Interval size measure
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**Interval size measure** means the //distance// between pitches. Intervals can be measured logarithmically or by frequancy ratios. ==Logarithmic== All logarithmic measures can be combined by adding and subtracting them. ===Gross=== Intervals are somtetimes expressed in the number of scale steps between them. These steps can be of different size, compare for example the names of the major scale in the classic music. For "atonal" music it was replaced by the number of 12edo-semitones. Proposal: The **relative interval measure** is the number of steps between two pitches of an [[equal]] tuning, sometimes called [[degree]]s (of an edo). For generators, the backslash notation is used d\edo, but it is also a ratio (of a logarithmic measure). For 1/16 octave (one step of [[16edo]]), the term [[eka]] has been coined (see [[Armodue theory]]). ===Fine=== The [[cent]] (¢), [[1200edo|1\1200 octave]], is the classic measure for intervals when more precision than 12edo is requied. Some people object to it on the grounds that it is too closely related to 12 equal. Other measures include the [[millioctave]] (mO), which is the [[1000edo|1000th part]] of an octave, or 1.2 cents; the [[Eptaméride]] or [[Savart]]: [[301edo|1\301 of an octave]]; the [[Jot]]: [[30103edo|1\30103 octave]]; the [[Morion]]: [[72edo|1\72 octave]]; the [[Farab]]: [[144edo|1\144 octave]]; the [[Flu]]: [[46032edo|1\46032 octave]]; the [[Purdal]]: [[9900edo|1\9900 octave]]; the [[Grad]]: 1/12 of a Pythagorean comma; the [[Mina]]: [[2460edo|1\2460 octave]]; the [[Mem]]: [[205edo|1\205 octave]] (used by [[http://www.h-pi.com/theory/measurement3.html|Hi-pi Instruments]]); the [[Tredek]]: [[270edo|1\270 octave]]; the [[Squb]]: [[494edo|1\494 octave]]; the [[Skisma]]: [[612edo|1\612 octave]]; the [[Woolhouse]]: [[730edo|1\730 octave]]; the [[Tina]]: [[8539edo|1\8539 octave]], the [[Prima]]: [[12276edo|1\12236 octave]], the [[Jinn]]: [16808edo|1\16808 octave]], and the [[Imp]]: [[31920edo|1\31920 octave]]. See [[http://www.huygens-fokker.org/docs/measures.html|Logarithmic Interval Measures]] Within a given [[equal]]-stepped tonal system, the [[Relative cent|relative cent]] (rct, r¢) can be used to describe properties of pitches (for instance the approximation of [[JI]] intervals). It is defined as on 100th (or 1 percent) of the interval between two neigbouring pitches in the used equal tuning. see also: Kirnberger Atom http://arxiv.org/abs/0907.5249 ==Ratio== Intervals can be measured also giving their [[http://en.wikipedia.org/wiki/Interval_ratio|(frequency) ratio]]. For instance the major third as [[5_4|5/4]] or the pure fifth [[3_2|3/2]]. When combining sizes given in ratios, you have to multiply oder divide: a pure fifth increased by a major third gives the major seventh 3/2*5/4 = 15/8, which is a diatonic semitone below an octave (2/1)/(15/8) = 2/1*8/15 = 16/15. Another notation for ratios is a vector of prime factor exponents, often called a [[monzo]], such as |-4 4 -1> (for the syntonic comma, 81/80 = 2^(-4) * 3^4 * 5^(-1)), which builds a bridge back to the logarithmic measure: intervals can be combined by component-wise addition or subtraction of their vectors.
Original HTML content:
<html><head><title>Interval size measure</title></head><body><strong>Interval size measure</strong> means the <em>distance</em> between pitches. Intervals can be measured logarithmically or by frequancy ratios.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Logarithmic"></a><!-- ws:end:WikiTextHeadingRule:0 -->Logarithmic</h2> All logarithmic measures can be combined by adding and subtracting them.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x-Logarithmic-Gross"></a><!-- ws:end:WikiTextHeadingRule:2 -->Gross</h3> Intervals are somtetimes expressed in the number of scale steps between them. These steps can be of different size, compare for example the names of the major scale in the classic music.<br /> <br /> For "atonal" music it was replaced by the number of 12edo-semitones.<br /> <br /> Proposal: The <strong>relative interval measure</strong> is the number of steps between two pitches of an <a class="wiki_link" href="/equal">equal</a> tuning, sometimes called <a class="wiki_link" href="/degree">degree</a>s (of an edo). For generators, the backslash notation is used d\edo, but it is also a ratio (of a logarithmic measure).<br /> <br /> For 1/16 octave (one step of <a class="wiki_link" href="/16edo">16edo</a>), the term <a class="wiki_link" href="/eka">eka</a> has been coined (see <a class="wiki_link" href="/Armodue%20theory">Armodue theory</a>).<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="x-Logarithmic-Fine"></a><!-- ws:end:WikiTextHeadingRule:4 -->Fine</h3> The <a class="wiki_link" href="/cent">cent</a> (¢), <a class="wiki_link" href="/1200edo">1\1200 octave</a>, is the classic measure for intervals when more precision than 12edo is requied. Some people object to it on the grounds that it is too closely related to 12 equal.<br /> <br /> Other measures include the <a class="wiki_link" href="/millioctave">millioctave</a> (mO), which is the <a class="wiki_link" href="/1000edo">1000th part</a> of an octave, or 1.2 cents; the <a class="wiki_link" href="/Eptam%C3%A9ride">Eptaméride</a> or <a class="wiki_link" href="/Savart">Savart</a>: <a class="wiki_link" href="/301edo">1\301 of an octave</a>; the <a class="wiki_link" href="/Jot">Jot</a>: <a class="wiki_link" href="/30103edo">1\30103 octave</a>; the <a class="wiki_link" href="/Morion">Morion</a>: <a class="wiki_link" href="/72edo">1\72 octave</a>; the <a class="wiki_link" href="/Farab">Farab</a>: <a class="wiki_link" href="/144edo">1\144 octave</a>; the <a class="wiki_link" href="/Flu">Flu</a>: <a class="wiki_link" href="/46032edo">1\46032 octave</a>; the <a class="wiki_link" href="/Purdal">Purdal</a>: <a class="wiki_link" href="/9900edo">1\9900 octave</a>; the <a class="wiki_link" href="/Grad">Grad</a>: 1/12 of a Pythagorean comma; the <a class="wiki_link" href="/Mina">Mina</a>: <a class="wiki_link" href="/2460edo">1\2460 octave</a>; the <a class="wiki_link" href="/Mem">Mem</a>: <a class="wiki_link" href="/205edo">1\205 octave</a> (used by <a class="wiki_link_ext" href="http://www.h-pi.com/theory/measurement3.html" rel="nofollow">Hi-pi Instruments</a>); the <a class="wiki_link" href="/Tredek">Tredek</a>: <a class="wiki_link" href="/270edo">1\270 octave</a>; the <a class="wiki_link" href="/Squb">Squb</a>: <a class="wiki_link" href="/494edo">1\494 octave</a>; the <a class="wiki_link" href="/Skisma">Skisma</a>: <a class="wiki_link" href="/612edo">1\612 octave</a>; the <a class="wiki_link" href="/Woolhouse">Woolhouse</a>: <a class="wiki_link" href="/730edo">1\730 octave</a>; the <a class="wiki_link" href="/Tina">Tina</a>: <a class="wiki_link" href="/8539edo">1\8539 octave</a>, the <a class="wiki_link" href="/Prima">Prima</a>: <a class="wiki_link" href="/12276edo">1\12236 octave</a>, the <a class="wiki_link" href="/Jinn">Jinn</a>: [16808edo|1\16808 octave]], and the <a class="wiki_link" href="/Imp">Imp</a>: <a class="wiki_link" href="/31920edo">1\31920 octave</a>.<br /> <br /> See <a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/measures.html" rel="nofollow">Logarithmic Interval Measures</a><br /> <br /> Within a given <a class="wiki_link" href="/equal">equal</a>-stepped tonal system, the <a class="wiki_link" href="/Relative%20cent">relative cent</a> (rct, r¢) can be used to describe properties of pitches (for instance the approximation of <a class="wiki_link" href="/JI">JI</a> intervals). It is defined as on 100th (or 1 percent) of the interval between two neigbouring pitches in the used equal tuning.<br /> <br /> see also: Kirnberger Atom <!-- ws:start:WikiTextUrlRule:84:http://arxiv.org/abs/0907.5249 --><a class="wiki_link_ext" href="http://arxiv.org/abs/0907.5249" rel="nofollow">http://arxiv.org/abs/0907.5249</a><!-- ws:end:WikiTextUrlRule:84 --><br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="x-Ratio"></a><!-- ws:end:WikiTextHeadingRule:6 -->Ratio</h2> Intervals can be measured also giving their <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Interval_ratio" rel="nofollow">(frequency) ratio</a>. For instance the major third as <a class="wiki_link" href="/5_4">5/4</a> or the pure fifth <a class="wiki_link" href="/3_2">3/2</a>. When combining sizes given in ratios, you have to multiply oder divide:<br /> a pure fifth increased by a major third gives the major seventh 3/2*5/4 = 15/8,<br /> which is a diatonic semitone below an octave (2/1)/(15/8) = 2/1*8/15 = 16/15.<br /> <br /> Another notation for ratios is a vector of prime factor exponents, often called a <a class="wiki_link" href="/monzo">monzo</a>, such as |-4 4 -1> (for the syntonic comma, 81/80 = 2^(-4) * 3^4 * 5^(-1)), which builds a bridge back to the logarithmic measure: intervals can be combined by component-wise addition or subtraction of their vectors.</body></html>