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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 as a [[generator]] of a [[cluster temperament]] which tempers out {{monzo|37 -16 -5}} (trisa-quingu comma). It has twelve clusters of notes in an octave. There are some extensions for 7 and higher limit including ''quintupole'' (12&121), ''quinticosiennic'' (12&145), and ''quintapole'' (12&85).
{{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 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.
 
== Scale tree ==
Ed5/2 scales can be approximated in [[EDO]]s by subdividing their approximations of 5/2.
 
{| class="wikitable"
|+
! colspan="4" |Major tenth
! Period
!Notes
|-
|9\7
|
|
|
|96.429
|Superpental Dorian mode ends, Mohajira Dorian-Mixolydian mode begins
|-
|
|31\24
|
|
|96.875
|
|-
|22\17
|
|
|
|97.059
|Mohajira Dorian-Mixolydian mode ends, Beatles Dorian-Mixolydian mode begins
|-
|
|35\27
|
|
|97.{{Overline|2}}
|
|-
| 13\10
|
|
|
|97.5
|Beatles Dorian-Mixolydian mode ends, Subpental Mixolydian mode begins
|-
|17\13
|
|
|
|98.077
|
|-
|21\16
|
|
|
|98.4375
|Subpental Mixolydian mode ends, Pental Mixolydian mode begins
|-
|
|25\19
|
|
|98.684
|
|-
|
|
|29\22
|
|98.8{{Overline|63}}
|
|-
|
|
|
|33\25
|99
|
|-
|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}}
|
|}


'''<font style="font-size: 1.35em">Trisa-quingu (12&amp;121)</font>'''<br>
== See also ==
'''<font style="font-size: 1.2em">5-limit</font>'''<br>
* [[12edo|12EDO]] - relative EDO
Comma: {{monzo|37 -16 -5}} = 137438953472/1345210003125<br>
* [[28ed5|28ED5]] - relative ED5
Mapping: [{{val|1 2 1}}, {{val|0 -5 16}}]<br>
* [[34ed7|34ED7]] - relative ED7
POTE generator: ~135/128 = 99.267<br>
* [[40ed10|40ED10]] - relative ED10
Vals: 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc<br>
* [[42ed11|42ED11]] - relative ED11
Badness: 0.444506<br><br>
* [[18/17 equal-step tuning|AS18/17]] - relative [[AS|ambitonal sequence]]
'''<font style="font-size: 1.35em">[[Octagar temperaments #Quintupole|Quintupole]] (12&amp;121)</font>'''<br>
{{See also|34ed7 #34ed7 as a generator}}
'''<font style="font-size: 1.2em">7-limit</font>'''<br>
Comma list: 4000/3969, 458752/455625<br>
Mapping: [{{val|1 2 1 0}}, {{val|0 -5 16 34}}]<br>
POTE generator: ~135/128 = 99.175<br>
Vals: 12, 97, 109, 121<br>
Badness: 0.111620<br><br>
'''<font style="font-size: 1.2em">11-limit</font>'''<br>
Comma list: 896/891, 1375/1372, 4375/4356<br>
Mapping: [{{val|1 2 1 0 -1}}, {{val|0 -5 16 34 54}}]<br>
POTE generator: ~132/125 = 99.156<br>
Vals: 12, 109, 121, 351bde, 472bdee<br>
Badness: 0.056501<br><br>
'''<font style="font-size: 1.35em">[[Hemifamity temperaments #Quinticosiennic|Quinticosiennic]] (12&amp;145)</font>'''<br>
'''<font style="font-size: 1.2em">7-limit</font>'''<br>
Comma list: 5120/5103, 395136/390625<br>
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>
'''<font style="font-size: 1.2em">11-limit</font>'''<br>
Comma list: 441/440, 896/891, 78408/78125<br>
Mapping: [{{val|1 2 1 -1 -2}}, {{val|0 -5 16 46 66}}]<br>
POTE generator: ~35/33 = 99.318<br>
Vals: 12, 133, 145<br>
Badness: 0.080674<br><br>
'''<font style="font-size: 1.35em">[[Marvel temperaments #Quintapole|Quintapole]] (12&amp;85)</font>'''<br>
{{See also|42ed11 #42ed11 as a generator}}
'''<font style="font-size: 1.2em">7-limit</font>'''<br>
Comma list: 225/224, 7812500/7411887<br>
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>
'''<font style="font-size: 1.2em">11-limit</font>'''<br>
Comma list: 100/99, 225/224, 85184/84035<br>
Mapping: [{{val|1 2 1 1 0}}, {{val|0 -5 16 22 42}}]<br>
POTE generator: ~21/20 = 98.954<br>
Vals: 12, 73ce, 85, 97d<br>
Badness: 0.104353<br><br>


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

Latest revision as of 18:40, 1 August 2025

← 15ed5/2 16ed5/2 17ed5/2 →
Prime factorization 24
Step size 99.1446 ¢ 
Octave 12\16ed5/2 (1189.74 ¢) (→ 3\4ed5/2)
Twelfth 19\16ed5/2 (1883.75 ¢)
(semiconvergent)
Consistency limit 11
Distinct consistency limit 6

16ED5/2 is the equal division of the 5/2 interval into 16 parts of 99.1446 cents each. 16 equal divisions of the just major tenth is not a "real" xenharmonic tuning; it is a slightly compressed version of the normal 12-tone scale.

Intervals

Degrees Enneatonic Cents
1 1#/2b F#/Gb 99.145
2 2 G 198.289
3 2#/3b G#/Jb G#/Ab 297.433
4 3 J A 396.578
5 3#/4b J#/Ab A#/Bb 495.723
6 4 A B 594.868
7 5 B H 694.012
8 5#/6b B#/Hb H#/Cb 793.157
9 6 H C 892.3015
10 6#/7b H#/Cb C#/Db 991.446
11 7 C D 1090.591
12 7#/8b C#/Db D#/Sb 1189.735
13 8 D S 1288.88
14 8#/9b D#/Eb S#/Eb 1388.0245
15 9 E 1487.169
16 1 F 1586.314

Harmonics

Approximation of harmonics in 16ed5/2
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -10.3 -18.2 -20.5 -10.3 -28.5 +2.1 -30.8 -36.4 -20.5 +12.8 -38.7
Relative (%) -10.4 -18.4 -20.7 -10.4 -28.7 +2.1 -31.1 -36.7 -20.7 +12.9 -39.1
Steps
(reduced) 		12
(12) 		19
(3) 		24
(8) 		28
(12) 		31
(15) 		34
(2) 		36
(4) 		38
(6) 		40
(8) 		42
(10) 		43
(11)
Approximation of harmonics in 16ed5/2
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +21.0 -8.2 -28.5 -41.1 -46.9 -46.7 -41.1 -30.8 -16.1 +2.5 +24.7
Relative (%) +21.2 -8.2 -28.7 -41.4 -47.3 -47.1 -41.5 -31.1 -16.3 +2.5 +24.9
Steps
(reduced) 		45
(13) 		46
(14) 		47
(15) 		48
(0) 		49
(1) 		50
(2) 		51
(3) 		52
(4) 		53
(5) 		54
(6) 		55
(7)

Regular temperaments

Main article: 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 12-, 109-, 121-, 133-, 145-, 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.

Scale tree

Ed5/2 scales can be approximated in EDOs by subdividing their approximations of 5/2.

Major tenth Period Notes
9\7 96.429 Superpental Dorian mode ends, Mohajira Dorian-Mixolydian mode begins
31\24 96.875
22\17 97.059 Mohajira Dorian-Mixolydian mode ends, Beatles Dorian-Mixolydian mode begins
35\27 97.2
13\10 97.5 Beatles Dorian-Mixolydian mode ends, Subpental Mixolydian mode begins
17\13 98.077
21\16 98.4375 Subpental Mixolydian mode ends, Pental Mixolydian mode begins
25\19 98.684
29\22 98.863
33\25 99
4\3 100 Pental Mixolydian mode ends, Soft Superpental Mixolydian mode begins
19\14 101.786
15\11 102.27 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\18 104.16

See also

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