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'''16ED5/2''' is the equal division of the [[5/2]] interval into 16 parts of 99.1446 [[cent]]s each. This is the scale which occurs as the dominant reformed Mixolydian mode tuned as an equal division of a just interval.
{{Infobox ET}}
 
'''16ED5/2''' is the equal division of the [[5/2]] interval into 16 parts of 99.1446 [[cent]]s each. 16 equal divisions of the just major tenth is not a "real" xenharmonic tuning; it is a slightly compressed version of the normal [[12edo|12-tone scale]].  
== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable"
|+
|+
!Degrees
! Degrees
! colspan="3" |Enneatonic
! colspan="3" | Enneatonic
!ed38\29
! Cents
!Golden
!ed5/2
!ed(7φ+6)\5(φ+1)
!ed4\3=''r¢''
|-
|-
|1
| 1
|1#/2b
| 1#/2b
| colspan="2" |F#/Gb
| colspan="2" | F#/Gb
|98.276
| 99.145
|98.3795
|99.145
|99.2705
|''100''
|-
|-
|2
| 2
|2
| 2
| colspan="2" |G
| colspan="2" | G
|196.552
| 198.289
|196.759
|198.289
|198.541
|''200''
|-
|-
|3
| 3
|2#/3b
| 2#/3b
|G#/Jb
| G#/Jb
|''G#/Ab''
| ''G#/Ab''
|294.828
| 297.433
|295.138
|297.433
|297.8115
|''300''
|-
|-
|4
| 4
|3
| 3
|J
| J
|''A''
| ''A''
|393.103
| 396.578
|393.518
|396.578
|397.082
|''400''
|-
|-
|5
| 5
|3#/4b
| 3#/4b
|J#/Ab
| J#/Ab
|''A#/Bb''
| ''A#/Bb''
|491.379
| 495.723
|491.897
|495.723
|496.3525
|''500''
|-
|-
|6
| 6
|4
| 4
|A
| A
|''B''
| ''B''
|589.655
| 594.868
|590.277
|594.868
|595.623
|''600''
|-
|-
|7
| 7
|5
| 5
|B
| B
|''H''
| ''H''
|687.931
| 694.012
|688.656
|694.012
|694.894
|''700''
|-
|-
|8
| 8
|5#/6b
| 5#/6b
|B#/Hb
| B#/Hb
|''H#/Cb''
| ''H#/Cb''
|786.207
| 793.157
|787.036
|793.157
|794.164
|''800''
|-
|-
|9
| 9
|6
| 6
|H
| H
|''C''
| ''C''
|884.483
| 892.3015
|885.415
|892.3015
|893.435
|''900''
|-
|-
|10
| 10
|6#/7b
| 6#/7b
|H#/Cb
| H#/Cb
|''C#/Db''
| ''C#/Db''
|982.759
| 991.446
|983.795
|991.446
|992.705
|''1000''
|-
|-
|11
| 11
|7
| 7
|C
| C
|''D''
| ''D''
|1081.0345
| 1090.591
|1082.174
|1090.591
|1091.976
|''1100''
|-
|-
|12
| 12
|7#/8b
| 7#/8b
|C#/Db
| C#/Db
|''D#/Sb''
| ''D#/Sb''
|1179.31
| 1189.735
|1180.554
|1189.735
|1191.246
|''1200''
|-
|-
|13
| 13
|8
| 8
|D
| D
|''S''
| ''S''
|1277.586
| 1288.88
|1278.933
|1288.88
|1290.517
|''1300''
|-
|-
|14
| 14
|8#/9b
| 8#/9b
|D#/Eb
| D#/Eb
|''S#/Eb''
| ''S#/Eb''
|1375.862
| 1388.0245
|1377.313
|1388.0245
|1389.787
|''1400''
|-
|-
|15
| 15
|9
| 9
| colspan="2" |E
| colspan="2" | E
|1474.138
| 1487.169
|1475.692
|1487.169
|1489.058
|''1500''
|-
|-
|16
| 16
|1
| 1
| colspan="2" |F
| colspan="2" | F
|1572.414
| 1586.314
|1574.0715
|1586.314
|1588.328
|''1600''
|}
|}
Coincidentally, 133 steps of the pyrite EDX of this size exceed 11 octaves by only 2.978¢.


== 16ed5/2 as a generator ==
== Harmonics ==
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.
{{Harmonics in equal
| steps = 16
| num = 5
| denom = 2
}}
{{Harmonics in equal
| steps = 16
| num = 5
| denom = 2
| start = 12
| collapsed = 1
}}
 
== Regular temperaments ==
{{Main| Quintaleap family }}
 
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 {{Optimal ET sequence| 12-, 109-, 121-, 133-, 145- }}, and [[157edo]].


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).
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 [[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">Quintaleap (12&amp;121)</font>
== Scale tree ==
'''5-limit'''<br>
Ed5/2 scales can be approximated in [[EDO]]s by subdividing their approximations of 5/2.
Comma: {{monzo|37 -16 -5}} = 137438953472/134521003125<br>
Mapping: [{{val|1 2 1}}, {{val|0 -5 16}}]<br>
POTE generator: ~135/128 = 99.267<br>
Vals: 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc<br>
Badness: 0.444506<br><br>
'''2.3.5.17.19 subgroup'''<br>
Comma list: 256/255, 361/360, 4624/4617<br>
Gencom: [2 18/17; 256/255 361/360 4624/4617]<br>
Gencom mapping: [{{val|1 2 1 5 4}}, {{val|0 -5 16 -11 3}}]<br>
POTE generator: ~18/17 = 99.276<br>
Vals: 12, 109, 121, 133<br>
RMS error: 0.3427 cents<br><br>


; <font style="font-size: 1.15em">[[Octagar temperaments #Quintupole|Quintupole]] (12&amp;121)</font>
{| class="wikitable"
{{See also|34ed7 #34ed7 as a generator}}
|+
'''7-limit'''<br>
! colspan="4" |Major tenth
Comma list: 4000/3969, 458752/455625<br>
! Period
Mapping: [{{val|1 2 1 0}}, {{val|0 -5 16 34}}]<br>
!Notes
POTE generator: ~135/128 = 99.175<br>
|-
Vals: 12, 97, 109, 121<br>
|9\7
Badness: 0.111620<br><br>
|
'''11-limit'''<br>
|
Comma list: 896/891, 1375/1372, 4375/4356<br>
|
Mapping: [{{val|1 2 1 0 -1}}, {{val|0 -5 16 34 54}}]<br>
|96.429
POTE generator: ~132/125 = 99.156<br>
|Superpental Dorian mode ends, Mohajira Dorian-Mixolydian mode begins
Vals: 12, 109, 121, 351bde, 472bdee<br>
|-
Badness: 0.056501<br><br>
|
'''13-limit'''<br>
|31\24
Comma list: 352/351, 364/363, 625/624, 2704/2695<br>
|
Mapping: [{{val|1 2 1 0 -1 -2}}, {{val|0 -5 16 34 54 69}}]<br>
|
POTE generator: ~55/52 = 99.165<br>
|96.875
Vals: 12f, 109, 121<br>
|
Badness: 0.038431<br><br>
|-
'''17-limit'''<br>
|22\17
Comma list: 256/255, 352/351, 364/363, 375/374, 442/441<br>
|
Mapping: [{{val|1 2 1 0 -1 -2 5}}, {{val|0 -5 16 34 54 69 -11}}]<br>
|
POTE generator: ~18/17 = 99.172<br>
|
Vals: 12f, 109, 121<br>
|97.059
Badness: 0.028721<br><br>
|Mohajira Dorian-Mixolydian mode ends, Beatles Dorian-Mixolydian mode begins
'''19-limit'''<br>
|-
Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374<br>
|
Mapping: [{{val|1 2 1 0 -1 -2 5 4}}, {{val|0 -5 16 34 54 69 -11 3}}]<br>
|35\27
POTE generator: ~18/17 = 99.164<br>
|
Vals: 12f, 109, 121<br>
|
Badness: 0.023818<br><br>
|97.{{Overline|2}}
; <font style="font-size: 1.15em">[[Hemifamity temperaments #Quinticosiennic|Quinticosiennic]] (12&amp;145)</font>
|
'''7-limit'''<br>
|-
Comma list: 5120/5103, 395136/390625<br>
| 13\10
Mapping: [{{val|1 2 1 -1}}, {{val|0 -5 16 46}}]<br>
|
POTE generator: ~135/128 = 99.345<br>
|
Vals: 12, 133, 145, 157, 302c, 459bcc<br>
|
Badness: 0.158041<br><br>
|97.5
'''11-limit'''<br>
|Beatles Dorian-Mixolydian mode ends, Subpental Mixolydian mode begins
Comma list: 441/440, 896/891, 78408/78125<br>
|-
Mapping: [{{val|1 2 1 -1 -2}}, {{val|0 -5 16 46 66}}]<br>
|17\13
POTE generator: ~35/33 = 99.318<br>
|
Vals: 12, 133, 145<br>
|
Badness: 0.080674<br><br>
|
'''13-limit'''<br>
|98.077
Comma list: 196/195, 352/351, 364/363, 78408/78125<br>
|
Mapping: [{{val|1 2 1 -1 -2 -3}}, {{val|0 -5 16 46 66 81}}]<br>
|-
POTE generator: ~35/33 = 99.307<br>
|21\16
Vals: 12f, 133, 145<br>
|
Badness: 0.052464<br><br>
|
'''17-limit'''<br>
|
Comma list: 196/195, 256/255, 352/351, 364/363, 3757/3750<br>
|98.4375
Mapping: [{{val|1 2 1 -1 -2 -3 5}}, {{val|0 -5 16 46 66 81 -11}}]<br>
|Subpental Mixolydian mode ends, Pental Mixolydian mode begins
POTE generator: ~18/17 = 99.308<br>
|-
Vals: 12f, 133, 145<br>
|
Badness: 0.037108<br><br>
|25\19
'''19-limit'''<br>
|
Comma list: 196/195, 256/255, 352/351, 361/360, 364/363, 476/475<br>
|
Mapping: [{{val|1 2 1 -1 -2 -3 5 4}}, {{val|0 -5 16 46 66 81 -11 3}}]<br>
|98.684
POTE generator: ~18/17 = 99.303<br>
|
Vals: 12f, 133, 145<br>
|-
Badness: 0.028440<br><br>
|
; <font style="font-size: 1.15em">[[Marvel temperaments #Quintapole|Quintapole]] (12&amp;85)</font>
|
{{See also|42ed11 #42ed11 as a generator}}
|29\22
'''7-limit'''<br>
|
Comma list: 225/224, 7812500/7411887<br>
|98.8{{Overline|63}}
Mapping: [{{val|1 2 1 1}}, {{val|0 -5 16 22}}]<br>
|
POTE generator: ~21/20 = 98.994<br>
|-
Vals: 12, 73c, 85, 97d<br>
|
Badness: 0.192498<br><br>
|
'''11-limit'''<br>
|
Comma list: 100/99, 225/224, 85184/84035<br>
|33\25
Mapping: [{{val|1 2 1 1 0}}, {{val|0 -5 16 22 42}}]<br>
|99
POTE generator: ~21/20 = 98.954<br>
|
Vals: 12, 73ce, 85, 97d<br>
|-
Badness: 0.104353<br><br>
|4\3
|
|
|
|100
|Pental Mixolydian mode ends, Soft Superpental Mixolydian mode begins
|-
|
| 19\14
|
|
|101.786
|
|-
|15\11
|
|
|
|102.{{Overline|27}}
|Soft Superpental Mixolydian mode ends, Intense Superpental Mixolydian mode begins
|-
|
|26\19
|
|
|102.632
|
|-
|11\8
|
|
|
|103.125
|Intense Superpental Mixolydian mode ends, Mixolydian-Ionian mode begins
|-
|
|18\13
|
|
|103.846
|
|-
|
|
|25\18
|
|104.1{{Overline|6}}
|
|}


== See also ==
== See also ==
* [[16edX|16EDX]]
* [[12edo|12EDO]] - relative EDO
* [[12edo|12EDO]] - relative EDO
* [[28ed5|28ED5]] - relative ED5
* [[28ed5|28ED5]] - relative ED5
Line 276: Line 259:
* [[40ed10|40ED10]] - relative ED10
* [[40ed10|40ED10]] - relative ED10
* [[42ed11|42ED11]] - relative ED11
* [[42ed11|42ED11]] - relative ED11
* [[18/17s equal temperament|AS18/17]] - relative [[AS|ambitonal sequence]]
* [[18/17 equal-step tuning|AS18/17]] - relative [[AS|ambitonal sequence]]


[[Category:EdX]]
[[Category:Nonoctave]]
[[Category:Nonoctave]]
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