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{{Infobox ET}}
{{Infobox ET}}
'''[[Edt|Division of the third harmonic]] into 30 equal parts''' (30edt) is related to [[19edo|19 edo]], but with the 3/1 rather than the 2/1 being just. The octave is about 4.5715 cents stretched and the step size is about 63.3985 cents. It is consistent to the [[9-odd-limit|10-integer-limit]].
{{ED intro}}


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.
== 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}}. Like 19edo, 30edt is [[consistent]] to the [[integer-limit|10-integer-limit]].
 
Because [[19edo]] has the [[3/1|3rd]], [[5/1|5th]], [[7/1|7th]], and [[13/1|13th]] [[harmonic]]s 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]].
While the fifth is just, the fourth is noticeably sharper and less accurate than in 19edo, being close to that of [[26edo]].


==Harmonics==
=== Harmonics ===
{{Harmonics in equal|30|3|1|prec=2|columns=15}}
{{Harmonics in equal|30|3|1|intervals=integer}}
{{Harmonics in equal|30|3|1|intervals=integer|columns=12|start=12|collapsed=1|title=Approximation of harmonics in 30edt (continued)}}


==Intervals of 30edt==
=== Subsets and supersets ===
{| class="wikitable center-all"
Since 30 factors into primes as 2 × 3 × 5, 30edt has subset edts {{EDs|equave=t| 2, 3, 5, 6, 10, and 15 }}.
 
== Intervals ==
{| class="wikitable center-all right-2 right-3 left-4"
|-
|-
! Degrees
! rowspan="2" | #
! Cents
! rowspan="2" | Cents
!Hekts
! rowspan="2" | [[Hekt]]s
! Approximate Ratios
! rowspan="2" | Approximate ratios
! Lambda scale name
! colspan="2" | Scale name
! Sigma scale name
|-
! Lambda
! Sigma
|-
|-
| 0
| 0
| 0
| 0
| 0
| 0
| <span style="color: #660000;">[[1/1|1/1]]</span>
| [[1/1]]
| colspan="2" | C
| colspan="2" | C
|-
|-
| | 1
| 1
| | 63.3985
| 63.4
| 43.333
| 43.3
| | 28/27, 27/26
| [[27/26]], [[28/27]]
| C^/Dbv
| C^/Dbv
| | C#/Dbb
| C#/Dbb
|-
|-
| | 2
| 2
| | 126.797
| 126.8
| 86.667
| 86.7
| | [[14/13|14/13]], [[15/14|15/14]], [[16/15|16/15]], 29/27
| [[14/13]], [[15/14]], [[16/15]], [[29/27]]
| Db
| Db
| | Cx/Db
| Cx/Db
|-
|-
| | 3
| 3
| | 190.1955
| 190.2
| 130
| 130.0
| | 10/9~9/8
| [[9/8]], [[10/9]]
| C#
| C#
| | D
| D
|-
|-
| | 4
| 4
| | 253.594
| 253.6
| 173.333
| 173.3
| | [[15/13|15/13]]
| [[15/13]]
| C#^/Dv
| C#^/Dv
| | D#/Ebb
| D#/Ebb
|-
|-
| | 5
| 5
| | 316.9925
| 317.0
| 216.667
| 216.7
| | 6/5
| [[6/5]]
| D
| D
| | Dx/Eb
| Dx/Eb
|-
|-
| | 6
| 6
| | 380.391
| 380.4
| 260
| 260.0
| | <span style="color: #660000;">[[5/4|5/4]]</span>
| [[5/4]]
| D^/Ev
| D^/Ev
| | E
| E
|-
|-
| | 7
| 7
| | 443.7895
| 443.8
| 303.333
| 303.3
| | 9/7
| [[9/7]]
| E
| E
| | E#/Fbb
| E#/Fbb
|-
|-
| | 8
| 8
| | 507.188
| 507.2
| 346.667
| 346.7
| | [[4/3|4/3]]
| [[4/3]]
| E^/Fbv
| E^/Fbv
| | Ex/Fb
| Ex/Fb
|-
|-
| | 9
| 9
| | 570.5865
| 570.6
| 390
| 390.0
| | 7/5
| [[7/5]]
| Fb
| Fb
| | F
| F
|-
|-
| | 10
| 10
| | 633.985
| 634.0
| 433.333
| 433.3
| | [[13/9|13/9]]
| [[13/9]]
| E#
| E#
| | F#/Gb
| F#/Gb
|-
|-
| | 11
| 11
| | 697.3835
| 697.4
| 476.667
| 476.7
| | 3/2
| [[3/2]]
| E#^/Fv
| E#^/Fv
| | G
| G
|-
|-
| | 12
| 12
| | 760.782
| 760.8
| 520
| 520.0
| | <span style="color: #660000;">[[14/9|14/9]]</span>
| [[14/9]]
| F
| F
| | G#/Hbb
| G#/Hbb
|-
|-
| | 13
| 13
| | 824.1805
| 824.2
| 563.333
| 563.3
| | 8/5
| [[8/5]]
| F^/Gv
| F^/Gv
| | Gx/Hb
| Gx/Hb
|-
|-
| | 14
| 14
| | 887.579
| 887.6
| 606.667
| 606.7
| | [[5/3|5/3]]
| [[5/3]]
| G
| G
| | H
| H
|-
|-
| | 15
| 15
| | 950.9775
| 951.0
| 650
| 650.0
| | 19/11
| [[19/11]]
| G^/Hbv
| G^/Hbv
| | H#/Jbb
| H#/Jbb
|-
|-
| | 16
| 16
| | 1014.376
| 1014.4
| 693.333
| 693.3
| | [[9/5|9/5]]
| [[9/5]]
| Hb
| Hb
| | Hx/Jb
| Hx/Jb
|-
|-
| | 17
| 17
| | 1077.7745
| 1077.8
| 736.667
| 736.7
| | 13/7
| [[13/7]]
| G#
| G#
| | J
| J
|-
|-
| | 18
| 18
| | 1141.173
| 1141.2
| 780
| 780.0
| | <span style="color: #660000;">[[27/14|27/14]]</span>
| [[27/14]]
| G#^/Hv
| G#^/Hv
| | J#/Kbb
| J#/Kbb
|-
|-
| | 19
| 19
| | 1204.5715
| 1204.6
| 823.333
| 823.3
| | 2/1
| [[2/1]]
|H
| H
| | Jx/Kb
| Jx/Kb
|-
|-
| | 20
| 20
| | 1267.97
| 1268.0
| 866.667
| 866.7
| | [[27/13|27/13]]
| [[27/13]]
| H^/Jv
| H^/Jv
| | K
| K
|-
|-
| | 21
| 21
| | 1331.3685
| 1331.4
| 910
| 910.0
| | 28/13
| [[28/13]]
| J
| J
| | K#/Lb
| K#/Lb
|-
|-
| | 22
| 22
| | 1394.767
| 1394.8
| 953.333
| 953.3
| | [[9/4|9/4]] ([[9/8|9/8]] plus an octave)
| [[9/4]]
| J^/Av
| J^/Av
| | L
| L
|-
|-
| | 23
| 23
| | 1458.1655
| 1458.2
| 996.667
| 996.7
| | 7/3
| [[7/3]]
| A
| A
| | L#/Abb
| L#/Abb
|-
|-
| | 24
| 24
| | 1521.564
| 1521.6
| 1040
| 1040.0
| | [[12/5|12/5]] (<span style="color: #660000;">[[6/5|6/5]]</span> plus an octave)
| [[12/5]]
| A^/Bbv
| A^/Bbv
| | Lx/Ab
| Lx/Ab
|-
|-
| | 25
| 25
| | 1584.9625
| 1585.0
| 1083.333
| 1083.3
| | 5/2
| [[5/2]]
| Bb
| Bb
| | A
| A
|-
|-
| | 26
| 26
| | 1648.361
| 1648.4
| 1126.667
| 1126.7
| | [[13/5|13/5]] ([[13/10|13/10]] plus an octave)
| [[13/5]]
| A#
| A#
| | A#/Bbb
| A#/Bbb
|-
|-
| | 27
| 27
| | 1711.7595
| 1711.8
| 1170
| 1170.0
| | 8/3
| [[8/3]]
| A#^/Bv
| A#^/Bv
| | Ax/Bb
| Ax/Bb
|-
|-
| | 28
| 28
| | 1775.158
| 1775.2
| 1213.333
| 1213.3
| | [[14/5|14/5]] ([[7/5|7/5]] plus an octave)
| [[14/5]]
| | B
| B
|-
|-
| | 29
| 29
| | 1838.5565
| 1838.6
| 1256.667
| 1256.7
| | 26/9
| [[26/9]]
| B^/Cv
| B^/Cv
| | B#/Cb
| B#/Cb
|-
|-
| | 30
| 30
| | 1901.955
| 1902.0
| 1300
| 1300.0
| | [[3/1|3/1]]
| [[3/1]]
| colspan="2" | C
| colspan="2" | C
|}
|}


30edt contains all [[19edo|19edo]] intervals within 3/1, all temepered progressively sharper. The accumulation of the .241 cent 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 cents 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 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 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 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 {{mos scalesig|6L 6s<3/1>|link=1}} 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 {{mos scalesig|8L 3s<3/1>|link=1}} and the major scale is LLLsLLLsLLs. The sharp 9/7 of 7 degrees, in addition to generating a Lambda mos will generate a {{mos scalesig|4L 9s<3/1>|link=1}} unfair "Superlambda" mos which does not border on being atonal as the 17edt rendition does.


== Music ==
== Music ==
Line 243: Line 252:


; [[Ray Perlner]]
; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=fEQ13hzs3fY ''Fugue for Piano in 30EDT Bohlen-Pierce-Stearns<nowiki>[</nowiki>9<nowiki>]</nowiki> 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)
 
== See also ==
* [[11edf]] – relative edf
* [[19edo]] – relative edo
* [[49ed6]] – relative ed6
* [[53ed7]] – relative ed7
* [[68ed12]] – relative ed12
* [[93ed30]] – relative ed30


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

Latest revision as of 11:19, 30 March 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 ¢. Like 19edo, 30edt 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.

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 (continued)
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)

Subsets and supersets

Since 30 factors into primes as 2 × 3 × 5, 30edt has subset edts 2, 3, 5, 6, 10, and 15.

Intervals

# Cents Hekts Approximate ratios Scale name
Lambda Sigma
0 0 0 1/1 C
1 63.4 43.3 27/26, 28/27 C^/Dbv C#/Dbb
2 126.8 86.7 14/13, 15/14, 16/15, 29/27 Db Cx/Db
3 190.2 130.0 9/8, 10/9 C# D
4 253.6 173.3 15/13 C#^/Dv D#/Ebb
5 317.0 216.7 6/5 D Dx/Eb
6 380.4 260.0 5/4 D^/Ev E
7 443.8 303.3 9/7 E E#/Fbb
8 507.2 346.7 4/3 E^/Fbv Ex/Fb
9 570.6 390.0 7/5 Fb F
10 634.0 433.3 13/9 E# F#/Gb
11 697.4 476.7 3/2 E#^/Fv G
12 760.8 520.0 14/9 F G#/Hbb
13 824.2 563.3 8/5 F^/Gv Gx/Hb
14 887.6 606.7 5/3 G H
15 951.0 650.0 19/11 G^/Hbv H#/Jbb
16 1014.4 693.3 9/5 Hb Hx/Jb
17 1077.8 736.7 13/7 G# J
18 1141.2 780.0 27/14 G#^/Hv J#/Kbb
19 1204.6 823.3 2/1 H Jx/Kb
20 1268.0 866.7 27/13 H^/Jv K
21 1331.4 910.0 28/13 J K#/Lb
22 1394.8 953.3 9/4 J^/Av L
23 1458.2 996.7 7/3 A L#/Abb
24 1521.6 1040.0 12/5 A^/Bbv Lx/Ab
25 1585.0 1083.3 5/2 Bb A
26 1648.4 1126.7 13/5 A# A#/Bbb
27 1711.8 1170.0 8/3 A#^/Bv Ax/Bb
28 1775.2 1213.3 14/5 B
29 1838.6 1256.7 26/9 B^/Cv B#/Cb
30 1902.0 1300.0 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 6L 6s⟨3/1⟩ 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 8L 3s⟨3/1⟩ and the major scale is LLLsLLLsLLs. The sharp 9/7 of 7 degrees, in addition to generating a Lambda mos will generate a 4L 9s⟨3/1⟩ unfair "Superlambda" mos which does not border on being atonal as the 17edt rendition does.

Music

Mason Green
Ray Perlner

See also

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