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== Theory ==
== 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}}. Like 19edo, 30edt is [[consistent]] to the [[integer-limit|10-integer-limit]].


Because [[19edo]] has the 3rd, 5th, 7th, and 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.
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]].
30edt is a [[Phoenix]] tuning and exhibits all the benefits of such tunings.


=== Harmonics ===
=== Harmonics ===
{{Harmonics in equal|30|3|1|intervals=integer}}
{{Harmonics in equal|30|3|1|intervals=integer}}
{{Harmonics in equal|30|3|1|intervals=integer|columns=12|start=12|collapsed=1|Approximation of harmonics in 30edt (continued)}}
{{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 ===
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"
{| class="wikitable center-all right-2 right-3 left-4"
|-
|-
! rowspan="2" | #
! rowspan="2" | #
! rowspan="2" | Cents
! rowspan="2" | Cents
! rowspan="2" | Hekts
! rowspan="2" | [[Hekt]]s
! rowspan="2" | Approximate ratios
! rowspan="2" | Approximate ratios
! colspan="2" | Scale name
! colspan="2" | Scale name
Line 34: Line 35:
|-
|-
| 1
| 1
| 63.3985
| 63.4
| 43.333
| 43.3
| 27/26, 28/27
| [[27/26]], [[28/27]]
| C^/Dbv
| C^/Dbv
| C#/Dbb
| C#/Dbb
|-
|-
| 2
| 2
| 126.797
| 126.8
| 86.667
| 86.7
| [[14/13]], [[15/14]], [[16/15]], [[29/27]]
| [[14/13]], [[15/14]], [[16/15]], [[29/27]]
| Db
| Db
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|-
|-
| 3
| 3
| 190.1955
| 190.2
| 130
| 130.0
| 9/8, 10/9
| [[9/8]], [[10/9]]
| C#
| C#
| D
| D
|-
|-
| 4
| 4
| 253.594
| 253.6
| 173.333
| 173.3
| [[15/13]]
| [[15/13]]
| C#^/Dv
| C#^/Dv
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|-
|-
| 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
| [[5/4]]
| [[5/4]]
| D^/Ev
| D^/Ev
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|-
|-
| 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]]
| E^/Fbv
| E^/Fbv
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|-
|-
| 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]]
| E#
| E#
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|-
|-
| 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
| [[14/9]]
| [[14/9]]
| F
| F
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|-
|-
| 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]]
| G
| G
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|-
|-
| 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]]
| Hb
| Hb
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|-
|-
| 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
| [[27/14]]
| [[27/14]]
| G#^/Hv
| G#^/Hv
Line 160: Line 161:
|-
|-
| 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]]
| H^/Jv
| H^/Jv
Line 174: Line 175:
|-
|-
| 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]]
| J^/Av
| J^/Av
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|-
|-
| 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]]
| A^/Bbv
| A^/Bbv
Line 202: Line 203:
|-
|-
| 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]]
| A#
| A#
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|-
|-
| 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]]
| 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]]
| colspan="2" | C
| colspan="2" | C
Line 244: Line 245:
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 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 {{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 252: Line 253:
; [[Ray Perlner]]
; [[Ray Perlner]]
* [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)
== See also ==
* [[11edf]] – relative edf
* [[19edo]] – relative edo
* [[49ed6]] – relative ed6
* [[53ed7]] – relative ed7
* [[68ed12]] – relative ed12
* [[93ed30]] – relative ed30


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