
; 11-limit: | ; 11-limit: | ||
: Tier-0: {{EDOs| 5, 7, 8d, 9, 12, 22, 31, 72, 270, 342, 1848, 6079, 6691 }}… | : Tier-0: {{EDOs| 5, 7, 8d, 9, 12, 22, 31, 72, 270, 342, 1848, 6079, 6691 }}… | ||
:: (continued) {{EDOs| 40006, 54624, 121524 }}… | |||
: Tier-1: {{EDOs| 7, 14c, 15, 27e, 41, 152, 1578, 8269, 14618, 14960 }}… | : Tier-1: {{EDOs| 7, 14c, 15, 27e, 41, 152, 1578, 8269, 14618, 14960 }}… | ||
: Tier-2: {{EDOs| 19, 46, 58, 118, 224, 494, 612, 5585, 12276, 12770 }}… | : Tier-2: {{EDOs| 19, 46, 58, 118, 224, 494, 612, 5585, 12276, 12770 }}… | ||
| Line 39: | Line 40: | ||
; 13-limit: | ; 13-limit: | ||
: Tier-0: {{EDOs| 5e, 7, 8d, 9, 12f, 19e, 19, 27e, 31, 41, 53, 58, 72, 224, 270, 494, 1506, 2190, 2684, 5585, 6079, 14618 }}… | : Tier-0: {{EDOs| 5e, 7, 8d, 9, 12f, 19e, 19, 27e, 31, 41, 53, 58, 72, 224, 270, 494, 1506, 2190, 2684, 5585, 6079, 14618 }}… | ||
:: (continued) {{EDOs| 73591, 81860, 87939, 96478 }}… | |||
: Tier-1: {{EDOs| 5, 7d, 15, 87, 130, 8269, 8539 }}… | : Tier-1: {{EDOs| 5, 7d, 15, 87, 130, 8269, 8539 }}… | ||
: Tier-2: {{EDOs| 10, 34d, 46, 198, 311, 684, 954, 1236 }}… | : Tier-2: {{EDOs| 10, 34d, 46, 198, 311, 684, 954, 1236 }}… | ||
| Line 46: | Line 48: | ||
; 17-limit: | ; 17-limit: | ||
: Tier-0: {{EDOs| 5eg, 8d, 10, 19eg, 22, 26, 27eg, 31, 46, 72, 494, 1506, 3395, 7033, 16808, | : Tier-0: {{EDOs| 5eg, 8d, 10, 19eg, 22, 26, 27eg, 31, 46, 72, 494, 1506, 3395, 7033, 16808 }}… | ||
:: (continued) {{EDOs| 20203, 102557 }}… | |||
: Tier-1: {{EDOs| 5g, 7dg, 7, 9, 12f, 41, 58, 111, 140, 183, 270, 581, 742, 764, 954, 1178, 1236, 1578, 2460, 8539, 14348 }}… | : Tier-1: {{EDOs| 5g, 7dg, 7, 9, 12f, 41, 58, 111, 140, 183, 270, 581, 742, 764, 954, 1178, 1236, 1578, 2460, 8539, 14348 }}… | ||
: Tier-2: {{EDOs| 7dfg, 9g, 22f, 29g, 94, 103, 121, 224, 311, 1817, 5855, 8269 }}… | : Tier-2: {{EDOs| 7dfg, 9g, 22f, 29g, 94, 103, 121, 224, 311, 1817, 5855, 8269 }}… | ||
| Line 54: | Line 57: | ||
; 19-limit: | ; 19-limit: | ||
: Tier-0: {{EDOs| 5egh, 7, 8d, 8dhh, 9g, 9, 12f, 19egh, 26, 27eg, 46, 72, 217, 270, 581, 742, 1178, 1578, 2000, 2460, 3395, 8269, 8539, 16808, | : Tier-0: {{EDOs| 5egh, 7, 8d, 8dhh, 9g, 9, 12f, 19egh, 26, 27eg, 46, 72, 217, 270, 581, 742, 1178, 1578, 2000, 2460, 3395, 8269, 8539, 16808 }}… | ||
:: (continued) {{EDOs| 20203, 128215 }}… | |||
: Tier-1: {{EDOs| 7dfgh, 10h, 14cf, 15g, 24, 29g, 31, 41, 53, 58h, 94, 111, 311 }}… | : Tier-1: {{EDOs| 7dfgh, 10h, 14cf, 15g, 24, 29g, 31, 41, 53, 58h, 94, 111, 311 }}… | ||
: Tier-2: {{EDOs| 7dgh, 12, 50, 103h, 121, 140, 183, 354, 422, 11934 }}… | : Tier-2: {{EDOs| 7dgh, 12, 50, 103h, 121, 140, 183, 354, 422, 11934 }}… | ||
| Line 63: | Line 67: | ||
; 23-limit: | ; 23-limit: | ||
: Tier-0: {{EDOs| 14cf, 15g, 26i, 29g, 46, 94, 190g, 193, 217, 270, 282, 311, 581, 1578, 2460, 8269, 11664, 16808 }}… | : Tier-0: {{EDOs| 14cf, 15g, 26i, 29g, 46, 94, 190g, 193, 217, 270, 282, 311, 581, 1578, 2460, 8269, 11664, 16808 }}… | ||
:: (continued) {{EDOs| 148418 }}… | |||
: Tier-1: {{EDOs| 15, 27egi, 31, 50, 72i, 130, 183, 243e, 373g, 422, 954h, 1889 }}… | : Tier-1: {{EDOs| 15, 27egi, 31, 50, 72i, 130, 183, 243e, 373g, 422, 954h, 1889 }}… | ||
: Tier-2: {{EDOs| 22fh, 34dh, 53, 72, 80, 111, 140, 152fg, 159, 525, 935, 1920, 2000, 3178, 8539 }}… | : Tier-2: {{EDOs| 22fh, 34dh, 53, 72, 80, 111, 140, 152fg, 159, 525, 935, 1920, 2000, 3178, 8539 }}… | ||
| Line 71: | Line 76: | ||
; 29-limit: | ; 29-limit: | ||
: Tier-0: {{EDOs| 26i, 46, 58hi, 80, 140, 159, 183, 243e, 311, 954hj, 1578, 16808 }}… | : Tier-0: {{EDOs| 26i, 46, 58hi, 80, 140, 159, 183, 243e, 311, 954hj, 1578, 16808 }}… | ||
:: (continued) {{EDOs| 148418 }}… | |||
; 31-limit: | ; 31-limit: | ||
: Tier-0: {{EDOs| 46, 99efk, 103h, 183, 190g, 217, 311, 954hj, 4501, 16808 }}… | : Tier-0: {{EDOs| 46, 99efk, 103h, 183, 190g, 217, 311, 954hj, 4501, 16808 }}… | ||
:: (continued) {{EDOs| 148418 }}… | |||
; 37-limit: | ; 37-limit: | ||
: Tier-0: {{EDOs| 80k, 99efk, 103hl, 270, 270i, 311, 1395, 1920, 4501, 16808 }}… | : Tier-0: {{EDOs| 80k, 99efk, 103hl, 270, 270i, 311, 1395, 1920, 4501, 16808 }}… | ||
:: (continued) {{EDOs| 26742, 58973, 65322, 67242, 83096, 148418, 241200 }}… | |||
; 2.3.5.11 subgroup: | ; 2.3.5.11 subgroup: | ||
Revision as of 19:10, 19 February 2023
For a method to iterate thru all GPVs, see Patent val/Properties.
The tier of an equal temperament is defined as the number of GPVs before it that is tuned more efficient than it. Efficiency can be determined in various ways, but the most common way is the TE relative error.
GPVs starting with 0 are not accounted as they would meddle with the results. For readability reasons, results smaller than 5 are not shown.
- 3-limit
- Tier-0: 5, 12, 41, 53, 306, 665, 15601…
- Tier-1: 7, 17, 24, 29, 94, 106, 253, 359, 971, 1330, 14936, 16266…
- Tier-2: 10, 147, 159, 200, 612, 1636, 1995, 14271…
- Tier-3: 65, 2301, 2660, 13606…
- Tier-4: 8, 9, 19, 36, 82, 412, 1024, 2966, 3325, 12941…
- Tier-5: 6b, 6, 14, 22, 212, 559, 718, 1277, 3631, 3990, 12276…
- 5-limit
- Tier-0: 7, 12, 19, 34, 53, 118, 441, 559, 612, 1171, 1783, 2513, 4296…
- Tier-1: 5, 171, 730, 6809, 8592, 16572…
- Tier-2: 15, 31, 99, 106, 1342, 3684, 11105, 12276…
- Tier-3: 10, 22, 46, 152, 236, 270, 289, 3125, 5467, 7980…
- Tier-4: 8, 9, 41, 224, 323, 494, 2954…
- Tier-5: 5c, 24, 27, 87, 205, 388, 1224, 1901, 9763, 12888…
- 7-limit
- Tier-0: 5, 10, 12, 19, 31, 72, 99, 171, 3125, 11664…
- Tier-1: 41, 270, 441, 2954, 6691, 14789…
- Tier-2: 7d, 8d, 9, 15, 22, 53, 130, 342, 612, 2019, 3566…
- Tier-3: 7, 27, 140, 1848, 3296, 11835…
- Tier-4: 14c, 46, 152, 301, 323, 3395, 6520…
- Tier-5: 5, 58, 118, 494, 513, 783, 1106, 3737, 4973, 5144, 6250, 9816, 15230…
- 11-limit
- Tier-0: 5, 7, 8d, 9, 12, 22, 31, 72, 270, 342, 1848, 6079, 6691…
- Tier-1: 7, 14c, 15, 27e, 41, 152, 1578, 8269, 14618, 14960…
- Tier-2: 19, 46, 58, 118, 224, 494, 612, 5585, 12276, 12770…
- Tier-3: 5e, 10, 19e, 130, 1106, 1236, 2190, 6349, 7927…
- Tier-4: 53, 80, 87, 311, 764, 836, 2684, 4843, 6421…
- Tier-5: 99e, 183, 400, 472, 684, 742, 8539…
- 13-limit
- Tier-0: 5e, 7, 8d, 9, 12f, 19e, 19, 27e, 31, 41, 53, 58, 72, 224, 270, 494, 1506, 2190, 2684, 5585, 6079, 14618…
- Tier-1: 5, 7d, 15, 87, 130, 8269, 8539…
- Tier-2: 10, 34d, 46, 198, 311, 684, 954, 1236…
- Tier-3: 7df, 29, 103, 581, 1178, 3395, 7033…
- Tier-4: 5f, 10e, 14cf, 111, 121, 140, 190, 764, 1448, 2460, 11664, 13112…
- Tier-5: 22, 26, 152f, 183, 373, 422, 742, 1012, 2954…
- 17-limit
- Tier-0: 5eg, 8d, 10, 19eg, 22, 26, 27eg, 31, 46, 72, 494, 1506, 3395, 7033, 16808…
- Tier-1: 5g, 7dg, 7, 9, 12f, 41, 58, 111, 140, 183, 270, 581, 742, 764, 954, 1178, 1236, 1578, 2460, 8539, 14348…
- Tier-2: 7dfg, 9g, 22f, 29g, 94, 103, 121, 224, 311, 1817, 5855, 8269…
- Tier-3: 14cf, 15g, 17cg, 50, 53, 87, 217, 422, 460, 814, 2000, 2684, 4573, 10428…
- Tier-4: 7g, 12, 34d, 400, 6079…
- Tier-5: 10e, 15, 282, 354, 1889, 9033…
- 19-limit
- Tier-0: 5egh, 7, 8d, 8dhh, 9g, 9, 12f, 19egh, 26, 27eg, 46, 72, 217, 270, 581, 742, 1178, 1578, 2000, 2460, 3395, 8269, 8539, 16808…
- Tier-1: 7dfgh, 10h, 14cf, 15g, 24, 29g, 31, 41, 53, 58h, 94, 111, 311…
- Tier-2: 7dgh, 12, 50, 103h, 121, 140, 183, 354, 422, 11934…
- Tier-3: 22fh, 72h, 99ef, 190g, 193, 243e, 460, 935, 6961, 11664…
- Tier-4: 5cf, 5g, 5eg, 10, 15, 34dh, 118f, 282, 954h, 1889, 4349, 6079…
- Tier-5: 22, 43, 80, 87, 130, 224, 373g, 400, 643, 3178, 3920, 4501…
- 23-limit
- Tier-0: 14cf, 15g, 26i, 29g, 46, 94, 190g, 193, 217, 270, 282, 311, 581, 1578, 2460, 8269, 11664, 16808…
- (continued) 148418…
- Tier-1: 15, 27egi, 31, 50, 72i, 130, 183, 243e, 373g, 422, 954h, 1889…
- Tier-2: 22fh, 34dh, 53, 72, 80, 111, 140, 152fg, 159, 525, 935, 1920, 2000, 3178, 8539…
- Tier-3: 19eghi, 58hi, 99ef, 121i, 624, 814, 4038, 4349, 6079, 6269…
- Tier-4: 27eg, 68, 718, 742i, 14348…
- Tier-5: 19egh, 19, 24, 41i, 43, 3395i, 7315…
- 29-limit
- Tier-0: 26i, 46, 58hi, 80, 140, 159, 183, 243e, 311, 954hj, 1578, 16808…
- (continued) 148418…
- 2.3.5.11 subgroup
- Tier-0: 7, 65, 152, 342, 494, 1171, 11105, 12276…
- Tier-1: 5, 5e, 8, 9, 12e, 12, 15, 22, 31, 53, 72, 87, 118, 270, 677, 1665, 2342, 9934…
- Tier-2: 19, 34, 183, 407, 559, 5520, 6691…
- Tier-3: 14, 24, 335, 612, 829, 901, 1513, 2836, 3513, 13347…
- Tier-4: 41, 46, 159, 224, 2684, 7862, 8763…
- Tier-5: 10, 19e, 137, 217, 836, 1019, 2072, 3855…
- sqrt (2).sqrt (3).5.7.11.13 subgroup
- Tier-0: 10e, 10, 34d, 58, 72, 270, 2684, 13112…
- Tier-1: 14cf, 130, 954, 10428, 16808…
- Tier-2: 24, 38df, 68, 198, 742, 814, 2000, 2954, 14124…
- Tier-3: 62, 140, 612, 1012, 1730…
- Tier-4: 14c, 34, 96d, 1084, 3696, 3768, 10158, 10356, 12158, 13040…
- Tier-5: 10f, 24d, 202, 212, 342f, 13854…
- sqrt (2).sqrt (3).5.7.11.13.17 subgroup
- Tier-0: 10e, 10, 34d, 58, 72, 742, 954, 2000, 2684, 10356, 10428, 16808…
- Tier-1: 14cf, 24, 68, 130, 140, 270, 814…
- Tier-2: 38df, 212g, 400, 2414, 3638, 12842…
- Tier-3: 10eg, 14c, 34, 58g, 130g, 328, 414, 684, 11170, 12428, 13112, 13170…
- Tier-4: 14cfgg, 62, 96d, 198g, 342f, 882…
- Tier-5: …