16ed5/2: Difference between revisions

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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.
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 tenths
! Period
!Notes
|-
| 6\5
|
|
|
|90
|Intense Aeolian-Subpental Dorian mode begins
|-
|
|17\14
|
|
| 91.071
|
|-
|11\9
|
|
|
|91.{{Overline|6}}
| Intense Aeolian-Subpental Dorian mode ends, Subpental Dorian mode begins
|-
|
|16\13
|
|
| 92.308
|
|-
|5\4
|
|
|
|93.75
|Subpental Dorian mode ends, Pental Dorian mode begins
|-
|
|
|24\19
|
|94.737
|
|-
|
|19\15
|
|
|95
|
|-
|14\11
|
|
|
|95.{{Overline|45}}
|Pental Dorian mode ends, Superpental Dorian mode begins
|-
|
| 23\18
|
|
|95.8{{Overline|3}}
|
|-
|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}}
|
|-
|7\5
|
|
|
|105
|Mixolydian-Ionian mode ends
|}Relative cents
{| class="wikitable" style="text-align: center"
!Degrees
! colspan="3" |Enneatonic
!Intense Aeolian-Subpental Dorian
!Dorian
|-
|1
|1#/2b
|G#/Jb
|''G#/Ab''
|90.625
|''93.75''
|-
|2
| 2
|J
|''A''
|181.25
|''187.5''
|-
|3
|2#/3b
|J#/Ab
|''A#/Bb''
|271.875
|''281.25''
|-
|4
|3
|J
|''B''
|362.5
|''375''
|-
|5
|3#/4b
|J#/Ab
|B#/Cb
|453.125
|''468.75''
|-
|6
|4
|A
|''C''
|543.75
|''562.5''
|-
|7
|5
|A#/Bb
|''C#/Qb''
|633.375
|''656.25''
|-
|8
|5#/6b
|B
|''Q''
|725
|''750''
|-
|9
|6
|C
|''D''
|815.625
|''843.75''
|-
|10
|6#/7b
| C#/Db
|D#/Sb
|906.25
|''937.5''
|-
|11
|7
|D
|''S''
|996.875
|''1031.25''
|-
|12
| 7#/8b
|D#/Eb
|''S#/Eb''
|1087.5
|''1125''
|-
|13
|8
| colspan="2" | E
|1178.125
|''1218.75''
|-
|14
|8#/9b
| colspan="2" |E#/Fb
|1268.75
|''1312.5''
|-
|15
|9
| colspan="2" |F
|1359.375
|''1406.25''
|-
|16
|1
| colspan="2" |G
|1450
|''1500''
|}
{| class="wikitable" style="text-align: center"
!Degrees
! colspan="3" |Enneatonic
!Subpental-Soft Superpental Mixolydian
!Intense Superpental Mixolydian ~ Mixolydian-Ionian
|-
| |1
|1#/2b
| colspan="2" |F#/Gb
|''100''
|''103.125''
|-
| |2
|2
| colspan="2" |G
|''200''
|''206.25''
|-
| |3
|2#/3b
|G#/Jb
|''G#/Ab''
|''300''
|''309.375''
|-
| |4
|3
|J
|''A''
|''400''
|''412.5''
|-
| | 5
|3#/4b
|J#/Ab
|''A#/Bb''
|''500''
|''515.625''
|-
| |6
|4
|A
|''B''
|''600''
|''618.75''
|-
| | 7
|5
|B
|''H''
|''700''
|''721.875''
|-
|8
|5#/6b
|B#/Hb
|''H#/Cb''
|''800''
|''825''
|-
|9
|6
|H
|''C''
|''900''
|''928.125''
|-
|10
|6#/7b
|H#/Cb
|''C#/Db''
|''1000''
|''1031.25''
|-
|11
|7
|C
|''D''
|''1100''
|''1134.375''
|-
|12
|7#/8b
|C#/Db
|''D#/Sb''
|''1200''
|''1237.5''
|-
|13
|8
|D
|''S''
|''1300''
|''1340.625''
|-
|14
|8#/9b
| D#/Eb
|''S#/Eb''
|''1400''
|''1443.75''
|-
| 15
|9
| colspan="2" |E
|''1500''
|''1546.875''
|-
|16
|1
| colspan="2" |F
|''1600''
|''1650''
|


== See also ==
== See also ==

Revision as of 00:20, 23 May 2023

← 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. This is the scale which occurs as the dominant reformed Mixolydian mode tuned as an equal division of a just interval.

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

Coincidentally, 133 steps of the pyrite EDX of this size exceed 11 octaves by only 2.978¢.

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 tenths Period Notes
6\5 90 Intense Aeolian-Subpental Dorian mode begins
17\14 91.071
11\9 91.6 Intense Aeolian-Subpental Dorian mode ends, Subpental Dorian mode begins
16\13 92.308
5\4 93.75 Subpental Dorian mode ends, Pental Dorian mode begins
24\19 94.737
19\15 95
14\11 95.45 Pental Dorian mode ends, Superpental Dorian mode begins
23\18 95.83
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
7\5 105 Mixolydian-Ionian mode ends

Relative cents

Degrees Enneatonic Intense Aeolian-Subpental Dorian Dorian
1 1#/2b G#/Jb G#/Ab 90.625 93.75
2 2 J A 181.25 187.5
3 2#/3b J#/Ab A#/Bb 271.875 281.25
4 3 J B 362.5 375
5 3#/4b J#/Ab B#/Cb 453.125 468.75
6 4 A C 543.75 562.5
7 5 A#/Bb C#/Qb 633.375 656.25
8 5#/6b B Q 725 750
9 6 C D 815.625 843.75
10 6#/7b C#/Db D#/Sb 906.25 937.5
11 7 D S 996.875 1031.25
12 7#/8b D#/Eb S#/Eb 1087.5 1125
13 8 E 1178.125 1218.75
14 8#/9b E#/Fb 1268.75 1312.5
15 9 F 1359.375 1406.25
16 1 G 1450 1500
Degrees Enneatonic Subpental-Soft Superpental Mixolydian Intense Superpental Mixolydian ~ Mixolydian-Ionian
1 1#/2b F#/Gb 100 103.125
2 2 G 200 206.25
3 2#/3b G#/Jb G#/Ab 300 309.375
4 3 J A 400 412.5
5 3#/4b J#/Ab A#/Bb 500 515.625
6 4 A B 600 618.75
7 5 B H 700 721.875
8 5#/6b B#/Hb H#/Cb 800 825
9 6 H C 900 928.125
10 6#/7b H#/Cb C#/Db 1000 1031.25
11 7 C D 1100 1134.375
12 7#/8b C#/Db D#/Sb 1200 1237.5
13 8 D S 1300 1340.625
14 8#/9b D#/Eb S#/Eb 1400 1443.75
15 9 E 1500 1546.875
16 1 F 1600 1650

See also
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