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'''30edo''', the '''30 equal division''', divides the octave into 30 equal steps of precisely 40 cents each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
{{Odd harmonics in edo|edo=30}}
30edo's [[patent val]] is a doubled version of the patent val for [[15edo]] through the 11-limit, so 30 can be viewed as a [[contorted]] version of 15. In the 13-limit it supplies the optimal patent val for [[quindecic]] temperament. If 15edo's mappings are still considered acceptable despite their low relative accuracy in this tuning, it can be seen as supplying an improved mapping of the 13th harmonic to 15edo, much like how 24edo supplies an improved 11 and 13 to 12edo.
[[File:Plot30.png|alt=plot30.png|thumb|A plot of the Z function around 30.]]
However, 5\30 is 200[[{{c}}]], which is a good (and familiar) approximation for 9/8, and hence 30edo can be viewed inconsistently, as having a 9/1 at 95\30 as well as 96\30.


Its [[patent val]] is a doubled version of the patent val for [[15edo]] through the 11-limit, so 30 can be viewed as a [[contorted]] (or [[enfactored]]) version of 15. In the 13-limit it supplies the optimal patent val for [[Trienstonic_clan#Quindecic|quindecic temperament]].
Instead of the 18\30 fifth of 720 cents, 30edo also makes available a 17\30 fifth of 680 cents. It is possible to interpret this fifth as [[mavila]] temperament using the 30bc [[val]], but the 360-cent [[5/4]] may be undesirable for some. When 30edo is used for pelogic, 5\30 can again be used inconsistently as a 9/8. An alternative option which uses the somewhat more accurate 400-cent [[5/4]] is [[shallowtone]] temperament using the 30b [[val]], although it is of very high [[badness]], being both high-[[error]] and high-[[complexity]]. [[Undecimation]] is also an option.


However, 5\30 is 200 cents, which is a good (and familiar) approximation for 9/8, and hence 30 can be viewed inconsistently, as having a 9' at 95\30 as well as a 9 at 96\30. Instead of the 18\30 fifth of 720 cents, 30 also makes available a 17\30 fifth of 680 cents. This is an ideal tuning for pelogic (5-limit mavila), which tempers out 135/128. When 30 is used for pelogic, 5\30 can again be used inconsistently as a 9/8.
=== Odd harmonics ===
{{Harmonics in equal|30}}


Below is a plot of the Z function around 30:
=== Subsets and supersets ===
30edo has subset edos {{EDOs| 1, 2, 3, 5, 6, 10, 15 }} and it is a [[largely composite]] edo.


[[File:plot30.png|alt=plot30.png|plot30.png]]
30edo is the 3rd {{w|primorial}} edo, being the product of first three primes and thus the smallest number with three distinct prime factors. As a corollary, 30edo is the smallest EDO that supports [[perfectly balanced]] scales that are minimal and not equally spaced. See the article on perfect balance.
 
Some multiples of 30edo have high consistency limits - [[270edo]], [[1920edo]], and others are simply useful like [[60edo]]. [[Zinc]] temperament is a rich augmentation of 30edo.


== Intervals ==
== Intervals ==
Inconsistent intervals are in ''italics''.
{| class="wikitable right-1 right-2"
|-
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Approximate ratios
|-
! 2.9.15.7.11.13 subgroup
! Ratios of 3 and 5<br>tending sharp
! Ratios of 3 and 5<br>tending flat
|-
| 0
| 0
| colspan="3" | [[1/1]]
|-
| 1
| 40
| [[40/39]]
|
| ''[[25/24]]'', [[36/35]], [[49/48]]
|-
| 2
| 80
| [[21/20]]
| ''[[16/15]]'', [[25/24]], ''[[36/35]]''
| ''[[15/14]]''
|-
| 3
| 120
| [[14/13]], [[15/14]], [[16/15]]
| [[13/12]]
| ''[[12/11]]''
|-
| 4
| 160
|
| ''[[10/9]]'', [[11/10]], [[12/11]], ''[[15/14]]''
| ''[[9/8]]'', ''[[13/12]]'', ''[[16/15]]''
|-
| 5
| 200
| [[9/8]]
|
| [[10/9]], ''[[11/10]]'', ''[[15/13]]''
|-
| 6
| 240
| [[8/7]], [[15/13]]
| ''[[7/6]]'', ''[[9/8]]''
|
|-
| 7
| 280
| [[13/11]]
| ''[[15/13]]''
| [[7/6]]
|-
| 8
| 320
| [[6/5]]
| [[6/5]], ''[[11/9]]''
| [[6/5]]
|-
| 9
| 360
| [[16/13]], [[11/9]]
|
| ''[[5/4]]''
|-
| 10
| 400
| [[14/11]]
| [[5/4]]
| ''[[11/9]]'', ''[[9/7]]''
|-
| 11
| 440
| [[9/7]], [[32/25]]
| [[13/10]]
|
|-
| 12
| 480
|
| ''[[9/7]]'', [[4/3]]
| ''[[13/10]]'', ''[[15/11]]''
|-
| 13
| 520
| [[27/20]], [[15/11]]
|
| ''[[4/3]]'', ''[[18/13]]''
|-
| 14
| 560
| [[11/8]], [[18/13]], [[25/18]]
| ''[[7/5]]'', ''[[15/11]]''
|
|-
| 15
| 600
|
| ''[[13/9]]'', ''[[18/13]]''
| [[7/5]], [[10/7]]
|-
| 16
| 640
| [[16/11]], [[13/9]], [[36/25]]
| ''[[10/7]]'', ''[[22/15]]''
|
|-
| 17
| 680
| [[40/27]], [[22/15]]
|
| ''[[3/2]]'', ''[[13/9]]''
|-
| 18
| 720
|
| ''[[14/9]]'', [[3/2]]
| ''[[20/13]]'', ''[[22/15]]''
|-
| 19
| 760
| [[14/9]], [[25/16]]
| [[20/13]]
|
|-
| 20
| 800
| [[11/7]]
| [[8/5]]
| ''[[14/9]]'', ''[[18/11]]''
|-
| 21
| 840
| [[13/8]], [[18/11]]
|
| ''[[8/5]]''
|-
| 22
| 880
| [[5/3]]
| [[5/3]], ''[[18/11]]''
| [[5/3]]
|-
| 23
| 920
| [[22/13]]
| ''[[26/15]]''
| [[12/7]]
|-
| 24
| 960
| [[7/4]], [[26/15]]
| ''[[12/7]]'', ''[[16/9]]''
|
|-
| 25
| 1000
| [[16/9]]
|
| [[9/5]], ''[[20/11]]'', ''[[26/15]]''
|-
| 26
| 1040
|
| ''[[9/5]]'', [[20/11]], [[11/6]], ''[[28/15]]''
| ''[[16/9]]'', ''[[24/13]]'', ''[[15/8]]''
|-
| 27
| 1080
| [[13/7]], [[28/15]], [[15/8]]
| [[24/13]]
| ''[[11/6]]''
|-
| 28
| 1120
| [[40/21]]
| ''[[15/8]]'', [[48/25]], ''[[35/18]]''
| ''[[28/15]]''
|-
| 29
| 1160
| [[39/20]]
|
| ''[[48/25]]'', [[35/18]], [[96/49]]
|-
| 30
| 1200
| colspan="3" | [[2/1]]
|}


{| class="wikitable"
== Notation ==
{| class="wikitable center-all"
|+ style="font-size: 105%" | Notation systems for 30edo
|-
|-
! Step
! Step
! Cents
! Cents
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs]][[Ups_and_Downs_Notation| notation]]
! colspan="3" | [[Ups and downs notation]]
! [[Armodue theory|Armodue notation]]
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:center;" | 0¢
| 0
| style="text-align:center;" | P1
| P1
| style="text-align:center;" | unison, minor 2nd
| unison, minor 2nd
| style="text-align:center;" | D, Eb
| D, Eb
| 1
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:center;" | 40
| 40
| style="text-align:center;" | ^1, ^m2
| ^1, ^m2
| style="text-align:center;" | up unison, upminor 2nd
| up unison, upminor 2nd
| style="text-align:center;" | ^D, ^Eb
| ^D, ^Eb
| 2b
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:center;" |80
| 80
| style="text-align:center;" | ^^1, v~2
| ^^1, v~2
| style="text-align:center;" | double-up unison, downmid 2nd
| dup unison, downmid 2nd
| style="text-align:center;" | ^^D, ^^Eb
| ^^D, ^^Eb
| 9#
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:center;" | 120
| 120
| style="text-align:center;" | ~2
| ~2
| style="text-align:center;" | mid 2nd
| mid 2nd
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>E
| v<span style="font-size: 90%; vertical-align: super;">3</span>E
| 1#
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:center;" | 160
| 160
| style="text-align:center;" | ^~2
| ^~2
| style="text-align:center;" | upmid 2nd
| upmid 2nd
| style="text-align:center;" | vvE
| vvE
| 2
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:center;" | 200
| 200
| style="text-align:center;" | vM2
| vM2
| style="text-align:center;" | downmajor 2nd
| downmajor 2nd
| style="text-align:center;" | vE
| vE
| 3b
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:center;" | 240
| 240
| style="text-align:center;" | M2, m3
| M2, m3
| style="text-align:center;" | major 2nd, minor 3rd
| major 2nd, minor 3rd
| style="text-align:center;" | E, F
| E, F
| 1x, 4bb
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | 280
| 280
| style="text-align:center;" | ^m3
| ^m3
| style="text-align:center;" | upminor 3rd
| upminor 3rd
| style="text-align:center;" | ^F
| ^F
| 2#
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:center;" | 320
| 320
| style="text-align:center;" | v~3
| v~3
| style="text-align:center;" | downmid 3rd
| downmid 3rd
| style="text-align:center;" | ^^F
| ^^F
| 3
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:center;" | 360
| 360
| style="text-align:center;" | ~3
| ~3
| style="text-align:center;" | mid 3rd
| mid 3rd
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>F, v<span style="font-size: 90%; vertical-align: super;">3</span>F#
| ^<span style="font-size: 90%; vertical-align: super;">3</span>F, v<span style="font-size: 90%; vertical-align: super;">3</span>F#
| 4b
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:center;" | 400
| 400
| style="text-align:center;" | ^~3
| ^~3
| style="text-align:center;" | upmid 3rd
| upmid 3rd
| style="text-align:center;" | vvF#
| vvF#
| 5b
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:center;" | 440
| 440
| style="text-align:center;" | vM3, v4
| vM3, v4
| style="text-align:center;" | downmajor 3rd, down 4th
| downmajor 3rd, down 4th
| style="text-align:center;" | vF#, vG
| vF#, vG
| 3#
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:center;" | 480
| 480
| style="text-align:center;" | P4
| M3, P4
| style="text-align:center;" | major 3rd, perfect 4th
| major 3rd, perfect 4th
| style="text-align:center;" | F#, G
| F#, G
| 4
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:center;" | 520
| 520
| style="text-align:center;" | ^4, ^d5
| ^4
| style="text-align:center;" | up 4th, updim 5th
| up 4th
| style="text-align:center;" | ^G, ^Ab
| ^G
| 5
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:center;" | 560
| 560
| style="text-align:center;" | v~4, v~d5
| v~4, v~d5
| style="text-align:center;" | downmid 4th, downmid 5th
| downmid 4th, downmid 5th
| style="text-align:center;" | ^^G, ^^Ab
| ^^G, ^^Ab
| 6b
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:center;" | 600
| 600
| style="text-align:center;" | ~4, ~5
| ~4, ~5
| style="text-align:center;" | mid 4th, mid 5th
| mid 4th, mid 5th
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>G, v<span style="font-size: 90%; vertical-align: super;">3</span>A
| ^<span style="font-size: 90%; vertical-align: super;">3</span>G, v<span style="font-size: 90%; vertical-align: super;">3</span>A
| 4#
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:center;" | 640
| 640
| style="text-align:center;" | ^~A4, ^~5
| ^~4, ^~5
| style="text-align:center;" | upmid 4th, upmid 5th
| upmid 4th, upmid 5th
| style="text-align:center;" | vvG#, vvA
| vvG#, vvA
| 5#
|-
|-
| style="text-align:center;" | 17
| 17
| style="text-align:center;" | 680
| 680
| style="text-align:center;" | vA4, v5
| v5
| style="text-align:center;" | downaug 4th, down 5th
| down 5th
| style="text-align:center;" | vG#, vA
| vA
| 6
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:center;" | 720
| 720
| style="text-align:center;" | P5
| P5, m6
| style="text-align:center;" | perfect 5th, minor 6th
| perfect 5th, minor 6th
| style="text-align:center;" | A, Bb
| A, Bb
| 7b
|-
|-
| style="text-align:center;" | 19
| 19
| style="text-align:center;" | 760
| 760
| style="text-align:center;" | ^5, ^m6
| ^5, ^m6
| style="text-align:center;" | up 5th, upminor 6th
| up 5th, upminor 6th
| style="text-align:center;" | ^A, ^Bb
| ^A, ^Bb
| 5x, 8bb
|-
|-
| style="text-align:center;" | 20
| 20
| style="text-align:center;" | 800
| 800
| style="text-align:center;" | v~6
| v~6
| style="text-align:center;" | downmid 6th
| downmid 6th
| style="text-align:center;" | ^^Bb
| ^^Bb
| 6#
|-
|-
| style="text-align:center;" | 21
| 21
| style="text-align:center;" | 840
| 840
| style="text-align:center;" | ~6
| ~6
| style="text-align:center;" | mid 6th
| mid 6th
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>B
| v<span style="font-size: 90%; vertical-align: super;">3</span>B
| 7
|-
|-
| style="text-align:center;" | 22
| 22
| style="text-align:center;" | 880
| 880
| style="text-align:center;" | ^~6
| ^~6
| style="text-align:center;" | upmid 6th
| upmid 6th
| style="text-align:center;" | vvB
| vvB
| 8b
|-
|-
| style="text-align:center;" | 23
| 23
| style="text-align:center;" | 920
| 920
| style="text-align:center;" | vM6
| vM6
| style="text-align:center;" | downmajor 6th
| downmajor 6th
| style="text-align:center;" | vB
| vB
| 6x, 9bb
|-
|-
| style="text-align:center;" | 24
| 24
| style="text-align:center;" | 960
| 960
| style="text-align:center;" | M6. m7
| M6. m7
| style="text-align:center;" | major 6th, minor 7th
| major 6th, minor 7th
| style="text-align:center;" | B, C
| B, C
| 7#
|-
|-
| style="text-align:center;" | 25
| 25
| style="text-align:center;" | 1000
| 1000
| style="text-align:center;" | ^m7
| ^m7
| style="text-align:center;" | upminor 7th
| upminor 7th
| style="text-align:center;" | ^C
| ^C
| 8
|-
|-
| style="text-align:center;" | 26
| 26
| style="text-align:center;" | 1040
| 1040
| style="text-align:center;" | v~7
| v~7
| style="text-align:center;" | downmid 7th
| downmid 7th
| style="text-align:center;" | ^^C
| ^^C
| 9b
|-
|-
| style="text-align:center;" | 27
| 27
| style="text-align:center;" | 1080
| 1080
| style="text-align:center;" | ~7
| ~7
| style="text-align:center;" | mid 7th
| mid 7th
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>C
| ^<span style="font-size: 90%; vertical-align: super;">3</span>C
| 1b
|-
|-
| style="text-align:center;" | 28
| 28
| style="text-align:center;" | 1120
| 1120
| style="text-align:center;" | ^~7, vv8
| ^~7, vv8
| style="text-align:center;" | upmid 7th, double-down 8ve
| upmid 7th, dud 8ve
| style="text-align:center;" | vvC#, vvD
| vvC#, vvD
| 8#
|-
|-
| style="text-align:center;" | 29
| 29
| style="text-align:center;" | 1160
| 1160
| style="text-align:center;" | vM7, v8
| vM7, v8
| style="text-align:center;" | downmajor 7th, down 8ve
| downmajor 7th, down 8ve
| style="text-align:center;" | vC#, vD
| vC#, vD
| 9
|-
|-
| style="text-align:center;" | 30
| 30
| style="text-align:center;" | 1200
| 1200
| style="text-align:center;" | P8
| P8
| style="text-align:center;" | major 7th, 8ve
| major 7th, 8ve
| style="text-align:center;" | C#, D
| C#, D
| 1
|}
|}


== Rank Two Temperaments ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6-szg}}
 
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt-szg}}
 
=== Kite's ups and downs notation ===
30edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
{{Sharpness-sharp6a}}
 
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}
 
=== Sagittal notation ===
==== Best fifth notation ====
This notation uses the same sagittal sequence as edos [[23edo #Second-best fifth notation|23b]], [[37edo #Sagittal notation|37]], and [[44edo #Sagittal notation|44]], and is a superset of the notations for edos [[15edo #Sagittal notation|15]], [[10edo #Sagittal notation|10]], and [[5edo #Sagittal notation|5]].


As 30edo is highly composite, only 7, 11 and 13 steps create mode of symmetry scales that cover every interval using one period per octave. 7/30 produces [[Chromatic_pairs#Lovecraft|Lovecraft]], in which 2 generators is a moderately sharp [[11/8]], 3 a near perfect [[13/8]] and 5 the familiar mildly flat [[9/8]] from [[12edo]], creating the possibility of ignoring the 3rd & 5th entirely to use those harmonics as the primary building blocks of harmony in a similar way to [[orgone]]. 11 produces a flat [[sensi]] scale. 13 is an excellent higher order [[Pelogic_family#Mavila|Mavila]] tuning that functions the closest to the familiar diatonic scale you can get in this edo.
===== Evo and Revo flavors =====


; MOS scales:
<imagemap>
File:30-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 519 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 519 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:30-EDO_Sagittal.svg]]
</imagemap>


* [[Lovecraft5|Lovecraft[5]]] - 77772
===== Evo-SZ flavor =====
* [[Lovecraft9|Lovecraft[9]]] - 525252522
 
* [[Lovecraft13|Lovecraft[13]]] - 3223223223222
<imagemap>
* Lovecraft[17] - 22221222122212221
File:30-EDO_Evo-SZ_Sagittal.svg
* [[Sensi5|Sensi[5]]] - 83838
desc none
* [[Sensi8|Sensi[8]]] - 53353353
rect 80 0 300 50 [[Sagittal_notation]]
* [[Sensi11|Sensi[11]]] - 33323332332
rect 495 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
* [[Sensi19|Sensi[19]]] - 2121212212121221212
rect 20 80 495 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
* Mavila[5] - 94944
default [[File:30-EDO_Evo-SZ_Sagittal.svg]]
* Mavila[7] - 5445444
</imagemap>
* Mavila[9] - 444414441
 
* Mavila[16] - 3131313113131311
==== Second-best fifth notation ====
* Mavila[23] - 21121121121112112112111
This notation uses the same sagittal sequence as edos [[35edo #Sagittal notation|35]] and [[40edo #Sagittal notation|40]].
 
<imagemap>
File:30b_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 479 0 639 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 479 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
default [[File:30b_Sagittal.svg]]
</imagemap>
 
== Regular temperament properties ==
=== Rank-2 temperaments ===
As 30edo is largely composite, only 7, 11 and 13 steps create [[MOS scale]]s that cover every interval using one period per octave. 
 
7/30 produces [[No-threes subgroup temperaments#Lovecraft|Lovecraft]], in which 2 generators is a moderately sharp [[11/8]], 3 a near perfect [[13/8]] and 5 the familiar mildly flat [[9/8]] from [[12edo]], creating the possibility of ignoring the 3rd & 5th entirely to use those harmonics as the primary building blocks of harmony in a similar way to [[orgone]]. 
 
11 produces a flat [[sensi]] scale. 13 is an excellent higher order [[Pelogic_family#Mavila|Mavila]] tuning that functions the closest to the familiar diatonic scale you can get in this edo.


== Commas ==
=== Commas ===
30 EDO [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 30 48 70 84 104 111 }}.)
30et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 30 48 70 84 104 111 }}.


{| class="commatable wikitable center-1 center-2 right-4 center-5"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
|-
! [[Harmonic Limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Monzo]]
Line 244: Line 518:
| 90.22
| 90.22
| Sawa
| Sawa
| Limma, Pythagorean minor sedond
| Blackwood comma, Pythagorean limma
|-
|-
| 5
| 5
Line 251: Line 525:
| 49.17
| 49.17
| Triyo
| Triyo
| Maximal diesis, Porcupine comma
| Porcupine comma, maximal diesis
|-
|-
| 5
| 5
Line 258: Line 532:
| 41.06
| 41.06
| Trigu
| Trigu
| Diesis, augmented comma
| Augmented comma, diesis
|-
|-
| 5
| 5
Line 279: Line 553:
| 35.70
| 35.70
| Zozo
| Zozo
| Slendro diesis
| Semaphoresma, slendro diesis
|-
|-
| 7
| 7
Line 293: Line 567:
| 22.23
| 22.23
| Laquadzo-atrigu
| Laquadzo-atrigu
| Squalentine
| Squalentine comma
|-
|-
| 7
| 7
Line 307: Line 581:
| 13.79
| 13.79
| Zotrigu
| Zotrigu
| Septimal semicomma, Starling comma
| Starling comma, septimal semicomma
|-
|-
| 7
| 7
Line 314: Line 588:
| 13.47
| 13.47
| Rurutriyo
| Rurutriyo
| Octagar
| Octagar comma
|-
|-
| 7
| 7
Line 328: Line 602:
| 5.36
| 5.36
| Saruru-atrigu
| Saruru-atrigu
| Porwell
| Porwell comma
|-
|-
| 7
| 7
Line 384: Line 658:
| 3.03
| 3.03
| Triluyo
| Triluyo
| Wizardharry
| Wizardharry comma
|-
|-
| 11
| 11
Line 394: Line 668:
|}
|}
<references/>
<references/>
== Octave stretch or compression ==
30edo's simple [[prime]]s with the most error - 3, 5 and 11 - are all tuned sharp, so it can benefit from [[octave shrinking]]. Some compressed-octave 30edo tunings (least to most compressed) include [[zpi|122zpi]], [[equal tuning|100ed10]], [[ed12|108ed12]] or [[ed6|78ed6]].
Alternatively, if one wishes to use 30edo as a [[dual-fifth]] tuning, [[equal tuning|95ed9]] is a good option, sharing the error equally between both fifths (20{{c}} error each). This does come at the cost of making most of 30edo's worst primes slightly worse, though not enough to affect their usability.
== Scales ==
=== MOS scales ===
{{Main|List of MOS scales in 30edo}}
* [[Lovecraft5|Lovecraft[5]]] - 77772
* [[Lovecraft9|Lovecraft[9]]] - 525252522
* [[Lovecraft13|Lovecraft[13]]] - 3223223223222
* Lovecraft[17] - 22221222122212221
* [[Sensi5|Sensi[5]]] - 83838
* [[Sensi8|Sensi[8]]] - 53353353
* [[Sensi11|Sensi[11]]] - 33323332332
* [[Sensi19|Sensi[19]]] - 2121212212121221212
* Mavila[5] - 94944
* Mavila[7] - 5445444
* Mavila[9] - 444414441
* Mavila[16] - 3131313113131311
* Mavila[23] - 21121121121112112112111
=== Subsets of [[mavila]][16] ===
{{Idiosyncratic terms|Most of these names were coined, and have so far been soley used by, [[Budjarn Lambeth]].}}
* Arcade (approximated from [[32afdo]]): 9 3 5 8 5
* [[Blackened Skies]] (approximated from [[Compton]] in [[72edo]]): 8 5 2 3 2 8 2
* Carousel (original/default tuning): 9 4 4 9 4
* Dewdrops (original/default tuning): 4 4 4 5 4 4 5
* Geode (approximated from [[6afdo]]): 7 6 4 9 4
* [[Lost Spirit]] (approximated from [[Meantone]] in [[31edo]]): 7 5 2 3 5 3 5
* Lost phantom (original/default tuning): 8 5 2 2 6 2 5
* Mechanical (approximated from [[16afdo]]): 7 2 8 8 5
* Mushroom (approximated from [[30afdo]]): 7 5 5 3 10
* Nightdrive (original/default tuning): 8 5 4 9 4
* Pelagic (original/default tuning): 8 4 2 4 7 5
* Bathypelagic (original/default tuning): 8 4 2 3 8 5
* Underpass (approximated from [[10afdo]]): 8 9 5 3 5
* Volcanic (approximated from [[16afdo]]): 3 6 8 8 5
=== Polymicrotonal scales ===
* 10-tone 5&6edo scale: 5 1 4 2 3 3 2 4 1 5
* 12-tone 6&10edo scale{{idio}}: 3 2 1 3 3 3 3 2 1 3 3 3
* 12-tone 6&15edo scale{{idio}}: 2 3 3 2 2 3 3 2 2 3 3 2
* 12-tone 10&15edo scale{{idio}}: 3 1 2 3 3 3 3 3 1 2 3 3
* 14-tone 6&10edo scale: 3 2 1 3 1 2 3 3 2 1 3 1 2 3
* 18-tone 6&15edo scale: 2 2 1 1 2 2 2 2 1 1 2 2 2 2 1 1 2 2
* 20-tone 10&15edo scale: 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2
=== Other notable scales ===
* Approximation of [[Pelog]] lima: 3 4 10 3 10
* Approximation of Hirajoshi for metallic/percussive timbres: 5 3 9 3 10
* [[Amiot]] scale: 6 1 6 1 6 1 6
* Augmented[6] (exact from [[15edo]]): 8 2 8 2 8 2
* Dusty{{idio}} (original tuning): 8 5 5 3 7 2
* [[Equipentatonic]] (exact from [[5edo]]): 6 6 6 6 6
* Iron filing{{idio}} (original tuning): 3 2 2 2 1 2 2 3 1 3 1 2 2 1 3
* [[Moon dust]] (approximated from [[JI]]): ''nonoctave''
* Rockpool{{idio}} (approximated from [[47zpi]]): 2 8 2 6 6 6
* ''More listed in: [[User:BudjarnLambeth/Quasipelog theory#Scales]]''
== Delta-rational harmony ==
The tables below show chords that approximate 3-integer-limit [[delta-rational]] chords with least-squares error less than 0.001.
=== Fully delta-rational triads ===
{| class="mw-collapsible mw-collapsed class="wikitable sortable"
|-
! Chord
! Delta signature
! Error
|-
| 0,1,2
| +1+1
| 0.00026
|-
| 0,1,3
| +1+2
| 0.00058
|-
| 0,1,4
| +1+3
| 0.00094
|-
| 0,2,3
| +2+1
| 0.00047
|-
| 0,3,4
| +3+1
| 0.00068
|-
| 0,3,11
| +1+3
| 0.00064
|-
| 0,4,11
| +1+2
| 0.00039
|-
| 0,5,8
| +3+2
| 0.00057
|-
| 0,6,16
| +1+2
| 0.00042
|-
| 0,7,13
| +1+1
| 0.00035
|-
| 0,7,23
| +1+3
| 0.00024
|-
| 0,10,25
| +1+2
| 0.00072
|-
| 0,11,17
| +3+2
| 0.00063
|-
| 0,11,27
| +1+2
| 0.00072
|-
| 0,13,23
| +1+1
| 0.00030
|-
| 0,14,29
| +2+3
| 0.00019
|-
| 0,15,19
| +3+1
| 0.00069
|-
| 0,20,25
| +3+1
| 0.00085
|}
=== Partially delta-rational tetrads ===
{| class="mw-collapsible mw-collapsed class="wikitable sortable"
|-
! Chord
! Delta signature
! Error
|-
| 0,1,2,3
| +1+?+1
| 0.00064
|-
| 0,1,3,4
| +1+?+1
| 0.00097
|-
| 0,1,15,16
| +2+?+3
| 0.00097
|-
| 0,1,15,17
| +1+?+3
| 0.00098
|-
| 0,1,16,17
| +2+?+3
| 0.00060
|-
| 0,1,16,18
| +1+?+3
| 0.00050
|-
| 0,1,17,18
| +2+?+3
| 0.00021
|-
| 0,1,17,19
| +1+?+3
| 0.00002
|-
| 0,1,18,19
| +2+?+3
| 0.00018
|-
| 0,1,18,20
| +1+?+3
| 0.00047
|-
| 0,1,19,20
| +2+?+3
| 0.00058
|-
| 0,1,19,21
| +1+?+3
| 0.00098
|-
| 0,1,20,21
| +2+?+3
| 0.00099
|-
| 0,1,28,29
| +1+?+2
| 0.00086
|-
| 0,2,3,4
| +2+?+1
| 0.00094
|-
| 0,2,6,11
| +1+?+3
| 0.00036
|-
| 0,2,7,12
| +1+?+3
| 0.00063
|-
| 0,2,11,12
| +3+?+2
| 0.00089
|-
| 0,2,11,14
| +1+?+2
| 0.00084
|-
| 0,2,12,13
| +3+?+2
| 0.00044
|-
| 0,2,12,15
| +1+?+2
| 0.00005
|-
| 0,2,13,14
| +3+?+2
| 0.00002
|-
| 0,2,13,16
| +1+?+2
| 0.00095
|-
| 0,2,14,15
| +3+?+2
| 0.00049
|-
| 0,2,15,16
| +3+?+2
| 0.00098
|-
| 0,2,16,20
| +1+?+3
| 0.00053
|-
| 0,2,17,19
| +2+?+3
| 0.00043
|-
| 0,2,17,21
| +1+?+3
| 0.00046
|-
| 0,2,18,20
| +2+?+3
| 0.00036
|-
| 0,3,4,8
| +2+?+3
| 0.00071
|-
| 0,3,5,9
| +2+?+3
| 0.00050
|-
| 0,3,7,12
| +1+?+2
| 0.00017
|-
| 0,3,9,16
| +1+?+3
| 0.00024
|-
| 0,3,16,22
| +1+?+3
| 0.00003
|-
| 0,3,17,18
| +2+?+1
| 0.00085
|-
| 0,3,17,19
| +1+?+1
| 0.00100
|-
| 0,3,17,20
| +2+?+3
| 0.00066
|-
| 0,3,17,21
| +1+?+2
| 0.00006
|-
| 0,3,18,19
| +2+?+1
| 0.00031
|-
| 0,3,18,20
| +1+?+1
| 0.00005
|-
| 0,3,18,21
| +2+?+3
| 0.00055
|-
| 0,3,19,20
| +2+?+1
| 0.00025
|-
| 0,3,19,21
| +1+?+1
| 0.00092
|-
| 0,3,20,21
| +2+?+1
| 0.00081
|-
| 0,3,24,29
| +1+?+3
| 0.00063
|-
| 0,4,5,15
| +1+?+3
| 0.00038
|-
| 0,4,7,12
| +2+?+3
| 0.00062
|-
| 0,4,10,19
| +1+?+3
| 0.00023
|-
| 0,4,11,17
| +1+?+2
| 0.00078
|-
| 0,4,12,13
| +3+?+1
| 0.00099
|-
| 0,4,13,14
| +3+?+1
| 0.00049
|-
| 0,4,13,15
| +3+?+2
| 0.00044
|-
| 0,4,13,16
| +1+?+1
| 0.00005
|-
| 0,4,14,15
| +3+?+1
| 0.00002
|-
| 0,4,14,16
| +3+?+2
| 0.00052
|-
| 0,4,15,16
| +3+?+1
| 0.00054
|-
| 0,4,17,21
| +2+?+3
| 0.00089
|-
| 0,4,18,22
| +2+?+3
| 0.00074
|-
| 0,4,20,25
| +1+?+2
| 0.00030
|-
| 0,4,22,29
| +1+?+3
| 0.00041
|-
| 0,5,6,9
| +3+?+2
| 0.00051
|-
| 0,5,6,18
| +1+?+3
| 0.00011
|-
| 0,5,8,16
| +1+?+2
| 0.00028
|-
| 0,5,9,15
| +2+?+3
| 0.00030
|-
| 0,5,10,14
| +1+?+1
| 0.00027
|-
| 0,5,10,21
| +1+?+3
| 0.00084
|-
| 0,5,11,13
| +2+?+1
| 0.00017
|-
| 0,5,12,14
| +2+?+1
| 0.00078
|-
| 0,5,14,21
| +1+?+2
| 0.00095
|-
| 0,5,15,25
| +1+?+3
| 0.00006
|-
| 0,5,18,23
| +2+?+3
| 0.00093
|-
| 0,5,20,29
| +1+?+3
| 0.00014
|-
| 0,5,22,28
| +1+?+2
| 0.00093
|-
| 0,5,23,24
| +3+?+1
| 0.00073
|-
| 0,5,23,25
| +3+?+2
| 0.00075
|-
| 0,5,23,26
| +1+?+1
| 0.00020
|-
| 0,5,24,25
| +3+?+1
| 0.00009
|-
| 0,5,24,26
| +3+?+2
| 0.00045
|-
| 0,5,25,26
| +3+?+1
| 0.00057
|-
| 0,6,7,21
| +1+?+3
| 0.00086
|-
| 0,6,8,13
| +1+?+1
| 0.00079
|-
| 0,6,10,17
| +2+?+3
| 0.00091
|-
| 0,6,11,20
| +1+?+2
| 0.00026
|-
| 0,6,14,17
| +3+?+2
| 0.00003
|-
| 0,6,19,21
| +2+?+1
| 0.00066
|-
| 0,6,19,23
| +1+?+1
| 0.00086
|-
| 0,6,20,22
| +2+?+1
| 0.00048
|-
| 0,7,8,11
| +2+?+1
| 0.00095
|-
| 0,7,8,12
| +3+?+2
| 0.00035
|-
| 0,7,9,11
| +3+?+1
| 0.00020
|-
| 0,7,9,12
| +2+?+1
| 0.00039
|-
| 0,7,10,12
| +3+?+1
| 0.00074
|-
| 0,7,11,19
| +2+?+3
| 0.00075
|-
| 0,7,13,23
| +1+?+2
| 0.00005
|-
| 0,7,14,28
| +1+?+3
| 0.00034
|-
| 0,7,16,21
| +1+?+1
| 0.00097
|-
| 0,7,18,27
| +1+?+2
| 0.00030
|-
| 0,7,21,24
| +3+?+2
| 0.00028
|-
| 0,7,27,29
| +2+?+1
| 0.00032
|-
| 0,8,10,27
| +1+?+3
| 0.00088
|-
| 0,8,12,21
| +2+?+3
| 0.00022
|-
| 0,8,14,18
| +3+?+2
| 0.00099
|-
| 0,8,15,17
| +3+?+1
| 0.00054
|-
| 0,8,15,18
| +2+?+1
| 0.00001
|-
| 0,8,16,18
| +3+?+1
| 0.00053
|-
| 0,8,22,27
| +1+?+1
| 0.00033
|-
| 0,9,10,15
| +3+?+2
| 0.00013
|-
| 0,9,10,29
| +1+?+3
| 0.00029
|-
| 0,9,12,19
| +1+?+1
| 0.00028
|-
| 0,9,12,25
| +1+?+2
| 0.00000
|-
| 0,9,16,28
| +1+?+2
| 0.00005
|-
| 0,9,19,25
| +1+?+1
| 0.00028
|-
| 0,9,20,24
| +3+?+2
| 0.00025
|-
| 0,9,21,23
| +3+?+1
| 0.00015
|-
| 0,9,21,24
| +2+?+1
| 0.00068
|-
| 0,10,13,17
| +2+?+1
| 0.00052
|-
| 0,10,13,24
| +2+?+3
| 0.00042
|-
| 0,10,15,20
| +3+?+2
| 0.00006
|-
| 0,10,17,24
| +1+?+1
| 0.00005
|-
| 0,10,25,29
| +3+?+2
| 0.00048
|-
| 0,10,26,28
| +3+?+1
| 0.00028
|-
| 0,10,26,29
| +2+?+1
| 0.00061
|-
| 0,11,13,16
| +3+?+1
| 0.00032
|-
| 0,11,17,21
| +2+?+1
| 0.00085
|-
| 0,11,20,25
| +3+?+2
| 0.00095
|-
| 0,12,14,23
| +1+?+1
| 0.00005
|-
| 0,12,17,20
| +3+?+1
| 0.00014
|-
| 0,12,22,26
| +2+?+1
| 0.00081
|-
| 0,12,24,29
| +3+?+2
| 0.00014
|-
| 0,13,18,27
| +1+?+1
| 0.00000
|-
| 0,13,21,24
| +3+?+1
| 0.00013
|-
| 0,14,16,23
| +3+?+2
| 0.00035
|-
| 0,14,19,24
| +2+?+1
| 0.00067
|-
| 0,14,25,28
| +3+?+1
| 0.00040
|-
| 0,15,23,28
| +2+?+1
| 0.00083
|-
| 0,16,19,23
| +3+?+1
| 0.00076
|-
| 0,17,20,28
| +3+?+2
| 0.00099
|-
| 0,17,21,27
| +2+?+1
| 0.00067
|-
| 0,17,22,26
| +3+?+1
| 0.00042
|-
| 0,18,25,29
| +3+?+1
| 0.00042
|-
| 0,19,20,29
| +3+?+2
| 0.00033
|-
| 0,21,23,28
| +3+?+1
| 0.00012
|}
== Instruments ==
[[Lumatone mapping for 30edo|Lumatone mappings for 30edo]] are available.


== Music ==
== Music ==
=== Modern renderings ===
; {{W|Evanescence}}
* [https://www.youtube.com/watch?v=ppHcUOpbnbI ''Bring Me To Life''] (2003) – microtonal cover in 30edo by [[Bryan Deister]] (2024)
; LouisF (Luis Clemente)
* [https://www.youtube.com/shorts/ujkKJekgDhM ''Balatro Main Theme''] via {{W|Balatro}} (2024) – microtonal cover in 30edo by [[Bryan Deister]] (2026)
; {{W|Mitski}}
* [https://www.youtube.com/shorts/4MI2opBMkd4 ''Eric''] (2012) – microtonal cover in 30edo by [[Bryan Deister]] (2025)
=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=uSpDz2Dmksw ''microtonal improvisation in 30edo''] (2023)
* [https://www.youtube.com/watch?v=NP3HGr3ZD70&lc=UgxFBmbxZa5dF4ZPj0F4AaABAg.AFzcn1LkVZNAG4JYIvXZvZ ''minuet in 30edo''] (2025)
* [https://www.youtube.com/watch?v=pa4YMCae2tE ''waltz in 30edo''] (2025)
* [https://www.youtube.com/watch?v=2TxCWDYUvYc ''30edo improv''] (2025)
* ''Ferris Wheel - 30edo'' (2026)
** [https://www.youtube.com/shorts/O6nOiLxYPdE <nowiki>[short]</nowiki>] (with Lumatone view)
** [https://www.youtube.com/watch?v=gyrb2-tt_m8 <nowiki>[full version]</nowiki>]
* [https://www.youtube.com/shorts/ZlXSZDSlH2c ''Fantasy in 30edo''] (2026)
; [[Todd Harrop]]
* [https://spectropolrecords.bandcamp.com/track/todd-harrop-fifteen-short-pieces ''Fifteen Short Pieces'']
; [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=XT2K75X79sE ''Mavila(7) improvisation''] (2026)
; [[Micronaive]]
* [https://youtu.be/tAxEetp1TaE ''No.27.62'']
; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=6gydbVD7Xdc ''Edolian - Shift''] (2020)


[http://spectropolrecords.bandcamp.com/track/todd-harrop-fifteen-short-pieces Fifteen Short Pieces] by Todd Harrop
== Related pages ==
* [[Mavila]]


[[Category:Equal divisions of the octave]]
[[Category:Inconsistent]]
[[Category:Pelogic]]
[[Category:Pelogic]]
[[Category:Zeta]]
[[Category:Todo:add rank 2 temperaments table]]
 
[[Category:Listen]]
{{Todo
| unify precision
| cleanup
}}
Retrieved from "https://en.xen.wiki/w/30edo"