Warped diatonic: Difference between revisions

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A **warped diatonic scale** is a scale (excluding the diatonic scale itself) that contains relatively long substrings of the [[5L 2s]] diatonic scale ("LLsLLLsLLsLLLs...") in its sequence of large and small steps, and such that the sizes of those steps are similar to those of the diatonic scale (namely, in the ballpark of 200 and 100 cents).

Such scales may mislead a diatonic-conditioned listener into assigning the intervals to diatonic scale categories, but the categorization will be violated when either (1) the part of the scale that doesn't agree with 5L2s is reached, or (2) the harmonic nature of the intervals is drastically different than what's expected from the diatonic scale.

Combinatorically, most [[distributionally even]] scales with more L steps than s steps do have significantly long substrings of the 5L2s diatonic scale in them. So, when searching for distributionally even warped diatonics, we can use the simple figure of merit that xLys is a good warped diatonic when x > y and 2x+y is in the vicinity of 12. If 2x+y is much less than 12, the steps will be too large to be recognized as the diatonic scale; conversely if 2x+y is much more than 12, the steps will be too small.

The scales at the top and bottom of this table are questionable as "warped diatonics", but the ones near 2x+y=12 are good examples.

||~ 2x+y ||~ Formula ||~ Temperaments/names ||~ 5L2s substrings ||~ Comments ||
|| 8 || [[3L 2s]] || Father || LsLLsL || ||
|| 9 || [[4L 1s]] || Bug, superpelog || LLLsLL, LLsLLL || ||
|| 10 || [[4L 2s]] || Decimal, lemba || LLsLLsLL || "Remove one L" ||
|| 11 || [[4L 3s]] || Orgone, keemun || LsLLsL || "Replace one L with s" ||
|| 11 || [[5L 1s]] || Machine, gorgo || LLLsLL, LLsLLL || "Remove one s" ||
|| **12** || **[[5L 2s]]** || **Ordinary diatonic** || (infinitely long) || ||
|| 13 || [[5L 3s]] || Oneirotonic || LsLLsLL, LLsLLsL || "Add one s" ||
|| 13 || [[6L 1s]] || Archeotonic, leantone || LLLsLL, LLsLLL || "Replace one s with L" ||
|| 14 || [[5L 4s]] || Superpelog, godzilla || LsLLsL || "Add two s's" ||
|| 14 || [[6L 2s]] || Hedgehog || LLsLLLsLL || "Add one L" ||
|| 15 || [[6L 3s]] || Triforce || LLsLLsLL || ||
|| 15 || [[7L 1s]] || Porcupine || LLLsLL, LLsLLL || ||
|| 16 || [[6L 4s]] || Lemba, antikythera || LsLLsL || ||
|| 16 || [[7L 2s]] || Mavila || LLsLLLsLL || ||

Original HTML content:

<html><head><title>Warped diatonic</title></head><body>A <strong>warped diatonic scale</strong> is a scale (excluding the diatonic scale itself) that contains relatively long substrings of the <a class="wiki_link" href="/5L%202s">5L 2s</a> diatonic scale (&quot;LLsLLLsLLsLLLs...&quot;) in its sequence of large and small steps, and such that the sizes of those steps are similar to those of the diatonic scale (namely, in the ballpark of 200 and 100 cents).<br />
<br />
Such scales may mislead a diatonic-conditioned listener into assigning the intervals to diatonic scale categories, but the categorization will be violated when either (1) the part of the scale that doesn't agree with 5L2s is reached, or (2) the harmonic nature of the intervals is drastically different than what's expected from the diatonic scale.<br />
<br />
Combinatorically, most <a class="wiki_link" href="/distributionally%20even">distributionally even</a> scales with more L steps than s steps do have significantly long substrings of the 5L2s diatonic scale in them. So, when searching for distributionally even warped diatonics, we can use the simple figure of merit that xLys is a good warped diatonic when x &gt; y and 2x+y is in the vicinity of 12. If 2x+y is much less than 12, the steps will be too large to be recognized as the diatonic scale; conversely if 2x+y is much more than 12, the steps will be too small.<br />
<br />
The scales at the top and bottom of this table are questionable as &quot;warped diatonics&quot;, but the ones near 2x+y=12 are good examples.<br />
<br />


<table class="wiki_table">
    <tr>
        <th>2x+y<br />
</th>
        <th>Formula<br />
</th>
        <th>Temperaments/names<br />
</th>
        <th>5L2s substrings<br />
</th>
        <th>Comments<br />
</th>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td><a class="wiki_link" href="/3L%202s">3L 2s</a><br />
</td>
        <td>Father<br />
</td>
        <td>LsLLsL<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td><a class="wiki_link" href="/4L%201s">4L 1s</a><br />
</td>
        <td>Bug, superpelog<br />
</td>
        <td>LLLsLL, LLsLLL<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td><a class="wiki_link" href="/4L%202s">4L 2s</a><br />
</td>
        <td>Decimal, lemba<br />
</td>
        <td>LLsLLsLL<br />
</td>
        <td>&quot;Remove one L&quot;<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td><a class="wiki_link" href="/4L%203s">4L 3s</a><br />
</td>
        <td>Orgone, keemun<br />
</td>
        <td>LsLLsL<br />
</td>
        <td>&quot;Replace one L with s&quot;<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td><a class="wiki_link" href="/5L%201s">5L 1s</a><br />
</td>
        <td>Machine, gorgo<br />
</td>
        <td>LLLsLL, LLsLLL<br />
</td>
        <td>&quot;Remove one s&quot;<br />
</td>
    </tr>
    <tr>
        <td><strong>12</strong><br />
</td>
        <td><strong><a class="wiki_link" href="/5L%202s">5L 2s</a></strong><br />
</td>
        <td><strong>Ordinary diatonic</strong><br />
</td>
        <td>(infinitely long)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td><a class="wiki_link" href="/5L%203s">5L 3s</a><br />
</td>
        <td>Oneirotonic<br />
</td>
        <td>LsLLsLL, LLsLLsL<br />
</td>
        <td>&quot;Add one s&quot;<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td><a class="wiki_link" href="/6L%201s">6L 1s</a><br />
</td>
        <td>Archeotonic, leantone<br />
</td>
        <td>LLLsLL, LLsLLL<br />
</td>
        <td>&quot;Replace one s with L&quot;<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td><a class="wiki_link" href="/5L%204s">5L 4s</a><br />
</td>
        <td>Superpelog, godzilla<br />
</td>
        <td>LsLLsL<br />
</td>
        <td>&quot;Add two s's&quot;<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td><a class="wiki_link" href="/6L%202s">6L 2s</a><br />
</td>
        <td>Hedgehog<br />
</td>
        <td>LLsLLLsLL<br />
</td>
        <td>&quot;Add one L&quot;<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td><a class="wiki_link" href="/6L%203s">6L 3s</a><br />
</td>
        <td>Triforce<br />
</td>
        <td>LLsLLsLL<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td><a class="wiki_link" href="/7L%201s">7L 1s</a><br />
</td>
        <td>Porcupine<br />
</td>
        <td>LLLsLL, LLsLLL<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td><a class="wiki_link" href="/6L%204s">6L 4s</a><br />
</td>
        <td>Lemba, antikythera<br />
</td>
        <td>LsLLsL<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td><a class="wiki_link" href="/7L%202s">7L 2s</a><br />
</td>
        <td>Mavila<br />
</td>
        <td>LLsLLLsLL<br />
</td>
        <td><br />
</td>
    </tr>
</table>

</body></html>

Warped diatonic: Difference between revisions

Revision as of 18:13, 7 December 2011

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