Superparticular ratio: Difference between revisions
Wikispaces>Sarzadoce **Imported revision 244980011 - Original comment: ** |
Wikispaces>Sarzadoce **Imported revision 244981789 - 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:Sarzadoce|Sarzadoce]] and made on <tt>2011-08-09 01: | : This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2011-08-09 01:53:17 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>244981789</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> | ||
| Line 10: | Line 10: | ||
These ratios have some peculiar properties: | These ratios have some peculiar properties: | ||
* The difference tone of the dyad is also the virtual fundamental. | * The difference tone of the dyad is also the virtual fundamental. | ||
* The first | * The first 6 such ratios ([[3_2|3/2]], [[4_3|4/3]], [[5_4|5/4]], [[6_5|6/5]], [[7_6|7/6]], [[8_7|8/7]]) are notable [[harmonic entropy]] minima. | ||
* The difference between two successive epimoric ratios is always an epimoric ratio. | * The difference between two successive epimoric ratios is always an epimoric ratio. | ||
* The sum of two successive epimoric ratios is either an epimoric ratio or an epimeric ratio. | * The sum of two successive epimoric ratios is either an epimoric ratio or an epimeric ratio. | ||
* Every epimoric ratio can be split into two or more smaller epimoric ratios via the arithmetic mean. | |||
Curiously enough, the ancient Greeks | Curiously enough, the ancient Greeks did not consider 2/1 to be superparticular because it is a harmonic. | ||
| Line 22: | Line 23: | ||
<br /> | <br /> | ||
These ratios have some peculiar properties:<br /> | These ratios have some peculiar properties:<br /> | ||
<ul><li>The difference tone of the dyad is also the virtual fundamental.</li><li>The first | <ul><li>The difference tone of the dyad is also the virtual fundamental.</li><li>The first 6 such ratios (<a class="wiki_link" href="/3_2">3/2</a>, <a class="wiki_link" href="/4_3">4/3</a>, <a class="wiki_link" href="/5_4">5/4</a>, <a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/7_6">7/6</a>, <a class="wiki_link" href="/8_7">8/7</a>) are notable <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> minima.</li><li>The difference between two successive epimoric ratios is always an epimoric ratio.</li><li>The sum of two successive epimoric ratios is either an epimoric ratio or an epimeric ratio.</li><li>Every epimoric ratio can be split into two or more smaller epimoric ratios via the arithmetic mean.</li></ul><br /> | ||
Curiously enough, the ancient Greeks | Curiously enough, the ancient Greeks did not consider 2/1 to be superparticular because it is a harmonic.<br /> | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<span style="background-color: initial;"><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow">http://en.wikipedia.org/wiki/Superparticular_number</a></span></body></html></pre></div> | <span style="background-color: initial;"><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow">http://en.wikipedia.org/wiki/Superparticular_number</a></span></body></html></pre></div> | ||