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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | A marvelous dwarf is a scale with the following attributes: |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-10 17:31:55 UTC</tt>.<br>
| | (1) It is a [[Marvel_family|marvel]] tempering of a 5-limit [[Dwarves|dwarf]]. |
| : The original revision id was <tt>169293443</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <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">A marvelous dwarf is a scale with the following attributes:
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| (1) It is a [[Marvel family|marvel]] tempering of a 5-limit [[Dwarves|dwarf]].
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| (2) It has the same number n of otonal tetrads, otonal pentads, utonal tetrads and utonal pentads. As a consequence of this it also has n subminor and n supermajor tetrads. | | (2) It has the same number n of otonal tetrads, otonal pentads, utonal tetrads and utonal pentads. As a consequence of this it also has n subminor and n supermajor tetrads. |
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| (3) It is covered by its pentads--that is, every note is harmonized by a pentad, and the scale is the union of its pentads. | | (3) It is covered by its pentads--that is, every note is harmonized by a pentad, and the scale is the union of its pentads. |
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| (4) It has more 5-limit triads than pentads. | | (4) It has more 5-limit triads than pentads. |
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| (5) It has no approximate tetrads deriving from anything but marvel; in the 5-limit scale which is tempered the smallest comma which produces approximate tetrads is 225/224. | | (5) It has no approximate tetrads deriving from anything but marvel; in the 5-limit scale which is tempered the smallest comma which produces approximate tetrads is 225/224. |
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| If every condition but the third--the covering condition--is | | If every condition but the third--the covering condition--is |
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| satisfied, I'm calling it a semimarvelous dwarf. Why there are these scales exhibiting such regularity as a result of finding the 5-limit dwarf is an interesting question. Whatever the reason for it, the marvelous dwarves--of size 12, 15, 18, 19, 20, 21, and 25--seem like excellent scales for instrumentalists and composers interested in 9-limit harmony and scales in this size range. The 25-note scale, whose 5-limit preimage we've discussed before as genus(15^4), is particularly striking from the point of view of the quantity of pentads it supplies. | | satisfied, I'm calling it a semimarvelous dwarf. Why there are these scales exhibiting such regularity as a result of finding the 5-limit dwarf is an interesting question. Whatever the reason for it, the marvelous dwarves--of size 12, 15, 18, 19, 20, 21, and 25--seem like excellent scales for instrumentalists and composers interested in 9-limit harmony and scales in this size range. The 25-note scale, whose 5-limit preimage we've discussed before as genus(15^4), is particularly striking from the point of view of the quantity of pentads it supplies. |
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| <11 17 26| 1 6-6 11 semimarvelous | | <11 17 26| 1 6-6 11 semimarvelous |
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| <12 19 28| 2 6-6 6 marvelous | | <12 19 28| 2 6-6 6 marvelous |
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| <13 20 30| 1 7-6 13 semimarvelous | | <13 20 30| 1 7-6 13 semimarvelous |
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| <14 22 33| 2 7-6 7 semimarvelous | | <14 22 33| 2 7-6 7 semimarvelous |
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| <15 24 35| 3 8-8 5 marvelous | | <15 24 35| 3 8-8 5 marvelous |
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| <16 25 37| 2 7-6 8 semimarvelous | | <16 25 37| 2 7-6 8 semimarvelous |
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| <17 27 40| 4 10-9 4.25 semimarvelous | | <17 27 40| 4 10-9 4.25 semimarvelous |
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| <18 29 42| 4 10-10 4.5 marvelous | | <18 29 42| 4 10-10 4.5 marvelous |
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| <19 30 44| 5 12-11 3.8 marvelous | | <19 30 44| 5 12-11 3.8 marvelous |
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| <20 32 47| 6 12-12 3.333 marvelous | | <20 32 47| 6 12-12 3.333 marvelous |
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| <21 33 49| 5 12-12 4.2 marvelous | | <21 33 49| 5 12-12 4.2 marvelous |
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| <22 35 51| 6 14-13 3.667 semimarvelous | | <22 35 51| 6 14-13 3.667 semimarvelous |
| <25 40 58| 9 16-16 2.778 marvelous</pre></div>
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| <h4>Original HTML content:</h4>
| | <25 40 58| 9 16-16 2.778 marvelous |
| <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>Marvelous dwarves</title></head><body>A marvelous dwarf is a scale with the following attributes:<br />
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| <br />
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| (1) It is a <a class="wiki_link" href="/Marvel%20family">marvel</a> tempering of a 5-limit <a class="wiki_link" href="/Dwarves">dwarf</a>.<br />
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| (2) It has the same number n of otonal tetrads, otonal pentads, utonal tetrads and utonal pentads. As a consequence of this it also has n subminor and n supermajor tetrads.<br />
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| (3) It is covered by its pentads--that is, every note is harmonized by a pentad, and the scale is the union of its pentads.<br />
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| (4) It has more 5-limit triads than pentads.<br />
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| (5) It has no approximate tetrads deriving from anything but marvel; in the 5-limit scale which is tempered the smallest comma which produces approximate tetrads is 225/224.<br />
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| <br />
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| If every condition but the third--the covering condition--is<br />
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| satisfied, I'm calling it a semimarvelous dwarf. Why there are these scales exhibiting such regularity as a result of finding the 5-limit dwarf is an interesting question. Whatever the reason for it, the marvelous dwarves--of size 12, 15, 18, 19, 20, 21, and 25--seem like excellent scales for instrumentalists and composers interested in 9-limit harmony and scales in this size range. The 25-note scale, whose 5-limit preimage we've discussed before as genus(15^4), is particularly striking from the point of view of the quantity of pentads it supplies.<br />
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| There is a marvelous or semimarvelous dwarf for each scale size from 11 to 22, and then the 25 note scale. So far as I know this is the complete list. <br />
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| Here is a brief description; the numbers are pentad number, numbers of major-minor triads, and size/pentad ratio.<br />
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| &lt;11 17 26| 1 6-6 11 semimarvelous<br />
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| &lt;12 19 28| 2 6-6 6 marvelous<br />
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| &lt;13 20 30| 1 7-6 13 semimarvelous<br />
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| &lt;14 22 33| 2 7-6 7 semimarvelous<br />
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| &lt;15 24 35| 3 8-8 5 marvelous<br />
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| &lt;16 25 37| 2 7-6 8 semimarvelous<br />
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| &lt;17 27 40| 4 10-9 4.25 semimarvelous<br />
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| &lt;18 29 42| 4 10-10 4.5 marvelous<br />
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| &lt;19 30 44| 5 12-11 3.8 marvelous<br />
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| &lt;20 32 47| 6 12-12 3.333 marvelous<br />
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| &lt;21 33 49| 5 12-12 4.2 marvelous<br />
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| &lt;22 35 51| 6 14-13 3.667 semimarvelous<br />
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| &lt;25 40 58| 9 16-16 2.778 marvelous</body></html></pre></div>
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