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'''[[EDF|Division of the just perfect fifth]] into 28 equal parts''' (28EDF) is related to [[48edo|48 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 3.3514 cents stretched and the step size is about 25.0698 cents (corresponding to 47.8663 [[edo]]). It is related to the regular temperament which tempers out |187 -159 28> in the 5-limit; 6656/6655, 256000/255879, and 38671875/38614472 in the 13-limit (2.3.5.11.13 subgroup), which is supported by 335, [[383edo|383]], 718, [[1053edo|1053]], and 1101 EDOs.
{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 28 equal parts''' (28EDF) is related to [[48edo|48 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 3.3514 cents stretched and the step size is about 25.0698 cents (corresponding to 47.8663 [[edo]]).
 
It is related to the regular temperament which tempers out |187 -159 28> in the 5-limit; [[6656/6655]], [[256000/255879]], and [[38671875/38614472]] in the 13-limit (2.3.5.11.13 subgroup), which is supported by {{EDOs| 335, 383, 718, 1053, and 1101}} EDOs.
 
== Harmonics ==
{{Harmonics in equal|28|3|2|intervals=prime}}


== Intervals ==
== Intervals ==
{| class="wikitable"
{{Interval table}}
! rowspan="2" |
 
! rowspan="2" |''ed233\420-5¢ (~101ed4!)''
 
! rowspan="2" |ed31\54 (~49edo!)
{{stub}}
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed696¢
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed708¢
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57 (~47edo!)
! rowspan="2" |''ed37\60+5¢ (~91ed4!)''
|-
!(~ed17\29)
!(~ed10\17)
|-
|1
|''23.5969-23.7755''
|24.6032
|24.81485
|24.8571
|25.0698
|25.1299
|25.2091
|25.2857
|25.3234
|25.5639
|''26.4286-26.6071''
|-
|2
|''47.1939-47.551''
|49.20635
|49.6297
|49.7143
|50.1396
|50.2597
|50.41825
|50.5714
|50.6475
|51.1278
|''52.8571-53.2143''
|-
|3
|''70.7908-71.3625''
|73.8095
|74.4446
|74.5714
|75.2095
|75.3896
|75.6274
|75.8571
|75.9712
|76.6917
|''79.2857-79.8574''
|-
|4
|''94.3878-95.102''
|98.4127
|99.2594
|99.4286
|100.2793
|100.5194
|100.8365
|101.1429
|101.295
|102.2556
|''105.7143-106.4286''
|-
|5
|''117.9847-118.87755''
|123.0159
|124.0743
|124.2857
|125.3491
|125.6493
|126.0456
|126.4286
|126.6187
|127.81955
|''132.1429-133.0357''
|-
|6
|''141.5816-142.653''
|147.61905
|148.8891
|149.1429
|150.4189
|150.77915
|151.2547
|151.7143
|151.9425
|153.3835
|''158.5714-159.6429''
|-
|7
|''165.1786-166.4286''
|172.2222
|173.704
|174
|175.48875
|175.909
|176.4639
|177
|177.2662
|178.947
|''185-186.25''
|-
|8
|''188.7755-190.2041''
|196.8254
|198.5188
|198.8571
|200.5586
|201.0389
|201.673
|202.2857
|202.5899
|204.5113
|''211.4286-212.8571''
|-
|9
|''212.3724-213.9796''
|221.4286
|223.3337
|223.7143
|225.6284
|226.1687
|226.8821
|227.5714
|227.9137
|230.0752
|''237.8571-239.4643''
|-
|10
|''235.9694-237.7551''
|246.03175
|248.1485
|248.5714
|250.6982
|251.2986
|252.0912
|252.8571
|253.2374
|255.6391
|''264.2857-266.0714''
|-
|11
|''259.5663-261.5306''
|270.6349
|272.9634
|273.4286
|275.768
|276.4284
|277.3004
|278.1429
|278.5612
|281.203
|''290.7143-292.6786''
|-
|12
|''283.1633-285.3061''
|295.2381
|297.782
|298.2857
|300.8379
|301.5583
|302.5095
|303.4286
|303.8849
|306.7669
|''317.1429-319.2857''
|-
|13
|''306.7602-309.0816''
|319.8413
|322.5931
|323.1429
|325.9077
|326.6682
|327.7186
|328.7143
|329.2087
|332.3308
|''343.5714-345.8929''
|-
|14
|''330.3571-332.857''1
|344.4444
|347.408
|348
|350.9775
|351.818
|352.9277
|354
|354.5324
|357.8947
|''370-372.5''
|-
|15
|''353.9541-356.63265''
|369.0476
|372.2228
|372.8571
|376.0473
|376.9479
|378.13685
|379.2857
|379.8561
|383.45865
|''396.4286-399.1071''
|-
|16
|''377.551-380.4082''
|393.6508
|397.03765
|397.7143
|401.1171
|402.0777
|403.346
|404.5714
|405.1799
|409.0226
|''422.8571-425.7143''
|-
|17
|''401.148-404.1837''
|418.254
|421.8525
|422.5714
|426.187
|427.2076
|428.5551
|429.8571
|430.5036
|434.5865
|''449.5714-452.3214''
|-
|18
|''424.7449-427.9592''
|442.8571
|446.66735
|447.4286
|451.2568
|452.33745
|453.7642
|455.1429
|455.8274
|460.1504
|''475.7143-478.9286''
|-
|19
|''448.3418-451.7347''
|467.460
|471.4822
|472.2857
|476.3266
|477.4673
|478.97333
|480.4286
|481.1511
|485.7143
|''502.1429-505.5357''
|-
|20
|''471.9388-475.5102''
|492.0635
|496.2971
|497.1429
|501.3964
|502.5972
|504.1825
|505.7143
|506.4749
|511.2781
|''528.5714-532.1429''
|-
|21
|''495.5357-499.2857''
|516.6667
|521.1119
|522
|526.46625
|527.727
|529.3916
|531
|531.7986
|536.8421
|''555-558.75''
|-
|22
|''519.13265-523.0612''
|541.2698
|545.9268
|546.8571
|551.536
|552.8569
|554.6007
|556.2857
|557.1223
|562.406
|''581.4286-585.3571''
|-
|23
|''542.7296-546.8367''
|565.873
|570.7416
|571.4286
|576.6059
|577.9867
|579.8098
|581.5714
|582.4461
|587.9699
|''607.8571-611.9643''
|-
|24
|''566.3265-570.6122''
|590.4762
|595.5565
|596.5714
|601.6757
|603.1166
|605.019
|606.8571
|607.7698
|613.5338
|''634.2857-638.5714''
|-
|25
|''589.9235-594.3878''
|615.0794
|620.3713
|621.4286
|626.7455
|628.2465
|630.2281
|632.1429
|633.0936
|639.0977
|''660.7143-665.1786''
|-
|26
|''613.5204-618.1633''
|639.6825
|645.1862
|646.2857
|651.8154
|653.3763
|655.4372
|657.4286
|658.4173
|664.66165
|''687.1429-691.7857''
|-
|27
|''637.1173-641.9388''
|664.2857
|670.001
|671.1429
|676.8852
|678.5062
|680.6463
|682.7143
|683.7411
|690.2256
|''713.5714-718.3929''
|-
|28
|''660.7143-665.714''3
|688.8889
|694.8158
|696
|701.955
|703.636
|705.85545
|708
|709.0648
|715.7895
|''740-745''
|-
|29
|''684.3112-689.4898''
|713.4921
|719.6307
|720.8571
|727.0248
|728.7659
|731.0646
|733.2857
|734.3885
|741.3534
|''766.4286-771.6071''
|-
|30
|''707.9082-713.2653''
|738.0952
|744.4456
|745.7143
|752.0946
|753.89575
|756.2736
|758.5714
|759.7123
|766.9173
|''792.8571-798.2143''
|-
|31
|''731.5051-737.0408''
|762.6984
|769.2604
|770.5714
|777.1645
|779.0256
|781.4828
|783.8571
|785.036
|792.4812
|''819.2857-824.8214''
|-
|32
|''755.102-760.8163''
|787.3016
|794.0753
|795.4286
|802.2343
|804.1555
|806.6919
|809.1429
|810.3598
|818.0451
|''845.7143-851.4286''
|-
|33
|''778.699-784.5918''
|811.9048
|818.89015
|820.2857
|827.3041
|829.2853
|831.9011
|834.4286
|835.6835
|843.609
|''872.1429-876.0357''
|-
|34
|''802.2959-808.36735''
|836.5079
|843.705
|845.1429
|852.3739
|854.4152
|857.1102
|859.7143
|961.0073
|869.1729
|''898.5714-904.6429''
|-
|35
|''825.8929-832.1429''
|861.1111
|868.51985
|870
|877.44375
|879.545
|882.3193
|885
|886.331
|894.7368
|''925-931.25''
|-
|36
|''849.4898-855.9184''
|885.7143
|893.3347
|894.8571
|902.5136
|904.6749
|907.5284
|910.2857
|911.6547
|920.30075
|''951.4286-957.8571''
|-
|37
|''873.0867-879.6939''
|910.3175
|918.1496
|919.7143
|927.5834
|929.8048
|932.3767
|935.5714
|936.9785
|945.8647
|''977.8571-984.4643''
|-
|38
|''896.6387-903.4694''
|934.9206
|942.9644
|944.5714
|952.6532
|954.9346
|957.9467
|960.8571
|962.3022
|971.4286
|''1004.2857-1011.0714''
|-
|39
|''920.2806-927.2449''
|959.5238
|967.7793
|969.4286
|977.723
|980.0645
|983.1558
|986.1429
|987.626
|996.9925
|''1030.7143-1037.6786''
|-
|40
|''943.8776-951.0204''
|984.127
|992.5941
|994.2857
|1002.7929
|1005.1943
|1008.3649
|1011.4286
|1012.9497
|1022.5564
|''1057.1429-1064.2857''
|-
|41
|''967.4745-974.7959''
|1008.7302
|1017.409
|1019.1429
|1027.8627
|1030.3242
|1033.57405
|1035.7143
|1038.2735
|1048.1203
|''1083.5714-1090.8929''
|-
|42
|''991.0714-998.5714''
|1033.333
|1042.2238
|1044
|1052.9325
|1055.454
|1058.7832
|1062
|1063.5972
|1073.6842
|''1110-1117.5''
|-
|43
|''1014.6684-1022.3469''
|1057.9365
|1067.0386
|1068.8571
|1078.0023
|1080.5839
|1083.9923
|1087.2857
|1088.92095
|1099.2481
|''1136.4286-1144.1071''
|-
|44
|''1038.2653-1046.12245''
|1082.5397
|1091.8535
|1093.7143
|1103.0721
|1105.7138
|1109.2014
|1112.5714
|1114.2447
|1124.812
|''1162.8571-1170.7143''
|-
|45
|''1061.8622-1069.898''
|1107.1429
|1116.6684
|1118.5714
|1128.142
|1130.8436
|1134.41055
|1137.8571
|1139.5684
|1150.3759
|''1189.2857-1197.3214''
|-
|46
|''1085.4592-1093.6735''
|1131.7646
|1141.4832
|1143.4286
|1153.2118
|1155.9735
|1159.6297
|1163.1429
|1164.8922
|1175.93985
|''1215.7143-1223.9286''
|-
|47
|''1109.0561-1117.449''
|1156.3492
|1166.2981
|1168.2857
|1178.2816
|1181.1033
|1184.8288
|1188.4286
|1190.2159
|1201.5038
|''1242.1429-1250.5357''
|-
|48
|''1132.6531-1141.2245''
|1180.9524
|1191.1129
|1193.1429
|1203.3514
|1206.2332
|1210.0379
|1213.7143
|1215.5397
|1227.0677
|''1268.5714-1277.1429''
|-
|49
|''1156.25-1165''
|1205.5556
|1215.9278
|1218
|1228.42125
|1231.6306
|1235.247
|1239
|1240.8634
|1252.6316
|''1295-1303.75''
|-
|50
|''1179.8469-1188.7755''
|1230.1587
|1240.74265
|1242.8571
|1253.4911
|1256.4929
|1260.4561
|1264.2857
|1266.18715
|1278.1955
|''1321.4286-1330.3571''
|-
|51
|''1203.4439-1212.551''
|1254.7619
|1265.5575
|1267.7143
|1278.5609
|1281.6228
|1285.6653
|1289.5714
|1291.5109
|1303.7594
|''1347.8571-1356.9643''
|-
|52
|''1227.0408-1236.3265''
|1279.3651
|1290.37235
|1292.5714
|1303.6307
|1306.7526
|1310.8744
|1314.8571
|1316.8346
|1329.3233
|''1374.2857-1383.5714''
|-
|53
|''1250.6378-1260.102''
|1303.96825
|1315.1872
|1317.4286
|1328.7005
|1331.8825
|1336.0835
|1340.1429
|1342.1584
|1354.8872
|''1400.7143-1410.1786''
|-
|54
|''1227.0408-1236.3265''
|1328.5714
|1340.0021
|1342.2857
|1353.7704
|1357.01235
|1361.2927
|1365.4286
|1367.4821
|1380.4511
|''1427.1429-1436.7857''
|-
|55
|''1297.8316-1307.6531''
|1353.1746
|1364.8169
|1367.1429
|1378.8418
|1382.1422
|1386.5018
|1390.7143
|1392.8059
|1406.015
|''1453.5714-1463.3929''
|-
|56
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1392
|1403.91
|1407.2721
|1411.7109
|1416
|1418.1296
|1431.57895
|''1480-1490''
|}
[[Category:Edf]]
[[Category:Edonoi]]

Latest revision as of 19:21, 1 August 2025

← 27edf 28edf 29edf →
Prime factorization 22 × 7
Step size 25.0698 ¢ 
Octave 48\28edf (1203.35 ¢) (→ 12\7edf)
Twelfth 76\28edf (1905.31 ¢) (→ 19\7edf)
Consistency limit 6
Distinct consistency limit 6

Division of the just perfect fifth into 28 equal parts (28EDF) is related to 48 edo, but with the 3/2 rather than the 2/1 being just. The octave is about 3.3514 cents stretched and the step size is about 25.0698 cents (corresponding to 47.8663 edo).

It is related to the regular temperament which tempers out |187 -159 28> in the 5-limit; 6656/6655, 256000/255879, and 38671875/38614472 in the 13-limit (2.3.5.11.13 subgroup), which is supported by 335, 383, 718, 1053, and 1101 EDOs.

Harmonics

Approximation of prime harmonics in 28edf
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +3.4 +3.4 -3.6 -9.5 +10.3 -3.2 +8.7 -8.3 +11.9 +11.7 -3.5
Relative (%) +13.4 +13.4 -14.2 -37.8 +41.0 -12.6 +34.8 -33.3 +47.4 +46.6 -13.9
Steps
(reduced) 		48
(20) 		76
(20) 		111
(27) 		134
(22) 		166
(26) 		177
(9) 		196
(0) 		203
(7) 		217
(21) 		233
(9) 		237
(13)

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 25.1
2 50.1
3 75.2 21/20, 23/22, 24/23, 26/25, 27/26
4 100.3 17/16, 18/17
5 125.3 14/13, 15/14
6 150.4 12/11
7 175.5 10/9, 11/10, 21/19
8 200.6 9/8
9 225.6 17/15
10 250.7 15/13, 23/20
11 275.8 20/17, 27/23
12 300.8 25/21
13 325.9 6/5
14 351 11/9, 27/22
15 376 5/4, 26/21
16 401.1 19/15
17 426.2 23/18
18 451.3 13/10, 22/17
19 476.3 25/19
20 501.4 4/3
21 526.5 15/11, 19/14, 23/17, 27/20
22 551.5 11/8, 26/19
23 576.6 7/5
24 601.7 17/12, 24/17
25 626.7 10/7, 13/9, 23/16
26 651.8 16/11, 19/13
27 676.9
28 702 3/2


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