Xenharmonic Wiki

30edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
Fredg999 category edits (talk | contribs)
autopatrolled, Bots, emailconfirmed
4,475 edits
m Categories, sort key, misc. edits
Added infobox ET, cleanup (WIP)
Line 1: Line 1:
'''30edo''', the '''30 equal division''', divides the octave into 30 equal steps of precisely 40 cents each.  
{{Infobox ET
| Prime factorization = 2 × 3 × 5
| Step size = 40¢
| Fifth = 18\30 = 720¢ (→[[5edo|3\5]])
| Major 2nd = 6\30 = 240¢
| Minor 2nd = 0\30 = 0¢
| Augmented 1sn = 6\30 = 240¢
}}
'''30 equal divisions of the octave''' ('''30edo''') is the [[tuning system]] derived by dividing the [[octave]] into 30 equal steps of 40{{cent}} each.


== Theory ==
== Theory ==
{{Odd harmonics in edo|edo=30}}
{{Harmonics in equal|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 [[Trienstonic_clan#Quindecic|quindecic temperament]].
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 [[Trienstonic_clan#Quindecic|quindecic temperament]].


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.
However, 5\30 is 200 cents, 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.


Below is a plot of the Z function around 30:
Below is a plot of the Z function around 30:
Line 13: Line 21:


== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! Step
! Step
! Cents
! [[Cent]]s
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs]][[Ups_and_Downs_Notation| notation]]
! colspan="3" | [[Ups and downs 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
|-
|-
| 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
|-
|-
| 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
| double-up unison, downmid 2nd
| style="text-align:center;" | ^^D, ^^Eb
| ^^D, ^^Eb
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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#
|-
|-
| 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#
|-
|-
| 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
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:center;" | 480
| 480
| style="text-align:center;" | P4
| P4
| style="text-align:center;" | major 3rd, perfect 4th
| major 3rd, perfect 4th
| style="text-align:center;" | F#, G
| F#, G
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:center;" | 520
| 520
| style="text-align:center;" | ^4, ^d5
| ^4, ^d5
| style="text-align:center;" | up 4th, updim 5th
| up 4th, updim 5th
| style="text-align:center;" | ^G, ^Ab
| ^G, ^Ab
|-
|-
| 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
|-
|-
| 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
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:center;" | 640
| 640
| style="text-align:center;" | ^~A4, ^~5
| ^~A4, ^~5
| style="text-align:center;" | upmid 4th, upmid 5th
| upmid 4th, upmid 5th
| style="text-align:center;" | vvG#, vvA
| vvG#, vvA
|-
|-
| style="text-align:center;" | 17
| 17
| style="text-align:center;" | 680
| 680
| style="text-align:center;" | vA4, v5
| vA4, v5
| style="text-align:center;" | downaug 4th, down 5th
| downaug 4th, down 5th
| style="text-align:center;" | vG#, vA
| vG#, vA
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:center;" | 720
| 720
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | perfect 5th, minor 6th
| perfect 5th, minor 6th
| style="text-align:center;" | A, Bb
| A, Bb
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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, double-down 8ve
| style="text-align:center;" | vvC#, vvD
| vvC#, vvD
|-
|-
| 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
|-
|-
| 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
|}
|}


== Rank Two Temperaments ==
== Rank-2 temperaments ==
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.  
As 30edo is highly composite, only 7, 11 and 13 steps create [[MOS scale]]s 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.  


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