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**Imported revision 621530565 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
The ''30 equal division'' divides the octave into 30 equal steps of precisely 40 cents each. Its [[Patent_val|patent val]] is a doubled version of the patent val for [[15edo|15edo]] through the 11-limit, so 30 can be viewed as a [[contorted|contorted]] version of 15. In the 13-limit it supplies the optimal patent val for [[Trienstonic_clan#Quindecic|quindecic temperament]].
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-13 05:16:49 UTC</tt>.<br>
: The original revision id was <tt>621530565</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The //30 equal division// divides the octave into 30 equal steps of precisely 40 cents each. 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 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.
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Below is a plot of the Z function around 30:
Below is a plot of the Z function around 30:


[[image:plot30.png]]
[[File:plot30.png|alt=plot30.png|plot30.png]]


==Intervals==


==Intervals==
{| class="wikitable"
||~ **Step** ||~ **Cents** ||||||~ [[xenharmonic/Ups and Downs Notation|ups and downs]][[xenharmonic/Ups and Downs Notation| notation]] ||
|-
||= 0 ||= 0¢ ||= P1 ||= unison, minor 2nd ||= D, Eb ||
! | '''Step'''
||= 1 ||= 40 ||= ^1, ^m2 ||= up unison, upminor 2nd ||= D^, Eb^ ||
! | '''Cents'''
||= 2 ||= 80 ||= ^^1, v~2 ||= double-up unison, downmid 2nd ||= D^^, Eb^^ ||
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs]][[Ups_and_Downs_Notation| notation]]
||= 3 ||= 120 ||= ~2 ||= mid 2nd ||= Ev&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
|-
||= 4 ||= 160 ||= ^~2 ||= upmid 2nd ||= Evv ||
| style="text-align:center;" | 0
||= 5 ||= 200 ||= vM2 ||= downmajor 2nd ||= Ev ||
| style="text-align:center;" | 0¢
||= 6 ||= 240 ||= M2, m3 ||= major 2nd, minor 3rd ||= E, F ||
| style="text-align:center;" | P1
||= 7 ||= 280 ||= ^m3 ||= upminor 3rd ||= F^ ||
| style="text-align:center;" | unison, minor 2nd
||= 8 ||= 320 ||= v~3 ||= downmid 3rd ||= F^^ ||
| style="text-align:center;" | D, Eb
||= 9 ||= 360 ||= ~3 ||= mid 3rd ||= F^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
|-
||= 10 ||= 400 ||= ^~3 ||= upmid 3rd ||= F#vv ||
| style="text-align:center;" | 1
||= 11 ||= 440 ||= vM3, v4 ||= downmajor 3rd, down 4th ||= F#v, Gv ||
| style="text-align:center;" | 40
||= 12 ||= 480 ||= P4 ||= major 3rd, perfect 4th ||= F#, G ||
| style="text-align:center;" | ^1, ^m2
||= 13 ||= 520 ||= ^4, ^d5 ||= up 4th, updim 5th ||= G^, Ab^ ||
| style="text-align:center;" | up unison, upminor 2nd
||= 14 ||= 560 ||= ^^4, ^^d5 ||= double-up 4th, double-up dim 5th ||= G^^, Ab^^ ||
| style="text-align:center;" | D^, Eb^
||= 15 ||= 600 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;4, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;5 ||= triple-up 4th, triple-down 5th ||= G^3, Av3 ||
|-
||= 16 ||= 640 ||= vvA4, vv5 ||= double-down aug 4th, double-down 5th ||= G#vv, Avv ||
| style="text-align:center;" | 2
||= 17 ||= 680 ||= vA4, v5 ||= downaug 4th, down 5th ||= G#v, Av ||
| style="text-align:center;" | 80
||= 18 ||= 720 ||= P5 ||= perfect 5th, minor 6th ||= A, Bb ||
| style="text-align:center;" | ^^1, v~2
||= 19 ||= 760 ||= ^5, ^m6 ||= up 5th, upminor 6th ||= A^, Bb^ ||
| style="text-align:center;" | double-up unison, downmid 2nd
||= 20 ||= 800 ||= v~6 ||= downmid 6th ||= Bb^^ ||
| style="text-align:center;" | D^^, Eb^^
||= 21 ||= 840 ||= ~6 ||= mid 6th ||= Bv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
|-
||= 22 ||= 880 ||= ^~6 ||= upmid 6th ||= Bvv ||
| style="text-align:center;" | 3
||= 23 ||= 920 ||= vM6 ||= downmajor 6th ||= Bv ||
| style="text-align:center;" | 120
||= 24 ||= 960 ||= M6. m7 ||= major 6th, minor 7th ||= B, C ||
| style="text-align:center;" | ~2
||= 25 ||= 1000 ||= ^m7 ||= upminor 7th ||= C^ ||
| style="text-align:center;" | mid 2nd
||= 26 ||= 1040 ||= v~7 ||= downmid 7th ||= C^^ ||
| style="text-align:center;" | Ev<span style="font-size: 90%; vertical-align: super;">3</span>
||= 27 ||= 1080 ||= ~7 ||= mid 7th ||= C^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
|-
||= 28 ||= 1120 ||= ^~7, vv8 ||= upmid 7th, double-down 8ve ||= C#vv, Dvv ||
| style="text-align:center;" | 4
||= 29 ||= 1160 ||= vM7, v8 ||= downmajor 7th, down 8ve ||= C#v, Dv ||
| style="text-align:center;" | 160
||= 30 ||= 1200 ||= P8 ||= major 7th, 8ve ||= C#, D ||
| style="text-align:center;" | ^~2
| style="text-align:center;" | upmid 2nd
| style="text-align:center;" | Evv
|-
| style="text-align:center;" | 5
| style="text-align:center;" | 200
| style="text-align:center;" | vM2
| style="text-align:center;" | downmajor 2nd
| style="text-align:center;" | Ev
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 240
| style="text-align:center;" | M2, m3
| style="text-align:center;" | major 2nd, minor 3rd
| style="text-align:center;" | E, F
|-
| style="text-align:center;" | 7
| style="text-align:center;" | 280
| style="text-align:center;" | ^m3
| style="text-align:center;" | upminor 3rd
| style="text-align:center;" | F^
|-
| style="text-align:center;" | 8
| style="text-align:center;" | 320
| style="text-align:center;" | v~3
| style="text-align:center;" | downmid 3rd
| style="text-align:center;" | F^^
|-
| style="text-align:center;" | 9
| style="text-align:center;" | 360
| style="text-align:center;" | ~3
| style="text-align:center;" | mid 3rd
| style="text-align:center;" | F^<span style="font-size: 90%; vertical-align: super;">3</span>
|-
| style="text-align:center;" | 10
| style="text-align:center;" | 400
| style="text-align:center;" | ^~3
| style="text-align:center;" | upmid 3rd
| style="text-align:center;" | F#vv
|-
| style="text-align:center;" | 11
| style="text-align:center;" | 440
| style="text-align:center;" | vM3, v4
| style="text-align:center;" | downmajor 3rd, down 4th
| style="text-align:center;" | F#v, Gv
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 480
| style="text-align:center;" | P4
| style="text-align:center;" | major 3rd, perfect 4th
| style="text-align:center;" | F#, G
|-
| style="text-align:center;" | 13
| style="text-align:center;" | 520
| style="text-align:center;" | ^4, ^d5
| style="text-align:center;" | up 4th, updim 5th
| style="text-align:center;" | G^, Ab^
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 560
| style="text-align:center;" | ^^4, ^^d5
| style="text-align:center;" | double-up 4th, double-up dim 5th
| style="text-align:center;" | G^^, Ab^^
|-
| style="text-align:center;" | 15
| style="text-align:center;" | 600
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>4, v<span style="font-size: 90%; vertical-align: super;">3</span>5
| style="text-align:center;" | triple-up 4th, triple-down 5th
| style="text-align:center;" | G^3, Av3
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 640
| style="text-align:center;" | vvA4, vv5
| style="text-align:center;" | double-down aug 4th, double-down 5th
| style="text-align:center;" | G#vv, Avv
|-
| style="text-align:center;" | 17
| style="text-align:center;" | 680
| style="text-align:center;" | vA4, v5
| style="text-align:center;" | downaug 4th, down 5th
| style="text-align:center;" | G#v, Av
|-
| style="text-align:center;" | 18
| style="text-align:center;" | 720
| style="text-align:center;" | P5
| style="text-align:center;" | perfect 5th, minor 6th
| style="text-align:center;" | A, Bb
|-
| style="text-align:center;" | 19
| style="text-align:center;" | 760
| style="text-align:center;" | ^5, ^m6
| style="text-align:center;" | up 5th, upminor 6th
| style="text-align:center;" | A^, Bb^
|-
| style="text-align:center;" | 20
| style="text-align:center;" | 800
| style="text-align:center;" | v~6
| style="text-align:center;" | downmid 6th
| style="text-align:center;" | Bb^^
|-
| style="text-align:center;" | 21
| style="text-align:center;" | 840
| style="text-align:center;" | ~6
| style="text-align:center;" | mid 6th
| style="text-align:center;" | Bv<span style="font-size: 90%; vertical-align: super;">3</span>
|-
| style="text-align:center;" | 22
| style="text-align:center;" | 880
| style="text-align:center;" | ^~6
| style="text-align:center;" | upmid 6th
| style="text-align:center;" | Bvv
|-
| style="text-align:center;" | 23
| style="text-align:center;" | 920
| style="text-align:center;" | vM6
| style="text-align:center;" | downmajor 6th
| style="text-align:center;" | Bv
|-
| style="text-align:center;" | 24
| style="text-align:center;" | 960
| style="text-align:center;" | M6. m7
| style="text-align:center;" | major 6th, minor 7th
| style="text-align:center;" | B, C
|-
| style="text-align:center;" | 25
| style="text-align:center;" | 1000
| style="text-align:center;" | ^m7
| style="text-align:center;" | upminor 7th
| style="text-align:center;" | C^
|-
| style="text-align:center;" | 26
| style="text-align:center;" | 1040
| style="text-align:center;" | v~7
| style="text-align:center;" | downmid 7th
| style="text-align:center;" | C^^
|-
| style="text-align:center;" | 27
| style="text-align:center;" | 1080
| style="text-align:center;" | ~7
| style="text-align:center;" | mid 7th
| style="text-align:center;" | C^<span style="font-size: 90%; vertical-align: super;">3</span>
|-
| style="text-align:center;" | 28
| style="text-align:center;" | 1120
| style="text-align:center;" | ^~7, vv8
| style="text-align:center;" | upmid 7th, double-down 8ve
| style="text-align:center;" | C#vv, Dvv
|-
| style="text-align:center;" | 29
| style="text-align:center;" | 1160
| style="text-align:center;" | vM7, v8
| style="text-align:center;" | downmajor 7th, down 8ve
| style="text-align:center;" | C#v, Dv
|-
| style="text-align:center;" | 30
| style="text-align:center;" | 1200
| style="text-align:center;" | P8
| style="text-align:center;" | major 7th, 8ve
| style="text-align:center;" | C#, D
|}


=Commas=  
=Commas=
30 EDO tempers out the following commas. (Note: This assumes the val &lt; 30 48 70 84 104 111 | .)
30 EDO tempers out the following commas. (Note: This assumes the val &lt; 30 48 70 84 104 111 | .)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 ||&lt; | 8 -5 &gt; ||&gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||=  ||
||= 250/243 ||&lt; | 1 -5 3 &gt; ||&gt; 49.17 ||= Maximal Diesis ||= Porcupine Comma ||=  ||
||= 128/125 ||&lt; | 7 0 -3 &gt; ||&gt; 41.06 ||= Diesis ||= Augmented Comma ||=  ||
||= 15625/15552 ||&lt; | -6 -5 6 &gt; ||&gt; 8.11 ||= Kleisma ||= Semicomma Majeur ||=  ||
||= 1029/1000 ||&lt; | -3 1 -3 3 &gt; ||&gt; 49.49 ||= Keega ||=  ||=  ||
||= 49/48 ||&lt; | -4 -1 0 2 &gt; ||&gt; 35.70 ||= Slendro Diesis ||=  ||=  ||
||= 64/63 ||&lt; | 6 -2 0 -1 &gt; ||&gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 64827/64000 ||&lt; | -9 3 -3 4 &gt; ||&gt; 22.23 ||= Squalentine ||=  ||=  ||
||= 875/864 ||&lt; | -5 -3 3 1 &gt; ||&gt; 21.90 ||= Keema ||=  ||=  ||
||= 126/125 ||&lt; | 1 2 -3 1 &gt; ||&gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=  ||
||= 4000/3969 ||&lt; | 5 -4 3 -2 &gt; ||&gt; 13.47 ||= Octagar ||=  ||=  ||
||= 1029/1024 ||&lt; | -10 1 0 3 &gt; ||&gt; 8.43 ||= Gamelisma ||=  ||=  ||
||= 6144/6125 ||&lt; | 11 1 -3 -2 &gt; ||&gt; 5.36 ||= Porwell ||=  ||=  ||
||= 250047/250000 ||&lt; | -4 6 -6 3 &gt; ||&gt; 0.33 ||= Landscape Comma ||=  ||=  ||
||= 100/99 ||&lt; | 2 -2 2 0 -1 &gt; ||&gt; 17.40 ||= Ptolemisma ||=  ||=  ||
||= 121/120 ||&lt; | -3 -1 -1 0 2 &gt; ||&gt; 14.37 ||= Biyatisma ||=  ||=  ||
||= 176/175 ||&lt; | 4 0 -2 -1 1 &gt; ||&gt; 9.86 ||= Valinorsma ||=  ||=  ||
||= 65536/65219 ||&lt; | 16 0 0 -2 -3 &gt; ||&gt; 8.39 ||= Orgonisma ||=  ||=  ||
||= 385/384 ||&lt; | -7 -1 1 1 1 &gt; ||&gt; 4.50 ||= Keenanisma ||=  ||=  ||
||= 441/440 ||&lt; | -3 2 -1 2 -1 &gt; ||&gt; 3.93 ||= Werckisma ||=  ||=  ||
||= 4000/3993 ||&lt; | 5 -1 3 0 -3 &gt; ||&gt; 3.03 ||= Wizardharry ||=  ||=  ||
||= 3025/3024 ||&lt; | -4 -3 2 -1 2 &gt; ||&gt; 0.57 ||= Lehmerisma ||=  ||=  ||


=Music=  
{| class="wikitable"
|-
! | Comma
! | Monzo
! | Value (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| | | 8 -5 &gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| style="text-align:center;" |
|-
| style="text-align:center;" | 250/243
| | | 1 -5 3 &gt;
| style="text-align:right;" | 49.17
| style="text-align:center;" | Maximal Diesis
| style="text-align:center;" | Porcupine Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 128/125
| | | 7 0 -3 &gt;
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 15625/15552
| | | -6 -5 6 &gt;
| style="text-align:right;" | 8.11
| style="text-align:center;" | Kleisma
| style="text-align:center;" | Semicomma Majeur
| style="text-align:center;" |
|-
| style="text-align:center;" | 1029/1000
| | | -3 1 -3 3 &gt;
| style="text-align:right;" | 49.49
| style="text-align:center;" | Keega
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 49/48
| | | -4 -1 0 2 &gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 64827/64000
| | | -9 3 -3 4 &gt;
| style="text-align:right;" | 22.23
| style="text-align:center;" | Squalentine
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 875/864
| | | -5 -3 3 1 &gt;
| style="text-align:right;" | 21.90
| style="text-align:center;" | Keema
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 126/125
| | | 1 2 -3 1 &gt;
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 4000/3969
| | | 5 -4 3 -2 &gt;
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 6144/6125
| | | 11 1 -3 -2 &gt;
| style="text-align:right;" | 5.36
| style="text-align:center;" | Porwell
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 250047/250000
| | | -4 6 -6 3 &gt;
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 100/99
| | | 2 -2 2 0 -1 &gt;
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 121/120
| | | -3 -1 -1 0 2 &gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 176/175
| | | 4 0 -2 -1 1 &gt;
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 65536/65219
| | | 16 0 0 -2 -3 &gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 385/384
| | | -7 -1 1 1 1 &gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 4000/3993
| | | 5 -1 3 0 -3 &gt;
| style="text-align:right;" | 3.03
| style="text-align:center;" | Wizardharry
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 3025/3024
| | | -4 -3 2 -1 2 &gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| style="text-align:center;" |
|}


[[@http://spectropolrecords.bandcamp.com/track/todd-harrop-fifteen-short-pieces|Fifteen Short Pieces]] by &lt;span class="wiki_link_ext"&gt;Todd Harrop&lt;/span&gt;</pre></div>
=Music=
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;30edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;30 equal division&lt;/em&gt; divides the octave into 30 equal steps of precisely 40 cents each. Its &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt; is a doubled version of the patent val for &lt;a class="wiki_link" href="/15edo"&gt;15edo&lt;/a&gt; through the 11-limit, so 30 can be viewed as a &lt;a class="wiki_link" href="/contorted"&gt;contorted&lt;/a&gt; version of 15. In the 13-limit it supplies the optimal patent val for &lt;a class="wiki_link" href="/Trienstonic%20clan#Quindecic"&gt;quindecic temperament&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
Below is a plot of the Z function around 30:&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:712:&amp;lt;img src=&amp;quot;/file/view/plot30.png/220060356/plot30.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/plot30.png/220060356/plot30.png" alt="plot30.png" title="plot30.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:712 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Intervals&lt;/h2&gt;


&lt;table class="wiki_table"&gt;
[http://spectropolrecords.bandcamp.com/track/todd-harrop-fifteen-short-pieces Fifteen Short Pieces] by <span style="">Todd Harrop</span>     [[Category:edo]]
    &lt;tr&gt;
[[Category:inconsistent]]
        &lt;th&gt;&lt;strong&gt;Step&lt;/strong&gt;&lt;br /&gt;
[[Category:pelogic]]
&lt;/th&gt;
[[Category:todo:unify_precision]]
        &lt;th&gt;&lt;strong&gt;Cents&lt;/strong&gt;&lt;br /&gt;
[[Category:z]]
&lt;/th&gt;
        &lt;th colspan="3"&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation"&gt; notation&lt;/a&gt;&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0¢&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;unison, minor 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D, Eb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;40&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^1, ^m2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;up unison, upminor 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D^, Eb^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;80&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^^1, v~2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-up unison, downmid 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D^^, Eb^^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;120&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;~2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;mid 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ev&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;160&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^~2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;upmid 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Evv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;200&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;vM2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;downmajor 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ev&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;240&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;M2, m3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;major 2nd, minor 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;E, F&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;280&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^m3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;upminor 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;320&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;v~3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;downmid 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F^^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;360&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;~3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;mid 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;10&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;400&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^~3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;upmid 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F#vv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;440&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;vM3, v4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;downmajor 3rd, down 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F#v, Gv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;12&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;480&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;major 3rd, perfect 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F#, G&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;520&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^4, ^d5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;up 4th, updim 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G^, Ab^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;14&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;560&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^^4, ^^d5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-up 4th, double-up dim 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G^^, Ab^^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;15&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;600&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;4, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;triple-up 4th, triple-down 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G^3, Av3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;16&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;640&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;vvA4, vv5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;double-down aug 4th, double-down 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G#vv, Avv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;17&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;680&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;vA4, v5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;downaug 4th, down 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G#v, Av&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;18&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;720&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perfect 5th, minor 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A, Bb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;19&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;760&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^5, ^m6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;up 5th, upminor 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A^, Bb^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;20&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;800&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;v~6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;downmid 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Bb^^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;21&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;840&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;~6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;mid 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Bv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;22&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;880&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^~6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;upmid 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Bvv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;23&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;920&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;vM6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;downmajor 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Bv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;24&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;960&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;M6. m7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;major 6th, minor 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;B, C&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;25&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1000&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^m7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;upminor 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;26&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1040&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;v~7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;downmid 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C^^&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;27&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1080&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;~7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;mid 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;28&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1120&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;^~7, vv8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;upmid 7th, double-down 8ve&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#vv, Dvv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;29&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1160&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;vM7, v8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;downmajor 7th, down 8ve&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#v, Dv&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;30&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1200&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;major 7th, 8ve&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#, D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Commas&lt;/h1&gt;
30 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 30 48 70 84 104 111 | .)&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;th&gt;Comma&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Monzo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Value (Cents)&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 1&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 2&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 3&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;256/243&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 8 -5 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;90.22&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Limma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Pythagorean Minor 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;250/243&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 1 -5 3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;49.17&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Maximal Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Porcupine Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;128/125&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 7 0 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;41.06&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Augmented Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;15625/15552&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -6 -5 6 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;8.11&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Kleisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Semicomma Majeur&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;1029/1000&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -3 1 -3 3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;49.49&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Keega&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;49/48&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -4 -1 0 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;35.70&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Slendro Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;64/63&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 6 -2 0 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;27.26&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Septimal Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Archytas' Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Leipziger Komma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;64827/64000&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -9 3 -3 4 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;22.23&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Squalentine&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;875/864&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -5 -3 3 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;21.90&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Keema&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;126/125&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 1 2 -3 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;13.79&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Septimal Semicomma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Starling Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;4000/3969&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 5 -4 3 -2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;13.47&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Octagar&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;1029/1024&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -10 1 0 3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;8.43&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Gamelisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;6144/6125&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 11 1 -3 -2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;5.36&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Porwell&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;250047/250000&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -4 6 -6 3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;0.33&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Landscape Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;100/99&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 2 -2 2 0 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;17.40&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ptolemisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;121/120&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -3 -1 -1 0 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;14.37&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Biyatisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;176/175&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 4 0 -2 -1 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;9.86&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Valinorsma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;65536/65219&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 16 0 0 -2 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;8.39&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Orgonisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;385/384&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -7 -1 1 1 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;4.50&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Keenanisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;441/440&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -3 2 -1 2 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;3.93&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Werckisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;4000/3993&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 5 -1 3 0 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;3.03&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Wizardharry&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;3025/3024&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -4 -3 2 -1 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;0.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Lehmerisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Music&lt;/h1&gt;
&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://spectropolrecords.bandcamp.com/track/todd-harrop-fifteen-short-pieces" rel="nofollow" target="_blank"&gt;Fifteen Short Pieces&lt;/a&gt; by &lt;span class="wiki_link_ext"&gt;Todd Harrop&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

The 30 equal division divides the octave into 30 equal steps of precisely 40 cents each. 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.

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.

Below is a plot of the Z function around 30:

plot30.png

Intervals

Step Cents ups and downs notation
0 0¢ P1 unison, minor 2nd D, Eb
1 40 ^1, ^m2 up unison, upminor 2nd D^, Eb^
2 80 ^^1, v~2 double-up unison, downmid 2nd D^^, Eb^^
3 120 ~2 mid 2nd Ev3
4 160 ^~2 upmid 2nd Evv
5 200 vM2 downmajor 2nd Ev
6 240 M2, m3 major 2nd, minor 3rd E, F
7 280 ^m3 upminor 3rd F^
8 320 v~3 downmid 3rd F^^
9 360 ~3 mid 3rd F^3
10 400 ^~3 upmid 3rd F#vv
11 440 vM3, v4 downmajor 3rd, down 4th F#v, Gv
12 480 P4 major 3rd, perfect 4th F#, G
13 520 ^4, ^d5 up 4th, updim 5th G^, Ab^
14 560 ^^4, ^^d5 double-up 4th, double-up dim 5th G^^, Ab^^
15 600 ^34, v35 triple-up 4th, triple-down 5th G^3, Av3
16 640 vvA4, vv5 double-down aug 4th, double-down 5th G#vv, Avv
17 680 vA4, v5 downaug 4th, down 5th G#v, Av
18 720 P5 perfect 5th, minor 6th A, Bb
19 760 ^5, ^m6 up 5th, upminor 6th A^, Bb^
20 800 v~6 downmid 6th Bb^^
21 840 ~6 mid 6th Bv3
22 880 ^~6 upmid 6th Bvv
23 920 vM6 downmajor 6th Bv
24 960 M6. m7 major 6th, minor 7th B, C
25 1000 ^m7 upminor 7th C^
26 1040 v~7 downmid 7th C^^
27 1080 ~7 mid 7th C^3
28 1120 ^~7, vv8 upmid 7th, double-down 8ve C#vv, Dvv
29 1160 vM7, v8 downmajor 7th, down 8ve C#v, Dv
30 1200 P8 major 7th, 8ve C#, D

Commas

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

Comma Monzo Value (Cents) Name 1 Name 2 Name 3
256/243 | 8 -5 > 90.22 Limma Pythagorean Minor 2nd
250/243 | 1 -5 3 > 49.17 Maximal Diesis Porcupine Comma
128/125 | 7 0 -3 > 41.06 Diesis Augmented Comma
15625/15552 | -6 -5 6 > 8.11 Kleisma Semicomma Majeur
1029/1000 | -3 1 -3 3 > 49.49 Keega
49/48 | -4 -1 0 2 > 35.70 Slendro Diesis
64/63 | 6 -2 0 -1 > 27.26 Septimal Comma Archytas' Comma Leipziger Komma
64827/64000 | -9 3 -3 4 > 22.23 Squalentine
875/864 | -5 -3 3 1 > 21.90 Keema
126/125 | 1 2 -3 1 > 13.79 Septimal Semicomma Starling Comma
4000/3969 | 5 -4 3 -2 > 13.47 Octagar
1029/1024 | -10 1 0 3 > 8.43 Gamelisma
6144/6125 | 11 1 -3 -2 > 5.36 Porwell
250047/250000 | -4 6 -6 3 > 0.33 Landscape Comma
100/99 | 2 -2 2 0 -1 > 17.40 Ptolemisma
121/120 | -3 -1 -1 0 2 > 14.37 Biyatisma
176/175 | 4 0 -2 -1 1 > 9.86 Valinorsma
65536/65219 | 16 0 0 -2 -3 > 8.39 Orgonisma
385/384 | -7 -1 1 1 1 > 4.50 Keenanisma
441/440 | -3 2 -1 2 -1 > 3.93 Werckisma
4000/3993 | 5 -1 3 0 -3 > 3.03 Wizardharry
3025/3024 | -4 -3 2 -1 2 > 0.57 Lehmerisma

Music

Fifteen Short Pieces by Todd Harrop

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