7/4: 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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-09-23 13:51:06 UTC</tt>.<br>

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-09-23 15:44:25 UTC</tt>.<br>

: The original revision id was <tt>~~164931099~~</tt>.<br>

: The original revision id was <tt>164972615</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">Frequency ratio 7:4, measuring approximately 968.8259064691249 cents, ~~often gets labeled~~ the "harmonic seventh." It represents the interval between the 4th and 7th harmonics. It is also called a "septimal subminor seventh" -- the word "septimal" referring to the presence of a 7 as the highest prime in the ratio and the word "subminor" referring to ~~its~~ narrowness compared with a traditional minor seventh (such as [[9_5|9:5]] or [[16_9|16:9]], ~~10 degrees of~~ [[12edo]] or a minor seventh found in a meantone system).

<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">Frequency ratio 7:4, measuring approximately 968.8259064691249 cents, has been given the name "harmonic seventh." It represents the interval between the 4th and 7th harmonics in the [[OverToneSeries|overtone series]]. It is also called a "septimal subminor seventh" -- the word "septimal" referring to the presence of a 7 as the highest prime in the ratio, and the word "subminor" referring to the harmonic seventh's narrowness compared with a traditional minor seventh (such as [[9_5|9:5]] or [[16_9|16:9]], [[12edo]]'s 1000-cent interval, or a minor seventh found in a meantone system).

7:4 has seen use in blues music and some musical traditions of the world, but has mostly not been recognized as a "consonance" in Western music theory. ~~However, in~~ most Just Intonation systems, it is treated as a consonance in its own right, with its own distinct quality.

7:4 has seen use in blues music, barbershop quartet music, and some musical traditions of the world, but has mostly not been recognized as a "consonance" in Western music theory. In most Just Intonation systems, the harmonic seventh is treated as a fundamental consonance in its own right, with its own distinct quality.

=Harmonic Seventh Chord=

7:4 appears in an otonal tetrad that forms the basis of much JI music, commonly called a "harmonic seventh chord." It consists of a major triad (4:5:6) plus a harmonic seventh: 4:5:6:7(:8). This tetrad, a hallmark of blues and barbershop harmony, not to mention modern Just Intonation practice, represents a sequence of overtones from the fourth to the seventh. (8, being a doubling of 4, represents an octave above the root.) The intervals between adjacent members of the chord decrease in size:

~~==Otonal Tetrad==~~

~~7:4 appears in the otonal tetrad that forms the basis of much JI music:~~

~~===4:5:6:7:8===~~

This triad represents a sequence of overtones from the fourth to the eighth. (8, being a doubling of 4, represents an octave above the root.) The intervals between adjacent members of the chord decrease in size:

[[5_4|5:4]] - approx. 386 cents - a major third

[[6_5|6:5]] - approx. 316 cents - a minor third

[[7_6|7:6]] - approx. 267 cents - a septimal subminor third

8:7 - approx. 231 cents - a septimal supermajor seventh

[[8_7|8:7]] - approx. 231 cents - a septimal supermajor second

This chord is similar to the "dominant seventh chord" in 12edo, the most significant difference being the mistuning of the harmonic seventh. In 12edo, the interval closest to the harmonic seventh is the minor seventh at 1000 cents. The difference of about 31 cents is striking, and especially noticable when the chords are presented next to one another. While the "dominant seventh chord" of 12edo is treated as a dissonance that needs to resolve (usually in the chord pattern V7 to I), the "harmonic seventh chord" has a much different flavor and is often treated by composers in Just Intonation as a consonance.

Another interval found in a harmonic seventh chord is the septimal tritone of [[7_5|7:5]], which represents the interval between the major third (5) and the harmonic seventh (7). This interval, at 583 cents, sounds distinct from 12edo's half-octave tritone of 600 cents. In just intonation, 7:5 is treated as a //consonant tritone//, and has a much mellower and sweeter sound than the 600-cent tritone we are used to hearing.

~~This chord is similar to the "dominant seventh chord" in~~ 12edo, the ~~most significant difference being the mistuning~~ of the ~~harmonic seventh. In 12edo,~~ the ~~interval closest to~~ the harmonic seventh is the minor seventh at 1000 cents. The difference of about 31 cents is striking, and especially noticable when the chords are presented next to one another. While the "dominant seventh chord" of 12edo is treated as a dissonance that needs to resolve (usually in the chord ~~pattern V7 to I), the "otonal tetrad" has a much different flavor and is often treated by composers in Just Intonation as a consonance.~~

Since 12edo does not distinguish between a minor and subminor third or a major and supermajor second, the intervals between adjacent members of the chord do not have the pattern of decreasing step size which characterizes the harmonic seventh chord:

Also, 12edo does not distinguish between a minor and subminor third or a major and supermajor second. Thus, the intervals between adjacent members of the chord do not have that pattern of decreasing step size:

5:4 becomes 400 cents.

6:5 becomes 300 cents.

7:6 becomes 300 cents.

8:7 becomes 200 cents.</pre></div>

8:7 becomes 200 cents.

=Meantone Augmented Sixth=

In meantone systems -- which are generated by repeatedly stacking a slightly flatted (from just) [[perfect fifth]] such that four fifths gives a near-just [[major third]] -- there is sometimes a good approximation of the harmonic seventh in the form of an "augmented sixth". Quarter-comma meantone (aurally identical, for most intents and purposes, to [[31edo]]) is one such system. In quarter-comma meantone, the interval of C to A# approximates a harmonic seventh, and is a distinct interval from C to Bb, a meantone minor seventh (falling somewhere between 16:9 and 9:5). The augmented sixth appears in tonal harmony in the "augmented sixth chord," and is treated as a rare and special dissonance. The so-called "German Sixth," in quarter-comma meantone, would approximate the harmonic seventh chord of 4:5:6:7(:8).

Note that a good approximation of the harmonic seventh is not available in every meantone system. In [[19edo]] (aurally identical, more or less, to 1/3-comma meantone), the "augmented sixth" is an interval of 947 cents -- about 22 cents flat of 7:4, and so less effective as a consonance.

See: [[http://en.wikipedia.org/wiki/Septimal_meantone_temperament|Septimal Meantone Temperament on Wikipedia]].</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>7_4</title></head><body>Frequency ratio 7:4, measuring approximately 968.8259064691249 cents, ~~often gets labeled~~ the &quot;harmonic seventh.&quot; It represents the interval between the 4th and 7th harmonics. It is also called a &quot;septimal subminor seventh&quot; -- the word &quot;septimal&quot; referring to the presence of a 7 as the highest prime in the ratio and the word &quot;subminor&quot; referring to ~~its~~ narrowness compared with a traditional minor seventh (such as <a class="wiki_link" href="/9_5">9:5</a> or <a class="wiki_link" href="/16_9">16:9</a>, ~~10 degrees of~~ <a class="wiki_link" href="/12edo">12edo</a> or a minor seventh found in a meantone system).<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>7_4</title></head><body>Frequency ratio 7:4, measuring approximately 968.8259064691249 cents, has been given the name &quot;harmonic seventh.&quot; It represents the interval between the 4th and 7th harmonics in the <a class="wiki_link" href="/OverToneSeries">overtone series</a>. It is also called a &quot;septimal subminor seventh&quot; -- the word &quot;septimal&quot; referring to the presence of a 7 as the highest prime in the ratio, and the word &quot;subminor&quot; referring to the harmonic seventh's narrowness compared with a traditional minor seventh (such as <a class="wiki_link" href="/9_5">9:5</a> or <a class="wiki_link" href="/16_9">16:9</a>, <a class="wiki_link" href="/12edo">12edo</a>'s 1000-cent interval, or a minor seventh found in a meantone system).<br />

<br />

7:4 has seen use in blues music, barbershop quartet music, and some musical traditions of the world, but has mostly not been recognized as a &quot;consonance&quot; in Western music theory. In most Just Intonation systems, the harmonic seventh is treated as a fundamental consonance in its own right, with its own distinct quality.<br />

<br />

7:4 ~~has seen use~~ in ~~blues~~ music ~~and some musical traditions of the world~~, ~~but has mostly not been recognized as~~ a &quot;~~consonance~~&quot; ~~in Western music theory~~. ~~However~~, ~~in most~~ Just Intonation ~~systems~~, ~~it is treated as~~ a ~~consonance in its own right~~, ~~with its own distinct quality~~.<br />

<h1 id="toc0"><a name="Harmonic Seventh Chord"></a>Harmonic Seventh Chord</h1>

7:4 appears in an otonal tetrad that forms the basis of much JI music, commonly called a &quot;harmonic seventh chord.&quot; It consists of a major triad (4:5:6) plus a harmonic seventh: 4:5:6:7(:8). This tetrad, a hallmark of blues and barbershop harmony, not to mention modern Just Intonation practice, represents a sequence of overtones from the fourth to the seventh. (8, being a doubling of 4, represents an octave above the root.) The intervals between adjacent members of the chord decrease in size:<br />

<br />

~~<h2 id="toc0"><a name="x-Otonal Tetrad"></a>Otonal Tetrad</h2>~~

~~7:4 appears in the otonal tetrad that forms the basis of much JI music:<br />~~

~~<h3 id="toc1"><a name="x-Otonal Tetrad-4:5:6:7:8"></a>4:5:6:7:8</h3>~~

This triad represents a sequence of overtones from the fourth to the eighth. (8, being a doubling of 4, represents an octave above the root.) The intervals between adjacent members of the chord decrease in size:<br />

<a class="wiki_link" href="/5_4">5:4</a> - approx. 386 cents - a major third<br />

<a class="wiki_link" href="/6_5">6:5</a> - approx. 316 cents - a minor third<br />

<a class="wiki_link" href="/7_6">7:6</a> - approx. 267 cents - a septimal subminor third<br />

8:7 - approx. 231 cents - a septimal supermajor seventh<br />

<a class="wiki_link" href="/8_7">8:7</a> - approx. 231 cents - a septimal supermajor second<br />

<br />

This chord is similar to the &quot;dominant seventh chord&quot; in 12edo, the most significant difference being the mistuning of the harmonic seventh. In 12edo, the interval closest to the harmonic seventh is the minor seventh at 1000 cents. The difference of about 31 cents is striking, and especially noticable when the chords are presented next to one another. While the &quot;dominant seventh chord&quot; of 12edo is treated as a dissonance that needs to resolve (usually in the chord pattern V7 to I), the &quot;harmonic seventh chord&quot; has a much different flavor and is often treated by composers in Just Intonation as a consonance.<br />

<br />

Another interval found in a harmonic seventh chord is the septimal tritone of <a class="wiki_link" href="/7_5">7:5</a>, which represents the interval between the major third (5) and the harmonic seventh (7). This interval, at 583 cents, sounds distinct from 12edo's half-octave tritone of 600 cents. In just intonation, 7:5 is treated as a <em>consonant tritone</em>, and has a much mellower and sweeter sound than the 600-cent tritone we are used to hearing.<br />

<br />

~~This chord is similar to the &quot;dominant seventh chord&quot; in~~ 12edo, the ~~most significant difference being the mistuning~~ of the ~~harmonic seventh. In 12edo,~~ the ~~interval closest to~~ the harmonic seventh is the minor seventh at 1000 cents. The difference of about 31 cents is striking, and especially noticable when the chords are presented next to one another. While the &quot;dominant seventh chord&quot; of 12edo is treated as a dissonance that needs to resolve (usually in the chord ~~pattern V7 to I), the &quot;otonal tetrad&quot; has a much different flavor and is often treated by composers in Just Intonation as a consonance.~~<br />

Since 12edo does not distinguish between a minor and subminor third or a major and supermajor second, the intervals between adjacent members of the chord do not have the pattern of decreasing step size which characterizes the harmonic seventh chord:<br />

<br />

Also, 12edo does not distinguish between a minor and subminor third or a major and supermajor second. Thus, the intervals between adjacent members of the chord do not have that pattern of decreasing step size:<br />

5:4 becomes 400 cents.<br />

6:5 becomes 300 cents.<br />

7:6 becomes 300 cents.<br />

8:7 becomes 200 cents.</body></html></pre></div>

8:7 becomes 200 cents.<br />

<br />

<h1 id="toc1"><a name="Meantone Augmented Sixth"></a>Meantone Augmented Sixth</h1>

In meantone systems -- which are generated by repeatedly stacking a slightly flatted (from just) <a class="wiki_link" href="/perfect%20fifth">perfect fifth</a> such that four fifths gives a near-just <a class="wiki_link" href="/major%20third">major third</a> -- there is sometimes a good approximation of the harmonic seventh in the form of an &quot;augmented sixth&quot;. Quarter-comma meantone (aurally identical, for most intents and purposes, to <a class="wiki_link" href="/31edo">31edo</a>) is one such system. In quarter-comma meantone, the interval of C to A# approximates a harmonic seventh, and is a distinct interval from C to Bb, a meantone minor seventh (falling somewhere between 16:9 and 9:5). The augmented sixth appears in tonal harmony in the &quot;augmented sixth chord,&quot; and is treated as a rare and special dissonance. The so-called &quot;German Sixth,&quot; in quarter-comma meantone, would approximate the harmonic seventh chord of 4:5:6:7(:8).<br />

<br />

Note that a good approximation of the harmonic seventh is not available in every meantone system. In <a class="wiki_link" href="/19edo">19edo</a> (aurally identical, more or less, to 1/3-comma meantone), the &quot;augmented sixth&quot; is an interval of 947 cents -- about 22 cents flat of 7:4, and so less effective as a consonance.<br />

<br />

See: <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_meantone_temperament" rel="nofollow">Septimal Meantone Temperament on Wikipedia</a>.</body></html></pre></div>

@@ 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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-09-23 13:51:06 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-09-23 15:44:25 UTC</tt>.<br>
-: The original revision id was <tt>164931099</tt>.<br>
+: The original revision id was <tt>164972615</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">Frequency ratio 7:4, measuring approximately 968.8259064691249 cents, often gets labeled the "harmonic seventh." It represents the interval between the 4th and 7th harmonics. It is also called a "septimal subminor seventh" -- the word "septimal" referring to the presence of a 7 as the highest prime in the ratio and the word "subminor" referring to its narrowness compared with a traditional minor seventh (such as [[9_5|9:5]] or [[16_9|16:9]], 10 degrees of [[12edo]] or a minor seventh found in a meantone system).
+<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">Frequency ratio 7:4, measuring approximately 968.8259064691249 cents, has been given the name "harmonic seventh." It represents the interval between the 4th and 7th harmonics in the [[OverToneSeries|overtone series]]. It is also called a "septimal subminor seventh" -- the word "septimal" referring to the presence of a 7 as the highest prime in the ratio, and the word "subminor" referring to the harmonic seventh's narrowness compared with a traditional minor seventh (such as [[9_5|9:5]] or [[16_9|16:9]], [[12edo]]'s 1000-cent interval, or a minor seventh found in a meantone system).
-:4 has seen use in blues music and some musical traditions of the world, but has mostly not been recognized as a "consonance" in Western music theory. However, in most Just Intonation systems, it is treated as a consonance in its own right, with its own distinct quality.
+:4 has seen use in blues music, barbershop quartet music, and some musical traditions of the world, but has mostly not been recognized as a "consonance" in Western music theory. In most Just Intonation systems, the harmonic seventh is treated as a fundamental consonance in its own right, with its own distinct quality.
+=Harmonic Seventh Chord=
+:4 appears in an otonal tetrad that forms the basis of much JI music, commonly called a "harmonic seventh chord." It consists of a major triad (4:5:6) plus a harmonic seventh: 4:5:6:7(:8). This tetrad, a hallmark of blues and barbershop harmony, not to mention modern Just Intonation practice, represents a sequence of overtones from the fourth to the seventh. (8, being a doubling of 4, represents an octave above the root.) The intervals between adjacent members of the chord decrease in size:
-==Otonal Tetrad==
-:4 appears in the otonal tetrad that forms the basis of much JI music:
-===4:5:6:7:8===
-This triad represents a sequence of overtones from the fourth to the eighth. (8, being a doubling of 4, represents an octave above the root.) The intervals between adjacent members of the chord decrease in size:
 [[5_4|5:4]] - approx. 386 cents - a major third
 [[6_5|6:5]] - approx. 316 cents - a minor third
 [[7_6|7:6]] - approx. 267 cents - a septimal subminor third
-:7 - approx. 231 cents - a septimal supermajor seventh
+[[8_7|8:7]] - approx. 231 cents - a septimal supermajor second
+This chord is similar to the "dominant seventh chord" in 12edo, the most significant difference being the mistuning of the harmonic seventh. In 12edo, the interval closest to the harmonic seventh is the minor seventh at 1000 cents. The difference of about 31 cents is striking, and especially noticable when the chords are presented next to one another. While the "dominant seventh chord" of 12edo is treated as a dissonance that needs to resolve (usually in the chord pattern V7 to I), the "harmonic seventh chord" has a much different flavor and is often treated by composers in Just Intonation as a consonance.
+Another interval found in a harmonic seventh chord is the septimal tritone of [[7_5|7:5]], which represents the interval between the major third (5) and the harmonic seventh (7). This interval, at 583 cents, sounds distinct from 12edo's half-octave tritone of 600 cents. In just intonation, 7:5 is treated as a //consonant tritone//, and has a much mellower and sweeter sound than the 600-cent tritone we are used to hearing.
-This chord is similar to the "dominant seventh chord" in 12edo, the most significant difference being the mistuning of the harmonic seventh. In 12edo, the interval closest to the harmonic seventh is the minor seventh at 1000 cents. The difference of about 31 cents is striking, and especially noticable when the chords are presented next to one another. While the "dominant seventh chord" of 12edo is treated as a dissonance that needs to resolve (usually in the chord pattern V7 to I), the "otonal tetrad" has a much different flavor and is often treated by composers in Just Intonation as a consonance.
+Since 12edo does not distinguish between a minor and subminor third or a major and supermajor second, the intervals between adjacent members of the chord do not have the pattern of decreasing step size which characterizes the harmonic seventh chord:
-Also, 12edo does not distinguish between a minor and subminor third or a major and supermajor second. Thus, the intervals between adjacent members of the chord do not have that pattern of decreasing step size:
 :4 becomes 400 cents.
 :5 becomes 300 cents.
 :6 becomes 300 cents.
-:7 becomes 200 cents.</pre></div>
+:7 becomes 200 cents.
+=Meantone Augmented Sixth=
+In meantone systems -- which are generated by repeatedly stacking a slightly flatted (from just) [[perfect fifth]] such that four fifths gives a near-just [[major third]] -- there is sometimes a good approximation of the harmonic seventh in the form of an "augmented sixth". Quarter-comma meantone (aurally identical, for most intents and purposes, to [[31edo]]) is one such system. In quarter-comma meantone, the interval of C to A# approximates a harmonic seventh, and is a distinct interval from C to Bb, a meantone minor seventh (falling somewhere between 16:9 and 9:5). The augmented sixth appears in tonal harmony in the "augmented sixth chord," and is treated as a rare and special dissonance. The so-called "German Sixth," in quarter-comma meantone, would approximate the harmonic seventh chord of 4:5:6:7(:8).
+Note that a good approximation of the harmonic seventh is not available in every meantone system. In [[19edo]] (aurally identical, more or less, to 1/3-comma meantone), the "augmented sixth" is an interval of 947 cents -- about 22 cents flat of 7:4, and so less effective as a consonance.
+See: [[http://en.wikipedia.org/wiki/Septimal_meantone_temperament|Septimal Meantone Temperament on Wikipedia]].</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;7_4&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Frequency ratio 7:4, measuring approximately 968.8259064691249 cents, often gets labeled the &amp;quot;harmonic seventh.&amp;quot; It represents the interval between the 4th and 7th harmonics. It is also called a &amp;quot;septimal subminor seventh&amp;quot; -- the word &amp;quot;septimal&amp;quot; referring to the presence of a 7 as the highest prime in the ratio and the word &amp;quot;subminor&amp;quot; referring to its narrowness compared with a traditional minor seventh (such as &lt;a class="wiki_link" href="/9_5"&gt;9:5&lt;/a&gt; or &lt;a class="wiki_link" href="/16_9"&gt;16:9&lt;/a&gt;, 10 degrees of &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; or a minor seventh found in a meantone system).&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;7_4&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Frequency ratio 7:4, measuring approximately 968.8259064691249 cents, has been given the name &amp;quot;harmonic seventh.&amp;quot; It represents the interval between the 4th and 7th harmonics in the &lt;a class="wiki_link" href="/OverToneSeries"&gt;overtone series&lt;/a&gt;. It is also called a &amp;quot;septimal subminor seventh&amp;quot; -- the word &amp;quot;septimal&amp;quot; referring to the presence of a 7 as the highest prime in the ratio, and the word &amp;quot;subminor&amp;quot; referring to the harmonic seventh's narrowness compared with a traditional minor seventh (such as &lt;a class="wiki_link" href="/9_5"&gt;9:5&lt;/a&gt; or &lt;a class="wiki_link" href="/16_9"&gt;16:9&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;'s 1000-cent interval, or a minor seventh found in a meantone system).&lt;br /&gt;
+&lt;br /&gt;
+:4 has seen use in blues music, barbershop quartet music, and some musical traditions of the world, but has mostly not been recognized as a &amp;quot;consonance&amp;quot; in Western music theory. In most Just Intonation systems, the harmonic seventh is treated as a fundamental consonance in its own right, with its own distinct quality.&lt;br /&gt;
 &lt;br /&gt;
-:4 has seen use in blues music and some musical traditions of the world, but has mostly not been recognized as a &amp;quot;consonance&amp;quot; in Western music theory. However, in most Just Intonation systems, it is treated as a consonance in its own right, with its own distinct quality.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Harmonic Seventh Chord"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Harmonic Seventh Chord&lt;/h1&gt;
+:4 appears in an otonal tetrad that forms the basis of much JI music, commonly called a &amp;quot;harmonic seventh chord.&amp;quot; It consists of a major triad (4:5:6) plus a harmonic seventh: 4:5:6:7(:8). This tetrad, a hallmark of blues and barbershop harmony, not to mention modern Just Intonation practice, represents a sequence of overtones from the fourth to the seventh. (8, being a doubling of 4, represents an octave above the root.) The intervals between adjacent members of the chord decrease in size:&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Otonal Tetrad"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Otonal Tetrad&lt;/h2&gt;
-:4 appears in the otonal tetrad that forms the basis of much JI music:&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Otonal Tetrad-4:5:6:7:8"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;4:5:6:7:8&lt;/h3&gt;
- This triad represents a sequence of overtones from the fourth to the eighth. (8, being a doubling of 4, represents an octave above the root.) The intervals between adjacent members of the chord decrease in size:&lt;br /&gt;
 &lt;a class="wiki_link" href="/5_4"&gt;5:4&lt;/a&gt; - approx. 386 cents - a major third&lt;br /&gt;
 &lt;a class="wiki_link" href="/6_5"&gt;6:5&lt;/a&gt; - approx. 316 cents - a minor third&lt;br /&gt;
 &lt;a class="wiki_link" href="/7_6"&gt;7:6&lt;/a&gt; - approx. 267 cents - a septimal subminor third&lt;br /&gt;
-:7 - approx. 231 cents - a septimal supermajor seventh&lt;br /&gt;
+&lt;a class="wiki_link" href="/8_7"&gt;8:7&lt;/a&gt; - approx. 231 cents - a septimal supermajor second&lt;br /&gt;
+&lt;br /&gt;
+This chord is similar to the &amp;quot;dominant seventh chord&amp;quot; in 12edo, the most significant difference being the mistuning of the harmonic seventh. In 12edo, the interval closest to the harmonic seventh is the minor seventh at 1000 cents. The difference of about 31 cents is striking, and especially noticable when the chords are presented next to one another. While the &amp;quot;dominant seventh chord&amp;quot; of 12edo is treated as a dissonance that needs to resolve (usually in the chord pattern V7 to I), the &amp;quot;harmonic seventh chord&amp;quot; has a much different flavor and is often treated by composers in Just Intonation as a consonance.&lt;br /&gt;
+&lt;br /&gt;
+Another interval found in a harmonic seventh chord is the septimal tritone of &lt;a class="wiki_link" href="/7_5"&gt;7:5&lt;/a&gt;, which represents the interval between the major third (5) and the harmonic seventh (7). This interval, at 583 cents, sounds distinct from 12edo's half-octave tritone of 600 cents. In just intonation, 7:5 is treated as a &lt;em&gt;consonant tritone&lt;/em&gt;, and has a much mellower and sweeter sound than the 600-cent tritone we are used to hearing.&lt;br /&gt;
 &lt;br /&gt;
-This chord is similar to the &amp;quot;dominant seventh chord&amp;quot; in 12edo, the most significant difference being the mistuning of the harmonic seventh. In 12edo, the interval closest to the harmonic seventh is the minor seventh at 1000 cents. The difference of about 31 cents is striking, and especially noticable when the chords are presented next to one another. While the &amp;quot;dominant seventh chord&amp;quot; of 12edo is treated as a dissonance that needs to resolve (usually in the chord pattern V7 to I), the &amp;quot;otonal tetrad&amp;quot; has a much different flavor and is often treated by composers in Just Intonation as a consonance.&lt;br /&gt;
+Since 12edo does not distinguish between a minor and subminor third or a major and supermajor second, the intervals between adjacent members of the chord do not have the pattern of decreasing step size which characterizes the harmonic seventh chord:&lt;br /&gt;
 &lt;br /&gt;
-Also, 12edo does not distinguish between a minor and subminor third or a major and supermajor second. Thus, the intervals between adjacent members of the chord do not have that pattern of decreasing step size:&lt;br /&gt;
 :4 becomes 400 cents.&lt;br /&gt;
 :5 becomes 300 cents.&lt;br /&gt;
 :6 becomes 300 cents.&lt;br /&gt;
-:7 becomes 200 cents.&lt;/body&gt;&lt;/html&gt;</pre></div>
+:7 becomes 200 cents.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Meantone Augmented Sixth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Meantone Augmented Sixth&lt;/h1&gt;
+ In meantone systems -- which are generated by repeatedly stacking a slightly flatted (from just) &lt;a class="wiki_link" href="/perfect%20fifth"&gt;perfect fifth&lt;/a&gt; such that four fifths gives a near-just &lt;a class="wiki_link" href="/major%20third"&gt;major third&lt;/a&gt; -- there is sometimes a good approximation of the harmonic seventh in the form of an &amp;quot;augmented sixth&amp;quot;. Quarter-comma meantone (aurally identical, for most intents and purposes, to &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;) is one such system. In quarter-comma meantone, the interval of C to A# approximates a harmonic seventh, and is a distinct interval from C to Bb, a meantone minor seventh (falling somewhere between 16:9 and 9:5). The augmented sixth appears in tonal harmony in the &amp;quot;augmented sixth chord,&amp;quot; and is treated as a rare and special dissonance. The so-called &amp;quot;German Sixth,&amp;quot; in quarter-comma meantone, would approximate the harmonic seventh chord of 4:5:6:7(:8).&lt;br /&gt;
+&lt;br /&gt;
+Note that a good approximation of the harmonic seventh is not available in every meantone system. In &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt; (aurally identical, more or less, to 1/3-comma meantone), the &amp;quot;augmented sixth&amp;quot; is an interval of 947 cents -- about 22 cents flat of 7:4, and so less effective as a consonance.&lt;br /&gt;
+&lt;br /&gt;
+See: &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_meantone_temperament" rel="nofollow"&gt;Septimal Meantone Temperament on Wikipedia&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>