TOP tuning
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-07-25 15:39:27 UTC.
- The original revision id was 242778957.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
[[toc|flat]]
=Proportional error=
A //tuning// for a regular temperament is defined by a vector T in [[Vals and Tuning Space#Vals and Monzos|Tenney tuning space]] whose entries are the size of the interval, in cents, which the k generators of the regular temperament (often the first k primes) are mapped to. T is denoted by a [[http://en.wikipedia.org/wiki/Bra-ket_notation|bra vector]], and if M is a monzo then <T|M> is the size, in cents, of the interval defined by M in the tuning T. If q is the rational number which M represents, then we may also write this quantity as T(q).
Given a tuning T and a rational number q in the domain of the regular temperament T is a tuning for, the //error// of T on q is define as
[[math]]
Err(q) = |T(q) - cents(q)|
[[math]]
that is, the absolute value of the difference between the value in cents T assigns to q and the actual size in cents of q. The //proportional error// is defined as 0 when q equals 1 and otherwise
[[math]]
PE(q) = \frac{Err(q)}{cents(Ben(q))}
[[math]]
where Ben(q) is the [[Benedetti height]], the product of the numerator and denominator of q.Original HTML content:
<html><head><title>TOP tuning</title></head><body><!-- ws:start:WikiTextTocRule:4:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:4 --><!-- ws:start:WikiTextTocRule:5: --><a href="#Proportional error">Proportional error</a><!-- ws:end:WikiTextTocRule:5 --><!-- ws:start:WikiTextTocRule:6: -->
<!-- ws:end:WikiTextTocRule:6 --><br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc0"><a name="Proportional error"></a><!-- ws:end:WikiTextHeadingRule:2 -->Proportional error</h1>
A <em>tuning</em> for a regular temperament is defined by a vector T in <a class="wiki_link" href="/Vals%20and%20Tuning%20Space#Vals and Monzos">Tenney tuning space</a> whose entries are the size of the interval, in cents, which the k generators of the regular temperament (often the first k primes) are mapped to. T is denoted by a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Bra-ket_notation" rel="nofollow">bra vector</a>, and if M is a monzo then <T|M> is the size, in cents, of the interval defined by M in the tuning T. If q is the rational number which M represents, then we may also write this quantity as T(q). <br />
<br />
Given a tuning T and a rational number q in the domain of the regular temperament T is a tuning for, the <em>error</em> of T on q is define as <br />
<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]<br/>
Err(q) = |T(q) - cents(q)|<br/>[[math]]
--><script type="math/tex">Err(q) = |T(q) - cents(q)|</script><!-- ws:end:WikiTextMathRule:0 --><br />
<br />
that is, the absolute value of the difference between the value in cents T assigns to q and the actual size in cents of q. The <em>proportional error</em> is defined as 0 when q equals 1 and otherwise<br />
<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]<br/>
PE(q) = \frac{Err(q)}{cents(Ben(q))}<br/>[[math]]
--><script type="math/tex">PE(q) = \frac{Err(q)}{cents(Ben(q))}</script><!-- ws:end:WikiTextMathRule:1 --><br />
<br />
where Ben(q) is the <a class="wiki_link" href="/Benedetti%20height">Benedetti height</a>, the product of the numerator and denominator of q.</body></html>