72edo: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>~~2016~~-12-~~28 05~~:09:07 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-02-24 17:48:44 UTC</tt>.

: The original revision id was <tt>~~602866264~~</tt>.

: The original revision id was <tt>607132219</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Line 18:

=Commas and Temperaments=

72et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and

the kleisma, 15625/15552,

the ampersand, 34171875/33554432 = |-25 7 6>,

the graviton, 129140163/128000000 = |-13 17 -6>, and

the ennealimma, 7629394531250/7625597484987 = |1 -27 18> in the 5-limit.

The 7-limit commas include 225/224, 1029/1024, 16875/16807, 19683/19600, 420175/419904, 2401/2400, 4375/4374 and 250047/250000.

72et shines in the 11-limit, with commas 243/242, 385/384, 441/440, 540/539, 1375/1372, 6250/6237, 4000/3993, 3025/3024 and 9801/9800.

For the 13-limit, it tempers out 169/168, 325/324, 351/350, 364/363, 625/624, 676/675, 729/728, 1001/1000, 1575/1573, 1716/1715, 2080/2079 and 6656/6655.

Line 47:

=Intervals=

|| degrees || cents value || approximate ratios (11-limit) ||||= [[xenharmonic/Ups and Downs Notation|ups and downs ]][[xenharmonic/Ups and Downs Notation|notation]] ||

|| degrees || cents value || approximate ratios (11-limit) ||||||= [[xenharmonic/Ups and Downs Notation|ups and downs ]][[xenharmonic/Ups and Downs Notation|notation]] ||

|| 0 || 0 || 1/1 ||= P1 ||= D ||

|| 0 || 0 || 1/1 ||= P1 ||= perfect unison ||= D ||

|| 1 || 16.667 || 81/80 ||= ^1 (up unison) ||= D^ ||

|| 1 || 16.667 || 81/80 ||= ^1 ||= up unison ||= D^ ||

|| 2 || 33.333 || 45/44 ||= ^^1(double-up unison) ||= D^^ ||

|| 2 || 33.333 || 45/44 ||= ^^ ||= double-up unison ||= D^^ ||

|| 3 || 50 || 33/32 ||= ^31, v3m2 || D^3, Ebv3 ||

|| 3 || 50 || 33/32 ||= ^31, v3m2 ||= triple-up unison,

|| 4 || 66.667 || 25/24 ||= vvm2 ||= Ebvv ||

triple-down minor 2nd ||= D^3, Ebv3 ||

|| 5 || 83.333 || 21/20 ||= vm2 (downminor 2nd) ||= Ebv ||

|| 4 || 66.667 || 25/24 ||= vvm2 ||= double-downminor 2nd ||= Ebvv ||

|| 6 || 100 || 35/33 ||= m2 ||= Eb ||

|| 5 || 83.333 || 21/20 ||= vm2 ||= downminor 2nd ||= Ebv ||

|| 7 || 116.667 || 15/14 ||= ^m2 (upminor 2nd) ||= Eb^ ||

|| 6 || 100 || 35/33 ||= m2 ||= minor 2nd ||= Eb ||

|| 8 || 133.333 || 27/25 ||= v~2 ||= Eb^^ ||

|| 7 || 116.667 || 15/14 ||= ^m2 ||= upminor 2nd ||= Eb^ ||

|| 9 || 150 || 12/11 ||= ~2 (mid 2nd) ||= Ev3 ||

|| 8 || 133.333 || 27/25 ||= v~2 ||= downmid 2nd ||= Eb^^ ||

|| 10 || 166.667 || 11/10 ||= ^~2 (upmid 2nd) ||= Evv ||

|| 9 || 150 || 12/11 ||= ~2 ||= mid 2nd ||= Ev3 ||

|| 11 || 183.333 || 10/9 ||= vM2 ||= Ev ||

|| 10 || 166.667 || 11/10 ||= ^~2 ||= upmid 2nd ||= Evv ||

|| 12 || 200 || 9/8 ||= M2 ||= E ||

|| 11 || 183.333 || 10/9 ||= vM2 ||= downmajor 2nd ||= Ev ||

|| 13 || 216.667 || 25/22 ||= ^M2 ||= E^ ||

|| 12 || 200 || 9/8 ||= M2 ||= major 2nd ||= E ||

|| 14 || 233.333 || 8/7 ||= ^^M2 ||= E^^ ||

|| 13 || 216.667 || 25/22 ||= ^M2 ||= upmajor 2nd ||= E^ ||

|| 15 || 250 || 81/70 ||= ^3M2, v3m3 ||= E^3, Fv3 ||

|| 14 || 233.333 || 8/7 ||= ^^M2 ||= double-upmajor 2nd ||= E^^ ||

|| 16 || 266.667 || 7/6 ||= vvm3 ||= Fvv ||

|| 15 || 250 || 81/70 ||= ^3M2, v3m3 ||= triple-up major 2nd,

|| 17 || 283.333 || 33/28 ||= vm3 ||= Fv ||

triple-down minor 3rd ||= E^3, Fv3 ||

|| 18 || 300 || 25/21 ||= m3 ||= F ||

|| 16 || 266.667 || 7/6 ||= vvm3 ||= double-downminor 3rd ||= Fvv ||

|| 19 || 316.667 || 6/5 ||= ^m3 ||= F^ ||

|| 17 || 283.333 || 33/28 ||= vm3 ||= downminor 3rd ||= Fv ||

|| 20 || 333.333 || 40/33 ||= v~3 ||= F^^ ||

|| 18 || 300 || 25/21 ||= m3 ||= minor 3rd ||= F ||

|| 21 || 350 || 11/9 ||= ~3 ||= F^3 ||

|| 19 || 316.667 || 6/5 ||= ^m3 ||= upminor 3rd ||= F^ ||

|| 22 || 366.667 || 99/80 ||= ^~3 ||= F#vv ||

|| 20 || 333.333 || 40/33 ||= v~3 ||= downmid 3rd ||= F^^ ||

|| 23 || 383.333 || 5/4 ||= vM3 ||= F#v ||

|| 21 || 350 || 11/9 ||= ~3 ||= mid 3rd ||= F^3 ||

|| 24 || 400 || 44/35 ||= M3 ||= F# ||

|| 22 || 366.667 || 99/80 ||= ^~3 ||= upmid 3rd ||= F#vv ||

|| 25 || 416.667 || 14/11 ||= ^M3 ||= F#^ ||

|| 23 || 383.333 || 5/4 ||= vM3 ||= downmajor 3rd ||= F#v ||

|| 26 || 433.333 || 9/7 ||= ^^M3 ||= F#^^ ||

|| 24 || 400 || 44/35 ||= M3 ||= major 3rd ||= F# ||

|| 27 || 450 || 35/27 ||= ^3M3, v34 ||= F#^3, Gv3 ||

|| 25 || 416.667 || 14/11 ||= ^M3 ||= upmajor 3rd ||= F#^ ||

|| 28 || 466.667 || 21/16 ||= vv4 ||= Gvv ||

|| 26 || 433.333 || 9/7 ||= ^^M3 ||= double-upmajor 3rd ||= F#^^ ||

|| 29 || 483.333 || 33/25 ||= v4 ||= Gv ||

|| 27 || 450 || 35/27 ||= ^3M3, v34 ||= triple-up major 3rd,

|| 30 || 500 || 4/3 ||= P4 ||= G ||

triple-down 4th ||= F#^3, Gv3 ||

|| 31 || 516.667 || 27/20 ||= ^4 ||= G^ ||

|| 28 || 466.667 || 21/16 ||= vv4 ||= double-down 4th ||= Gvv ||

|| 32 || 533.333 || 15/11 ||= ^^4 ||= G^^ ||

|| 29 || 483.333 || 33/25 ||= v4 ||= down 4th ||= Gv ||

|| 33 || 550 || 11/8 ||= ^34 ||= G^3 ||

|| 30 || 500 || 4/3 ||= P4 ||= perfect 4th ||= G ||

|| 34 || 566.667 || 25/18 ||= vvA4 ||= G#vv ||

|| 31 || 516.667 || 27/20 ||= ^4 ||= up 4th ||= G^ ||

|| 35 || 583.333 || 7/5 ||= vA4, vd5 ||= G#v, Abv ||

|| 32 || 533.333 || 15/11 ||= ^^4 ||= double-up 4th ||= G^^ ||

|| 36 || 600 || 99/70 ||= A4, d5 ||= G#, Ab ||

|| 33 || 550 || 11/8 ||= ^34 ||= triple-up 4th ||= G^3 ||

|| 37 || 616.667 || 10/7 ||= ^A4, ^d5 ||= G#^, Ab^ ||

|| 34 || 566.667 || 25/18 ||= vvA4 ||= double-down aug 4th ||= G#vv ||

|| 38 || 633.333 || 36/25 ||= ^^d5 ||= Ab^^ ||

|| 35 || 583.333 || 7/5 ||= vA4, vd5 ||= downaug 4th, updim 5th ||= G#v, Abv ||

|| 39 || 650 || 16/11 ||= v35 ||= Av3 ||

|| 36 || 600 || 99/70 ||= A4, d5 ||= aug 4th, dim 5th ||= G#, Ab ||

|| 40 || 666.667 || 22/15 ||= vv5 ||= Avv ||

|| 37 || 616.667 || 10/7 ||= ^A4, ^d5 ||= upaug 4th, downdim 5th ||= G#^, Ab^ ||

|| 41 || 683.333 || 40/27 ||= v5 ||= Av ||

|| 38 || 633.333 || 36/25 ||= ^^d5 ||= double-updim 5th ||= Ab^^ ||

|| 42 || 700 || 3/2 ||= P5 ||= A ||

|| 39 || 650 || 16/11 ||= v35 ||= triple-down 5th ||= Av3 ||

|| 43 || 716.667 || 50/33 ||= ^5 ||= A^ ||

|| 40 || 666.667 || 22/15 ||= vv5 ||= double-down 5th ||= Avv ||

|| 44 || 733.333 || 32/21 ||= ^^5 ||= A^^ ||

|| 41 || 683.333 || 40/27 ||= v5 ||= down 5th ||= Av ||

|| 45 || 750 || 54/35 ||= ^35, v3m6 ||= A^3, Bbv3 ||

|| 42 || 700 || 3/2 ||= P5 ||= perfect 5th ||= A ||

|| 46 || 766.667 || 14/9 ||= vvm6 ||= Bbvv ||

|| 43 || 716.667 || 50/33 ||= ^5 ||= up 5th ||= A^ ||

|| 47 || 783.333 || 11/7 ||= vm6 ||= ~~Bbv~~ ||

|| 44 || 733.333 || 32/21 ||= ^^5 ||= double-up 5th ||= A^^ ||

|| 48 || 800 || 35/22 ||= m6 ||= Bb ||

|| 45 || 750 || 54/35 ||= ^35, v3m6 ||= triple-up 5th,

|| 49 || 816.667 || 8/5 ||= ^m6 ||= Bb^ ||

triple-down minor 6th ||= A^3, Bbv3 ||

|| 50 || 833.333 || 81/50 ||= v~6 ||= Bb^^ ||

|| 46 || 766.667 || 14/9 ||= vvm6 ||= double-downminor 6th ||= Bbvv ||

|| 51 || 850 || 18/11 ||= ~6 ||= Bv3 ||

|| 47 || 783.333 || 11/7 ||= vm6 ||= downminor 6th ||= Bbv ||

|| 52 || 866.667 || 33/20 ||= ^~6 ||= Bvv ||

|| 48 || 800 || 35/22 ||= m6 ||= minor 6th ||= Bb ||

|| 53 || 883.333 || 5/3 ||= vM6 ||= Bv ||

|| 49 || 816.667 || 8/5 ||= ^m6 ||= upminor 6th ||= Bb^ ||

|| 54 || 900 || 27/16 ||= M6 ||= B ||

|| 50 || 833.333 || 81/50 ||= v~6 ||= downmid 6th ||= Bb^^ ||

|| 55 || 916.667 || 56/33 ||= ^M6 ||= B^ ||

|| 51 || 850 || 18/11 ||= ~6 ||= mid 6th ||= Bv3 ||

|| 56 || 933.333 || 12/7 ||= ^^M6 ||= B^^ ||

|| 52 || 866.667 || 33/20 ||= ^~6 ||= upmid 6th ||= Bvv ||

|| 57 || 950 || 121/70 ||= ^3M6, v3m7 ||= B^3, Cv3 ||

|| 53 || 883.333 || 5/3 ||= vM6 ||= downmajor 6th ||= Bv ||

|| 58 || 966.667 || 7/4 ||= vvm7 ||= Cvv ||

|| 54 || 900 || 27/16 ||= M6 ||= major 6th ||= B ||

|| 59 || 983.333 || 44/25 ||= vm7 ||= Cv ||

|| 55 || 916.667 || 56/33 ||= ^M6 ||= upmajor 6th ||= B^ ||

|| 60 || 1000 || 16/9 ||= m7 ||= C ||

|| 56 || 933.333 || 12/7 ||= ^^M6 ||= double-upmajor 6th ||= B^^ ||

|| 61 || 1016.667 || 9/5 ||= ^m7 ||= C^ ||

|| 57 || 950 || 121/70 ||= ^3M6, v3m7 ||= triple-up major 6th,

|| 62 || 1033.333 || 20/11 ||= v~7 ||= C^^ ||

triple-down minor 7th ||= B^3, Cv3 ||

|| 63 || 1050 || 11/6 ||= ~7 ||= C^3 ||

|| 58 || 966.667 || 7/4 ||= vvm7 ||= double-downminor 7th ||= Cvv ||

|| 64 || 1066.667 || 50/27 ||= ^~7 ||= C#vv ||

|| 59 || 983.333 || 44/25 ||= vm7 ||= downminor 7th ||= Cv ||

|| 65 || 1083.333 || 15/8 ||= vM7 ||= C#v ||

|| 60 || 1000 || 16/9 ||= m7 ||= minor 7th ||= C ||

|| 66 || 1100 || 66/35 ||= M7 ||= C# ||

|| 61 || 1016.667 || 9/5 ||= ^m7 ||= upminor 7th ||= C^ ||

|| 67 || 1116.667 || 21/11 ||= ^M7 ||= C#^ ||

|| 62 || 1033.333 || 20/11 ||= v~7 ||= downmid 7th ||= C^^ ||

|| 68 || 1133.333 || 27/14 ||= ^^M7 ||= C#^^ ||

|| 63 || 1050 || 11/6 ||= ~7 ||= mid 7th ||= C^3 ||

|| 69 || 1150 || 35/18 ||= ^3M7, v38 ||= C#^3, Dv3 ||

|| 64 || 1066.667 || 50/27 ||= ^~7 ||= upmin 7th ||= C#vv ||

|| 70 || 1166.667 || 49/25 ||= vv8 ||= Dvv ||

|| 65 || 1083.333 || 15/8 ||= vM7 ||= downmajor 7th ||= C#v ||

|| 71 || 1183.333 || 99/50 ||= v8 ||= Dv ||

|| 66 || 1100 || 66/35 ||= M7 ||= major 7th ||= C# ||

|| 72 || 1200 || 2/1 ||= P8 ||= D ||

|| 67 || 1116.667 || 21/11 ||= ^M7 ||= upmajor 7th ||= C#^ ||

|| 68 || 1133.333 || 27/14 ||= ^^M7 ||= double-upmajor 7th ||= C#^^ ||

|| 69 || 1150 || 35/18 ||= ^3M7, v38 ||= triple-up major 7th,

triple-down octave ||= C#^3, Dv3 ||

|| 70 || 1166.667 || 49/25 ||= vv8 ||= double-down octave ||= Dvv ||

|| 71 || 1183.333 || 99/50 ||= v8 ||= down octave ||= Dv ||

|| 72 || 1200 || 2/1 ||= P8 ||= perfect octave ||= D ||

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[xenharmonic/Ups and Downs Notation#Chord%20names%20in%20other%20EDOs|Ups and Downs Notation - Chord names in other EDOs]].

Line 200:

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<h1 id="toc0"><a name="Commas and Temperaments"></a>Commas and Temperaments</h1>

72et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and

72et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and

the kleisma, 15625/15552,

the kleisma, 15625/15552,

the ampersand, 34171875/33554432 = |-25 7 6&gt;,

the ampersand, 34171875/33554432 = |-25 7 6&gt;,

the graviton, 129140163/128000000 = |-13 17 -6&gt;, and

the graviton, 129140163/128000000 = |-13 17 -6&gt;, and

the ennealimma, 7629394531250/7625597484987 = |1 -27 18&gt; in the 5-limit.

the ennealimma, 7629394531250/7625597484987 = |1 -27 18&gt; in the 5-limit.

The 7-limit commas include 225/224, 1029/1024, 16875/16807, 19683/19600, 420175/419904, 2401/2400, 4375/4374 and 250047/250000.

72et shines in the 11-limit, with commas 243/242, 385/384, 441/440, 540/539, 1375/1372, 6250/6237, 4000/3993, 3025/3024 and 9801/9800.

72et shines in the 11-limit, with commas 243/242, 385/384, 441/440, 540/539, 1375/1372, 6250/6237, 4000/3993, 3025/3024 and 9801/9800.

For the 13-limit, it tempers out 169/168, 325/324, 351/350, 364/363, 625/624, 676/675, 729/728, 1001/1000, 1575/1573, 1716/1715, 2080/2079 and 6656/6655.

Line 577:

Line 583:

<td>approximate ratios (11-limit)

</td>

<td colspan="2" style="text-align: center;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">ups and downs </a><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">notation</a>

<td colspan="3" style="text-align: center;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">ups and downs </a><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">notation</a>

</td>

</tr>

Line 588:

Line 594:

</td>

<td style="text-align: center;">P1

</td>

<td style="text-align: center;">perfect unison

</td>

<td style="text-align: center;">D

Line 599:

Line 607:

<td>81/80

</td>

<td style="text-align: center;">^1 (up unison)

<td style="text-align: center;">^1

</td>

<td style="text-align: center;">up unison

</td>

<td style="text-align: center;">D^

Line 611:

Line 621:

<td>45/44

</td>

<td style="text-align: center;">^^1(double-up unison)

<td style="text-align: center;">^^

</td>

<td style="text-align: center;">double-up unison

</td>

<td style="text-align: center;">D^^

Line 625:

Line 637:

<td style="text-align: center;">^31, v3m2

</td>

<td>D^3, Ebv3

<td style="text-align: center;">triple-up unison,

triple-down minor 2nd

</td>

<td style="text-align: center;">D^3, Ebv3

</td>

</tr>

Line 636:

Line 651:

</td>

<td style="text-align: center;">vvm2

</td>

<td style="text-align: center;">double-downminor 2nd

</td>

<td style="text-align: center;">Ebvv

Line 647:

Line 664:

<td>21/20

</td>

<td style="text-align: center;">vm2 (downminor 2nd)

<td style="text-align: center;">vm2

</td>

<td style="text-align: center;">downminor 2nd

</td>

<td style="text-align: center;">Ebv

Line 660:

Line 679:

</td>

<td style="text-align: center;">m2

</td>

<td style="text-align: center;">minor 2nd

</td>

<td style="text-align: center;">Eb

Line 671:

Line 692:

<td>15/14

</td>

<td style="text-align: center;">^m2 (upminor 2nd)

<td style="text-align: center;">^m2

</td>

<td style="text-align: center;">upminor 2nd

</td>

<td style="text-align: center;">Eb^

Line 684:

Line 707:

</td>

<td style="text-align: center;">v~2

</td>

<td style="text-align: center;">downmid 2nd

</td>

<td style="text-align: center;">Eb^^

Line 695:

Line 720:

<td>12/11

</td>

<td style="text-align: center;">~2 (mid 2nd)

<td style="text-align: center;">~2

</td>

<td style="text-align: center;">mid 2nd

</td>

<td style="text-align: center;">Ev3

Line 707:

Line 734:

<td>11/10

</td>

<td style="text-align: center;">^~2 (upmid 2nd)

<td style="text-align: center;">^~2

</td>

<td style="text-align: center;">upmid 2nd

</td>

<td style="text-align: center;">Evv

Line 720:

Line 749:

</td>

<td style="text-align: center;">vM2

</td>

<td style="text-align: center;">downmajor 2nd

</td>

<td style="text-align: center;">Ev

Line 732:

Line 763:

</td>

<td style="text-align: center;">M2

</td>

<td style="text-align: center;">major 2nd

</td>

<td style="text-align: center;">E

Line 744:

Line 777:

</td>

<td style="text-align: center;">^M2

</td>

<td style="text-align: center;">upmajor 2nd

</td>

<td style="text-align: center;">E^

Line 756:

Line 791:

</td>

<td style="text-align: center;">^^M2

</td>

<td style="text-align: center;">double-upmajor 2nd

</td>

<td style="text-align: center;">E^^

Line 768:

Line 805:

</td>

<td style="text-align: center;">^3M2, v3m3

</td>

<td style="text-align: center;">triple-up major 2nd,

triple-down minor 3rd

</td>

<td style="text-align: center;">E^3, Fv3

Line 780:

Line 820:

</td>

<td style="text-align: center;">vvm3

</td>

<td style="text-align: center;">double-downminor 3rd

</td>

<td style="text-align: center;">Fvv

Line 792:

Line 834:

</td>

<td style="text-align: center;">vm3

</td>

<td style="text-align: center;">downminor 3rd

</td>

<td style="text-align: center;">Fv

Line 804:

Line 848:

</td>

<td style="text-align: center;">m3

</td>

<td style="text-align: center;">minor 3rd

</td>

<td style="text-align: center;">F

Line 816:

Line 862:

</td>

<td style="text-align: center;">^m3

</td>

<td style="text-align: center;">upminor 3rd

</td>

<td style="text-align: center;">F^

Line 828:

Line 876:

</td>

<td style="text-align: center;">v~3

</td>

<td style="text-align: center;">downmid 3rd

</td>

<td style="text-align: center;">F^^

Line 840:

Line 890:

</td>

<td style="text-align: center;">~3

</td>

<td style="text-align: center;">mid 3rd

</td>

<td style="text-align: center;">F^3

Line 852:

Line 904:

</td>

<td style="text-align: center;">^~3

</td>

<td style="text-align: center;">upmid 3rd

</td>

<td style="text-align: center;">F#vv

Line 864:

Line 918:

</td>

<td style="text-align: center;">vM3

</td>

<td style="text-align: center;">downmajor 3rd

</td>

<td style="text-align: center;">F#v

Line 876:

Line 932:

</td>

<td style="text-align: center;">M3

</td>

<td style="text-align: center;">major 3rd

</td>

<td style="text-align: center;">F#

Line 888:

Line 946:

</td>

<td style="text-align: center;">^M3

</td>

<td style="text-align: center;">upmajor 3rd

</td>

<td style="text-align: center;">F#^

Line 900:

Line 960:

</td>

<td style="text-align: center;">^^M3

</td>

<td style="text-align: center;">double-upmajor 3rd

</td>

<td style="text-align: center;">F#^^

Line 912:

Line 974:

</td>

<td style="text-align: center;">^3M3, v34

</td>

<td style="text-align: center;">triple-up major 3rd,

triple-down 4th

</td>

<td style="text-align: center;">F#^3, Gv3

Line 924:

Line 989:

</td>

<td style="text-align: center;">vv4

</td>

<td style="text-align: center;">double-down 4th

</td>

<td style="text-align: center;">Gvv

Line 936:

Line 1,003:

</td>

<td style="text-align: center;">v4

</td>

<td style="text-align: center;">down 4th

</td>

<td style="text-align: center;">Gv

Line 948:

Line 1,017:

</td>

<td style="text-align: center;">P4

</td>

<td style="text-align: center;">perfect 4th

</td>

<td style="text-align: center;">G

Line 960:

Line 1,031:

</td>

<td style="text-align: center;">^4

</td>

<td style="text-align: center;">up 4th

</td>

<td style="text-align: center;">G^

Line 972:

Line 1,045:

</td>

<td style="text-align: center;">^^4

</td>

<td style="text-align: center;">double-up 4th

</td>

<td style="text-align: center;">G^^

Line 984:

Line 1,059:

</td>

<td style="text-align: center;">^34

</td>

<td style="text-align: center;">triple-up 4th

</td>

<td style="text-align: center;">G^3

Line 996:

Line 1,073:

</td>

<td style="text-align: center;">vvA4

</td>

<td style="text-align: center;">double-down aug 4th

</td>

<td style="text-align: center;">G#vv

Line 1,008:

Line 1,087:

</td>

<td style="text-align: center;">vA4, vd5

</td>

<td style="text-align: center;">downaug 4th, updim 5th

</td>

<td style="text-align: center;">G#v, Abv

Line 1,020:

Line 1,101:

</td>

<td style="text-align: center;">A4, d5

</td>

<td style="text-align: center;">aug 4th, dim 5th

</td>

<td style="text-align: center;">G#, Ab

Line 1,032:

Line 1,115:

</td>

<td style="text-align: center;">^A4, ^d5

</td>

<td style="text-align: center;">upaug 4th, downdim 5th

</td>

<td style="text-align: center;">G#^, Ab^

Line 1,044:

Line 1,129:

</td>

<td style="text-align: center;">^^d5

</td>

<td style="text-align: center;">double-updim 5th

</td>

<td style="text-align: center;">Ab^^

Line 1,056:

Line 1,143:

</td>

<td style="text-align: center;">v35

</td>

<td style="text-align: center;">triple-down 5th

</td>

<td style="text-align: center;">Av3

Line 1,068:

Line 1,157:

</td>

<td style="text-align: center;">vv5

</td>

<td style="text-align: center;">double-down 5th

</td>

<td style="text-align: center;">Avv

Line 1,080:

Line 1,171:

</td>

<td style="text-align: center;">v5

</td>

<td style="text-align: center;">down 5th

</td>

<td style="text-align: center;">Av

Line 1,092:

Line 1,185:

</td>

<td style="text-align: center;">P5

</td>

<td style="text-align: center;">perfect 5th

</td>

<td style="text-align: center;">A

Line 1,104:

Line 1,199:

</td>

<td style="text-align: center;">^5

</td>

<td style="text-align: center;">up 5th

</td>

<td style="text-align: center;">A^

Line 1,116:

Line 1,213:

</td>

<td style="text-align: center;">^^5

</td>

<td style="text-align: center;">double-up 5th

</td>

<td style="text-align: center;">A^^

Line 1,128:

Line 1,227:

</td>

<td style="text-align: center;">^35, v3m6

</td>

<td style="text-align: center;">triple-up 5th,

triple-down minor 6th

</td>

<td style="text-align: center;">A^3, Bbv3

Line 1,140:

Line 1,242:

</td>

<td style="text-align: center;">vvm6

</td>

<td style="text-align: center;">double-downminor 6th

</td>

<td style="text-align: center;">Bbvv

Line 1,152:

Line 1,256:

</td>

<td style="text-align: center;">vm6

</td>

<td style="text-align: center;">downminor 6th

</td>

<td style="text-align: center;">Bbv

Line 1,164:

Line 1,270:

</td>

<td style="text-align: center;">m6

</td>

<td style="text-align: center;">minor 6th

</td>

<td style="text-align: center;">Bb

Line 1,176:

Line 1,284:

</td>

<td style="text-align: center;">^m6

</td>

<td style="text-align: center;">upminor 6th

</td>

<td style="text-align: center;">Bb^

Line 1,188:

Line 1,298:

</td>

<td style="text-align: center;">v~6

</td>

<td style="text-align: center;">downmid 6th

</td>

<td style="text-align: center;">Bb^^

Line 1,200:

Line 1,312:

</td>

<td style="text-align: center;">~6

</td>

<td style="text-align: center;">mid 6th

</td>

<td style="text-align: center;">Bv3

Line 1,212:

Line 1,326:

</td>

<td style="text-align: center;">^~6

</td>

<td style="text-align: center;">upmid 6th

</td>

<td style="text-align: center;">Bvv

Line 1,224:

Line 1,340:

</td>

<td style="text-align: center;">vM6

</td>

<td style="text-align: center;">downmajor 6th

</td>

<td style="text-align: center;">Bv

Line 1,236:

Line 1,354:

</td>

<td style="text-align: center;">M6

</td>

<td style="text-align: center;">major 6th

</td>

<td style="text-align: center;">B

Line 1,248:

Line 1,368:

</td>

<td style="text-align: center;">^M6

</td>

<td style="text-align: center;">upmajor 6th

</td>

<td style="text-align: center;">B^

Line 1,260:

Line 1,382:

</td>

<td style="text-align: center;">^^M6

</td>

<td style="text-align: center;">double-upmajor 6th

</td>

<td style="text-align: center;">B^^

Line 1,272:

Line 1,396:

</td>

<td style="text-align: center;">^3M6, v3m7

</td>

<td style="text-align: center;">triple-up major 6th,

triple-down minor 7th

</td>

<td style="text-align: center;">B^3, Cv3

Line 1,284:

Line 1,411:

</td>

<td style="text-align: center;">vvm7

</td>

<td style="text-align: center;">double-downminor 7th

</td>

<td style="text-align: center;">Cvv

Line 1,296:

Line 1,425:

</td>

<td style="text-align: center;">vm7

</td>

<td style="text-align: center;">downminor 7th

</td>

<td style="text-align: center;">Cv

Line 1,308:

Line 1,439:

</td>

<td style="text-align: center;">m7

</td>

<td style="text-align: center;">minor 7th

</td>

<td style="text-align: center;">C

Line 1,320:

Line 1,453:

</td>

<td style="text-align: center;">^m7

</td>

<td style="text-align: center;">upminor 7th

</td>

<td style="text-align: center;">C^

Line 1,332:

Line 1,467:

</td>

<td style="text-align: center;">v~7

</td>

<td style="text-align: center;">downmid 7th

</td>

<td style="text-align: center;">C^^

Line 1,344:

Line 1,481:

</td>

<td style="text-align: center;">~7

</td>

<td style="text-align: center;">mid 7th

</td>

<td style="text-align: center;">C^3

Line 1,356:

Line 1,495:

</td>

<td style="text-align: center;">^~7

</td>

<td style="text-align: center;">upmin 7th

</td>

<td style="text-align: center;">C#vv

Line 1,368:

Line 1,509:

</td>

<td style="text-align: center;">vM7

</td>

<td style="text-align: center;">downmajor 7th

</td>

<td style="text-align: center;">C#v

Line 1,380:

Line 1,523:

</td>

<td style="text-align: center;">M7

</td>

<td style="text-align: center;">major 7th

</td>

<td style="text-align: center;">C#

Line 1,392:

Line 1,537:

</td>

<td style="text-align: center;">^M7

</td>

<td style="text-align: center;">upmajor 7th

</td>

<td style="text-align: center;">C#^

Line 1,404:

Line 1,551:

</td>

<td style="text-align: center;">^^M7

</td>

<td style="text-align: center;">double-upmajor 7th

</td>

<td style="text-align: center;">C#^^

Line 1,416:

Line 1,565:

</td>

<td style="text-align: center;">^3M7, v38

</td>

<td style="text-align: center;">triple-up major 7th,

triple-down octave

</td>

<td style="text-align: center;">C#^3, Dv3

Line 1,428:

Line 1,580:

</td>

<td style="text-align: center;">vv8

</td>

<td style="text-align: center;">double-down octave

</td>

<td style="text-align: center;">Dvv

Line 1,440:

Line 1,594:

</td>

<td style="text-align: center;">v8

</td>

<td style="text-align: center;">down octave

</td>

<td style="text-align: center;">Dv

Line 1,452:

Line 1,608:

</td>

<td style="text-align: center;">P8

</td>

<td style="text-align: center;">perfect octave

</td>

<td style="text-align: center;">D

Line 1,766:

Line 1,924:

72edo is the ninth <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>, as well as being a peak and gap edo, and the maximum value of the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#The%20Z%20function">Z function</a> in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.

<img src="http://xenharmonic.wikispaces.com/file/view/plot72.png/219772696/plot72.png" alt="plot72.png" title="plot72.png" />

<img src="http://xenharmonic.wikispaces.com/file/view/plot72.png/219772696/plot72.png" alt="plot72.png" title="plot72.png" />

<h1 id="toc5"><a name="Music"></a>Music</h1>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2016-12-28 05:09:07 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-02-24 17:48:44 UTC</tt>.<br>
-: The original revision id was <tt>602866264</tt>.<br>
+: The original revision id was <tt>607132219</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>
@@ Line 18: / Line 18: @@
 =Commas and Temperaments=
 et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and
 the kleisma, 15625/15552,
 the ampersand, 34171875/33554432 = |-25 7 6&gt;,
 the graviton, 129140163/128000000 = |-13 17 -6&gt;, and
 the ennealimma, 7629394531250/7625597484987 = |1 -27 18&gt; in the 5-limit.
 The 7-limit commas include 225/224, 1029/1024, 16875/16807, 19683/19600, 420175/419904, 2401/2400, 4375/4374 and 250047/250000.
 et shines in the 11-limit, with commas 243/242, 385/384, 441/440, 540/539, 1375/1372, 6250/6237, 4000/3993, 3025/3024 and 9801/9800.
 For the 13-limit, it tempers out 169/168, 325/324, 351/350, 364/363, 625/624, 676/675, 729/728, 1001/1000, 1575/1573, 1716/1715, 2080/2079 and 6656/6655.
@@ Line 47: / Line 47: @@
 =Intervals=
-|| degrees || cents value || approximate ratios (11-limit) ||||= [[xenharmonic/Ups and Downs Notation|ups and downs ]][[xenharmonic/Ups and Downs Notation|notation]] ||
+|| degrees || cents value || approximate ratios (11-limit) ||||||= [[xenharmonic/Ups and Downs Notation|ups and downs ]][[xenharmonic/Ups and Downs Notation|notation]] ||
-|| 0 || 0 || 1/1 ||= P1 ||= D ||
+|| 0 || 0 || 1/1 ||= P1 ||= perfect unison ||= D ||
-|| 1 || 16.667 || 81/80 ||= ^1 (up unison) ||= D^ ||
+|| 1 || 16.667 || 81/80 ||= ^1 ||= up unison ||= D^ ||
-|| 2 || 33.333 || 45/44 ||= ^^1(double-up unison) ||= D^^ ||
+|| 2 || 33.333 || 45/44 ||= ^^ ||= double-up unison ||= D^^ ||
-|| 3 || 50 || 33/32 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;1, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;m2 || D^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Ebv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
+|| 3 || 50 || 33/32 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;1, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= triple-up unison,
-|| 4 || 66.667 || 25/24 ||= vvm2 ||= Ebvv ||
+triple-down minor 2nd ||= D^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Ebv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
-|| 5 || 83.333 || 21/20 ||= vm2 (downminor 2nd) ||= Ebv ||
+|| 4 || 66.667 || 25/24 ||= vvm2 ||= double-downminor 2nd ||= Ebvv ||
-|| 6 || 100 || 35/33 ||= m2 ||= Eb ||
+|| 5 || 83.333 || 21/20 ||= vm2 ||= downminor 2nd ||= Ebv ||
-|| 7 || 116.667 || 15/14 ||= ^m2 (upminor 2nd) ||= Eb^ ||
+|| 6 || 100 || 35/33 ||= m2 ||= minor 2nd ||= Eb ||
-|| 8 || 133.333 || 27/25 ||= v~2 ||= Eb^^ ||
+|| 7 || 116.667 || 15/14 ||= ^m2 ||= upminor 2nd ||= Eb^ ||
-|| 9 || 150 || 12/11 ||= ~2 (mid 2nd) ||= Ev&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
+|| 8 || 133.333 || 27/25 ||= v~2 ||= downmid 2nd ||= Eb^^ ||
-|| 10 || 166.667 || 11/10 ||= ^~2 (upmid 2nd) ||= Evv ||
+|| 9 || 150 || 12/11 ||= ~2 ||= mid 2nd ||= Ev&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
-|| 11 || 183.333 || 10/9 ||= vM2 ||= Ev ||
+|| 10 || 166.667 || 11/10 ||= ^~2 ||= upmid 2nd ||= Evv ||
-|| 12 || 200 || 9/8 ||= M2 ||= E ||
+|| 11 || 183.333 || 10/9 ||= vM2 ||= downmajor 2nd ||= Ev ||
-|| 13 || 216.667 || 25/22 ||= ^M2 ||= E^ ||
+|| 12 || 200 || 9/8 ||= M2 ||= major 2nd ||= E ||
-|| 14 || 233.333 || 8/7 ||= ^^M2 ||= E^^ ||
+|| 13 || 216.667 || 25/22 ||= ^M2 ||= upmajor 2nd ||= E^ ||
-|| 15 || 250 || 81/70 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;M2, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;m3 ||= E^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Fv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
+|| 14 || 233.333 || 8/7 ||= ^^M2 ||= double-upmajor 2nd ||= E^^ ||
-|| 16 || 266.667 || 7/6 ||= vvm3 ||= Fvv ||
+|| 15 || 250 || 81/70 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;M2, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;m3 ||= triple-up major 2nd,
-|| 17 || 283.333 || 33/28 ||= vm3 ||= Fv ||
+triple-down minor 3rd ||= E^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Fv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
-|| 18 || 300 || 25/21 ||= m3 ||= F ||
+|| 16 || 266.667 || 7/6 ||= vvm3 ||= double-downminor 3rd ||= Fvv ||
-|| 19 || 316.667 || 6/5 ||= ^m3 ||= F^ ||
+|| 17 || 283.333 || 33/28 ||= vm3 ||= downminor 3rd ||= Fv ||
-|| 20 || 333.333 || 40/33 ||= v~3 ||= F^^ ||
+|| 18 || 300 || 25/21 ||= m3 ||= minor 3rd ||= F ||
-|| 21 || 350 || 11/9 ||= ~3 ||= F^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
+|| 19 || 316.667 || 6/5 ||= ^m3 ||= upminor 3rd ||= F^ ||
-|| 22 || 366.667 || 99/80 ||= ^~3 ||= F#vv ||
+|| 20 || 333.333 || 40/33 ||= v~3 ||= downmid 3rd ||= F^^ ||
-|| 23 || 383.333 || 5/4 ||= vM3 ||= F#v ||
+|| 21 || 350 || 11/9 ||= ~3 ||= mid 3rd ||= F^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
-|| 24 || 400 || 44/35 ||= M3 ||= F# ||
+|| 22 || 366.667 || 99/80 ||= ^~3 ||= upmid 3rd ||= F#vv ||
-|| 25 || 416.667 || 14/11 ||= ^M3 ||= F#^ ||
+|| 23 || 383.333 || 5/4 ||= vM3 ||= downmajor 3rd ||= F#v ||
-|| 26 || 433.333 || 9/7 ||= ^^M3 ||= F#^^ ||
+|| 24 || 400 || 44/35 ||= M3 ||= major 3rd ||= F# ||
-|| 27 || 450 || 35/27 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;M3, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;4 ||= F#^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Gv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
+|| 25 || 416.667 || 14/11 ||= ^M3 ||= upmajor 3rd ||= F#^ ||
-|| 28 || 466.667 || 21/16 ||= vv4 ||= Gvv ||
+|| 26 || 433.333 || 9/7 ||= ^^M3 ||= double-upmajor 3rd ||= F#^^ ||
-|| 29 || 483.333 || 33/25 ||= v4 ||= Gv ||
+|| 27 || 450 || 35/27 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;M3, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;4 ||= triple-up major 3rd,
-|| 30 || 500 || 4/3 ||= P4 ||= G ||
+triple-down 4th ||= F#^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Gv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
-|| 31 || 516.667 || 27/20 ||= ^4 ||= G^ ||
+|| 28 || 466.667 || 21/16 ||= vv4 ||= double-down 4th ||= Gvv ||
-|| 32 || 533.333 || 15/11 ||= ^^4 ||= G^^ ||
+|| 29 || 483.333 || 33/25 ||= v4 ||= down 4th ||= Gv ||
-|| 33 || 550 || 11/8 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;4 ||= G^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
+|| 30 || 500 || 4/3 ||= P4 ||= perfect 4th ||= G ||
-|| 34 || 566.667 || 25/18 ||= vvA4 ||= G#vv ||
+|| 31 || 516.667 || 27/20 ||= ^4 ||= up 4th ||= G^ ||
-|| 35 || 583.333 || 7/5 ||= vA4, vd5 ||= G#v, Abv ||
+|| 32 || 533.333 || 15/11 ||= ^^4 ||= double-up 4th ||= G^^ ||
-|| 36 || 600 || 99/70 ||= A4, d5 ||= G#, Ab ||
+|| 33 || 550 || 11/8 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;4 ||= triple-up 4th ||= G^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
-|| 37 || 616.667 || 10/7 ||= ^A4, ^d5 ||= G#^, Ab^ ||
+|| 34 || 566.667 || 25/18 ||= vvA4 ||= double-down aug 4th ||= G#vv ||
-|| 38 || 633.333 || 36/25 ||= ^^d5 ||= Ab^^ ||
+|| 35 || 583.333 || 7/5 ||= vA4, vd5 ||= downaug 4th, updim 5th ||= G#v, Abv ||
-|| 39 || 650 || 16/11 ||= v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;5 ||= Av&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
+|| 36 || 600 || 99/70 ||= A4, d5 ||= aug 4th, dim 5th ||= G#, Ab ||
-|| 40 || 666.667 || 22/15 ||= vv5 ||= Avv ||
+|| 37 || 616.667 || 10/7 ||= ^A4, ^d5 ||= upaug 4th, downdim 5th ||= G#^, Ab^ ||
-|| 41 || 683.333 || 40/27 ||= v5 ||= Av ||
+|| 38 || 633.333 || 36/25 ||= ^^d5 ||= double-updim 5th ||= Ab^^ ||
-|| 42 || 700 || 3/2 ||= P5 ||= A ||
+|| 39 || 650 || 16/11 ||= v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;5 ||= triple-down 5th ||= Av&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
-|| 43 || 716.667 || 50/33 ||= ^5 ||= A^ ||
+|| 40 || 666.667 || 22/15 ||= vv5 ||= double-down 5th ||= Avv ||
-|| 44 || 733.333 || 32/21 ||= ^^5 ||= A^^ ||
+|| 41 || 683.333 || 40/27 ||= v5 ||= down 5th ||= Av ||
-|| 45 || 750 || 54/35 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;5, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;m6 ||= A^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Bbv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
+|| 42 || 700 || 3/2 ||= P5 ||= perfect 5th ||= A ||
-|| 46 || 766.667 || 14/9 ||= vvm6 ||= Bbvv ||
+|| 43 || 716.667 || 50/33 ||= ^5 ||= up 5th ||= A^ ||
-|| 47 || 783.333 || 11/7 ||= vm6 ||= Bbv ||
+|| 44 || 733.333 || 32/21 ||= ^^5 ||= double-up 5th ||= A^^ ||
-|| 48 || 800 || 35/22 ||= m6 ||= Bb ||
+|| 45 || 750 || 54/35 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;5, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;m6 ||= triple-up 5th,
-|| 49 || 816.667 || 8/5 ||= ^m6 ||= Bb^ ||
+triple-down minor 6th ||= A^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Bbv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
-|| 50 || 833.333 || 81/50 ||= v~6 ||= Bb^^ ||
+|| 46 || 766.667 || 14/9 ||= vvm6 ||= double-downminor 6th ||= Bbvv ||
-|| 51 || 850 || 18/11 ||= ~6 ||= Bv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
+|| 47 || 783.333 || 11/7 ||= vm6 ||= downminor 6th ||= Bbv ||
-|| 52 || 866.667 || 33/20 ||= ^~6 ||= Bvv ||
+|| 48 || 800 || 35/22 ||= m6 ||= minor 6th ||= Bb ||
-|| 53 || 883.333 || 5/3 ||= vM6 ||= Bv ||
+|| 49 || 816.667 || 8/5 ||= ^m6 ||= upminor 6th ||= Bb^ ||
-|| 54 || 900 || 27/16 ||= M6 ||= B ||
+|| 50 || 833.333 || 81/50 ||= v~6 ||= downmid 6th ||= Bb^^ ||
-|| 55 || 916.667 || 56/33 ||= ^M6 ||= B^ ||
+|| 51 || 850 || 18/11 ||= ~6 ||= mid 6th ||= Bv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
-|| 56 || 933.333 || 12/7 ||= ^^M6 ||= B^^ ||
+|| 52 || 866.667 || 33/20 ||= ^~6 ||= upmid 6th ||= Bvv ||
-|| 57 || 950 || 121/70 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;M6, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;m7 ||= B^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Cv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
+|| 53 || 883.333 || 5/3 ||= vM6 ||= downmajor 6th ||= Bv ||
-|| 58 || 966.667 || 7/4 ||= vvm7 ||= Cvv ||
+|| 54 || 900 || 27/16 ||= M6 ||= major 6th ||= B ||
-|| 59 || 983.333 || 44/25 ||= vm7 ||= Cv ||
+|| 55 || 916.667 || 56/33 ||= ^M6 ||= upmajor 6th ||= B^ ||
-|| 60 || 1000 || 16/9 ||= m7 ||= C ||
+|| 56 || 933.333 || 12/7 ||= ^^M6 ||= double-upmajor 6th ||= B^^ ||
-|| 61 || 1016.667 || 9/5 ||= ^m7 ||= C^ ||
+|| 57 || 950 || 121/70 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;M6, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;m7 ||= triple-up major 6th,
-|| 62 || 1033.333 || 20/11 ||= v~7 ||= C^^ ||
+triple-down minor 7th ||= B^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Cv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
-|| 63 || 1050 || 11/6 ||= ~7 ||= C^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
+|| 58 || 966.667 || 7/4 ||= vvm7 ||= double-downminor 7th ||= Cvv ||
-|| 64 || 1066.667 || 50/27 ||= ^~7 ||= C#vv ||
+|| 59 || 983.333 || 44/25 ||= vm7 ||= downminor 7th ||= Cv ||
-|| 65 || 1083.333 || 15/8 ||= vM7 ||= C#v ||
+|| 60 || 1000 || 16/9 ||= m7 ||= minor 7th ||= C ||
-|| 66 || 1100 || 66/35 ||= M7 ||= C# ||
+|| 61 || 1016.667 || 9/5 ||= ^m7 ||= upminor 7th ||= C^ ||
-|| 67 || 1116.667 || 21/11 ||= ^M7 ||= C#^ ||
+|| 62 || 1033.333 || 20/11 ||= v~7 ||= downmid 7th ||= C^^ ||
-|| 68 || 1133.333 || 27/14 ||= ^^M7 ||= C#^^ ||
+|| 63 || 1050 || 11/6 ||= ~7 ||= mid 7th ||= C^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
-|| 69 || 1150 || 35/18 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;M7, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;8 ||= C#^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Dv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
+|| 64 || 1066.667 || 50/27 ||= ^~7 ||= upmin 7th ||= C#vv ||
-|| 70 || 1166.667 || 49/25 ||= vv8 ||= Dvv ||
+|| 65 || 1083.333 || 15/8 ||= vM7 ||= downmajor 7th ||= C#v ||
-|| 71 || 1183.333 || 99/50 ||= v8 ||= Dv ||
+|| 66 || 1100 || 66/35 ||= M7 ||= major 7th ||= C# ||
-|| 72 || 1200 || 2/1 ||= P8 ||= D ||
+|| 67 || 1116.667 || 21/11 ||= ^M7 ||= upmajor 7th ||= C#^ ||
+|| 68 || 1133.333 || 27/14 ||= ^^M7 ||= double-upmajor 7th ||= C#^^ ||
+|| 69 || 1150 || 35/18 ||= ^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;M7, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;8 ||= triple-up major 7th,
+triple-down octave ||= C#^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Dv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; ||
+|| 70 || 1166.667 || 49/25 ||= vv8 ||= double-down octave ||= Dvv ||
+|| 71 || 1183.333 || 99/50 ||= v8 ||= down octave ||= Dv ||
+|| 72 || 1200 || 2/1 ||= P8 ||= perfect octave ||= D ||
 Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[xenharmonic/Ups and Downs Notation#Chord%20names%20in%20other%20EDOs|Ups and Downs Notation - Chord names in other EDOs]].
@@ Line 200: / Line 206: @@
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Commas and Temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Commas and Temperaments&lt;/h1&gt;
   &lt;br /&gt;
-et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and &lt;br /&gt;
+et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and&lt;br /&gt;
-the kleisma, 15625/15552, &lt;br /&gt;
+the kleisma, 15625/15552,&lt;br /&gt;
-the ampersand, 34171875/33554432 = |-25 7 6&amp;gt;, &lt;br /&gt;
+the ampersand, 34171875/33554432 = |-25 7 6&amp;gt;,&lt;br /&gt;
-the graviton, 129140163/128000000 = |-13 17 -6&amp;gt;, and &lt;br /&gt;
+the graviton, 129140163/128000000 = |-13 17 -6&amp;gt;, and&lt;br /&gt;
-the ennealimma, 7629394531250/7625597484987 = |1 -27 18&amp;gt; in the 5-limit. &lt;br /&gt;
+the ennealimma, 7629394531250/7625597484987 = |1 -27 18&amp;gt; in the 5-limit.&lt;br /&gt;
 &lt;br /&gt;
 The 7-limit commas include 225/224, 1029/1024, 16875/16807, 19683/19600, 420175/419904, 2401/2400, 4375/4374 and 250047/250000.&lt;br /&gt;
 &lt;br /&gt;
-et shines in the 11-limit, with commas 243/242, 385/384, 441/440, 540/539, 1375/1372, 6250/6237, 4000/3993, 3025/3024 and 9801/9800. &lt;br /&gt;
+et shines in the 11-limit, with commas 243/242, 385/384, 441/440, 540/539, 1375/1372, 6250/6237, 4000/3993, 3025/3024 and 9801/9800.&lt;br /&gt;
 &lt;br /&gt;
 For the 13-limit, it tempers out 169/168, 325/324, 351/350, 364/363, 625/624, 676/675, 729/728, 1001/1000, 1575/1573, 1716/1715, 2080/2079 and 6656/6655.&lt;br /&gt;
@@ Line 577: / Line 583: @@
          &lt;td&gt;approximate ratios (11-limit)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td colspan="2" style="text-align: center;"&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;td colspan="3" style="text-align: center;"&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;/td&gt;
      &lt;/tr&gt;
@@ Line 588: / Line 594: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;P1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;perfect unison&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
@@ Line 599: / Line 607: @@
          &lt;td&gt;81/80&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;^1 (up unison)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;^1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;up unison&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;D^&lt;br /&gt;
@@ Line 611: / Line 621: @@
          &lt;td&gt;45/44&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;^^1(double-up unison)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;^^&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-up unison&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;D^^&lt;br /&gt;
@@ Line 625: / Line 637: @@
          &lt;td style="text-align: center;"&gt;^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;1, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;m2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;D^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Ebv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;triple-up unison,&lt;br /&gt;
+triple-down minor 2nd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Ebv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 636: / Line 651: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;vvm2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-downminor 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Ebvv&lt;br /&gt;
@@ Line 647: / Line 664: @@
          &lt;td&gt;21/20&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;vm2 (downminor 2nd)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;vm2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;downminor 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Ebv&lt;br /&gt;
@@ Line 660: / Line 679: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;m2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;minor 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Eb&lt;br /&gt;
@@ Line 671: / Line 692: @@
          &lt;td&gt;15/14&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;^m2 (upminor 2nd)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;^m2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;upminor 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Eb^&lt;br /&gt;
@@ Line 684: / Line 707: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;v~2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;downmid 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Eb^^&lt;br /&gt;
@@ Line 695: / Line 720: @@
          &lt;td&gt;12/11&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;~2 (mid 2nd)&lt;br /&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;
@@ Line 707: / Line 734: @@
          &lt;td&gt;11/10&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;^~2 (upmid 2nd)&lt;br /&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;
@@ Line 720: / Line 749: @@
 &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;
@@ Line 732: / Line 763: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;M2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;major 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;E&lt;br /&gt;
@@ Line 744: / Line 777: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^M2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;upmajor 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;E^&lt;br /&gt;
@@ Line 756: / Line 791: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^^M2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-upmajor 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;E^^&lt;br /&gt;
@@ Line 768: / Line 805: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;M2, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;m3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;triple-up major 2nd,&lt;br /&gt;
+triple-down minor 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;E^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Fv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
@@ Line 780: / Line 820: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;vvm3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-downminor 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Fvv&lt;br /&gt;
@@ Line 792: / Line 834: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;vm3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;downminor 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Fv&lt;br /&gt;
@@ Line 804: / Line 848: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;m3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;minor 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;F&lt;br /&gt;
@@ Line 816: / Line 862: @@
 &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;
@@ Line 828: / Line 876: @@
 &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;
@@ Line 840: / Line 890: @@
 &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;
@@ Line 852: / Line 904: @@
 &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;
@@ Line 864: / Line 918: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;vM3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;downmajor 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;F#v&lt;br /&gt;
@@ Line 876: / Line 932: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;M3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;major 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;F#&lt;br /&gt;
@@ Line 888: / Line 946: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^M3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;upmajor 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;F#^&lt;br /&gt;
@@ Line 900: / Line 960: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^^M3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-upmajor 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;F#^^&lt;br /&gt;
@@ Line 912: / Line 974: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;M3, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;triple-up major 3rd,&lt;br /&gt;
+triple-down 4th&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;, Gv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
@@ Line 924: / Line 989: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;vv4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-down 4th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Gvv&lt;br /&gt;
@@ Line 936: / Line 1,003: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;v4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;down 4th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Gv&lt;br /&gt;
@@ Line 948: / Line 1,017: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;P4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;perfect 4th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;G&lt;br /&gt;
@@ Line 960: / Line 1,031: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;up 4th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;G^&lt;br /&gt;
@@ Line 972: / Line 1,045: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^^4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-up 4th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;G^^&lt;br /&gt;
@@ Line 984: / Line 1,059: @@
 &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&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;triple-up 4th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;G^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
@@ Line 996: / Line 1,073: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;vvA4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-down aug 4th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;G#vv&lt;br /&gt;
@@ Line 1,008: / Line 1,087: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;vA4, vd5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;downaug 4th, updim 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;G#v, Abv&lt;br /&gt;
@@ Line 1,020: / Line 1,101: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;A4, d5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;aug 4th, dim 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;G#, Ab&lt;br /&gt;
@@ Line 1,032: / Line 1,115: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^A4, ^d5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;upaug 4th, downdim 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;G#^, Ab^&lt;br /&gt;
@@ Line 1,044: / Line 1,129: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^^d5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-updim 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Ab^^&lt;br /&gt;
@@ Line 1,056: / Line 1,143: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;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-down 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Av&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
@@ Line 1,068: / Line 1,157: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;vv5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-down 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Avv&lt;br /&gt;
@@ Line 1,080: / Line 1,171: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;v5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;down 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Av&lt;br /&gt;
@@ Line 1,092: / Line 1,185: @@
 &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&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;A&lt;br /&gt;
@@ Line 1,104: / Line 1,199: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;up 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;A^&lt;br /&gt;
@@ Line 1,116: / Line 1,213: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^^5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-up 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;A^^&lt;br /&gt;
@@ Line 1,128: / Line 1,227: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;5, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;m6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;triple-up 5th,&lt;br /&gt;
+triple-down minor 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;A^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Bbv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
@@ Line 1,140: / Line 1,242: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;vvm6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-downminor 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Bbvv&lt;br /&gt;
@@ Line 1,152: / Line 1,256: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;vm6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;downminor 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Bbv&lt;br /&gt;
@@ Line 1,164: / Line 1,270: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;m6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;minor 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Bb&lt;br /&gt;
@@ Line 1,176: / Line 1,284: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^m6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;upminor 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Bb^&lt;br /&gt;
@@ Line 1,188: / Line 1,298: @@
 &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;
@@ Line 1,200: / Line 1,312: @@
 &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;
@@ Line 1,212: / Line 1,326: @@
 &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;
@@ Line 1,224: / Line 1,340: @@
 &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;
@@ Line 1,236: / Line 1,354: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;M6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;major 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;B&lt;br /&gt;
@@ Line 1,248: / Line 1,368: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^M6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;upmajor 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;B^&lt;br /&gt;
@@ Line 1,260: / Line 1,382: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^^M6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-upmajor 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;B^^&lt;br /&gt;
@@ Line 1,272: / Line 1,396: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;M6, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;m7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;triple-up major 6th,&lt;br /&gt;
+triple-down minor 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;B^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;, Cv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
@@ Line 1,284: / Line 1,411: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;vvm7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-downminor 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Cvv&lt;br /&gt;
@@ Line 1,296: / Line 1,425: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;vm7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;downminor 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Cv&lt;br /&gt;
@@ Line 1,308: / Line 1,439: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;m7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;minor 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
@@ Line 1,320: / Line 1,453: @@
 &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;
@@ Line 1,332: / Line 1,467: @@
 &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;
@@ Line 1,344: / Line 1,481: @@
 &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;
@@ Line 1,356: / Line 1,495: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^~7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;upmin 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C#vv&lt;br /&gt;
@@ Line 1,368: / Line 1,509: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;vM7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;downmajor 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C#v&lt;br /&gt;
@@ Line 1,380: / Line 1,523: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;M7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;major 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
@@ Line 1,392: / Line 1,537: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^M7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;upmajor 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C#^&lt;br /&gt;
@@ Line 1,404: / Line 1,551: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^^M7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-upmajor 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C#^^&lt;br /&gt;
@@ Line 1,416: / Line 1,565: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;M7, v&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;triple-up major 7th,&lt;br /&gt;
+triple-down octave&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;, Dv&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
@@ Line 1,428: / Line 1,580: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;vv8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-down octave&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Dvv&lt;br /&gt;
@@ Line 1,440: / Line 1,594: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;v8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;down octave&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Dv&lt;br /&gt;
@@ Line 1,452: / Line 1,608: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;perfect octave&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
@@ Line 1,766: / Line 1,924: @@
 edo is the ninth &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt;, as well as being a peak and gap edo, and the maximum value of the &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#The%20Z%20function"&gt;Z function&lt;/a&gt; in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:1571:&amp;lt;img src=&amp;quot;http://xenharmonic.wikispaces.com/file/view/plot72.png/219772696/plot72.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://xenharmonic.wikispaces.com/file/view/plot72.png/219772696/plot72.png" alt="plot72.png" title="plot72.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1571 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:1717:&amp;lt;img src=&amp;quot;http://xenharmonic.wikispaces.com/file/view/plot72.png/219772696/plot72.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://xenharmonic.wikispaces.com/file/view/plot72.png/219772696/plot72.png" alt="plot72.png" title="plot72.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1717 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Music&lt;/h1&gt;