30edt: Difference between revisions

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{{ED intro}}
{{ED intro}}


== Theory ==
30edt is related to [[19edo]], but with the [[3/1]] rather than the [[2/1]] being [[just]], which results in octaves being is [[stretched and compressed tuning|stretched]] by about 4.5715{{cent}} and the step size is about. It is [[consistent]] to the 10-[[integer-limit]].
30edt is related to [[19edo]], but with the [[3/1]] rather than the [[2/1]] being [[just]], which results in octaves being is [[stretched and compressed tuning|stretched]] by about 4.5715{{cent}} and the step size is about. It is [[consistent]] to the 10-[[integer-limit]].


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30edt is a [[Phoenix]] tuning and exhibits all the benefits of such tunings.
30edt is a [[Phoenix]] tuning and exhibits all the benefits of such tunings.


== Harmonics ==
=== Harmonics ===
{{Harmonics in equal
{{Harmonics in equal|30|3|1|intervals=integer}}
| steps = 30
{{Harmonics in equal|30|3|1|intervals=integer|columns=12|start=12|collapsed=1|Approximation of harmonics in 30edt (continued)}}
| num = 3
| denom = 1
| intervals = integer
}}
{{Harmonics in equal
| steps = 30
| num = 3
| denom = 1
| start = 12
| collapsed = 1
| intervals = integer
}}


== Intervals of 30edt ==
== Intervals of 30edt ==
{| class="wikitable center-all"
{| class="wikitable center-all right-2 right-3 left-4"
|-
|-
! rowspan="2" | Degrees
! rowspan="2" | #
! rowspan="2" | Cents
! rowspan="2" | Cents
! rowspan="2" | Hekts
! rowspan="2" | Hekts
! rowspan="2" | Approximate Ratios
! rowspan="2" | Approximate ratios
! colspan="2" | Scale name
! colspan="2" | Scale name
|-
|-
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| 0
| 0
| 0
| 0
| <span style="color: #660000;">[[1/1]]</span>
| [[1/1]]
| colspan="2" | C
| colspan="2" | C
|-
|-
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| 63.3985
| 63.3985
| 43.333
| 43.333
| 28/27, 27/26
| 27/26, 28/27
| C^/Dbv
| C^/Dbv
| C#/Dbb
| C#/Dbb
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| 126.797
| 126.797
| 86.667
| 86.667
| [[14/13]], [[15/14]], [[16/15]], 29/27
| [[14/13]], [[15/14]], [[16/15]], [[29/27]]
| Db
| Db
| Cx/Db
| Cx/Db
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| 190.1955
| 190.1955
| 130
| 130
| 10/9~9/8
| 9/8, 10/9
| C#
| C#
| D
| D
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| 380.391
| 380.391
| 260
| 260
| <span style="color: #660000;">[[5/4]]</span>
| [[5/4]]
| D^/Ev
| D^/Ev
| E
| E
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| 760.782
| 760.782
| 520
| 520
| <span style="color: #660000;">[[14/9]]</span>
| [[14/9]]
| F
| F
| G#/Hbb
| G#/Hbb
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| 1141.173
| 1141.173
| 780
| 780
| <span style="color: #660000;">[[27/14]]</span>
| [[27/14]]
| G#^/Hv
| G#^/Hv
| J#/Kbb
| J#/Kbb
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| 1394.767
| 1394.767
| 953.333
| 953.333
| [[9/4]] ([[9/8]] plus an octave)
| [[9/4]]
| J^/Av
| J^/Av
| L
| L
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| 1521.564
| 1521.564
| 1040
| 1040
| [[12/5]] (<span style="color: #660000;">[[6/5]]</span> plus an octave)
| [[12/5]]
| A^/Bbv
| A^/Bbv
| Lx/Ab
| Lx/Ab
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| 1648.361
| 1648.361
| 1126.667
| 1126.667
| [[13/5]] ([[13/10]] plus an octave)
| [[13/5]]
| A#
| A#
| A#/Bbb
| A#/Bbb
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| 1775.158
| 1775.158
| 1213.333
| 1213.333
| [[14/5]] ([[7/5]] plus an octave)
| [[14/5]]
| B
| B
|-
|-
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|}
|}


30edt contains all [[19edo]] intervals within 3/1, all temepered progressively sharper. The accumulation of the 0.241{{c}} sharpening of the unit step relative to 19edo leads to the excellent 6edt approximations of 6/5 and 5/2. Non-redundantly with simpler edts, the 41 degree ~9/2 is only .6615{{c}} flatter than that in 6edo.
30edt contains all [[19edo]] intervals within 3/1, all tempered progressively sharper. The accumulation of the 0.241{{c}} sharpening of the unit step relative to 19edo leads to the excellent 6edt approximations of 6/5 and 5/2. Non-redundantly with simpler edts, the 41 degree ~9/2 is only .6615{{c}} flatter than that in 6edo.


30edt also contains all the MOS contained in 15edt, being the double of this equal division. Being even, 30edt introduces MOS with an even number of periods per tritave such as a {{sl|6L 6s}} similar to Hexe Dodecatonic. This MOS has a period of 1/6 of the tritave and the generator is a single or double step. The major scale is sLsLsLsLsLsL, and the minor scale is LsLsLsLsLsLs. Being a "real" 3/2, the interval of 11 degrees generates an unfair Sigma scale of {{sl|8L 3s}} and the major scale is LLLsLLLsLLs. The sharp 9/7 of 7 degrees, in addition to generating a Lambda MOS will generate a {{sl|4L 9s}} unfair "Superlambda" MOS which does not border on being atonal as the 17edt rendition does.
30edt also contains all the mos contained in [[15edt]], being the double of this equal division. Being even, 30edt introduces mos with an even number of periods per tritave such as a {{sl|6L 6s}} similar to Hexe Dodecatonic. This mos has a period of 1/6 of the tritave and the generator is a single or double step. The major scale is sLsLsLsLsLsL, and the minor scale is LsLsLsLsLsLs. Being a "real" 3/2, the interval of 11 degrees generates an [[unfair]] [[Sigma]] scale of {{sl|8L 3s}} and the major scale is LLLsLLLsLLs. The sharp 9/7 of 7 degrees, in addition to generating a Lambda mos will generate a {{sl|4L 9s}} unfair "Superlambda" mos which does not border on being atonal as the 17edt rendition does.


== Music ==
== Music ==
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* [https://www.youtube.com/watch?v=fEQ13hzs3fY ''Fugue for Piano in 30EDT Bohlen-Pierce-Stearns{{lbrack}}9{{rbrack}} sLsLssLsL "Dur I"''] (2024)
* [https://www.youtube.com/watch?v=fEQ13hzs3fY ''Fugue for Piano in 30EDT Bohlen-Pierce-Stearns{{lbrack}}9{{rbrack}} sLsLssLsL "Dur I"''] (2024)


[[Category:Edt]]
[[Category:Listen]]
[[Category:Listen]]

Revision as of 11:44, 23 January 2025

← 29edt 30edt 31edt →
Prime factorization 2 × 3 × 5
Step size 63.3985 ¢ 
Octave 19\30edt (1204.57 ¢)
Consistency limit 10
Distinct consistency limit 7

30 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 30edt or 30ed3), is a nonoctave tuning system that divides the interval of 3/1 into 30 equal parts of about 63.4 ¢ each. Each step represents a frequency ratio of 31/30, or the 30th root of 3.

Theory

30edt is related to 19edo, but with the 3/1 rather than the 2/1 being just, which results in octaves being is stretched by about 4.5715 ¢ and the step size is about. It is consistent to the 10-integer-limit.

Because 19edo has the 3rd, 5th, 7th, and 13th harmonics all flat (the latter two very flat), it benefits greatly from octave stretching. 30edt is one possible alternative; at the cost of sharpening the octave, it achieves much better matches to the odd harmonics; the 3 (tritave) is by definition just, the 5 slightly sharp, and the 7 and 13 slightly flat.

While the fifth is just, the fourth is noticeably sharper and less accurate than in 19edo, being close to that of 26edo.

30edt is a Phoenix tuning and exhibits all the benefits of such tunings.

Harmonics

Approximation of harmonics in 30edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +4.6 +0.0 +9.1 +3.2 +4.6 -8.7 +13.7 +0.0 +7.8 -30.4 +9.1
Relative (%) +7.2 +0.0 +14.4 +5.1 +7.2 -13.7 +21.6 +0.0 +12.3 -48.0 +14.4
Steps
(reduced) 		19
(19) 		30
(0) 		38
(8) 		44
(14) 		49
(19) 		53
(23) 		57
(27) 		60
(0) 		63
(3) 		65
(5) 		68
(8)
Approximation of harmonics in 30edt
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -2.6 -4.1 +3.2 +18.3 -23.3 +4.6 -25.6 +12.4 -8.7 -25.8 +24.0 +13.7
Relative (%) -4.2 -6.5 +5.1 +28.8 -36.7 +7.2 -40.4 +19.5 -13.7 -40.8 +37.9 +21.6
Steps
(reduced) 		70
(10) 		72
(12) 		74
(14) 		76
(16) 		77
(17) 		79
(19) 		80
(20) 		82
(22) 		83
(23) 		84
(24) 		86
(26) 		87
(27)

Intervals of 30edt

# Cents Hekts Approximate ratios Scale name
Lambda Sigma
0 0 0 1/1 C
1 63.3985 43.333 27/26, 28/27 C^/Dbv C#/Dbb
2 126.797 86.667 14/13, 15/14, 16/15, 29/27 Db Cx/Db
3 190.1955 130 9/8, 10/9 C# D
4 253.594 173.333 15/13 C#^/Dv D#/Ebb
5 316.9925 216.667 6/5 D Dx/Eb
6 380.391 260 5/4 D^/Ev E
7 443.7895 303.333 9/7 E E#/Fbb
8 507.188 346.667 4/3 E^/Fbv Ex/Fb
9 570.5865 390 7/5 Fb F
10 633.985 433.333 13/9 E# F#/Gb
11 697.3835 476.667 3/2 E#^/Fv G
12 760.782 520 14/9 F G#/Hbb
13 824.1805 563.333 8/5 F^/Gv Gx/Hb
14 887.579 606.667 5/3 G H
15 950.9775 650 19/11 G^/Hbv H#/Jbb
16 1014.376 693.333 9/5 Hb Hx/Jb
17 1077.7745 736.667 13/7 G# J
18 1141.173 780 27/14 G#^/Hv J#/Kbb
19 1204.5715 823.333 2/1 H Jx/Kb
20 1267.97 866.667 27/13 H^/Jv K
21 1331.3685 910 28/13 J K#/Lb
22 1394.767 953.333 9/4 J^/Av L
23 1458.1655 996.667 7/3 A L#/Abb
24 1521.564 1040 12/5 A^/Bbv Lx/Ab
25 1584.9625 1083.333 5/2 Bb A
26 1648.361 1126.667 13/5 A# A#/Bbb
27 1711.7595 1170 8/3 A#^/Bv Ax/Bb
28 1775.158 1213.333 14/5 B
29 1838.5565 1256.667 26/9 B^/Cv B#/Cb
30 1901.955 1300 3/1 C

30edt contains all 19edo intervals within 3/1, all tempered progressively sharper. The accumulation of the 0.241 ¢ sharpening of the unit step relative to 19edo leads to the excellent 6edt approximations of 6/5 and 5/2. Non-redundantly with simpler edts, the 41 degree ~9/2 is only .6615 ¢ flatter than that in 6edo.

30edt also contains all the mos contained in 15edt, being the double of this equal division. Being even, 30edt introduces mos with an even number of periods per tritave such as a Template:Sl similar to Hexe Dodecatonic. This mos has a period of 1/6 of the tritave and the generator is a single or double step. The major scale is sLsLsLsLsLsL, and the minor scale is LsLsLsLsLsLs. Being a "real" 3/2, the interval of 11 degrees generates an unfair Sigma scale of Template:Sl and the major scale is LLLsLLLsLLs. The sharp 9/7 of 7 degrees, in addition to generating a Lambda mos will generate a Template:Sl unfair "Superlambda" mos which does not border on being atonal as the 17edt rendition does.

Music

Mason Green
Ray Perlner
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