Quarter-comma meantone: Difference between revisions

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<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:~~xenwolf~~|~~xenwolf~~]] and made on <tt>2011-06-~~22 15~~:14:44 UTC</tt>.<br>

: This revision was by author [[User:guest|guest]] and made on <tt>2011-11-07 12:37:48 UTC</tt>.<br>

: The original revision id was <tt>~~238235303~~</tt>.<br>

: The original revision id was <tt>272769030</tt>.<br>

: The revision comment was: <tt>~~interval table: precision, typogr., col. heads~~</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">Quarter comma meantone is the tuning of [[Meantone family|meantone temperament]] which makes the fifth the fourth root of 5, or in other words 696.~~478~~ cents. This means the fifth is flattened by 1/4 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 5.377 cents, hence the name quarter-comma or 1/4-comma meantone. It is the minimax tuning for 5-limit meantone, meaning the maximum error on the 5-limit [[Diamonds|diamond]] is minimized. It is also the minimax tuning for septimal meantone in the 7- and 9-limit odd limits, and for meanpop, the version of 11-limit meantone which tunes 11/8 to the doubly diminished fifth, C-Gbb. Moreover, historically it was the predominant tuning of Western common-practice music in the latter part of the Renaissance and the early modern (17th century) era.

<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">Quarter comma meantone is the tuning of [[Meantone family|meantone temperament]] which makes the fifth the fourth root of 5, or in other words 696.578 cents. This means the fifth is flattened by 1/4 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 5.377 cents, hence the name quarter-comma or 1/4-comma meantone. It is the minimax tuning for 5-limit meantone, meaning the maximum error on the 5-limit [[Diamonds|diamond]] is minimized. It is also the minimax tuning for septimal meantone in the 7- and 9-limit odd limits, and for meanpop, the version of 11-limit meantone which tunes 11/8 to the doubly diminished fifth, C-Gbb. Moreover, historically it was the predominant tuning of Western common-practice music in the latter part of the Renaissance and the early modern (17th century) era.

Because of all of these features, it has a certain claim to be considered the canonical meantone tuning and often "meantone" is taken to mean quarter-comma specifically. The traditional interval names, listed below, may be considered to have their older traditional tuning in the quarter-comma meantone tunings listed below. However, other tunings (besides [[12edo]] which is ubiquitous but not an especially good tuning for meantone) may be preferred. Note, for instance, that the doubly augmented second and the doubly diminished fourth are both neutral thirds, about six cents apart. If we want a tuning which will equate the two, we need to use [[31edo]] instead.

=Intervals of quarter-comma meantone=

Below is a table listing 36 of the notes of quarter-comma meantone, giving the size in cents, the number of fifths up or down on the [[chain of fifths]], the [[http://en.wikipedia.org/wiki/Letter_notation|letter notation]] for the interval relative to C as the tonic, the traditional name for the interval, a shorthand form of the name, and the size of the interval in [[Fractional monzos|fractional monzo]] notation.

||~ Size

in [[cent]]s ||~ Fifths

dist. ||~ Note

C-based ||~ Traditional

Interval name ||~ Int.

(short)||~ Fract.

(short) ||~ Fract.

[[Fractional monzos|Monzo]] ||

||= 0 ||= 0 || C || Unison ||= P1 || [0 0 0> ||

Line 24:

|| 76.0490 ||= 7 || C# || Augmented unison, chromatic semitone ||= A1 || [-4 0 7/4> ||

|| 117.108 ||= -5 || Db || Minor second, diatonic semitone ||= m2 || [3 0 -5/4> ||

|| 152.098 ||= 14 || C## || Augmented chromatic semitone || || [-8 0 7/2> ||

|| 158.167 ||= -17 || Ebbb || Doubly diminished third || || [10 0 -17/4> ||

|| 193.157 ||= 2 || D || Major second ||= M2 || [-1 0 1/2> ||

|| 234.216 ||= -10 || Ebb || Diminished third ||= d3 || [6 0 -5/2> ||

|| 269.206 ||= 9 || D# || Augmented second ||= A2 || [-5 0 9/4> ||

|| 310.265 ||= -3 || Eb || Minor third ||= m3 || [2 0 -3/4> ||

|| 345.255 ||= 16 || D## || Doubly augmented second || || [-9 0 4> ||

|| 351.324 ||= -15 || Fbb || Doubly diminished fourth || || [9 0 -15/4> ||

|| 386.314 ||= 4 || E || Major third ||= M3 || [-2 0 1> ||

|| 427.373 ||= -8 || Fb || Diminished fourth ||= d4 || [5 0 -2> ||

|| 462.363 ||= 11 || E# || Augmented third ||= A3 || [-6 0 11/4> ||

|| 503.422 ||= -1 || F || "Perfect" fourth ||= P4 || [1 0 -1/4> ||

|| 544.480 ||= -13 || Gbb || Doubly diminished fifth || || [8 0 -13/4> ||

|| 579.471 ||= 6 || F# || Augmented fourth ||= A4 || [-3 0 3/2> ||

|| 620.529 ||= -6 || Gb || Diminished fifth ||= d5 || [4 0 -3/2> ||

|| 655.520 ||= 13 || F## || Doubly augmented fourth || || [-7 0 13/4> ||

|| 696.578 ||= 1 || G || "Perfect" fifth ||= P5 || [0 0 1/4> ||

|| 737.637 ||= -11 || Abb || Diminshed sixth||= d6 || [7 0 -11/4> ||

|| 737.637 ||= -11 || Abb || Diminshed sixth ||= d6 || [7 0 -11/4> ||

|| 772.627 ||= 8 || G# || Augmented fifth ||= A5 || [-4 0 2> ||

|| 813.686 ||= -4 || Ab || Minor sixth ||= m6 || [3 0 -1> ||

|| 848.676 ||= 15 || G## || Doubly augmented fifth || || [-8 0 15/4> ||

|| 854.745 ||= -16 || Bbbb || Doubly diminished seventh || || [10 0 -4> ||

|| 889.735 ||= 3 || A || Major sixth ||= M6 || [-1 0 3/4> ||

|| 930.794 ||= -9 || Bbb || Diminished seventh ||= d7 || [6 0 -9/4> ||

|| 965.784 ||= 10 || A# || Augmented sixth ||= A6 || [-5 0 5/2> ||

|| 1006.84 ||= -2 || Bb || Minor seventh ||= m7 || [2 0 -1/2> ||

|| 1041.83 ||= 17 || A## || Doubly augmented sixth || || [-9 0 17/4> ||

|| 1047.90 ||= -14 || C'bb || Doubly diminished octave || || [9 0 -7/2> ||

|| 1082.89 ||= 5 || B || Major seventh ||= M7 || [-2 0 5/4> ||

|| 1123.95 ||= -7 || C'b || Diminished octave ||= d8 || [5 0 -7/4> ||

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See [[http://en.wikipedia.org/wiki/Quarter-comma_meantone|Quarter-comma meantone]]

==Music==

[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/reelrock.mp3|Reelin' and Rockin']] by [[http://en.wikipedia.org/wiki/Chuck_Berry|Chuck Berry]]

[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Le%20Tic-Toc%20Choc.mp3|Le Tic-Toc Choc]] by [[http://en.wikipedia.org/wiki/Fran%C3%A7ois_Couperin|François Couperin]]

[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Prelude%20in%20g%20Op%2023%20%235.mp3|Prelude in g minor Op 23 #5]] by [[http://en.wikipedia.org/wiki/Sergei_Rachmaninoff|Sergei Rachmaninoff]]

[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/At%20the%20Hop.mp3|At the Hop]] by [[http://en.wikipedia.org/wiki/Arthur_Singer|Singer]], [[http://en.wikipedia.org/wiki/John_Medora|Medora]] and [[http://en.wikipedia.org/wiki/David_White_~~(musician)~~|White]]

[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/At%20the%20Hop.mp3|At the Hop]] by [[http://en.wikipedia.org/wiki/Arthur_Singer|Singer]], [[http://en.wikipedia.org/wiki/John_Medora|Medora]] and [[http://en.wikipedia.org/wiki/David_White_%28musician%29|White]]

[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Meistersinger%20Prelude.mp3|Prelude to Die Meistersinger von Nürnberg]] by [[http://en.wikipedia.org/wiki/Richard_Wagner|Richard Wagner]]

[[http://www.youtube.com/watch?v=b68Ek5hDgDo|Laura che il verde lauro]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Nicola%20Vicentino%20~~(1511~~-%20c.1576)%20-%20Laura%20che%20il%20verde%20lauro.mp3|play]] by [[http://en.wikipedia.org/wiki/Nicola_Vicentino|Nicola Vicentino]]

[[http://www.youtube.com/watch?v=b68Ek5hDgDo|Laura che il verde lauro]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Nicola%20Vicentino%20%281511-%20c.1576%29%20-%20Laura%20che%20il%20verde%20lauro.mp3|play]] by [[http://en.wikipedia.org/wiki/Nicola_Vicentino|Nicola Vicentino]]

[[http://youtu.be/6t5PdeC1Trw|Country Bumpkin]] by [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Country%20Bumpkin%20Live%20At%20The%20Turks%2023,9,2010ce.mp3|play]] [[Mark Allan Barnes]]

Five canons by [[Claudi Meneghin]]:

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[[http://www.youtube.com/watch?v=6_YVhwArGTI|Canon 2 in 1 on "Ah! vous dirai-je, maman" at the diminished fifth]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Claudi%20Meneghin_%20Canon%202%20in%201%20on%20_Ah_%20vous%20dirai-je,%20maman_%20.mp3|play]]</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"><html><head><title>Quarter-comma meantone</title></head><body>Quarter comma meantone is the tuning of <a class="wiki_link" href="/Meantone%20family">meantone temperament</a> which makes the fifth the fourth root of 5, or in other words 696.~~478~~ cents. This means the fifth is flattened by 1/4 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 5.377 cents, hence the name quarter-comma or 1/4-comma meantone. It is the minimax tuning for 5-limit meantone, meaning the maximum error on the 5-limit <a class="wiki_link" href="/Diamonds">diamond</a> is minimized. It is also the minimax tuning for septimal meantone in the 7- and 9-limit odd limits, and for meanpop, the version of 11-limit meantone which tunes 11/8 to the doubly diminished fifth, C-Gbb. Moreover, historically it was the predominant tuning of Western common-practice music in the latter part of the Renaissance and the early modern (17th century) era. <br />

<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"><html><head><title>Quarter-comma meantone</title></head><body>Quarter comma meantone is the tuning of <a class="wiki_link" href="/Meantone%20family">meantone temperament</a> which makes the fifth the fourth root of 5, or in other words 696.578 cents. This means the fifth is flattened by 1/4 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 5.377 cents, hence the name quarter-comma or 1/4-comma meantone. It is the minimax tuning for 5-limit meantone, meaning the maximum error on the 5-limit <a class="wiki_link" href="/Diamonds">diamond</a> is minimized. It is also the minimax tuning for septimal meantone in the 7- and 9-limit odd limits, and for meanpop, the version of 11-limit meantone which tunes 11/8 to the doubly diminished fifth, C-Gbb. Moreover, historically it was the predominant tuning of Western common-practice music in the latter part of the Renaissance and the early modern (17th century) era.<br />

<br />

Because of all of these features, it has a certain claim to be considered the canonical meantone tuning and often &quot;meantone&quot; is taken to mean quarter-comma specifically. The traditional interval names, listed below, may be considered to have their older traditional tuning in the quarter-comma meantone tunings listed below. However, other tunings (besides <a class="wiki_link" href="/12edo">12edo</a> which is ubiquitous but not an especially good tuning for meantone) may be preferred. Note, for instance, that the doubly augmented second and the doubly diminished fourth are both neutral thirds, about six cents apart. If we want a tuning which will equate the two, we need to use <a class="wiki_link" href="/31edo">31edo</a> instead.<br />

<br />

<h1 id="toc0"><a name="Intervals of quarter-comma meantone"></a>Intervals of quarter-comma meantone</h1>

Below is a table listing 36 of the notes of quarter-comma meantone, giving the size in cents, the number of fifths up or down on the <a class="wiki_link" href="/chain%20of%20fifths">chain of fifths</a>, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Letter_notation" rel="nofollow">letter notation</a> for the interval relative to C as the tonic, the traditional name for the interval, a shorthand form of the name, and the size of the interval in <a class="wiki_link" href="/Fractional%20monzos">fractional monzo</a> notation.<br />

<br />

Line 85:

<tr>

in <a class="wiki_link" href="/cent">cent</a>s<br />

</th>

<th>Fifths <br />

<th>Fifths<br />

dist.<br />

</th>

Line 94:

C-based<br />

</th>

<th>Traditional <br />

<th>Traditional<br />

Interval name<br />

</th>

(short)<br />

</th>

Line 614:

<br />

<h2 id="toc1"><a name="Intervals of quarter-comma meantone-Music"></a>Music</h2>

<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/reelrock.mp3" rel="nofollow">Reelin' and Rockin'</a> by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Chuck_Berry" rel="nofollow">Chuck Berry</a><br />

<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Le%20Tic-Toc%20Choc.mp3" rel="nofollow">Le Tic-Toc Choc</a> by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Fran%C3%A7ois_Couperin" rel="nofollow">François Couperin</a><br />

<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Prelude%20in%20g%20Op%2023%20%235.mp3" rel="nofollow">Prelude in g minor Op 23 #5</a> by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Sergei_Rachmaninoff" rel="nofollow">Sergei Rachmaninoff</a><br />

<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/At%20the%20Hop.mp3" rel="nofollow">At the Hop</a> by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Arthur_Singer" rel="nofollow">Singer</a>, <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/John_Medora" rel="nofollow">Medora</a> and <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/David_White_~~(musician)~~" rel="nofollow">White</a><br />

<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/At%20the%20Hop.mp3" rel="nofollow">At the Hop</a> by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Arthur_Singer" rel="nofollow">Singer</a>, <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/John_Medora" rel="nofollow">Medora</a> and <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/David_White_%28musician%29" rel="nofollow">White</a><br />

<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Meistersinger%20Prelude.mp3" rel="nofollow">Prelude to Die Meistersinger von Nürnberg</a> by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Richard_Wagner" rel="nofollow">Richard Wagner</a><br />

<a class="wiki_link_ext" href="http://www.youtube.com/watch?v=b68Ek5hDgDo" rel="nofollow">Laura che il verde lauro</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Nicola%20Vicentino%20~~(1511~~-%20c.1576)%20-%20Laura%20che%20il%20verde%20lauro.mp3" rel="nofollow">play</a> by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Nicola_Vicentino" rel="nofollow">Nicola Vicentino</a><br />

<a class="wiki_link_ext" href="http://www.youtube.com/watch?v=b68Ek5hDgDo" rel="nofollow">Laura che il verde lauro</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Nicola%20Vicentino%20%281511-%20c.1576%29%20-%20Laura%20che%20il%20verde%20lauro.mp3" rel="nofollow">play</a> by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Nicola_Vicentino" rel="nofollow">Nicola Vicentino</a><br />

<a class="wiki_link_ext" href="http://youtu.be/6t5PdeC1Trw" rel="nofollow">Country Bumpkin</a> by <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Country%20Bumpkin%20Live%20At%20The%20Turks%2023,9,2010ce.mp3" rel="nofollow">play</a> <a class="wiki_link" href="/Mark%20Allan%20Barnes">Mark Allan Barnes</a><br />

Five canons by <a class="wiki_link" href="/Claudi%20Meneghin">Claudi Meneghin</a>:<br />

@@ 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:xenwolf|xenwolf]] and made on <tt>2011-06-22 15:14:44 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2011-11-07 12:37:48 UTC</tt>.<br>
-: The original revision id was <tt>238235303</tt>.<br>
+: The original revision id was <tt>272769030</tt>.<br>
-: The revision comment was: <tt>interval table: precision, typogr., col. heads</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">Quarter comma meantone is the tuning of [[Meantone family|meantone temperament]] which makes the fifth the fourth root of 5, or in other words 696.478 cents. This means the fifth is flattened by 1/4 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 5.377 cents, hence the name quarter-comma or 1/4-comma meantone. It is the minimax tuning for 5-limit meantone, meaning the maximum error on the 5-limit [[Diamonds|diamond]] is minimized. It is also the minimax tuning for septimal meantone in the 7- and 9-limit odd limits, and for meanpop, the version of 11-limit meantone which tunes 11/8 to the doubly diminished fifth, C-Gbb. Moreover, historically it was the predominant tuning of Western common-practice music in the latter part of the Renaissance and the early modern (17th century) era.
+<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">Quarter comma meantone is the tuning of [[Meantone family|meantone temperament]] which makes the fifth the fourth root of 5, or in other words 696.578 cents. This means the fifth is flattened by 1/4 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 5.377 cents, hence the name quarter-comma or 1/4-comma meantone. It is the minimax tuning for 5-limit meantone, meaning the maximum error on the 5-limit [[Diamonds|diamond]] is minimized. It is also the minimax tuning for septimal meantone in the 7- and 9-limit odd limits, and for meanpop, the version of 11-limit meantone which tunes 11/8 to the doubly diminished fifth, C-Gbb. Moreover, historically it was the predominant tuning of Western common-practice music in the latter part of the Renaissance and the early modern (17th century) era.
 Because of all of these features, it has a certain claim to be considered the canonical meantone tuning and often "meantone" is taken to mean quarter-comma specifically. The traditional interval names, listed below, may be considered to have their older traditional tuning in the quarter-comma meantone tunings listed below. However, other tunings (besides [[12edo]] which is ubiquitous but not an especially good tuning for meantone) may be preferred. Note, for instance, that the doubly augmented second and the doubly diminished fourth are both neutral thirds, about six cents apart. If we want a tuning which will equate the two, we need to use [[31edo]] instead.
 =Intervals of quarter-comma meantone=
 Below is a table listing 36 of the notes of quarter-comma meantone, giving the size in cents, the number of fifths up or down on the [[chain of fifths]], the [[http://en.wikipedia.org/wiki/Letter_notation|letter notation]] for the interval relative to C as the tonic, the traditional name for the interval, a shorthand form of the name, and the size of the interval in [[Fractional monzos|fractional monzo]] notation.
 ||~ Size
 in [[cent]]s ||~ Fifths
 dist. ||~ Note
 C-based ||~ Traditional
 Interval name ||~ Int.
-(short)||~ Fract.
+(short) ||~ Fract.
 [[Fractional monzos|Monzo]] ||
 ||= 0 ||= 0 || C || Unison ||= P1 || [0 0 0&gt; ||
@@ Line 24: / Line 24: @@
 || 76.0490 ||= 7 || C# || Augmented unison, chromatic semitone ||= A1 || [-4 0 7/4&gt; ||
 || 117.108 ||= -5 || Db || Minor second, diatonic semitone ||= m2 || [3 0 -5/4&gt; ||
-|| 152.098 ||= 14 || C## || Augmented chromatic semitone || || [-8 0 7/2&gt; ||
+|| 152.098 ||= 14 || C## || Augmented chromatic semitone ||   || [-8 0 7/2&gt; ||
-|| 158.167 ||= -17 || Ebbb || Doubly diminished third || || [10 0 -17/4&gt; ||
+|| 158.167 ||= -17 || Ebbb || Doubly diminished third ||   || [10 0 -17/4&gt; ||
 || 193.157 ||= 2 || D || Major second ||= M2 || [-1 0 1/2&gt; ||
 || 234.216 ||= -10 || Ebb || Diminished third ||= d3 || [6 0 -5/2&gt; ||
 || 269.206 ||= 9 || D# || Augmented second ||= A2 || [-5 0 9/4&gt; ||
 || 310.265 ||= -3 || Eb || Minor third ||= m3 || [2 0 -3/4&gt; ||
-|| 345.255 ||= 16 || D## || Doubly augmented second || || [-9 0 4&gt; ||
+|| 345.255 ||= 16 || D## || Doubly augmented second ||   || [-9 0 4&gt; ||
-|| 351.324 ||= -15 || Fbb || Doubly diminished fourth || || [9 0 -15/4&gt; ||
+|| 351.324 ||= -15 || Fbb || Doubly diminished fourth ||   || [9 0 -15/4&gt; ||
 || 386.314 ||= 4 || E || Major third ||= M3 || [-2 0 1&gt; ||
 || 427.373 ||= -8 || Fb || Diminished fourth ||= d4 || [5 0 -2&gt; ||
 || 462.363 ||= 11 || E# || Augmented third ||= A3 || [-6 0 11/4&gt; ||
 || 503.422 ||= -1 || F || "Perfect" fourth ||= P4 || [1 0 -1/4&gt; ||
-|| 544.480 ||= -13 || Gbb || Doubly diminished fifth || || [8 0 -13/4&gt; ||
+|| 544.480 ||= -13 || Gbb || Doubly diminished fifth ||   || [8 0 -13/4&gt; ||
 || 579.471 ||= 6 || F# || Augmented fourth ||= A4 || [-3 0 3/2&gt; ||
 || 620.529 ||= -6 || Gb || Diminished fifth ||= d5 || [4 0 -3/2&gt; ||
-|| 655.520 ||= 13 || F## || Doubly augmented fourth || || [-7 0 13/4&gt; ||
+|| 655.520 ||= 13 || F## || Doubly augmented fourth ||   || [-7 0 13/4&gt; ||
 || 696.578 ||= 1 || G || "Perfect" fifth ||= P5 || [0 0 1/4&gt; ||
-|| 737.637 ||= -11 || Abb || Diminshed sixth||= d6 || [7 0 -11/4&gt; ||
+|| 737.637 ||= -11 || Abb || Diminshed sixth ||= d6 || [7 0 -11/4&gt; ||
 || 772.627 ||= 8 || G# || Augmented fifth ||= A5 || [-4 0 2&gt; ||
 || 813.686 ||= -4 || Ab || Minor sixth ||= m6 || [3 0 -1&gt; ||
-|| 848.676 ||= 15 || G## || Doubly augmented fifth || || [-8 0 15/4&gt; ||
+|| 848.676 ||= 15 || G## || Doubly augmented fifth ||   || [-8 0 15/4&gt; ||
-|| 854.745 ||= -16 || Bbbb || Doubly diminished seventh || || [10 0 -4&gt; ||
+|| 854.745 ||= -16 || Bbbb || Doubly diminished seventh ||   || [10 0 -4&gt; ||
 || 889.735 ||= 3 || A || Major sixth ||= M6 || [-1 0 3/4&gt; ||
 || 930.794 ||= -9 || Bbb || Diminished seventh ||= d7 || [6 0 -9/4&gt; ||
 || 965.784 ||= 10 || A# || Augmented sixth ||= A6 || [-5 0 5/2&gt; ||
 || 1006.84 ||= -2 || Bb || Minor seventh ||= m7 || [2 0 -1/2&gt; ||
-|| 1041.83 ||= 17 || A## || Doubly augmented sixth || || [-9 0 17/4&gt; ||
+|| 1041.83 ||= 17 || A## || Doubly augmented sixth ||   || [-9 0 17/4&gt; ||
-|| 1047.90 ||= -14 || C'bb || Doubly diminished octave || || [9 0 -7/2&gt; ||
+|| 1047.90 ||= -14 || C'bb || Doubly diminished octave ||   || [9 0 -7/2&gt; ||
 || 1082.89 ||= 5 || B || Major seventh ||= M7 || [-2 0 5/4&gt; ||
 || 1123.95 ||= -7 || C'b || Diminished octave ||= d8 || [5 0 -7/4&gt; ||
@@ Line 59: / Line 59: @@
 See [[http://en.wikipedia.org/wiki/Quarter-comma_meantone|Quarter-comma meantone]]
 ==Music==
 [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/reelrock.mp3|Reelin' and Rockin']] by [[http://en.wikipedia.org/wiki/Chuck_Berry|Chuck Berry]]
 [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Le%20Tic-Toc%20Choc.mp3|Le Tic-Toc Choc]] by [[http://en.wikipedia.org/wiki/Fran%C3%A7ois_Couperin|François Couperin]]
 [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Prelude%20in%20g%20Op%2023%20%235.mp3|Prelude in g minor Op 23 #5]] by [[http://en.wikipedia.org/wiki/Sergei_Rachmaninoff|Sergei Rachmaninoff]]
-[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/At%20the%20Hop.mp3|At the Hop]] by [[http://en.wikipedia.org/wiki/Arthur_Singer|Singer]], [[http://en.wikipedia.org/wiki/John_Medora|Medora]] and [[http://en.wikipedia.org/wiki/David_White_(musician)|White]]
+[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/At%20the%20Hop.mp3|At the Hop]] by [[http://en.wikipedia.org/wiki/Arthur_Singer|Singer]], [[http://en.wikipedia.org/wiki/John_Medora|Medora]] and [[http://en.wikipedia.org/wiki/David_White_%28musician%29|White]]
 [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Meistersinger%20Prelude.mp3|Prelude to Die Meistersinger von Nürnberg]] by [[http://en.wikipedia.org/wiki/Richard_Wagner|Richard Wagner]]
-[[http://www.youtube.com/watch?v=b68Ek5hDgDo|Laura che il verde lauro]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Nicola%20Vicentino%20(1511-%20c.1576)%20-%20Laura%20che%20il%20verde%20lauro.mp3|play]] by [[http://en.wikipedia.org/wiki/Nicola_Vicentino|Nicola Vicentino]]
+[[http://www.youtube.com/watch?v=b68Ek5hDgDo|Laura che il verde lauro]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Nicola%20Vicentino%20%281511-%20c.1576%29%20-%20Laura%20che%20il%20verde%20lauro.mp3|play]] by [[http://en.wikipedia.org/wiki/Nicola_Vicentino|Nicola Vicentino]]
 [[http://youtu.be/6t5PdeC1Trw|Country Bumpkin]] by [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Country%20Bumpkin%20Live%20At%20The%20Turks%2023,9,2010ce.mp3|play]] [[Mark Allan Barnes]]
 Five canons by [[Claudi Meneghin]]:
@@ Line 74: / Line 74: @@
 [[http://www.youtube.com/watch?v=6_YVhwArGTI|Canon 2 in 1 on "Ah! vous dirai-je, maman" at the diminished fifth]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Claudi%20Meneghin_%20Canon%202%20in%201%20on%20_Ah_%20vous%20dirai-je,%20maman_%20.mp3|play]]</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;Quarter-comma meantone&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Quarter comma meantone is the tuning of &lt;a class="wiki_link" href="/Meantone%20family"&gt;meantone temperament&lt;/a&gt; which makes the fifth the fourth root of 5, or in other words 696.478 cents. This means the fifth is flattened by 1/4 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 5.377 cents, hence the name quarter-comma or 1/4-comma meantone. It is the minimax tuning for 5-limit meantone, meaning the maximum error on the 5-limit &lt;a class="wiki_link" href="/Diamonds"&gt;diamond&lt;/a&gt; is minimized. It is also the minimax tuning for septimal meantone in the 7- and 9-limit odd limits, and for meanpop, the version of 11-limit meantone which tunes 11/8 to the doubly diminished fifth, C-Gbb. Moreover, historically it was the predominant tuning of Western common-practice music in the latter part of the Renaissance and the early modern (17th century) era. &lt;br /&gt;
+<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;Quarter-comma meantone&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Quarter comma meantone is the tuning of &lt;a class="wiki_link" href="/Meantone%20family"&gt;meantone temperament&lt;/a&gt; which makes the fifth the fourth root of 5, or in other words 696.578 cents. This means the fifth is flattened by 1/4 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 5.377 cents, hence the name quarter-comma or 1/4-comma meantone. It is the minimax tuning for 5-limit meantone, meaning the maximum error on the 5-limit &lt;a class="wiki_link" href="/Diamonds"&gt;diamond&lt;/a&gt; is minimized. It is also the minimax tuning for septimal meantone in the 7- and 9-limit odd limits, and for meanpop, the version of 11-limit meantone which tunes 11/8 to the doubly diminished fifth, C-Gbb. Moreover, historically it was the predominant tuning of Western common-practice music in the latter part of the Renaissance and the early modern (17th century) era.&lt;br /&gt;
 &lt;br /&gt;
 Because of all of these features, it has a certain claim to be considered the canonical meantone tuning and often &amp;quot;meantone&amp;quot; is taken to mean quarter-comma specifically. The traditional interval names, listed below, may be considered to have their older traditional tuning in the quarter-comma meantone tunings listed below. However, other tunings (besides &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; which is ubiquitous but not an especially good tuning for meantone) may be preferred. Note, for instance, that the doubly augmented second and the doubly diminished fourth are both neutral thirds, about six cents apart. If we want a tuning which will equate the two, we need to use &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; instead.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Intervals of quarter-comma meantone"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Intervals of quarter-comma meantone&lt;/h1&gt;
-Below is a table listing 36 of the notes of quarter-comma meantone, giving the size in cents, the number of fifths up or down on the &lt;a class="wiki_link" href="/chain%20of%20fifths"&gt;chain of fifths&lt;/a&gt;, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Letter_notation" rel="nofollow"&gt;letter notation&lt;/a&gt; for the interval relative to C as the tonic, the traditional name for the interval, a shorthand form of the name, and the size of the interval in &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;fractional monzo&lt;/a&gt; notation.&lt;br /&gt;
+ Below is a table listing 36 of the notes of quarter-comma meantone, giving the size in cents, the number of fifths up or down on the &lt;a class="wiki_link" href="/chain%20of%20fifths"&gt;chain of fifths&lt;/a&gt;, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Letter_notation" rel="nofollow"&gt;letter notation&lt;/a&gt; for the interval relative to C as the tonic, the traditional name for the interval, a shorthand form of the name, and the size of the interval in &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;fractional monzo&lt;/a&gt; notation.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 85: / Line 85: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;th&gt;Size &lt;br /&gt;
+         &lt;th&gt;Size&lt;br /&gt;
 in &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;Fifths &lt;br /&gt;
+         &lt;th&gt;Fifths&lt;br /&gt;
 dist.&lt;br /&gt;
 &lt;/th&gt;
@@ Line 94: / Line 94: @@
 C-based&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;Traditional &lt;br /&gt;
+         &lt;th&gt;Traditional&lt;br /&gt;
 Interval name&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;Int. &lt;br /&gt;
+         &lt;th&gt;Int.&lt;br /&gt;
 (short)&lt;br /&gt;
 &lt;/th&gt;
@@ Line 614: / Line 614: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Intervals of quarter-comma meantone-Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Music&lt;/h2&gt;
-&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/reelrock.mp3" rel="nofollow"&gt;Reelin' and Rockin'&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Chuck_Berry" rel="nofollow"&gt;Chuck Berry&lt;/a&gt;&lt;br /&gt;
+ &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/reelrock.mp3" rel="nofollow"&gt;Reelin' and Rockin'&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Chuck_Berry" rel="nofollow"&gt;Chuck Berry&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Le%20Tic-Toc%20Choc.mp3" rel="nofollow"&gt;Le Tic-Toc Choc&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Fran%C3%A7ois_Couperin" rel="nofollow"&gt;François Couperin&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Prelude%20in%20g%20Op%2023%20%235.mp3" rel="nofollow"&gt;Prelude in g minor Op 23 #5&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Sergei_Rachmaninoff" rel="nofollow"&gt;Sergei Rachmaninoff&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/At%20the%20Hop.mp3" rel="nofollow"&gt;At the Hop&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Arthur_Singer" rel="nofollow"&gt;Singer&lt;/a&gt;, &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/John_Medora" rel="nofollow"&gt;Medora&lt;/a&gt; and &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/David_White_(musician)" rel="nofollow"&gt;White&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/At%20the%20Hop.mp3" rel="nofollow"&gt;At the Hop&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Arthur_Singer" rel="nofollow"&gt;Singer&lt;/a&gt;, &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/John_Medora" rel="nofollow"&gt;Medora&lt;/a&gt; and &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/David_White_%28musician%29" rel="nofollow"&gt;White&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Meistersinger%20Prelude.mp3" rel="nofollow"&gt;Prelude to Die Meistersinger von Nürnberg&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Richard_Wagner" rel="nofollow"&gt;Richard Wagner&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=b68Ek5hDgDo" rel="nofollow"&gt;Laura che il verde lauro&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Nicola%20Vicentino%20(1511-%20c.1576)%20-%20Laura%20che%20il%20verde%20lauro.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Nicola_Vicentino" rel="nofollow"&gt;Nicola Vicentino&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=b68Ek5hDgDo" rel="nofollow"&gt;Laura che il verde lauro&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Nicola%20Vicentino%20%281511-%20c.1576%29%20-%20Laura%20che%20il%20verde%20lauro.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Nicola_Vicentino" rel="nofollow"&gt;Nicola Vicentino&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://youtu.be/6t5PdeC1Trw" rel="nofollow"&gt;Country Bumpkin&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Country%20Bumpkin%20Live%20At%20The%20Turks%2023,9,2010ce.mp3" rel="nofollow"&gt;play&lt;/a&gt; &lt;a class="wiki_link" href="/Mark%20Allan%20Barnes"&gt;Mark Allan Barnes&lt;/a&gt;&lt;br /&gt;
 Five canons by &lt;a class="wiki_link" href="/Claudi%20Meneghin"&gt;Claudi Meneghin&lt;/a&gt;:&lt;br /&gt;