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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This table shows all the simple intervals of [[Meantone_family#Septimal meantone|septimal meantone]], which includes the entire 7-limit tonality diamond. Other relevant tables of meantone intervals are the table of [[Quarter-comma_meantone|quarter-comma meantone]] intervals and the table of [[31edo#Intervals|31 edo intervals]].
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2012-07-01 07:48:37 UTC</tt>.<br>
: The original revision id was <tt>349330032</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">This table shows all the simple intervals of [[Meantone family#Septimal%20meantone|septimal meantone]], which includes the entire 7-limit tonality diamond. Other relevant tables of meantone intervals are the table of [[quarter-comma meantone]] intervals and the table of [[31edo#Intervals|31 edo intervals]].


In [[12edo]] the diminished second vanishes, so this cornucopia of intervals collapses to a mere 12. None of the intervals is inherently septimal in 12edo, because they all have simpler 5-limit descriptions.
In [[12edo|12edo]] the diminished second vanishes, so this cornucopia of intervals collapses to a mere 12. None of the intervals is inherently septimal in 12edo, because they all have simpler 5-limit descriptions.


In [[19edo]], in contrast, the *double* diminished second vanishes, so the equivalences are A1~d2, A2~d3, A3~d4, A4~dd5, AA4~d5, A5~d6, A6~d7, and A7~d8. Thus some intervals are undeniably septimal, but ambiguously so because 49/48 vanishes.
In [[19edo|19edo]], in contrast, the *double* diminished second vanishes, so the equivalences are A1~d2, A2~d3, A3~d4, A4~dd5, AA4~d5, A5~d6, A6~d7, and A7~d8. Thus some intervals are undeniably septimal, but ambiguously so because 49/48 vanishes.


More complex meantone edos such as [[31edo]] distinguish all intervals listed on this table.
More complex meantone edos such as [[31edo|31edo]] distinguish all intervals listed on this table.


See also [[http://en.wikipedia.org/wiki/List_of_meantone_intervals|Wikipedia's list of meantone intervals]]
See also [http://en.wikipedia.org/wiki/List_of_meantone_intervals Wikipedia's list of meantone intervals]
||~ Name ||~ Size* ||~ Ratios ||
||||||~ Unisons ||
|| Perfect unison (P1) || 0 || 1/1 ||
|| Augmented unison (A1) || 75.5 || 28/27~25/24~21/20 ||
||||||~ Seconds ||
|| Diminished second (d2) || 42.0 || 128/125~64/63~50/49~36/35 ||
|| Minor second (m2) || 117.5 || 16/15~15/14 ||
|| Major second (M2) || 193.0 || 10/9~9/8 ||
|| Augmented second (A2) || 268.5 || 7/6 ||
||||||~ Thirds ||
|| Diminished third (d3) || 235.0 || 8/7 ||
|| Minor third (m3) || 310.5 || 6/5 ||
|| Major third (M3) || 386.0 || 5/4 ||
|| Augmented third (A3) || 461.5 || 21/16 ||
||||||~ Fourths ||
|| Diminished fourth (d4) || 428.0 || 9/7 (a bit 14/11-ish) ||
|| Perfect fourth (P4) || 503.5 || 4/3 ||
|| Augmented fourth (A4) || 579.0 || 7/5 ||
|| Double augmented fourth (AA4) || 654.5 || Close to 16/11 ||
||||||~ Fifths ||
|| Double diminished fifth (dd5) || 545.5 || Close to 11/8 ||
|| Diminished fifth (d5) || 621.0 || 10/7 ||
|| Perfect fifth (P5) || 696.5 || 3/2 ||
|| Augmented fifth (A5) || 772.0 || 14/9 (a bit 11/7-ish) ||
||||||~ Sixths ||
|| Diminished sixth (d6) || 738.5 || 32/21 ||
|| Minor sixth (m6) || 814.0 || 8/5 ||
|| Major sixth (M6) || 889.5 || 5/3 ||
|| Augmented sixth (A6) || 965.0 || 7/4 ||
||||||~ Sevenths ||
|| Diminished seventh (d7) || 931.5 || 12/7 ||
|| Minor seventh (m7) || 1007.0 || 16/9~9/5 ||
|| Major seventh (M7) || 1082.5 || 15/8 ||
|| Augmented seventh (A7) || 1158.0 || 35/18~49/25~63/32 ||
||||||~ Octaves ||
|| Diminished octave (d8) || 1124.5 || 27/14~48/25~40/21 ||
|| Perfect octave (P8) || 1200 || 2/1 ||
|| Augmented octave (A8) || 1275.5 || 25/12~21/10 ||
``*`` In POTE septimal meantone</pre></div>
<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;Meantone intervals&lt;/title&gt;&lt;/head&gt;&lt;body&gt;This table shows all the simple intervals of &lt;a class="wiki_link" href="/Meantone%20family#Septimal%20meantone"&gt;septimal meantone&lt;/a&gt;, which includes the entire 7-limit tonality diamond. Other relevant tables of meantone intervals are the table of &lt;a class="wiki_link" href="/quarter-comma%20meantone"&gt;quarter-comma meantone&lt;/a&gt; intervals and the table of &lt;a class="wiki_link" href="/31edo#Intervals"&gt;31 edo intervals&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
In &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; the diminished second vanishes, so this cornucopia of intervals collapses to a mere 12. None of the intervals is inherently septimal in 12edo, because they all have simpler 5-limit descriptions.&lt;br /&gt;
&lt;br /&gt;
In &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, in contrast, the *double* diminished second vanishes, so the equivalences are A1~d2, A2~d3, A3~d4, A4~dd5, AA4~d5, A5~d6, A6~d7, and A7~d8. Thus some intervals are undeniably septimal, but ambiguously so because 49/48 vanishes.&lt;br /&gt;
&lt;br /&gt;
More complex meantone edos such as &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; distinguish all intervals listed on this table.&lt;br /&gt;
&lt;br /&gt;
See also &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/List_of_meantone_intervals" rel="nofollow"&gt;Wikipedia's list of meantone intervals&lt;/a&gt;&lt;br /&gt;


 
{| class="wikitable"
&lt;table class="wiki_table"&gt;
|-
    &lt;tr&gt;
! | Name
        &lt;th&gt;Name&lt;br /&gt;
! | Size*
&lt;/th&gt;
! | Ratios
        &lt;th&gt;Size*&lt;br /&gt;
|-
&lt;/th&gt;
! colspan="3" | Unisons
        &lt;th&gt;Ratios&lt;br /&gt;
|-
&lt;/th&gt;
| | Perfect unison (P1)
    &lt;/tr&gt;
| | 0
    &lt;tr&gt;
| | 1/1
        &lt;th colspan="3"&gt;Unisons&lt;br /&gt;
|-
&lt;/th&gt;
| | Augmented unison (A1)
    &lt;/tr&gt;
| | 75.5
    &lt;tr&gt;
| | 28/27~25/24~21/20
        &lt;td&gt;Perfect unison (P1)&lt;br /&gt;
|-
&lt;/td&gt;
! colspan="3" | Seconds
        &lt;td&gt;0&lt;br /&gt;
|-
&lt;/td&gt;
| | Diminished second (d2)
        &lt;td&gt;1/1&lt;br /&gt;
| | 42.0
&lt;/td&gt;
| | 128/125~64/63~50/49~36/35
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| | Minor second (m2)
        &lt;td&gt;Augmented unison (A1)&lt;br /&gt;
| | 117.5
&lt;/td&gt;
| | 16/15~15/14
        &lt;td&gt;75.5&lt;br /&gt;
|-
&lt;/td&gt;
| | Major second (M2)
        &lt;td&gt;28/27~25/24~21/20&lt;br /&gt;
| | 193.0
&lt;/td&gt;
| | 10/9~9/8
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| | Augmented second (A2)
        &lt;th colspan="3"&gt;Seconds&lt;br /&gt;
| | 268.5
&lt;/th&gt;
| | 7/6
    &lt;/tr&gt;
|-
    &lt;tr&gt;
! colspan="3" | Thirds
        &lt;td&gt;Diminished second (d2)&lt;br /&gt;
|-
&lt;/td&gt;
| | Diminished third (d3)
        &lt;td&gt;42.0&lt;br /&gt;
| | 235.0
&lt;/td&gt;
| | 8/7
        &lt;td&gt;128/125~64/63~50/49~36/35&lt;br /&gt;
|-
&lt;/td&gt;
| | Minor third (m3)
    &lt;/tr&gt;
| | 310.5
    &lt;tr&gt;
| | 6/5
        &lt;td&gt;Minor second (m2)&lt;br /&gt;
|-
&lt;/td&gt;
| | Major third (M3)
        &lt;td&gt;117.5&lt;br /&gt;
| | 386.0
&lt;/td&gt;
| | 5/4
        &lt;td&gt;16/15~15/14&lt;br /&gt;
|-
&lt;/td&gt;
| | Augmented third (A3)
    &lt;/tr&gt;
| | 461.5
    &lt;tr&gt;
| | 21/16
        &lt;td&gt;Major second (M2)&lt;br /&gt;
|-
&lt;/td&gt;
! colspan="3" | Fourths
        &lt;td&gt;193.0&lt;br /&gt;
|-
&lt;/td&gt;
| | Diminished fourth (d4)
        &lt;td&gt;10/9~9/8&lt;br /&gt;
| | 428.0
&lt;/td&gt;
| | 9/7 (a bit 14/11-ish)
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| | Perfect fourth (P4)
        &lt;td&gt;Augmented second (A2)&lt;br /&gt;
| | 503.5
&lt;/td&gt;
| | 4/3
        &lt;td&gt;268.5&lt;br /&gt;
|-
&lt;/td&gt;
| | Augmented fourth (A4)
        &lt;td&gt;7/6&lt;br /&gt;
| | 579.0
&lt;/td&gt;
| | 7/5
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| | Double augmented fourth (AA4)
        &lt;th colspan="3"&gt;Thirds&lt;br /&gt;
| | 654.5
&lt;/th&gt;
| | Close to 16/11
    &lt;/tr&gt;
|-
    &lt;tr&gt;
! colspan="3" | Fifths
        &lt;td&gt;Diminished third (d3)&lt;br /&gt;
|-
&lt;/td&gt;
| | Double diminished fifth (dd5)
        &lt;td&gt;235.0&lt;br /&gt;
| | 545.5
&lt;/td&gt;
| | Close to 11/8
        &lt;td&gt;8/7&lt;br /&gt;
|-
&lt;/td&gt;
| | Diminished fifth (d5)
    &lt;/tr&gt;
| | 621.0
    &lt;tr&gt;
| | 10/7
        &lt;td&gt;Minor third (m3)&lt;br /&gt;
|-
&lt;/td&gt;
| | Perfect fifth (P5)
        &lt;td&gt;310.5&lt;br /&gt;
| | 696.5
&lt;/td&gt;
| | 3/2
        &lt;td&gt;6/5&lt;br /&gt;
|-
&lt;/td&gt;
| | Augmented fifth (A5)
    &lt;/tr&gt;
| | 772.0
    &lt;tr&gt;
| | 14/9 (a bit 11/7-ish)
        &lt;td&gt;Major third (M3)&lt;br /&gt;
|-
&lt;/td&gt;
! colspan="3" | Sixths
        &lt;td&gt;386.0&lt;br /&gt;
|-
&lt;/td&gt;
| | Diminished sixth (d6)
        &lt;td&gt;5/4&lt;br /&gt;
| | 738.5
&lt;/td&gt;
| | 32/21
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| | Minor sixth (m6)
        &lt;td&gt;Augmented third (A3)&lt;br /&gt;
| | 814.0
&lt;/td&gt;
| | 8/5
        &lt;td&gt;461.5&lt;br /&gt;
|-
&lt;/td&gt;
| | Major sixth (M6)
        &lt;td&gt;21/16&lt;br /&gt;
| | 889.5
&lt;/td&gt;
| | 5/3
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| | Augmented sixth (A6)
        &lt;th colspan="3"&gt;Fourths&lt;br /&gt;
| | 965.0
&lt;/th&gt;
| | 7/4
    &lt;/tr&gt;
|-
    &lt;tr&gt;
! colspan="3" | Sevenths
        &lt;td&gt;Diminished fourth (d4)&lt;br /&gt;
|-
&lt;/td&gt;
| | Diminished seventh (d7)
        &lt;td&gt;428.0&lt;br /&gt;
| | 931.5
&lt;/td&gt;
| | 12/7
        &lt;td&gt;9/7 (a bit 14/11-ish)&lt;br /&gt;
|-
&lt;/td&gt;
| | Minor seventh (m7)
    &lt;/tr&gt;
| | 1007.0
    &lt;tr&gt;
| | 16/9~9/5
        &lt;td&gt;Perfect fourth (P4)&lt;br /&gt;
|-
&lt;/td&gt;
| | Major seventh (M7)
        &lt;td&gt;503.5&lt;br /&gt;
| | 1082.5
&lt;/td&gt;
| | 15/8
        &lt;td&gt;4/3&lt;br /&gt;
|-
&lt;/td&gt;
| | Augmented seventh (A7)
    &lt;/tr&gt;
| | 1158.0
    &lt;tr&gt;
| | 35/18~49/25~63/32
        &lt;td&gt;Augmented fourth (A4)&lt;br /&gt;
|-
&lt;/td&gt;
! colspan="3" | Octaves
        &lt;td&gt;579.0&lt;br /&gt;
|-
&lt;/td&gt;
| | Diminished octave (d8)
        &lt;td&gt;7/5&lt;br /&gt;
| | 1124.5
&lt;/td&gt;
| | 27/14~48/25~40/21
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| | Perfect octave (P8)
        &lt;td&gt;Double augmented fourth (AA4)&lt;br /&gt;
| | 1200
&lt;/td&gt;
| | 2/1
        &lt;td&gt;654.5&lt;br /&gt;
|-
&lt;/td&gt;
| | Augmented octave (A8)
        &lt;td&gt;Close to 16/11&lt;br /&gt;
| | 1275.5
&lt;/td&gt;
| | 25/12~21/10
    &lt;/tr&gt;
|}
    &lt;tr&gt;
* In POTE septimal meantone
        &lt;th colspan="3"&gt;Fifths&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Double diminished fifth (dd5)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;545.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Close to 11/8&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Diminished fifth (d5)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;621.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Perfect fifth (P5)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;696.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Augmented fifth (A5)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;772.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9 (a bit 11/7-ish)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th colspan="3"&gt;Sixths&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Diminished sixth (d6)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;738.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;32/21&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Minor sixth (m6)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;814.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Major sixth (M6)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;889.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Augmented sixth (A6)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;965.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th colspan="3"&gt;Sevenths&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Diminished seventh (d7)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;931.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Minor seventh (m7)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1007.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/9~9/5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Major seventh (M7)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1082.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/8&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Augmented seventh (A7)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1158.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;35/18~49/25~63/32&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th colspan="3"&gt;Octaves&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Diminished octave (d8)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1124.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/14~48/25~40/21&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Perfect octave (P8)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1200&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2/1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Augmented octave (A8)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1275.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;25/12~21/10&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;!-- ws:start:WikiTextRawRule:00:``*`` --&gt;*&lt;!-- ws:end:WikiTextRawRule:00 --&gt; In POTE septimal meantone&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

This table shows all the simple intervals of septimal meantone, which includes the entire 7-limit tonality diamond. Other relevant tables of meantone intervals are the table of quarter-comma meantone intervals and the table of 31 edo intervals.

In 12edo the diminished second vanishes, so this cornucopia of intervals collapses to a mere 12. None of the intervals is inherently septimal in 12edo, because they all have simpler 5-limit descriptions.

In 19edo, in contrast, the *double* diminished second vanishes, so the equivalences are A1~d2, A2~d3, A3~d4, A4~dd5, AA4~d5, A5~d6, A6~d7, and A7~d8. Thus some intervals are undeniably septimal, but ambiguously so because 49/48 vanishes.

More complex meantone edos such as 31edo distinguish all intervals listed on this table.

See also Wikipedia's list of meantone intervals

Name Size* Ratios
Unisons
Perfect unison (P1) 0 1/1
Augmented unison (A1) 75.5 28/27~25/24~21/20
Seconds
Diminished second (d2) 42.0 128/125~64/63~50/49~36/35
Minor second (m2) 117.5 16/15~15/14
Major second (M2) 193.0 10/9~9/8
Augmented second (A2) 268.5 7/6
Thirds
Diminished third (d3) 235.0 8/7
Minor third (m3) 310.5 6/5
Major third (M3) 386.0 5/4
Augmented third (A3) 461.5 21/16
Fourths
Diminished fourth (d4) 428.0 9/7 (a bit 14/11-ish)
Perfect fourth (P4) 503.5 4/3
Augmented fourth (A4) 579.0 7/5
Double augmented fourth (AA4) 654.5 Close to 16/11
Fifths
Double diminished fifth (dd5) 545.5 Close to 11/8
Diminished fifth (d5) 621.0 10/7
Perfect fifth (P5) 696.5 3/2
Augmented fifth (A5) 772.0 14/9 (a bit 11/7-ish)
Sixths
Diminished sixth (d6) 738.5 32/21
Minor sixth (m6) 814.0 8/5
Major sixth (M6) 889.5 5/3
Augmented sixth (A6) 965.0 7/4
Sevenths
Diminished seventh (d7) 931.5 12/7
Minor seventh (m7) 1007.0 16/9~9/5
Major seventh (M7) 1082.5 15/8
Augmented seventh (A7) 1158.0 35/18~49/25~63/32
Octaves
Diminished octave (d8) 1124.5 27/14~48/25~40/21
Perfect octave (P8) 1200 2/1
Augmented octave (A8) 1275.5 25/12~21/10
  • In POTE septimal meantone
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