User:FloraC/Sequence of equal temperaments by error/Supplement: Difference between revisions

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For a method to iterate thru all GPVs, see [[Patent val/Properties]]. GPVs starting with a zero are excluded as they would meddle with the results. 
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.  
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 in the range of 5 and 16808.  


; 3-limit:  
; 3-limit:  
: Tier-0: {{EDOs| 1b, 1, 2, 5, 12, 41, 53, 306, 665, 15601, 31867 }}…
: Tier-0: {{EDOs| 5, 12, 41, 53, 306, 665, 15601 }}…
: Tier-1: {{EDOs| 3, 7, 17, 24, 29, 94, 106, 253, 359, 971, 1330 }}…
: Tier-1: {{EDOs| 7, 17, 24, 29, 94, 106, 253, 359, 971, 1330, 14936, 16266 }}…
: Tier-2: {{EDOs| 2bb, 4, 10, 147, 159, 200, 612, 1636, 1995 }}…
: Tier-2: {{EDOs| 10, 147, 159, 200, 612, 1636, 1995, 14271 }}…
: Tier-3: {{EDOs| 2b, 3b, 65, 2301, 2660 }}…
: Tier-3: {{EDOs| 65, 2301, 2660, 13606 }}…
: Tier-4: {{EDOs| 8, 9, 19, 36, 82, 412, 1024, 2966, 3325 }}…
: Tier-4: {{EDOs| 8, 9, 19, 36, 82, 412, 1024, 2966, 3325, 12941 }}…
: Tier-5: {{EDOs| 4b, 6b, 6, 14, 22, 212, 559, 718, 1277, 3631, 3990 }}…
: Tier-5: {{EDOs| 6b, 6, 14, 22, 212, 559, 718, 1277, 3631, 3990, 12276 }}…


; 5-limit:  
; 5-limit:  
: Tier-0: {{EDOs| 1bcc, 1b, 1c, 2c, 2, 3, 7, 12, 19, 34, 53, 118, 441, 559, 612, 1171, 1783, 2513, 4296 }}…
: Tier-0: {{EDOs| 7, 12, 19, 34, 53, 118, 441, 559, 612, 1171, 1783, 2513, 4296 }}…
: Tier-1: {{EDOs| 1, 4, 5, 171, 730, 6809 }}…
: Tier-1: {{EDOs| 5, 171, 730, 6809, 8592, 16572 }}…
: Tier-2: {{EDOs| 15, 31, 99, 106, 1342, 3684 }}…
: Tier-2: {{EDOs| 15, 31, 99, 106, 1342, 3684, 11105, 12276 }}…
: Tier-3: {{EDOs| 10, 22, 46, 152, 236, 270, 289, 3125 }}…
: Tier-3: {{EDOs| 10, 22, 46, 152, 236, 270, 289, 3125, 5467, 7980 }}…
: Tier-4: {{EDOs| 2bbccc, 2bbc, 8, 9, 41, 224, 323, 494, 2954 }}…
: Tier-4: {{EDOs| 8, 9, 41, 224, 323, 494, 2954 }}…
: Tier-5: {{EDOs| 2b, 3bc, 5c, 24, 27, 87, 205, 388, 1224, 1901 }}…
: Tier-5: {{EDOs| 5c, 24, 27, 87, 205, 388, 1224, 1901, 9763, 12888 }}…


; 7-limit:  
; 7-limit:  
: Tier-0: {{EDOs| 1bccddd, 1bccd, 1bd, 1c, 2cd, 3d, 4, 5, 10, 12, 19, 31, 72, 99, 171, 3125, 11664, 18355 }}…
: Tier-0: {{EDOs| 5, 10, 12, 19, 31, 72, 99, 171, 3125, 11664 }}…
: Tier-1: {{EDOs| 1b, 1, 2, 41, 270, 441, 2954 }}…
: Tier-1: {{EDOs| 41, 270, 441, 2954, 6691, 14789 }}…
: Tier-2: {{EDOs| 2d, 3, 7d, 8d, 9, 15, 22, 53, 130, 342, 612, 2019, 3566 }}…
: Tier-2: {{EDOs| 7d, 8d, 9, 15, 22, 53, 130, 342, 612, 2019, 3566 }}…
: Tier-3: {{EDOs| 1ccd, 7, 27, 140, 1848 }}…
: Tier-3: {{EDOs| 7, 27, 140, 1848, 3296, 11835 }}…
: Tier-4: {{EDOs| 14c, 46, 152, 301, 323, 3395 }}…
: Tier-4: {{EDOs| 14c, 46, 152, 301, 323, 3395, 6520 }}…
: Tier-5: {{EDOs| 5, 58, 118, 494, 513, 783, 1106, 3737 }}…
: Tier-5: {{EDOs| 5, 58, 118, 494, 513, 783, 1106, 3737, 4973, 5144, 6250, 9816, 15230 }}…


; 11-limit:  
; 11-limit:  
: Tier-0: {{EDOs| 1ce, 2cde, 2, 3de, 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 }}…
: 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-3: {{EDOs| 5e, 10, 19e, 130, 1106, 1236, 2190, 6349, 7927 }}…
: Tier-4: {{EDOs| 53, 80, 87, 311, 764, 836, 2684, 4843, 6421 }}…
: Tier-5: {{EDOs| 99e, 183, 400, 472, 684, 742, 8539 }}…


; 13-limit:  
; 13-limit:  
: Tier-0: {{EDOs| 2cde, 2f, 3, 3de, 3def, 4ef, 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 }}…
: Tier-1: {{EDOs| 5, 7d, 15, 87, 130, 8269, 8539 }}…
: Tier-2: {{EDOs| 10, 34d, 46, 198, 311, 684, 954, 1236 }}…
: Tier-3: {{EDOs| 7df, 29, 103, 581, 1178, 3395, 7033 }}…
: Tier-4: {{EDOs| 5f, 10e, 14cf, 111, 121, 140, 190, 764, 1448, 2460, 11664, 13112 }}…
: Tier-5: {{EDOs| 22, 26, 152f, 183, 373, 422, 742, 1012, 2954 }}…


; 17-limit:  
; 17-limit:  
: Tier-0: {{EDOs| 4ef, 5eg, 8d, 10, 19eg, 22, 26, 27eg, 31, 46, 72, 494, 1506, 3395, 7033, 16808, 20203 }}…
: Tier-0: {{EDOs| 5eg, 8d, 10, 19eg, 22, 26, 27eg, 31, 46, 72, 494, 1506, 3395, 7033, 16808, 20203 }}…
: 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-3: {{EDOs| 14cf, 15g, 17cg, 50, 53, 87, 217, 422, 460, 814, 2000, 2684, 4573, 10428 }}…
: Tier-4: {{EDOs| 7g, 12, 34d, 400, 6079 }}…
: Tier-5: {{EDOs| 10e, 15, 282, 354, 1889, 9033 }}…


; 19-limit:  
; 19-limit:  
Line 48: Line 65:
: 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 }}…


; 7-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 }}…


Revision as of 14:52, 9 June 2022

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 are shown in the range of 5 and 16808.

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, 20203…
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: 7, 8d, 8dhh, 9g, 9, 12f, 19egh, 26, 27eg, 46, 72, 217, 270, 581, 742, 1178, 1578, 2000, 2460, 3395, 8269, 8539, 16808, 20203…
23-limit
Tier-0: 14cf, 15g, 26i, 29g, 46, 94, 190g, 193, 217, 270, 282, 311, 581, 1578, 2460, 8269, 11664, 16808…
29-limit
Tier-0: 26i, 46, 58hi, 80, 140, 159, 183, 243e, 311, 954hj, 1578, 16808…
31-limit
Tier-0: 46, 99efk, 103h, 183, 190g, 217, 311, 954hj, 4501, 16808…
37-limit
Tier-0: 80k, 99efk, 103hl, 270, 270i, 311, 1395, 1920, 4501, 16808…
sqrt (2).sqrt (3).5.7.11.13 subgroup
Tier-0: 10e, 10, 34d, 58, 72, 270, 2684, 13112…
sqrt (2).sqrt (3).5.7.11.13.17 subgroup
Tier-0: 10e, 10, 34d, 58, 72, 742, 954, 2000, 2684, 10356, 10427, 16808…
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