Meantone intervals: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Wikipedia|List of meantone intervals}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-10-31 19:40:52 UTC</tt>.<br>
: The original revision id was <tt>270455766</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">||~ Name ||~ Size* ||~ Ratios ||
||||||~ Unisons ||
|| Perfect unison (P1) || 0 || 1/1 ||
|| Augmented unison (A1) || 75.5 || 25/24 ||
||||||~ Seconds ||
|| Diminished second (d2) || 42.0 || 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 ||
|| Perfect fourth (P4) || 503.5 || 4/3 ||
|| Augmented fourth (A4) || 579.0 || 7/6 ||
|| 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 ||
||||||~ 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 ||
||||||~ Octaves ||
|| Diminished octave (d8) || 1124.5 || 48/25 ||
|| Perfect octave (P8) || 1200.0 || 2/1 ||
|| Augmented octave (A8) || 1275.5 || 25/12 ||</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;


&lt;table class="wiki_table"&gt;
This table shows all the simple intervals of [[POTE]] [[Meantone family #Septimal meantone|septimal meantone]], which includes the entire [[7-odd-limit]] [[tonality diamond]]. Other relevant tables of meantone intervals are the table of [[quarter-comma meantone]] intervals and the table of [[31edo #Intervals|31edo intervals]]. Intervals in limits higher than 7 are in brackets.
    &lt;tr&gt;
        &lt;th&gt;Name&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Size*&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Ratios&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th colspan="3"&gt;Unisons&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Perfect unison (P1)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Augmented unison (A1)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;75.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;25/24&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th colspan="3"&gt;Seconds&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Diminished second (d2)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;42.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;36/35&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Minor second (m2)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;117.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/15~15/14&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Major second (M2)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;193.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/9~9/8&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Augmented second (A2)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;268.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th colspan="3"&gt;Thirds&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Diminished third (d3)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;235.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Minor third (m3)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;310.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Major third (M3)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;386.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Augmented third (A3)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;461.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;21/16&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th colspan="3"&gt;Fourths&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Diminished fourth (d4)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;428.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Perfect fourth (P4)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;503.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Augmented fourth (A4)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;579.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;Double augmented fourth (AA4)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;654.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Close to 16/11&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &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&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&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;48/25&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;1200.0&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&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
In [[12edo]] the diminished second vanishes, so this cornucopia of intervals collapses to a mere 12. Except arguably for the tritone, none of the intervals are inherently septimal in 12edo, because they all have simpler 5-limit interpretations.
 
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 tunings such as [[31edo]] distinguish all intervals listed on this table.
 
{| class="wikitable right-2"
|-
! Name
! Size<br>(cents)
! Ratios
|-
! colspan="3" | Unisons
|-
| Perfect unison (P1)
| 0.00
| [[1/1]]
|-
| Augmented unison (A1)
| 75.46
| [[21/20]], [[25/24]], [[28/27]]
|-
| Double-augmented unison (AA1)
| 150.93
| [[35/32]], [[12/11|(12/11)]]
|-
! colspan="3" | Seconds
|-
| Diminished second (d2)
| 42.06
| [[36/35]], [[50/49]], [[64/63]], [[128/125]]
|-
| Minor second (m2)
| 117.53
| [[15/14]], [[16/15]]
|-
| Major second (M2)
| 192.99
| [[9/8]], [[10/9]], [[28/25]]
|-
| Augmented second (A2)
| 268.45
| [[7/6]]
|-
! colspan="3" | Thirds
|-
| Diminished third (d3)
| 235.05
| [[8/7]]
|-
| Minor third (m3)
| 310.52
| [[6/5]]
|-
| Major third (M3)
| 385.98
| [[5/4]]
|-
| Augmented third (A3)
| 461.44
| [[21/16]]
|-
! colspan="3" | Fourths
|-
| Double-diminished fourth (dd4)
| 352.58
| [[49/40]], [[11/9|(11/9)]]
|-
| Diminished fourth (d4)
| 428.04
| [[9/7]], [[14/11|(14/11)]]
|-
| Perfect fourth (P4)
| 503.51
| [[4/3]]
|-
| Augmented fourth (A4)
| 578.97
| [[7/5]]
|-
| Double-augmented fourth (AA4)
| 654.43
| [[35/24]], [[16/11|(16/11)]]
|-
! colspan="3" | Fifths
|-
| Double-diminished fifth (dd5)
| 545.57
| [[48/35]], [[11/8|(11/8)]]
|-
| Diminished fifth (d5)
| 621.03
| [[10/7]]
|-
| Perfect fifth (P5)
| 696.49
| [[3/2]]
|-
| Augmented fifth (A5)
| 771.96
| [[14/9]], [[11/7|(11/7)]]
|-
| Double-augmented fifth (AA5)
| 847.42
| [[80/49]], [[18/11|(18/11)]]
|-
! colspan="3" | Sixths
|-
| Diminished sixth (d6)
| 738.56
| [[32/21]]
|-
| Minor sixth (m6)
| 814.02
| [[8/5]]
|-
| Major sixth (M6)
| 889.48
| [[5/3]]
|-
| Augmented sixth (A6)
| 964.95
| [[7/4]]
|-
! colspan="3" | Sevenths
|-
| Diminished seventh (d7)
| 931.55
| [[12/7]]
|-
| Minor seventh (m7)
| 1007.01
| [[9/5]], [[16/9]], [[25/14]]
|-
| Major seventh (M7)
| 1082.47
| [[15/8]], [[28/15]]
|-
| Augmented seventh (A7)
| 1157.94
| [[35/18]], [[49/25]], [[63/32]]
|-
! colspan="3" | Octaves
|-
| Double-diminished octave (dd8)
| 1049.07
| [[64/35]], [[11/6|(11/6)]]
|-
| Diminished octave (d8)
| 1124.54
| [[27/14]], [[40/21]], [[48/25]]
|-
| Perfect octave (P8)
| 1200.00
| [[2/1]]
|}
 
[[Category:Meantone]]
[[Category:Lists of intervals]]

Latest revision as of 04:43, 27 February 2026

English Wikipedia has an article on:

This table shows all the simple intervals of POTE septimal meantone, which includes the entire 7-odd-limit tonality diamond. Other relevant tables of meantone intervals are the table of quarter-comma meantone intervals and the table of 31edo intervals. Intervals in limits higher than 7 are in brackets.

In 12edo the diminished second vanishes, so this cornucopia of intervals collapses to a mere 12. Except arguably for the tritone, none of the intervals are inherently septimal in 12edo, because they all have simpler 5-limit interpretations.

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 tunings such as 31edo distinguish all intervals listed on this table.

Name Size
(cents) 		Ratios
Unisons
Perfect unison (P1) 0.00 1/1
Augmented unison (A1) 75.46 21/20, 25/24, 28/27
Double-augmented unison (AA1) 150.93 35/32, (12/11)
Seconds
Diminished second (d2) 42.06 36/35, 50/49, 64/63, 128/125
Minor second (m2) 117.53 15/14, 16/15
Major second (M2) 192.99 9/8, 10/9, 28/25
Augmented second (A2) 268.45 7/6
Thirds
Diminished third (d3) 235.05 8/7
Minor third (m3) 310.52 6/5
Major third (M3) 385.98 5/4
Augmented third (A3) 461.44 21/16
Fourths
Double-diminished fourth (dd4) 352.58 49/40, (11/9)
Diminished fourth (d4) 428.04 9/7, (14/11)
Perfect fourth (P4) 503.51 4/3
Augmented fourth (A4) 578.97 7/5
Double-augmented fourth (AA4) 654.43 35/24, (16/11)
Fifths
Double-diminished fifth (dd5) 545.57 48/35, (11/8)
Diminished fifth (d5) 621.03 10/7
Perfect fifth (P5) 696.49 3/2
Augmented fifth (A5) 771.96 14/9, (11/7)
Double-augmented fifth (AA5) 847.42 80/49, (18/11)
Sixths
Diminished sixth (d6) 738.56 32/21
Minor sixth (m6) 814.02 8/5
Major sixth (M6) 889.48 5/3
Augmented sixth (A6) 964.95 7/4
Sevenths
Diminished seventh (d7) 931.55 12/7
Minor seventh (m7) 1007.01 9/5, 16/9, 25/14
Major seventh (M7) 1082.47 15/8, 28/15
Augmented seventh (A7) 1157.94 35/18, 49/25, 63/32
Octaves
Double-diminished octave (dd8) 1049.07 64/35, (11/6)
Diminished octave (d8) 1124.54 27/14, 40/21, 48/25
Perfect octave (P8) 1200.00 2/1
Retrieved from "https://en.xen.wiki/index.php?title=Meantone_intervals&oldid=224842"