18ed5/2: Difference between revisions

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{{Infobox ET}}
{{ED intro}}
This scale is a neutral Mixolydian mode of the Middletown relative of [[Whitewood]] temperament, or the White Oak scale. The Middletown Valley relatives of [[Blackwood]] temperament are:
This scale is a neutral Mixolydian mode of the Middletown relative of [[Whitewood]] temperament, or the White Oak scale. The Middletown Valley relatives of [[Blackwood]] temperament are:


Fair Terra Rubra: bi-equal tetrad
*Fair Terra Rubra: bi-equal tetrad


Fair Tritetrachordal: Middletown Valley Double Augmented
*Fair Tritetrachordal: Middletown Valley Double Augmented


Fair Montrose: Vesuvius Marcodiminished
*Fair Montrose: Vesuvius Marcodiminished


Unfair Montrose: Annapolis~Oriole Macro-Blackwood
*Unfair Montrose: Annapolis~Oriole Macro-Blackwood


Unfair Tritetrachordal: Mt. Airy
*Unfair Tritetrachordal: Mt. Airy


Unfair Terra Rubra: Westminster
*Unfair Terra Rubra: Westminster
 
== Intervals ==


{| class="wikitable"
{| class="wikitable"
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! colspan="6" |Enneatonic
! colspan="6" |Enneatonic
!Dodecatonic
!Dodecatonic
!ed38\29
!Cents
!Pyrite
!ed5/2
!ed(7φ+6)\5(φ+1) (~[[68ed8]]!)
!ed4\3=''r¢''
|-
|-
| |1
| |1
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| colspan="4" |G
| colspan="4" |G
|1#/2b
|1#/2b
|87.356
|87.448
|88.1285
|88.1285
|88.2405
|88.889
|-
|-
| |2
| |2
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|''A''
|''A''
|2
|2
|174.713
|174.897
|176.257
|176.257
|176.481
|177.778
|-
|-
| |3
| |3
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|''Av/B\\''
|''Av/B\\''
|3
|3
|262.069
|262.345
|264.386
|264.386
|264.721
|266.667
|-
|-
| |4
| |4
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|''Aw/B^^''
|''Aw/B^^''
|3#/4b
|3#/4b
|349.425
|349.794
|352.514
|352.514
|352.962
|355.556
|-
|-
| |5
| |5
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|''Ap/B^''
|''Ap/B^''
|4
|4
|436.782
|437.242
|440.643
|440.643
|441.202
|444.444
|-
|-
| |6
| |6
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|''B''
|''B''
|5
|5
|524.138
|524.6905
|528.771
|528.771
|529.443
|533.333
|-
|-
| |7
| |7
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|''H''
|''H''
|5#/6b
|5#/6b
|611.494
|612.139
|616.9
|616.9
|617.683
|622.222
|-
|-
|8
|8
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|''C''
|''C''
|6
|6
|698.851
|699.587
|705.028
|705.028
|705.924
|711.111
|-
|-
|9
|9
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|''Cv/D\\''
|''Cv/D\\''
|7
|7
|786.207
|787.036
|793.157
|793.157
|794.164
|''800''
|-
|-
|10
|10
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|''Cw/D^^''
|''Cw/D^^''
|7#/8b
|7#/8b
|873.563
|874.484
|881.285
|881.285
|882.4045
|888.888
|-
|-
|11
|11
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|''Cp/D^''
|''Cp/D^''
|8
|8
|960.9195
|961.9325
|969.414
|969.414
|970.645
|977.778
|-
|-
|12
|12
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|''D''
|''D''
|9
|9
|1048.276
|1049.381
|1057.5425
|1057.5425
|1058.885
|1066.667
|-
|-
|13
|13
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|''S''
|''S''
|9#/Xb
|9#/Xb
|1135.632
|1136.829
|1145.671
|1145.671
|1147.126
|1155.556
|-
|-
|14
|14
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| colspan="4" |E
| colspan="4" |E
|X
|X
|1222.9885
|1224.278
|1233.8
|1233.8
|1235.366
|1244.444
|-
|-
|15
|15
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| colspan="2" |Ev/F\\
| colspan="2" |Ev/F\\
|E
|E
|1310.345
|1311.726
|1321.928
|1321.928
|1323.607
|1333.333
|-
|-
|16
|16
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| colspan="2" |Ew/F^^
| colspan="2" |Ew/F^^
|E#/0b
|E#/0b
|1397.701
|1399.175
|1410.0566
|1410.0566
|1411.847
|1422.222
|-
|-
|17
|17
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| colspan="2" |Ep/F^
| colspan="2" |Ep/F^
|0
|0
|1485.0575
|1486.30.
|1498.185
|1498.185
|1500.088
|1511.111
|-
|-
|18
|18
| colspan="6" |F
| colspan="6" |F
|1
|1
|1572.414
|1574.0715
|1586.314
|1586.314
|1588.328
|''1600''
|}
|}
== Harmonics ==
{{Harmonics in equal
| steps = 18
| num = 5
| denom = 2
}}
{{Harmonics in equal
| steps = 18
| num = 5
| denom = 2
| start = 12
| collapsed = 1
}}

Latest revision as of 08:39, 4 October 2024

← 17ed5/2 18ed5/2 19ed5/2 →
Prime factorization 2 × 32
Step size 88.1285 ¢ 
Octave 14\18ed5/2 (1233.8 ¢) (→ 7\9ed5/2)
Twelfth 22\18ed5/2 (1938.83 ¢) (→ 11\9ed5/2)
Consistency limit 3
Distinct consistency limit 3

18 equal divisions of 5/2 (abbreviated 18ed5/2) is a nonoctave tuning system that divides the interval of 5/2 into 18 equal parts of about 88.1 ¢ each. Each step represents a frequency ratio of (5/2)1/18, or the 18th root of 5/2.

This scale is a neutral Mixolydian mode of the Middletown relative of Whitewood temperament, or the White Oak scale. The Middletown Valley relatives of Blackwood temperament are:

  • Fair Terra Rubra: bi-equal tetrad
  • Fair Tritetrachordal: Middletown Valley Double Augmented
  • Fair Montrose: Vesuvius Marcodiminished
  • Unfair Montrose: Annapolis~Oriole Macro-Blackwood
  • Unfair Tritetrachordal: Mt. Airy
  • Unfair Terra Rubra: Westminster

Intervals

Degrees Enneatonic Dodecatonic Cents
1 FG G 1#/2b 88.1285
2 G J A J A 2 176.257
3 GJ GA J^/Ab A^/Bb Jv/A\\ Av/B\\ 3 264.386
4 J A J^^/Aw A^^/Bw Jw/A^^ Aw/B^^ 3#/4b 352.514
5 JA AB J#/Av A#/Bv Jp/A^ Ap/B^ 4 440.643
6 A B A B A B 5 528.771
7 AB BH B H B H 5#/6b 616.9
8 B H H C H C 6 705.028
9 BH HC H^/Cb C^/Db Hv/C\\ Cv/D\\ 7 793.157
10 H C H^^/Cw C^^/Dw Hw/C^^ Cw/D^^ 7#/8b 881.285
11 HC CD H#/Cv C#/Dv Hp/C^ Cp/D^ 8 969.414
12 C D C D C D 9 1057.5425
13 CD DS D S D S 9#/Xb 1145.671
14 D S E X 1233.8
15 DE SE E^/Fb Ev/F\\ E 1321.928
16 E E^^/Fw Ew/F^^ E#/0b 1410.0566
17 EF E#/Fv Ep/F^ 0 1498.185
18 F 1 1586.314

Harmonics

Approximation of harmonics in 18ed5/2
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +33.8 +36.9 -20.5 +33.8 -17.5 -19.9 +13.3 -14.4 -20.5 -9.3 +16.3
Relative (%) +38.4 +41.8 -23.3 +38.4 -19.8 -22.6 +15.1 -16.3 -23.3 -10.5 +18.5
Steps
(reduced) 		14
(14) 		22
(4) 		27
(9) 		32
(14) 		35
(17) 		38
(2) 		41
(5) 		43
(7) 		45
(9) 		47
(11) 		49
(13)
Approximation of harmonics in 18ed5/2
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -34.1 +13.9 -17.5 -41.1 +30.2 +19.4 +13.9 +13.3 +16.9 +24.5 +35.7
Relative (%) -38.7 +15.7 -19.8 -46.6 +34.3 +22.0 +15.8 +15.1 +19.2 +27.8 +40.5
Steps
(reduced) 		50
(14) 		52
(16) 		53
(17) 		54
(0) 		56
(2) 		57
(3) 		58
(4) 		59
(5) 		60
(6) 		61
(7) 		62
(8)
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