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| == Regular temperaments == | | == Regular temperaments == |
| 16ED5/2 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[Subgroup temperaments|subgroup temperament]] which tempers out 256/255, 361/360, and 4624/4617, which is a [[cluster temperament]] with 12 clusters of notes in an octave (''quintaleap'' temperament). This temperament is supported by {{Val list|12, 109, 121, 133, 145}}, and [[157edo|157]] EDOs.
| | {{Main| Quintaleap family }} |
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| Tempering out 400/399 (equating 20/19 and 21/20) leads ''[[Octagar temperaments #Quintupole|quintupole]]'' (12&121), and tempering out 476/475 (equating 19/17 with 28/25) leads ''[[Hemifamity temperaments #Quinticosiennic|quinticosiennic]]'' (12&145).
| | 16ed5/2 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[subgroup temperament]] which tempers out 256/255, 361/360, and 4624/4617, which is a [[cluster temperament]] with 12 clusters of notes in an octave (''quintaleap'' temperament). This temperament is supported by {{Val list| 12-, 109-, 121-, 133-, 145- }}, and [[157edo]]. |
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| Another temperament related to 16ED5/2 is ''[[Marvel temperaments #Quintapole|quintapole]]'' (12&85). It is practically identical to the [[18/17s equal temperament #Related temperament|Galilei tuning]], which is generated by the ratios 2/1 and 18/17.
| | Tempering out 400/399 (equating 20/19 and 21/20) leads to ''[[quintupole]]'' (12&121), and tempering out 476/475 (equating 19/17 with 28/25) leads to ''[[quinticosiennic]]'' (12&145). |
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| | | Another temperament related to 16ed5/2 is ''[[quintapole]]'' (12&85). It is practically identical to the [[18/17s equal temperament #Related temperament|Galilei tuning]], which is generated by the ratios 2/1 and 18/17. |
| ; <font style="font-size: 1.15em">[[High badness temperaments #Quintaleap|Quintaleap]] (12&121)</font>
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| '''5-limit'''<br>
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| Comma: {{monzo|37 -16 -5}} = 137438953472/134521003125<br>
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| Mapping: [{{val|1 2 1}}, {{val|0 -5 16}}]<br>
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| POTE generator: ~135/128 = 99.267<br>
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| Vals: 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc<br>
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| Badness: 0.444506<br><br>
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| '''2.3.5.17.19 subgroup'''<br>
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| Comma list: 256/255, 361/360, 4624/4617<br>
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| Gencom: [2 18/17; 256/255 361/360 4624/4617]<br>
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| Gencom mapping: [{{val|1 2 1 5 4}}, {{val|0 -5 16 -11 3}}]<br>
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| POTE generator: ~18/17 = 99.276<br>
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| Vals: 12, 109, 121, 133<br>
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| RMS error: 0.3427 cents<br><br>
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| ; <font style="font-size: 1.15em">[[Octagar temperaments #Quintupole|Quintupole]] (12&121)</font>
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| {{See also|34ed7 #34ed7 as a generator}}
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| '''7-limit'''<br>
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| Comma list: 4000/3969, 458752/455625<br>
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| Mapping: [{{val|1 2 1 0}}, {{val|0 -5 16 34}}]<br>
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| POTE generator: ~135/128 = 99.175<br>
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| Vals: 12, 97, 109, 121<br>
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| Badness: 0.111620<br><br>
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| '''11-limit'''<br>
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| Comma list: 896/891, 1375/1372, 4375/4356<br>
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| Mapping: [{{val|1 2 1 0 -1}}, {{val|0 -5 16 34 54}}]<br>
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| POTE generator: ~35/33 = 99.156<br>
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| Vals: 12, 109, 121, 351bde, 472bdee<br>
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| Badness: 0.056501<br><br>
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| '''13-limit'''<br>
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| Comma list: 352/351, 364/363, 625/624, 2704/2695<br>
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| Mapping: [{{val|1 2 1 0 -1 -2}}, {{val|0 -5 16 34 54 69}}]<br>
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| POTE generator: ~35/33 = 99.165<br>
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| Vals: 12f, 109, 121<br>
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| Badness: 0.038431<br><br>
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| '''17-limit'''<br>
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| Comma list: 256/255, 352/351, 364/363, 375/374, 442/441<br>
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| Mapping: [{{val|1 2 1 0 -1 -2 5}}, {{val|0 -5 16 34 54 69 -11}}]<br>
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| POTE generator: ~18/17 = 99.172<br>
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| Vals: 12f, 109, 121<br>
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| Badness: 0.028721<br><br>
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| '''19-limit'''<br>
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| Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374<br>
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| Mapping: [{{val|1 2 1 0 -1 -2 5 4}}, {{val|0 -5 16 34 54 69 -11 3}}]<br>
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| POTE generator: ~18/17 = 99.164<br>
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| Vals: 12f, 109, 121<br>
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| Badness: 0.023818<br><br>
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| ; <font style="font-size: 1.15em">[[Hemifamity temperaments #Quinticosiennic|Quinticosiennic]] (12&145)</font>
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| '''7-limit'''<br>
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| Comma list: 5120/5103, 395136/390625<br>
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| Mapping: [{{val|1 2 1 -1}}, {{val|0 -5 16 46}}]<br>
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| POTE generator: ~135/128 = 99.345<br>
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| Vals: 12, 133, 145, 157, 302c, 459bcc<br>
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| Badness: 0.158041<br><br>
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| '''11-limit'''<br>
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| Comma list: 441/440, 896/891, 78408/78125<br>
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| Mapping: [{{val|1 2 1 -1 -2}}, {{val|0 -5 16 46 66}}]<br>
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| POTE generator: ~35/33 = 99.318<br>
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| Vals: 12, 133, 145<br>
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| Badness: 0.080674<br><br>
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| '''13-limit'''<br>
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| Comma list: 196/195, 352/351, 364/363, 78408/78125<br>
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| Mapping: [{{val|1 2 1 -1 -2 -3}}, {{val|0 -5 16 46 66 81}}]<br>
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| POTE generator: ~35/33 = 99.307<br>
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| Vals: 12f, 133, 145<br>
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| Badness: 0.052464<br><br>
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| '''17-limit'''<br>
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| Comma list: 196/195, 256/255, 352/351, 364/363, 3757/3750<br>
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| Mapping: [{{val|1 2 1 -1 -2 -3 5}}, {{val|0 -5 16 46 66 81 -11}}]<br>
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| POTE generator: ~18/17 = 99.308<br>
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| Vals: 12f, 133, 145<br>
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| Badness: 0.037108<br><br>
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| '''19-limit'''<br>
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| Comma list: 196/195, 256/255, 352/351, 361/360, 364/363, 476/475<br>
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| Mapping: [{{val|1 2 1 -1 -2 -3 5 4}}, {{val|0 -5 16 46 66 81 -11 3}}]<br>
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| POTE generator: ~18/17 = 99.303<br>
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| Vals: 12f, 133, 145<br>
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| Badness: 0.028440<br><br>
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| ; <font style="font-size: 1.15em">[[Marvel temperaments #Quintapole|Quintapole]] (12&85)</font>
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| {{See also|42ed11 #42ed11 as a generator}}
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| '''7-limit'''<br>
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| Comma list: 225/224, 7812500/7411887<br>
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| Mapping: [{{val|1 2 1 1}}, {{val|0 -5 16 22}}]<br>
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| POTE generator: ~21/20 = 98.994<br>
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| Vals: 12, 73c, 85, 97d<br>
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| Badness: 0.192498<br><br>
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| '''11-limit'''<br>
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| Comma list: 100/99, 225/224, 85184/84035<br>
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| Mapping: [{{val|1 2 1 1 0}}, {{val|0 -5 16 22 42}}]<br>
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| POTE generator: ~21/20 = 98.954<br>
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| Vals: 12, 73ce, 85, 97d<br>
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| Badness: 0.104353<br><br>
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| == See also == | | == See also == |
16ED5/2 is the equal division of the 5/2 interval into 16 parts of 99.1446 cents each. This is the scale which occurs as the dominant reformed Mixolydian mode tuned as an equal division of a just interval.
Intervals
| Degrees
|
Enneatonic
|
ED38\29
|
Golden
|
ED5/2
|
ED(7φ+6)\5(φ+1)
|
ED4\3=r¢
|
| 1
|
1#/2b
|
F#/Gb
|
98.276
|
98.3795
|
99.145
|
99.2705
|
100
|
| 2
|
2
|
G
|
196.552
|
196.759
|
198.289
|
198.541
|
200
|
| 3
|
2#/3b
|
G#/Jb
|
G#/Ab
|
294.828
|
295.138
|
297.433
|
297.8115
|
300
|
| 4
|
3
|
J
|
A
|
393.103
|
393.518
|
396.578
|
397.082
|
400
|
| 5
|
3#/4b
|
J#/Ab
|
A#/Bb
|
491.379
|
491.897
|
495.723
|
496.3525
|
500
|
| 6
|
4
|
A
|
B
|
589.655
|
590.277
|
594.868
|
595.623
|
600
|
| 7
|
5
|
B
|
H
|
687.931
|
688.656
|
694.012
|
694.894
|
700
|
| 8
|
5#/6b
|
B#/Hb
|
H#/Cb
|
786.207
|
787.036
|
793.157
|
794.164
|
800
|
| 9
|
6
|
H
|
C
|
884.483
|
885.415
|
892.3015
|
893.435
|
900
|
| 10
|
6#/7b
|
H#/Cb
|
C#/Db
|
982.759
|
983.795
|
991.446
|
992.705
|
1000
|
| 11
|
7
|
C
|
D
|
1081.0345
|
1082.174
|
1090.591
|
1091.976
|
1100
|
| 12
|
7#/8b
|
C#/Db
|
D#/Sb
|
1179.31
|
1180.554
|
1189.735
|
1191.246
|
1200
|
| 13
|
8
|
D
|
S
|
1277.586
|
1278.933
|
1288.88
|
1290.517
|
1300
|
| 14
|
8#/9b
|
D#/Eb
|
S#/Eb
|
1375.862
|
1377.313
|
1388.0245
|
1389.787
|
1400
|
| 15
|
9
|
E
|
1474.138
|
1475.692
|
1487.169
|
1489.058
|
1500
|
| 16
|
1
|
F
|
1572.414
|
1574.0715
|
1586.314
|
1588.328
|
1600
|
Coincidentally, 133 steps of the pyrite EDX of this size exceed 11 octaves by only 2.978¢.
Regular temperaments
16ed5/2 can also be thought of as a generator of the 2.3.5.17.19 subgroup temperament which tempers out 256/255, 361/360, and 4624/4617, which is a cluster temperament with 12 clusters of notes in an octave (quintaleap temperament). This temperament is supported by Template:Val list, and 157edo.
Tempering out 400/399 (equating 20/19 and 21/20) leads to quintupole (12&121), and tempering out 476/475 (equating 19/17 with 28/25) leads to quinticosiennic (12&145).
Another temperament related to 16ed5/2 is quintapole (12&85). It is practically identical to the Galilei tuning, which is generated by the ratios 2/1 and 18/17.
See also