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{{Infobox ET}}
{{Infobox ET}}
{{ED intro}}
{{ED intro}}
== Theory ==
== Theory ==
{{Harmonics in equal|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]] version of 15. In the 13-limit it supplies the optimal patent val for [[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.
[[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.  


Instead of the 18\30 fifth of 720 cents, 30edo 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 30edo is used for pelogic, 5\30 can again be used inconsistently as a 9/8.
=== Odd harmonics ===
{{Harmonics in equal|30}}


=== Subsets and supersets ===
=== Subsets and supersets ===
30edo has subset edos {{EDOs|1, 2, 3, 5, 6, 10, 15}} and it is a [[largely composite]] edo.
30edo has subset edos {{EDOs| 1, 2, 3, 5, 6, 10, 15 }} and it is a [[largely composite]] edo.


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.
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.


== 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]]
|}
== Notation ==
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style="font-size: 105%" | Notation systems for 30edo
|-
|-
! Step
! Step
! [[Cent]]s
! Cents
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[Ups and downs notation]]
! [[Armodue_theory|Armodue Notation]]
! [[Armodue theory|Armodue notation]]
|-
|-
| 0
| 0
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|}
|}


== Notation ==
=== Stein–Zimmermann–Gould notation ===
===Sagittal notation===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
====Best fifth notation====
{{Sharpness-sharp6-szg}}
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]].
 
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]].


=====Evo and Revo flavors=====
===== Evo and Revo flavors =====


<imagemap>
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</imagemap>
</imagemap>


=====Evo-SZ flavor=====
===== Evo-SZ flavor =====


<imagemap>
<imagemap>
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</imagemap>
</imagemap>


====Second-best fifth notation====
==== Second-best fifth notation ====
This notation uses the same sagittal sequence as EDOs [[35edo#Sagittal notation|35]] and [[40edo#Sagittal notation|40]].
This notation uses the same sagittal sequence as edos [[35edo #Sagittal notation|35]] and [[40edo #Sagittal notation|40]].


<imagemap>
<imagemap>
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|}
|}
<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 ==
== Scales ==
=== MOS scales ===
=== MOS scales ===
* [[Lovecraft5|Lovecraft[5]]] - 77772
* [[Lovecraft5|Lovecraft[5]]] - 77772
* [[Lovecraft9|Lovecraft[9]]] - 525252522
* [[Lovecraft9|Lovecraft[9]]] - 525252522
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* Mavila[23] - 21121121121112112112111
* Mavila[23] - 21121121121112112112111


=== Subsets of [[Mavila]][16] ===
=== 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
* Arcade (approximated from [[32afdo]]): 9 3 5 8 5
* [[Blackened skies]] (approximated from [[Compton]] in [[72edo]]): 8 5 2 3 2 8 2
* [[Blackened Skies]] (approximated from [[Compton]] in [[72edo]]): 8 5 2 3 2 8 2
* Carousel (this is the original/default tuning): 9 4 4 9 4
* Carousel (original/default tuning): 9 4 4 9 4
* Dewdrops (this is the original/default tuning): 4 4 4 5 4 4 5
* Dewdrops (original/default tuning): 4 4 4 5 4 4 5
* Geode (approximated from [[6afdo]]): 7 6 4 9 4
* Geode (approximated from [[6afdo]]): 7 6 4 9 4
* [[Lost spirit]] (approximated from [[Meantone]] in [[31edo]]): 7 5 2 3 5 3 5
* [[Lost Spirit]] (approximated from [[Meantone]] in [[31edo]]): 7 5 2 3 5 3 5
* Lost phantom (this is the original/default tuning): 8 5 2 2 6 2 5
* Lost phantom (original/default tuning): 8 5 2 2 6 2 5
* Mechanical (approximated from [[16afdo]]): 7 2 8 8 5
* Mechanical (approximated from [[16afdo]]): 7 2 8 8 5
* Mushroom (approximated from [[30afdo]]): 7 5 5 3 10
* Mushroom (approximated from [[30afdo]]): 7 5 5 3 10
* Nightdrive (this is the original/default tuning): 8 5 4 9 4
* Nightdrive (original/default tuning): 8 5 4 9 4
* Pelagic (this is the original/default tuning): 8 4 2 4 7 5
* Pelagic (original/default tuning): 8 4 2 4 7 5
* Bathypelagic (this is the original/default tuning): 8 4 2 3 8 5
* Bathypelagic (original/default tuning): 8 4 2 3 8 5
* Underpass (approximated from [[10afdo]]): 8 9 5 3 5
* Underpass (approximated from [[10afdo]]): 8 9 5 3 5
* Volcanic (approximated from [[16afdo]]): 3 6 8 8 5
* Volcanic (approximated from [[16afdo]]): 3 6 8 8 5


=== Subsets of [[15edo]] ===
=== Polymicrotonal scales ===
* Augmented[6] MOS: 8 2 8 2 8 2
* 10-tone 5&6edo scale: 5 1 4 2 3 3 2 4 1 5
* Equipentatonic (exact from [[5edo]]): 6 6 6 6 6
* 12-tone 6&10edo scale{{idio}}: 3 2 1 3 3 3 3 2 1 3 3 3
* Rockpool (approximated from [[47zpi]]): 2 8 2 6 6 6
* 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 ===
=== Other notable scales ===
* Approximation of [[Pelog]] lima: 3 4 10 3 10
* Approximation of [[Pelog]] lima: 3 4 10 3 10
* [[Amiot]] scale
* 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 ==
== Delta-rational harmony ==
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| 0.00012
| 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)
; {{W|Mitski}}
* [https://www.youtube.com/shorts/4MI2opBMkd4 ''Eric''] (2012) – microtonal cover in 30edo by [[Bryan Deister]] (2025)
=== 21st century ===
; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=uSpDz2Dmksw ''microtonal improvisation in 30edo''] (2023)
* [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]]
; [[Todd Harrop]]
* [https://spectropolrecords.bandcamp.com/track/todd-harrop-fifteen-short-pieces ''Fifteen Short Pieces'']
* [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]]
; [[Micronaive]]
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== Related pages ==
== Related pages ==
* [[Lumatone mapping for 30edo]]
* [[Mavila]]
* [[Mavila]]


Retrieved from "https://en.xen.wiki/w/30edo"