30edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 29edo30edo31edo →
Prime factorization 2 × 3 × 5
Step size 40¢ 
Fifth 18\30 (720¢) (→3\5)
Semitones (A1:m2) 6:0 (240¢ : 0¢)
Dual sharp fifth 18\30 (720¢) (→3\5)
Dual flat fifth 17\30 (680¢)
Dual major 2nd 5\30 (200¢) (→1\6)
Consistency limit 5
Distinct consistency limit 5

30 equal divisions of the octave (abbreviated 30edo or 30ed2), also called 30-tone equal temperament (30tet) or 30 equal temperament (30et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 30 equal parts of exactly 40 ¢ each. Each step represents a frequency ratio of 21/30, or the 30th root of 2.

Theory

Approximation of odd harmonics in 30edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +18.0 +13.7 -8.8 -3.9 +8.7 -0.5 -8.3 +15.0 -17.5 +9.2 +11.7
Relative (%) +45.1 +34.2 -22.1 -9.8 +21.7 -1.3 -20.7 +37.6 -43.8 +23.0 +29.3
Steps
(reduced)
48
(18)
70
(10)
84
(24)
95
(5)
104
(14)
111
(21)
117
(27)
123
(3)
127
(7)
132
(12)
136
(16)

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.

plot30.png
A plot of the Z function around 30.

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.

Subsets and supersets

30edo has subset edos 1, 2, 3, 5, 6, 10, 15 and it is a largely composite edo.

30edo is the 3rd 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

Step Cents Ups and downs notation Armodue Notation
0 0 P1 unison, minor 2nd D, Eb 1
1 40 ^1, ^m2 up unison, upminor 2nd ^D, ^Eb 2b
2 80 ^^1, v~2 dup unison, downmid 2nd ^^D, ^^Eb 9#
3 120 ~2 mid 2nd v3E 1#
4 160 ^~2 upmid 2nd vvE 2
5 200 vM2 downmajor 2nd vE 3b
6 240 M2, m3 major 2nd, minor 3rd E, F 1x, 4bb
7 280 ^m3 upminor 3rd ^F 2#
8 320 v~3 downmid 3rd ^^F 3
9 360 ~3 mid 3rd ^3F, v3F# 4b
10 400 ^~3 upmid 3rd vvF# 5b
11 440 vM3, v4 downmajor 3rd, down 4th vF#, vG 3#
12 480 M3, P4 major 3rd, perfect 4th F#, G 4
13 520 ^4 up 4th ^G 5
14 560 v~4, v~d5 downmid 4th, downmid 5th ^^G, ^^Ab 6b
15 600 ~4, ~5 mid 4th, mid 5th ^3G, v3A 4#
16 640 ^~4, ^~5 upmid 4th, upmid 5th vvG#, vvA 5#
17 680 v5 down 5th vA 6
18 720 P5, m6 perfect 5th, minor 6th A, Bb 7b
19 760 ^5, ^m6 up 5th, upminor 6th ^A, ^Bb 5x, 8bb
20 800 v~6 downmid 6th ^^Bb 6#
21 840 ~6 mid 6th v3B 7
22 880 ^~6 upmid 6th vvB 8b
23 920 vM6 downmajor 6th vB 6x, 9bb
24 960 M6. m7 major 6th, minor 7th B, C 7#
25 1000 ^m7 upminor 7th ^C 8
26 1040 v~7 downmid 7th ^^C 9b
27 1080 ~7 mid 7th ^3C 1b
28 1120 ^~7, vv8 upmid 7th, dud 8ve vvC#, vvD 8#
29 1160 vM7, v8 downmajor 7th, down 8ve vC#, vD 9
30 1200 P8 major 7th, 8ve C#, D 1

Commas

30 EDO tempers out the following commas. (Note: This assumes the val 30 48 70 84 104 111].)

Prime
Limit
Ratio[1] Monzo Cents Color name Name(s)
3 256/243 [8 -5 90.22 Sawa Limma, Pythagorean minor sedond
5 250/243 [1 -5 3 49.17 Triyo Maximal diesis, Porcupine comma
5 128/125 [7 0 -3 41.06 Trigu Diesis, augmented comma
5 15625/15552 [-6 -5 6 8.11 Tribiyo Kleisma, semicomma majeur
7 1029/1000 [-3 1 -3 3 49.49 Trizogu Keega
7 49/48 [-4 -1 0 2 35.70 Zozo Slendro diesis
7 64/63 [6 -2 0 -1 27.26 Ru Septimal comma, Archytas' comma, Leipziger Komma
7 64827/64000 [-9 3 -3 4 22.23 Laquadzo-atrigu Squalentine
7 875/864 [-5 -3 3 1 21.90 Zotriyo Keema
7 126/125 [1 2 -3 1 13.79 Zotrigu Septimal semicomma, Starling comma
7 4000/3969 [5 -4 3 -2 13.47 Rurutriyo Octagar
7 1029/1024 [-10 1 0 3 8.43 Latrizo Gamelisma
7 6144/6125 [11 1 -3 -2 5.36 Saruru-atrigu Porwell
7 (12 digits) [-4 6 -6 3 0.33 Trizogugu Landscape comma
11 100/99 [2 -2 2 0 -1 17.40 Luyoyo Ptolemisma
11 121/120 [-3 -1 -1 0 2 14.37 Lologu Biyatisma
11 176/175 [4 0 -2 -1 1 9.86 Lorugugu Valinorsma
11 65536/65219 [16 0 0 -2 -3 8.39 Satrilu-aruru Orgonisma
11 385/384 [-7 -1 1 1 1 4.50 Lozoyo Keenanisma
11 441/440 [-3 2 -1 2 -1 3.93 Luzozogu Werckisma
11 4000/3993 [5 -1 3 0 -3 3.03 Triluyo Wizardharry
11 3025/3024 [-4 -3 2 -1 2 0.57 Loloruyoyo Lehmerisma
  1. Ratios longer than 10 digits are presented by placeholders with informative hints

Rank-2 temperaments

As 30edo is largely composite, only 7, 11 and 13 steps create MOS scales that cover every interval using one period per octave.

7/30 produces 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 Mavila tuning that functions the closest to the familiar diatonic scale you can get in this edo.

MOS scales

Subsets of Mavila[16]

  • Arcade (approximated from 32afdo): 9 3 5 8 5
  • 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
  • Dewdrops (this is the 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 (this is the 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 (this is the original/default tuning): 8 5 4 9 4
  • Pelagic (this is the original/default tuning): 8 4 2 4 7 5
  • Bathypelagic (this is the 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

Subsets of 15edo

  • Augmented[6] MOS: 8 2 8 2 8 2
  • Equipentatonic (exact from 5edo): 6 6 6 6 6
  • Rockpool (approximated from 47zpi): 2 8 2 6 6 6

Other notable scales

  • Approximation of Pelog lima: 3 4 10 3 10

Music

Bryan Deister
Todd Harrop
Micronaive
NullPointerException Music

Related pages