7/4: Difference between revisions
Wikispaces>xenwolf **Imported revision 433608212 - Original comment: ** |
Wikispaces>spt3125 **Imported revision 513182980 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-06-07 11:59:06 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>513182980</tt>.<br> | ||
: The revision comment was: <tt></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> | 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> | <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. | <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">**7/4** | ||
|-2 0 0 1> | |||
968.82591 cents | |||
[[media type="file" key="jid_7_4_pluck_adu_dr220.mp3"]] | |||
Frequency ratio **7:4**, measuring approximately 968.8 [[Cent|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, 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. | 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. | ||
| Line 42: | Line 47: | ||
|| [[26edo]] || 0.40486 [[cent|¢]] || 0.87720 [[relative cent|r¢]] || [[78edo]] || | || [[26edo]] || 0.40486 [[cent|¢]] || 0.87720 [[relative cent|r¢]] || [[78edo]] || | ||
|| [[83edo]] || 0.15121 ¢ || 1.0459 r¢ || [[166edo]] || | || [[83edo]] || 0.15121 ¢ || 1.0459 r¢ || [[166edo]] || | ||
|| [[57edo]] || 0.40485 ¢ || 1.9231 r¢ || || | || [[57edo]] || 0.40485 ¢ || 1.9231 r¢ || || | ||
|| [[31edo]] || | || [[31edo]] || 1.0839 ¢ || 2.8003 r¢ || || | ||
|| [[5edo]] | || [[5edo]] || 8.8259 ¢ || 3.6775 r¢ || [[10edo]], [[15edo]], [[20edo]], [[25edo]] || | ||
|| [[21edo]] || | || [[21edo]] || 2.6026 ¢ || 4.5547 r¢ || || | ||
|| [[88edo]] || | || [[88edo]] || 0.6441 ¢ || 4.7233 r¢ || || | ||
|| [[47edo]] || | || [[47edo]] || 1.3868 ¢ || 5.4319 r¢ || [[94edo]] || | ||
|| [[73edo]] || | || [[73edo]] || 1.0371 ¢ || 6.3091 r¢ || || | ||
|| [[36edo]] || | || [[36edo]] || 2.1592 ¢ || 6.4777 r¢ || [[72edo]] || | ||
|| [[16edo]] || | || [[16edo]] || 6.1741 ¢ || 8.2321 r¢ || || | ||
|| [[68edo]] || 0.14686 ¢ || 9.9865 r¢ || || | || [[68edo]] || 0.14686 ¢ || 9.9865 r¢ || || | ||
|| [[11edo]] || | || [[11edo]] || 12.992 ¢ || 11.910 r¢ || [[22edo]] || | ||
|| [[89edo]] || | || [[89edo]] || 1.9606 ¢ || 14.541 r¢ || || | ||
[[http://en.wikipedia.org/wiki/Harmonic_seventh]]</pre></div> | [[http://en.wikipedia.org/wiki/Harmonic_seventh]]</pre></div> | ||
<h4>Original HTML content:</h4> | <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 <strong>7:4</strong>, measuring approximately 968. | <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><strong>7/4</strong><br /> | ||
|-2 0 0 1&gt;<br /> | |||
968.82591 cents<br /> | |||
<!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_7_4_pluck_adu_dr220.mp3?h=20&amp;w=240&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;jid_7_4_pluck_adu_dr220.mp3&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;20&quot; width=&quot;240&quot;/&gt; --><embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_7_4_pluck_adu_dr220.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:0 --><br /> | |||
<br /> | |||
Frequency ratio <strong>7:4</strong>, measuring approximately 968.8 <a class="wiki_link" href="/Cent">cents</a>, has been given the name <strong>&quot;harmonic seventh.&quot;</strong> 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 <a class="wiki_link" href="/prime">prime</a> 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 /> | <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;<a class="wiki_link" href="/consonance">consonance</a>&quot; in Western music theory. In most <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a> systems, the harmonic seventh is treated as a fundamental consonance in its own right, with its own distinct quality.<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;<a class="wiki_link" href="/consonance">consonance</a>&quot; in Western music theory. In most <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a> systems, the harmonic seventh is treated as a fundamental consonance in its own right, with its own distinct quality.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Harmonic Seventh Chord"></a><!-- ws:end:WikiTextHeadingRule:1 -->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 /> | 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 /> | <br /> | ||
| Line 80: | Line 90: | ||
<a class="wiki_link" href="/8_7">8/7</a> becomes 200 cents.<br /> | <a class="wiki_link" href="/8_7">8/7</a> becomes 200 cents.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="Meantone Augmented Sixth"></a><!-- ws:end:WikiTextHeadingRule:3 -->Meantone Augmented Sixth</h1> | ||
In <a class="wiki_link" href="/Meantone%20family">meantone systems</a> -- 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;. <a class="wiki_link" href="/Quarter-comma%20meantone">Quarter-comma meantone</a> (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 /> | In <a class="wiki_link" href="/Meantone%20family">meantone systems</a> -- 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;. <a class="wiki_link" href="/Quarter-comma%20meantone">Quarter-comma meantone</a> (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 /> | <br /> | ||
| Line 87: | Line 97: | ||
See: <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_meantone_temperament" rel="nofollow">Septimal Meantone Temperament on Wikipedia</a>.<br /> | See: <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_meantone_temperament" rel="nofollow">Septimal Meantone Temperament on Wikipedia</a>.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc2"><a name="Approximations"></a><!-- ws:end:WikiTextHeadingRule:5 -->Approximations</h1> | ||
EDOs containing good approximations of the interval 7:4 are (ordered by relative delta):<br /> | EDOs containing good approximations of the interval 7:4 are (ordered by relative delta):<br /> | ||
<br /> | <br /> | ||