Catalog of rank-4 temperaments: Difference between revisions

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add unchanged-intervals along with eigenmonzo, per prior agreement on Xenwiki Work Group
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,160 edits
"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
Line 10: Line 10:
[[Optimal tuning]] ([[POTE]]): ~3/2 = 704.9532, ~5/4 = 384.0675, ~7/4 = 970.8803
[[Optimal tuning]] ([[POTE]]): ~3/2 = 704.9532, ~5/4 = 384.0675, ~7/4 = 970.8803


{{Val list|legend=1| 7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=1| 7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e }} <nowiki>*</nowiki>


<nowiki>*</nowiki> [[optimal patent val]]: [[104edo|104]]
<nowiki>*</nowiki> [[optimal patent val]]: [[104edo|104]]
Line 27: Line 27:
[[Optimal tuning]] ([[POTE]]): ~3/2 = 701.4578, ~11/10 = 157.7466, ~7/4 = 966.9589
[[Optimal tuning]] ([[POTE]]): ~3/2 = 701.4578, ~11/10 = 157.7466, ~7/4 = 966.9589


{{Val list|legend=1| 14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee }}
{{Optimal ET sequence|legend=1| 14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee }}


[[Badness]]: 0.0345 × 10<sup>-6</sup>
[[Badness]]: 0.0345 × 10<sup>-6</sup>
Line 42: Line 42:
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113


{{Val list|legend=1| 22, 31, 46, 53, 58, 80, 111 }}
{{Optimal ET sequence|legend=1| 22, 31, 46, 53, 58, 80, 111 }}


[[Badness]]: 0.0186 × 10<sup>-6</sup>
[[Badness]]: 0.0186 × 10<sup>-6</sup>
Line 55: Line 55:
[[Optimal tuning]] ([[POTE]]): ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464
[[Optimal tuning]] ([[POTE]]): ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464


{{Val list|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }}
{{Optimal ET sequence|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }}


[[Badness]]: 0.0509 × 10<sup>-6</sup>
[[Badness]]: 0.0509 × 10<sup>-6</sup>
Line 70: Line 70:
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.9075, ~5/4 = 387.0723, ~11/8 = 551.4538
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.9075, ~5/4 = 387.0723, ~11/8 = 551.4538


{{Val list|legend=1| 12f, 17c, 24d, 29, 41, 46, 53, 58, 87, 111, 140, 152f, 198, 350f, 437f, 490f }}
{{Optimal ET sequence|legend=1| 12f, 17c, 24d, 29, 41, 46, 53, 58, 87, 111, 140, 152f, 198, 350f, 437f, 490f }}


[[Badness]]: 2.550 × 10<sup>-6</sup>
[[Badness]]: 2.550 × 10<sup>-6</sup>
Line 83: Line 83:
Mapping generators:~2, ~3, ~5, ~7
Mapping generators:~2, ~3, ~5, ~7


{{Val list|legend=1| 10, 12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d }}
{{Optimal ET sequence|legend=1| 10, 12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d }}


=== Commas 364/363, 441/440 ===
=== Commas 364/363, 441/440 ===
Line 107: Line 107:
: [[Eigenmonzo]]s (unchanged-intervals): 2, 14/13, 6/5, 11/9
: [[Eigenmonzo]]s (unchanged-intervals): 2, 14/13, 6/5, 11/9


{{Val list|legend=1| 12f, 14cf, 15, 17c, 26, 29, 31f, 41, 46, 58, 72, 87, 130, 217, 289 }}
{{Optimal ET sequence|legend=1| 12f, 14cf, 15, 17c, 26, 29, 31f, 41, 46, 58, 72, 87, 130, 217, 289 }}


=== Commas 351/350, 441/440 ===
=== Commas 351/350, 441/440 ===
Line 118: Line 118:
Mapping generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 12f, 14cf, 19e, 26, 27e, 31, 45ef, 46, 58, 72, 103, 130, 233, 279, 409, 512bf, 642bf }}
{{Optimal ET sequence|legend=1| 12f, 14cf, 19e, 26, 27e, 31, 45ef, 46, 58, 72, 103, 130, 233, 279, 409, 512bf, 642bf }}


=== Commas 196/195, 352/351 ===
=== Commas 196/195, 352/351 ===
Line 129: Line 129:
Mapping generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 10, 12f, 17c, 19e, 27e, 29, 31, 41, 46, 58, 87, 118, 145, 232 }}
{{Optimal ET sequence|legend=1| 10, 12f, 17c, 19e, 27e, 29, 31, 41, 46, 58, 87, 118, 145, 232 }}


=== Tannic ===
=== Tannic ===
Line 140: Line 140:
Mapping generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 17c, 26, 29, 31, 43, 46, 60e, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff }}
{{Optimal ET sequence|legend=1| 17c, 26, 29, 31, 43, 46, 60e, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff }}


==== 17-limit ====
==== 17-limit ====
Line 151: Line 151:
Mapping generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef }}
{{Optimal ET sequence|legend=1| 17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef }}


=== Commas 441/440, 847/845 ===
=== Commas 441/440, 847/845 ===
Line 162: Line 162:
Mapping generators: ~2, ~3, ~5, ~13/11
Mapping generators: ~2, ~3, ~5, ~13/11


{{Val list|legend=1| 12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d }}
{{Optimal ET sequence|legend=1| 12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d }}


== Keenanismic (385/384) ==
== Keenanismic (385/384) ==
Line 182: Line 182:
: Eigenmonzos (unchanged-intervals): 2, 3, 7/5, 11/5
: Eigenmonzos (unchanged-intervals): 2, 3, 7/5, 11/5


{{Val list|legend=1| 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de }}
{{Optimal ET sequence|legend=1| 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de }}


[[Badness]]: 15.159 × 10<sup>-9</sup>
[[Badness]]: 15.159 × 10<sup>-9</sup>
Line 205: Line 205:
: Eigenmonzos (unchanged-intervals): 2, 14/11, 13/10, 4/3
: Eigenmonzos (unchanged-intervals): 2, 14/11, 13/10, 4/3


{{Val list|legend=1| 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff }}
{{Optimal ET sequence|legend=1| 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff }}


[[Badness]]: 2.206 × 10<sup>-6</sup>
[[Badness]]: 2.206 × 10<sup>-6</sup>
Line 218: Line 218:
[[Transpose]]: [2 3 5 7 385/35 625/48]
[[Transpose]]: [2 3 5 7 385/35 625/48]


{{Val list|legend=1| 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def }}
{{Optimal ET sequence|legend=1| 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def }}


[[Badness]]: 2.573 × 10<sup>-6</sup>
[[Badness]]: 2.573 × 10<sup>-6</sup>
Line 231: Line 231:
Mapping generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff }}
{{Optimal ET sequence|legend=1| 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff }}


=== Commas 352/351, 385/384 ===
=== Commas 352/351, 385/384 ===
Line 242: Line 242:
Mapping generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
{{Optimal ET sequence|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}


=== Commas 364/363, 385/384 ===
=== Commas 364/363, 385/384 ===
Line 253: Line 253:
Mapping generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159 }}
{{Optimal ET sequence|legend=1| 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159 }}


=== Commas 385/384, 847/845 ===
=== Commas 385/384, 847/845 ===
Line 264: Line 264:
Mapping generators: ~2, ~3, ~5, ~13/11
Mapping generators: ~2, ~3, ~5, ~13/11


{{Val list|legend=1| 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee }}
{{Optimal ET sequence|legend=1| 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee }}


== Swetismic (540/539) ==
== Swetismic (540/539) ==
Line 275: Line 275:
Mapping generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 8d, 9, 10, 12e, 14c, 17c, 19, 22, 27e, 31, 41, 53, 58, 72, 130, 152, 224, 354, 506e, 578, 730de, 761d, 985d, 1115de, 1267dde }}
{{Optimal ET sequence|legend=1| 8d, 9, 10, 12e, 14c, 17c, 19, 22, 27e, 31, 41, 53, 58, 72, 130, 152, 224, 354, 506e, 578, 730de, 761d, 985d, 1115de, 1267dde }}


=== Commas 540/539, 847/845 ===
=== Commas 540/539, 847/845 ===
Line 286: Line 286:
Mapping generators: ~2, ~3, ~5, ~13/11
Mapping generators: ~2, ~3, ~5, ~13/11


{{Val list|legend=1| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}
{{Optimal ET sequence|legend=1| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}


=== Commas 540/539, 625/624 ===
=== Commas 540/539, 625/624 ===
Line 297: Line 297:
Mapping generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 19, 22, 31, 49f, 50, 53, 72, 103, 121, 152f, 193, 224 }}
{{Optimal ET sequence|legend=1| 19, 22, 31, 49f, 50, 53, 72, 103, 121, 152f, 193, 224 }}


=== Commas 540/539, 676/675 ===
=== Commas 540/539, 676/675 ===
Line 308: Line 308:
Mapping generators: ~2, ~26/15, ~5, ~7
Mapping generators: ~2, ~26/15, ~5, ~7


{{Val list|legend=1| 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de }}
{{Optimal ET sequence|legend=1| 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de }}


== Pentacircle (896/891) ==
== Pentacircle (896/891) ==
Line 319: Line 319:
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722


{{Val list|legend=1| 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee }}
{{Optimal ET sequence|legend=1| 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee }}


[[Badness]]: 0.0658 × 10<sup>-6</sup>
[[Badness]]: 0.0658 × 10<sup>-6</sup>
Line 330: Line 330:
[[Mapping]]: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 0 0 -4 -7 }}, {{val| 0 0 1 0 0 0 }}, {{val| 0 0 0 1 1 1 }}]
[[Mapping]]: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 0 0 -4 -7 }}, {{val| 0 0 1 0 0 0 }}, {{val| 0 0 0 1 1 1 }}]


{{Val list|legend=1| 12f, 17c, 22, 29, 34d, 41, 46, 58, 80, 87, 121, 167, 179ef, 208, 266ef, 433bceef, 641bbceeeff, 699bbcdeeeff }}
{{Optimal ET sequence|legend=1| 12f, 17c, 22, 29, 34d, 41, 46, 58, 80, 87, 121, 167, 179ef, 208, 266ef, 433bceef, 641bbceeeff, 699bbcdeeeff }}


[[Badness]]: 3.375 × 10<sup>-6</sup>
[[Badness]]: 3.375 × 10<sup>-6</sup>
Line 343: Line 343:
Mapping generators: ~2, ~3, ~5, ~13/10
Mapping generators: ~2, ~3, ~5, ~13/10


{{Val list|legend=1| 16, 21, 24d, 29, 37, 45ef, 50, 53, 58, 87, 103, 111, 140, 190, 198, 301, 388, 689e }}
{{Optimal ET sequence|legend=1| 16, 21, 24d, 29, 37, 45ef, 50, 53, 58, 87, 103, 111, 140, 190, 198, 301, 388, 689e }}


== Lehmerismic (3025/3024) ==
== Lehmerismic (3025/3024) ==
Line 354: Line 354:
Mapping generators: ~2, ~3, ~5, ~55/36
Mapping generators: ~2, ~3, ~5, ~55/36


{{Val list|legend=1| 7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
{{Optimal ET sequence|legend=1| 7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}


== Trimitone (8019/8000) ==
== Trimitone (8019/8000) ==
Line 365: Line 365:
Mapping generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 12, 19, 26, 39d, 46, 53, 58, 72, 118, 130, 183, 190, 248, 255, 301, 373, 804, 876, 1177be }}
{{Optimal ET sequence|legend=1| 12, 19, 26, 39d, 46, 53, 58, 72, 118, 130, 183, 190, 248, 255, 301, 373, 804, 876, 1177be }}


== Kalismic (9801/9800) ==
== Kalismic (9801/9800) ==
Line 376: Line 376:
Mapping generators: ~99/70, ~3, ~5, ~7
Mapping generators: ~99/70, ~3, ~5, ~7


{{Val list|legend=1| 8d, 10, 12, 22, 34d, 46, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 4038, 4190, 4532, 11254, 15786e, 21896e }}
{{Optimal ET sequence|legend=1| 8d, 10, 12, 22, 34d, 46, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 4038, 4190, 4532, 11254, 15786e, 21896e }}


=== Commas 1716/1715, 2080/2079 ===
=== Commas 1716/1715, 2080/2079 ===
Line 395: Line 395:
: [[Eigenmonzo]]s (unchanged-intervals): 2, 6/5, 16/13, 9/7
: [[Eigenmonzo]]s (unchanged-intervals): 2, 6/5, 16/13, 9/7


{{Val list|legend=1| 10e, 12f, 14cf, 22f, 26, 36ce, 46, 58, 72, 130, 198, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d }}
{{Optimal ET sequence|legend=1| 10e, 12f, 14cf, 22f, 26, 36ce, 46, 58, 72, 130, 198, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d }}


== Semicanousmic (14641/14580) ==
== Semicanousmic (14641/14580) ==
Line 408: Line 408:
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.2503, ~18/11 = 854.5421, ~7/4 = 968.6866
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.2503, ~18/11 = 854.5421, ~7/4 = 968.6866


{{Val list|legend=1| 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e }}
{{Optimal ET sequence|legend=1| 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e }}


[[Badness]]: 0.351 × 10<sup>-6</sup>
[[Badness]]: 0.351 × 10<sup>-6</sup>
Line 421: Line 421:
Optimal tuning (POTE): ~3/2 = 702.4931, ~18/11 = 854.6400, ~7/4 = 969.0099
Optimal tuning (POTE): ~3/2 = 702.4931, ~18/11 = 854.6400, ~7/4 = 969.0099


Optimal GPV sequence: {{val list| 87, 111, 181, 198, 323, 410 }}
{{Optimal ET sequence|legend=1| 87, 111, 181, 198, 323, 410 }}


Badness: 17.1 × 10<sup>-6</sup>
Badness: 17.1 × 10<sup>-6</sup>
Line 434: Line 434:
Optimal tuning (POTE): ~3/2 = 702.4099, ~18/11 = 854.6338, ~7/4 = 969.0228
Optimal tuning (POTE): ~3/2 = 702.4099, ~18/11 = 854.6338, ~7/4 = 969.0228


Optimal GPV sequence: {{val list| 87, 94, 111, 181, 198g, 212g, 292, 323, 410 }}
{{Optimal ET sequence|legend=1| 87, 94, 111, 181, 198g, 212g, 292, 323, 410 }}


Badness: 34.0 × 10<sup>-6</sup>
Badness: 34.0 × 10<sup>-6</sup>
Line 447: Line 447:
Optimal tuning (POTE): ~3/2 = 702.3413, ~18/11 = 854.6472, ~7/4 = 968.9734
Optimal tuning (POTE): ~3/2 = 702.3413, ~18/11 = 854.6472, ~7/4 = 968.9734


Optimal GPV sequence: {{val list| 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef }}
{{Optimal ET sequence|legend=1| 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef }}


Badness: 41.9 × 10<sup>-6</sup>
Badness: 41.9 × 10<sup>-6</sup>
Line 465: Line 465:
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.2681, ~5/4 = 386.4785, ~7/4 = 968.9552
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.2681, ~5/4 = 386.4785, ~7/4 = 968.9552


{{Val list|legend=1| 17c, 19e, 24, 34d, 41, 53, 58, 94, 99e, 118, 152, 270, 581, 733, 851, 1003, 1273, 1854, 2124b }}
{{Optimal ET sequence|legend=1| 17c, 19e, 24, 34d, 41, 53, 58, 94, 99e, 118, 152, 270, 581, 733, 851, 1003, 1273, 1854, 2124b }}


[[Badness]]: 0.120 × 10<sup>-6</sup>
[[Badness]]: 0.120 × 10<sup>-6</sup>
Line 478: Line 478:
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.2721, ~5/4 = 386.4790, ~7/4 = 968.9705
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.2721, ~5/4 = 386.4790, ~7/4 = 968.9705


{{Val list|legend=1| 17c, 34dff, 36ce, 41, 53, 58, 94, 111, 152f, 212, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bf }}
{{Optimal ET sequence|legend=1| 17c, 34dff, 36ce, 41, 53, 58, 94, 111, 152f, 212, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bf }}


[[Badness]]: 3.31 × 10<sup>-6</sup>
[[Badness]]: 3.31 × 10<sup>-6</sup>
Line 491: Line 491:
Mapping generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 41, 53, 84, 87, 130, 183, 224, 270, 494, 764, 1164, 1205, 1475, 1969, 2239, 3133de, 4608cde, 5102bcde, 10474bbccdddeee }}
{{Optimal ET sequence|legend=1| 41, 53, 84, 87, 130, 183, 224, 270, 494, 764, 1164, 1205, 1475, 1969, 2239, 3133de, 4608cde, 5102bcde, 10474bbccdddeee }}


=== Tridecimal olympic ===
=== Tridecimal olympic ===
Line 502: Line 502:
Mapping generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e }} *
{{Optimal ET sequence|legend=1| 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e }} *


<nowiki>*</nowiki> [[optimal patent val]]: [[3044edo|3044]]
<nowiki>*</nowiki> [[optimal patent val]]: [[3044edo|3044]]

Revision as of 17:53, 7 May 2023

A rank-4 temperament has a period and three additional independent generators. Typical examples include 7-limit JI, full 11-limit temperament with a one-dimensional comma basis, and full 13-limit temperament with a two-dimensional comma basis.

Ptolemismic (100/99)

Subgroup: 2.3.5.7.11

Comma list: 100/99

Mapping: [⟨1 0 0 0 2], ⟨0 1 0 0 -2], ⟨0 0 1 0 2], ⟨0 0 0 1 0]]

Optimal tuning (POTE): ~3/2 = 704.9532, ~5/4 = 384.0675, ~7/4 = 970.8803

Optimal ET sequence: 7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e *

* optimal patent val: 104

Badness: 0.0225 × 10-6

Biyatismic (121/120)

Subgroup: 2.3.5.7.11

Comma list: 121/120

Mapping: [⟨1 0 1 0 2], ⟨0 1 1 0 1], ⟨0 0 -2 0 -1], ⟨0 0 0 1 0]]

Mapping generators: ~2, ~3, ~11/10, ~7

Optimal tuning (POTE): ~3/2 = 701.4578, ~11/10 = 157.7466, ~7/4 = 966.9589

Optimal ET sequence: 14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee

Badness: 0.0345 × 10-6

Valinorsmic (176/175)

Subgroup: 2.3.5.7.11

Comma list: 176/175

Mapping: [⟨1 0 0 0 -4], ⟨0 1 0 0 0], ⟨0 0 1 0 2], ⟨0 0 0 1 1]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal tuning (POTE): ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113

Optimal ET sequence: 22, 31, 46, 53, 58, 80, 111

Badness: 0.0186 × 10-6

Rastmic (243/242)

Subgroup: 2.3.5.7.11

Comma list: 243/242

Mapping: [⟨1 1 0 0 2], ⟨0 2 0 0 5], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]

Optimal tuning (POTE): ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464

Optimal ET sequence: 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202

Badness: 0.0509 × 10-6

Akua (352/351, 847/845)

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845

Mapping: [⟨1 0 0 10 0 5], ⟨0 1 0 -6 0 -3], ⟨0 0 1 1 0 0], ⟨0 0 0 0 1 1]]

Mapping generators: ~2, ~3, ~5, ~11

Optimal tuning (POTE): ~3/2 = 702.9075, ~5/4 = 387.0723, ~11/8 = 551.4538

Optimal ET sequence: 12f, 17c, 24d, 29, 41, 46, 53, 58, 87, 111, 140, 152f, 198, 350f, 437f, 490f

Badness: 2.550 × 10-6

Werckismic (441/440)

Subgroup: 2.3.5.7.11

Comma list: 441/440

Mapping: [⟨1 0 0 0 -3], ⟨0 1 0 0 2], ⟨0 0 1 0 -1], ⟨0 0 0 1 2]]

Mapping generators:~2, ~3, ~5, ~7

Optimal ET sequence: 10, 12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d

Commas 364/363, 441/440

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440

Mapping: [⟨1 0 0 0 -3 -8], ⟨0 1 0 0 2 5], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 2 3]]

Mapping generators: ~2, ~3, ~5, ~7

Mapping to lattice: [⟨0 1 1 -1 -1 0], ⟨0 0 1 0 -1 -2], ⟨0 0 1 1 1 1]]

Lattice basis:

3/2 length = 1.2263, 14/11 length = 1.4629, 21/16 length = 1.4657

Minimax tuning:

[[1 0 0 0 0 0⟩, [5/3 0 1/3 -1/3 -1/3 1/3⟩, [1/6 0 5/6 2/3 -5/6 1/3⟩, [0 0 0 1 0 0⟩, [1/6 0 -1/6 2/3 1/6 1/3⟩, [0 0 0 0 0 1⟩]
Eigenmonzos (unchanged-intervals): 2, 11/10, 8/7, 16/13
[[1 0 0 0 0 0⟩, [5/4 1/4 1/4 -1/4 -1/4 1/4⟩, [5/4 -3/4 5/4 -1/4 -1/4 1/4⟩, [17/8 -11/8 5/8 -1/8 3/8 1/8⟩, [5/2 -3/2 1/2 -1/2 1/2 1/2⟩, [17/8 -11/8 5/8 -9/8 3/8 9/8⟩]
Eigenmonzos (unchanged-intervals): 2, 14/13, 6/5, 11/9

Optimal ET sequence: 12f, 14cf, 15, 17c, 26, 29, 31f, 41, 46, 58, 72, 87, 130, 217, 289

Commas 351/350, 441/440

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440

Mapping: [⟨1 0 0 0 -3 1], ⟨0 1 0 0 2 -3], ⟨0 0 1 0 -1 2], ⟨0 0 0 1 2 1]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal ET sequence: 12f, 14cf, 19e, 26, 27e, 31, 45ef, 46, 58, 72, 103, 130, 233, 279, 409, 512bf, 642bf

Commas 196/195, 352/351

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351

Mapping: [⟨1 0 0 0 -3 2], ⟨0 1 0 0 2 -1], ⟨0 0 1 0 -1 -1], ⟨0 0 0 1 2 2]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal ET sequence: 10, 12f, 17c, 19e, 27e, 29, 31, 41, 46, 58, 87, 118, 145, 232

Tannic

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1287/1280

Mapping: [⟨1 0 0 0 -3 11], ⟨0 1 0 0 2 -4], ⟨0 0 1 0 -1 2], ⟨0 0 0 1 2 -2]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal ET sequence: 17c, 26, 29, 31, 43, 46, 60e, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 441/440, 561/560

Mapping: [⟨1 0 0 0 -3 11 7], ⟨0 1 0 0 2 -4 -3], ⟨0 0 1 0 -1 2 2], ⟨0 0 0 1 2 -2 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal ET sequence: 17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef

Commas 441/440, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 847/845

Mapping: [⟨1 0 0 0 -3 -3], ⟨0 1 0 0 2 2], ⟨0 0 1 1 1 1], ⟨0 0 0 2 4 5]]

Mapping generators: ~2, ~3, ~5, ~13/11

Optimal ET sequence: 12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d

Keenanismic (385/384)

Main article: Keenanismic family

Subgroup: 2.3.5.7.11

Comma list: 385/384

Mapping: [⟨1 0 0 0 7], ⟨0 1 0 0 1], ⟨0 0 1 0 -1], ⟨0 0 0 1 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Transpose: [2 3 5 7 385/35]

Minimax tuning:

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [7/3 1/3 2/3 -1/3 -1/3⟩, [7/3 1/3 -1/3 2/3 -1/3⟩, [7/3 1/3 -1/3 -1/3 2/3⟩]
Eigenmonzos (unchanged-intervals): 2, 3, 7/5, 11/5

Optimal ET sequence: 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de

Badness: 15.159 × 10-9

Martwin

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384

Mapping: [⟨1 0 0 0 7 2], ⟨0 1 0 0 1 4], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 -1 0]]

Mapping generators: ~2, ~3, ~5, ~7

Transpose: [2 3 5 7 385/35 324/25]

Lattice basis:

4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247

Minimax tuning: [to be confirmed]

[⟨1 0 0 0 0 0], ⟨0 1 0 0 0 0], ⟨2/3 4/3 1/3 0 0 -1/3], ⟨19/6 -1/6 -1/6 1/2 -1/2 1/6], ⟨19/6 -1/6 -1/6 -1/2 1/2 1/6], ⟨2/3 4/3 -2/3 0 0 2/3]]
Eigenmonzos (unchanged-intervals): 2, 14/11, 13/10, 4/3

Optimal ET sequence: 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff

Badness: 2.206 × 10-6

Ancient

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624

Mapping: [⟨1 0 0 0 7 -4], ⟨0 1 0 0 1 -1], ⟨0 0 1 0 -1 4], ⟨0 0 0 1 -1 0]]

Transpose: [2 3 5 7 385/35 625/48]

Optimal ET sequence: 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def

Badness: 2.573 × 10-6

Commas 351/350, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 385/384

Mapping: [⟨1 0 0 0 7 1], ⟨0 1 0 0 1 -3], ⟨0 0 1 0 -1 2], ⟨0 0 0 1 -1 1]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal ET sequence: 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff

Commas 352/351, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 1 -2], ⟨0 0 1 0 -1 -1], ⟨0 0 0 1 -1 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal ET sequence: 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e

Commas 364/363, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 385/384

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 1 3], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 -1 -3]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal ET sequence: 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159

Commas 385/384, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 847/845

Mapping: [⟨1 0 0 0 7 7], ⟨0 1 0 0 1 1], ⟨0 0 1 1 -2 -2], ⟨0 0 0 2 -2 -1]]

Mapping generators: ~2, ~3, ~5, ~13/11

Optimal ET sequence: 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee

Swetismic (540/539)

Subgroup: 2.3.5.7.11

Comma list: 540/539

Mapping: [⟨1 0 0 0 2], ⟨0 1 0 0 3], ⟨0 0 1 0 1], ⟨0 0 0 1 -2]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal ET sequence: 8d, 9, 10, 12e, 14c, 17c, 19, 22, 27e, 31, 41, 53, 58, 72, 130, 152, 224, 354, 506e, 578, 730de, 761d, 985d, 1115de, 1267dde

Commas 540/539, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 847/845

Mapping: [⟨1 0 0 0 2 2], ⟨0 1 0 0 3 3], ⟨0 0 1 1 -1 -1], ⟨0 0 0 2 -4 -3]]

Mapping generators: ~2, ~3, ~5, ~13/11

Optimal ET sequence: 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f

Commas 540/539, 625/624

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624

Mapping: [⟨1 0 0 0 2 -4], ⟨0 1 0 0 3 -1], ⟨0 0 1 0 1 4], ⟨0 0 0 1 -2 0]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal ET sequence: 19, 22, 31, 49f, 50, 53, 72, 103, 121, 152f, 193, 224

Commas 540/539, 676/675

Subgroup: 2.3.5.7.11

Comma list: 540/539, 676/675

Mapping: [⟨1 0 0 0 2 -1], ⟨0 2 0 0 6 3], ⟨0 0 1 0 1 1], ⟨0 0 0 1 -2 0]]

Mapping generators: ~2, ~26/15, ~5, ~7

Optimal ET sequence: 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de

Pentacircle (896/891)

Subgroup: 2.3.5.7.11

Comma list: 896/891

Mapping: [⟨1 0 0 0 7], ⟨0 1 0 0 -4], ⟨0 0 1 0 0], ⟨0 0 0 1 1]]

Optimal tuning (POTE): ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722

Optimal ET sequence: 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee

Badness: 0.0658 × 10-6

Commas 352/351, 364/363

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 -4 -7], ⟨0 0 1 0 0 0], ⟨0 0 0 1 1 1]]

Optimal ET sequence: 12f, 17c, 22, 29, 34d, 41, 46, 58, 80, 87, 121, 167, 179ef, 208, 266ef, 433bceef, 641bbceeeff, 699bbcdeeeff

Badness: 3.375 × 10-6

Topsy (847/845, 1001/1000)

Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1001/1000

Mapping: [⟨1 0 0 2 0 1], ⟨0 1 0 0 0 0], ⟨0 0 1 1 1 1], ⟨0 0 0 4 -3 1]]

Mapping generators: ~2, ~3, ~5, ~13/10

Optimal ET sequence: 16, 21, 24d, 29, 37, 45ef, 50, 53, 58, 87, 103, 111, 140, 190, 198, 301, 388, 689e

Lehmerismic (3025/3024)

Subgroup: 2.3.5.7.11

Comma list: 3025/3024

Mapping: [⟨1 0 0 0 2], ⟨0 1 0 1 2], ⟨0 0 1 0 -1], ⟨0 0 0 2 1]]

Mapping generators: ~2, ~3, ~5, ~55/36

Optimal ET sequence: 7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee

Trimitone (8019/8000)

Subgroup: 2.3.5.7.11

Comma list: 8019/8000

Mapping: [⟨1 0 0 0 6], ⟨0 1 0 0 -6], ⟨0 0 1 0 3], ⟨0 0 0 1 0]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal ET sequence: 12, 19, 26, 39d, 46, 53, 58, 72, 118, 130, 183, 190, 248, 255, 301, 373, 804, 876, 1177be

Kalismic (9801/9800)

Subgroup: 2.3.5.7.11

Comma list: 9801/9800

Mapping: [⟨2 0 0 0 3], ⟨0 1 0 0 -2], ⟨0 0 1 0 1], ⟨0 0 0 1 1]]

Mapping generators: ~99/70, ~3, ~5, ~7

Optimal ET sequence: 8d, 10, 12, 22, 34d, 46, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 4038, 4190, 4532, 11254, 15786e, 21896e

Commas 1716/1715, 2080/2079

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079

Mapping: [⟨2 0 0 0 3 -7], ⟨0 1 0 0 -2 1], ⟨0 0 1 0 1 0], ⟨0 0 0 1 1 2]]

Mapping generators: ~99/70, ~3, ~5, ~7

Lattice basis:

3/2 length = 1.1956, 7/4 length = 1.4506, 14/13 length = 1.8299

Minimax tuning:

[[1 0 0 0 0 0⟩, [7/10 4/5 0 -2/5 0 1/5⟩, [7/10 -1/5 1 -2/5 0 1/5⟩, [7/5 -2/5 0 1/5 0 2/5⟩, [11/5 -11/5 1 3/5 0 1/5⟩, [0 0 0 0 0 1⟩]
Eigenmonzos (unchanged-intervals): 2, 6/5, 16/13, 9/7

Optimal ET sequence: 10e, 12f, 14cf, 22f, 26, 36ce, 46, 58, 72, 130, 198, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d

Semicanousmic (14641/14580)

Subgroup: 2.3.5.7.11

Comma list: 14641/14580

Mapping: [⟨1 0 2 0 1], ⟨0 1 2 0 2], ⟨0 0 -4 0 -1], ⟨0 0 0 1 0]]

Mapping generators: ~2, ~3, ~18/11, ~7

Optimal tuning (POTE): ~3/2 = 702.2503, ~18/11 = 854.5421, ~7/4 = 968.6866

Optimal ET sequence: 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e

Badness: 0.351 × 10-6

Tridecimal semicanousmic

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 14641/14580

Mapping: [⟨1 0 2 0 1 -6], ⟨0 1 2 0 2 3], ⟨0 0 -4 0 -1 3], ⟨0 0 0 1 0 1]]

Optimal tuning (POTE): ~3/2 = 702.4931, ~18/11 = 854.6400, ~7/4 = 969.0099

Optimal ET sequence: 87, 111, 181, 198, 323, 410

Badness: 17.1 × 10-6

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 1089/1088, 14641/14580

Mapping: [⟨1 0 2 0 1 -6 -4], ⟨0 1 2 0 2 3 6], ⟨0 0 -4 0 -1 3 -2], ⟨0 0 0 1 0 1 0]]

Optimal tuning (POTE): ~3/2 = 702.4099, ~18/11 = 854.6338, ~7/4 = 969.0228

Optimal ET sequence: 87, 94, 111, 181, 198g, 212g, 292, 323, 410

Badness: 34.0 × 10-6

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 1089/1088, 1216/1215, 1445/1444

Mapping: [⟨1 0 2 0 1 -6 -4 -4], ⟨0 1 2 0 2 3 6 7], ⟨0 0 -4 0 -1 3 -2 -4], ⟨0 0 0 1 0 1 0 0]]

Optimal tuning (POTE): ~3/2 = 702.3413, ~18/11 = 854.6472, ~7/4 = 968.9734

Optimal ET sequence: 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef

Badness: 41.9 × 10-6

Semiporwellismic (16384/16335)

See Semiporwellismic clan #Semiporwellismic.

Symbiotic (19712/19683)

Subgroup: 2.3.5.7.11

Comma list: 19712/19683

Mapping: [⟨1 0 0 0 -8], ⟨0 1 0 0 9], ⟨0 0 1 0 0], ⟨0 0 0 1 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal tuning (POTE): ~3/2 = 702.2681, ~5/4 = 386.4785, ~7/4 = 968.9552

Optimal ET sequence: 17c, 19e, 24, 34d, 41, 53, 58, 94, 99e, 118, 152, 270, 581, 733, 851, 1003, 1273, 1854, 2124b

Badness: 0.120 × 10-6

Tridecimal symbiotic

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 19712/19683

Mapping: [⟨1 0 0 0 -8 -13], ⟨0 1 0 0 9 12], ⟨0 0 1 0 0 -1], ⟨0 0 0 1 -1 0]]

Optimal tuning (POTE): ~3/2 = 702.2721, ~5/4 = 386.4790, ~7/4 = 968.9705

Optimal ET sequence: 17c, 34dff, 36ce, 41, 53, 58, 94, 111, 152f, 212, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bf

Badness: 3.31 × 10-6

Olympic (131072/130977)

Subgroup: 2.3.5.7.11

Comma list: 131072/130977

Mapping: [⟨1 0 0 0 17], ⟨0 1 0 0 -5], ⟨0 0 1 0 0], ⟨0 0 0 1 -2]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal ET sequence: 41, 53, 84, 87, 130, 183, 224, 270, 494, 764, 1164, 1205, 1475, 1969, 2239, 3133de, 4608cde, 5102bcde, 10474bbccdddeee

Tridecimal olympic

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095

Mapping: [⟨1 0 0 0 17 12], ⟨0 1 0 0 -5 -2], ⟨0 0 1 0 0 -1], ⟨0 0 0 1 -2 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal ET sequence: 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e *

* optimal patent val: 3044

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