1953 scale

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author MasonGreen1 and made on 2015-11-23 19:25:14 UTC.

The original revision id was 567551435.

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:

The **1953 scale** is my (Mason Green's) name for a nineteen-note subset of [[53edo]] (a MOS). It is a 15L+4S scale (it has 15 long intervals 3 [[Holdrian comma|Holdrian commas]] wide, plus 4 short intervals 2 Holdrians wide).

This scale offers a possible alternative to [[19edo]] as a way of expanding beyond 12edo. Unlike 19edo, whose fifths are all significantly flat, the 1953 scale has nearly perfect fifths, making it sound potentially more like 12edo. Its major thirds are also significantly better than those of 19edo.

However, only thirteen of the nineteen fifths are near-perfect; the other 6 fifths are flat at 679 cents, which is close to the Pythagorean wolf fifth.

Although the abundance of wolf fifths might seem like a disadvantage, composers may use them strategically to add expression to a piece (as [[https://en.wikipedia.org/wiki/Blue_note|blue note]]s). Another option is to use them as melodic intervals (in melodic lines or modulation) rather than played together harmonically, which lessens the unpleasant effect of the "wolfiness" and can also be used to add expression.

The generator for this scale is a just minor third, making this a [[Hanson]] MOS. One thing that 19edo and 53edo have in common is that both have near-just minor thirds; thus, using minor thirds to generate a 19-note scale in 53edo is a natural option.

Because it has only nineteen notes, the 1953 scale is ideal for keyboard instruments (19-keys-per-octave pianos do exist and have been made for centuries, although they are less common than their 12-key cousins). Since it's non-equally-tempered, it might be more difficult to design a guitar to play in this scale although it's probably possible.

This scale's 5-limit performance is amazing; in the 7-limit department it does not perform as well unless we choose to use a non-patent val for 7 (which tempers out the septimal comma, much as 19edo itself does). The harmonic seventh tetrad can only be voiced using this alternate val, but it's possible to voice the subminor triad (6:7:9) using the //patent// 7 as well.

The eleventh harmonic does not appear (although this might not necessarily be a bad thing for those who consider it [[Naughty and nice harmonics|naughty]]). The thirteenth harmonic does appear and the half-suspended triad 10:13:15 is common. This makes the 1953 scale an ideal scale for exploring the possibilities of the 13th harmonic alongside the more familiar 5 and 7, and it can be considered a tuning of the 3.5.7.13 subset.

The distances (in 53edo scale degrees) between adjacent notes of the 1953 scale are 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2.
In terms of absolute positions within 53edo, this translates to 0, 3, 6, 9, 12, 14, 17, 20, 23, 26, 28, 31, 34, 37, 40, 42, 45, 48, 51.

|| Interval || Multiplicity || Generic interval || Width in degrees of 53edo || Approximate ratios ||
|| Quarter tone || 4 || 1 || 2 || <span style="background-color: #ffffff;">49/48, 36/35, 33/32</span> ||
|| Third-tone || 15 || 1 || 3 || <span style="background-color: #ffffff;">27/26, 26/25, 25/24</span> ||
|| Diatonic semitone || 8 || 2 || 5 || 16/15, 15/14 ||
|| 2/3-tone || 11 || 2 || 6 || <span style="background-color: #ffffff;">14/13, 13/12, 27/25</span> ||
|| Minor whole tone || 12 || <span style="background-color: #ffffff;">3</span> || 8 || <span style="background-color: #ffffff;">10/9</span> ||
|| Major whole tone || 7 || 3 || 9 || <span style="background-color: #ffffff;">9/8</span> ||
|| 5/4-tone, tempered septimal whole tone, tempered septimal minor third || 16 || 4 || 11 || <span style="background-color: #ffffff;">15/13, 8/7*, 7/6*</span> ||
|| Patent septimal minor third || 3 || 4 || 12 || <span style="background-color: #ffffff;">7/6</span> ||
|| Pythagorean minor third || 1 || 5 || 13 || 32/27 ||
|| Pental minor third || 18 || 5 || 14 || 6/5 ||
|| Tridecimal neutral third || 5 || 6 || 16 || 16/13 ||
|| Pental major third || 14 || 6 || 17 || 5/4 ||
|| Patent septimal major third || 9 || 7 || 19 || 9/7* ||
|| Tempered septimal major third, Barbados third, father || 10 || 7 || 20 || 9/7, <span style="background-color: #ffffff;">13/10</span> ||
|| Perfect fourth || 13 || 8 || 22 || 4/3 ||
|| Wolf fourth || 6 || 8 || 23 || 27/20 ||
|| Tridecimal augmented fourth, tempered lesser septimal tritone || 17 || 9 || 25 || 7/5*, 18/13 ||
|| Patent lesser septimal tritone || 2 || 9 || 26 || 7/5 ||
|| Patent greater septimal tritone || 2 || 10 || 27 || 10/7 ||
|| Tridecimal diminished fifth, tempered greater septimal tritone || 17 || 10 || 28 || 10/7*, 13/9 ||
|| Wolf fifth || 6 || 11 || 30 || 40/27 ||
|| Perfect fifth || 13 || 11 || 31 || 3/2 ||
Approximations using the alternate (non-patent) val for 7 are shown with an asterisk (*).

The 1953 scale, like the diatonic scale, possesses [[https://en.wikipedia.org/wiki/Myhill%27s_property|Myhill's property]]. It is also strictly proper (whereas the 12-equal-tempered diatonic scale is proper, but not strictly proper).

Original HTML content:

<html><head><title>1953 scale</title></head><body>The <strong>1953 scale</strong> is my (Mason Green's) name for a nineteen-note subset of <a class="wiki_link" href="/53edo">53edo</a> (a MOS). It is a 15L+4S scale (it has 15 long intervals 3 <a class="wiki_link" href="/Holdrian%20comma">Holdrian commas</a> wide, plus 4 short intervals 2 Holdrians wide).<br />
<br />
This scale offers a possible alternative to <a class="wiki_link" href="/19edo">19edo</a> as a way of expanding beyond 12edo. Unlike 19edo, whose fifths are all significantly flat, the 1953 scale has nearly perfect fifths, making it sound potentially more like 12edo. Its major thirds are also significantly better than those of 19edo.<br />
<br />
However, only thirteen of the nineteen fifths are near-perfect; the other 6 fifths are flat at 679 cents, which is close to the Pythagorean wolf fifth.<br />
<br />
Although the abundance of wolf fifths might seem like a disadvantage, composers may use them strategically to add expression to a piece (as <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Blue_note" rel="nofollow">blue note</a>s). Another option is to use them as melodic intervals (in melodic lines or modulation) rather than played together harmonically, which lessens the unpleasant effect of the &quot;wolfiness&quot; and can also be used to add expression.<br />
<br />
The generator for this scale is a just minor third, making this a <a class="wiki_link" href="/Hanson">Hanson</a> MOS. One thing that 19edo and 53edo have in common is that both have near-just minor thirds; thus, using minor thirds to generate a 19-note scale in 53edo is a natural option.<br />
<br />
Because it has only nineteen notes, the 1953 scale is ideal for keyboard instruments (19-keys-per-octave pianos do exist and have been made for centuries, although they are less common than their 12-key cousins). Since it's non-equally-tempered, it might be more difficult to design a guitar to play in this scale although it's probably possible.<br />
<br />
This scale's 5-limit performance is amazing; in the 7-limit department it does not perform as well unless we choose to use a non-patent val for 7 (which tempers out the septimal comma, much as 19edo itself does). The harmonic seventh tetrad can only be voiced using this alternate val, but it's possible to voice the subminor triad (6:7:9) using the <em>patent</em> 7 as well.<br />
<br />
The eleventh harmonic does not appear (although this might not necessarily be a bad thing for those who consider it <a class="wiki_link" href="/Naughty%20and%20nice%20harmonics">naughty</a>). The thirteenth harmonic does appear and the half-suspended triad 10:13:15 is common. This makes the 1953 scale an ideal scale for exploring the possibilities of the 13th harmonic alongside the more familiar 5 and 7, and it can be considered a tuning of the 3.5.7.13 subset.<br />
<br />
The distances (in 53edo scale degrees) between adjacent notes of the 1953 scale are 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2.<br />
In terms of absolute positions within 53edo, this translates to 0, 3, 6, 9, 12, 14, 17, 20, 23, 26, 28, 31, 34, 37, 40, 42, 45, 48, 51.<br />
<br />


<table class="wiki_table">
    <tr>
        <td>Interval<br />
</td>
        <td>Multiplicity<br />
</td>
        <td>Generic interval<br />
</td>
        <td>Width in degrees of 53edo<br />
</td>
        <td>Approximate ratios<br />
</td>
    </tr>
    <tr>
        <td>Quarter tone<br />
</td>
        <td>4<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><span style="background-color: #ffffff;">49/48, 36/35, 33/32</span><br />
</td>
    </tr>
    <tr>
        <td>Third-tone<br />
</td>
        <td>15<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><span style="background-color: #ffffff;">27/26, 26/25, 25/24</span><br />
</td>
    </tr>
    <tr>
        <td>Diatonic semitone<br />
</td>
        <td>8<br />
</td>
        <td>2<br />
</td>
        <td>5<br />
</td>
        <td>16/15, 15/14<br />
</td>
    </tr>
    <tr>
        <td>2/3-tone<br />
</td>
        <td>11<br />
</td>
        <td>2<br />
</td>
        <td>6<br />
</td>
        <td><span style="background-color: #ffffff;">14/13, 13/12, 27/25</span><br />
</td>
    </tr>
    <tr>
        <td>Minor whole tone<br />
</td>
        <td>12<br />
</td>
        <td><span style="background-color: #ffffff;">3</span><br />
</td>
        <td>8<br />
</td>
        <td><span style="background-color: #ffffff;">10/9</span><br />
</td>
    </tr>
    <tr>
        <td>Major whole tone<br />
</td>
        <td>7<br />
</td>
        <td>3<br />
</td>
        <td>9<br />
</td>
        <td><span style="background-color: #ffffff;">9/8</span><br />
</td>
    </tr>
    <tr>
        <td>5/4-tone, tempered septimal whole tone, tempered septimal minor third<br />
</td>
        <td>16<br />
</td>
        <td>4<br />
</td>
        <td>11<br />
</td>
        <td><span style="background-color: #ffffff;">15/13, 8/7*, 7/6*</span><br />
</td>
    </tr>
    <tr>
        <td>Patent septimal minor third<br />
</td>
        <td>3<br />
</td>
        <td>4<br />
</td>
        <td>12<br />
</td>
        <td><span style="background-color: #ffffff;">7/6</span><br />
</td>
    </tr>
    <tr>
        <td>Pythagorean minor third<br />
</td>
        <td>1<br />
</td>
        <td>5<br />
</td>
        <td>13<br />
</td>
        <td>32/27<br />
</td>
    </tr>
    <tr>
        <td>Pental minor third<br />
</td>
        <td>18<br />
</td>
        <td>5<br />
</td>
        <td>14<br />
</td>
        <td>6/5<br />
</td>
    </tr>
    <tr>
        <td>Tridecimal neutral third<br />
</td>
        <td>5<br />
</td>
        <td>6<br />
</td>
        <td>16<br />
</td>
        <td>16/13<br />
</td>
    </tr>
    <tr>
        <td>Pental major third<br />
</td>
        <td>14<br />
</td>
        <td>6<br />
</td>
        <td>17<br />
</td>
        <td>5/4<br />
</td>
    </tr>
    <tr>
        <td>Patent septimal major third<br />
</td>
        <td>9<br />
</td>
        <td>7<br />
</td>
        <td>19<br />
</td>
        <td>9/7*<br />
</td>
    </tr>
    <tr>
        <td>Tempered septimal major third, Barbados third, father<br />
</td>
        <td>10<br />
</td>
        <td>7<br />
</td>
        <td>20<br />
</td>
        <td>9/7, <span style="background-color: #ffffff;">13/10</span><br />
</td>
    </tr>
    <tr>
        <td>Perfect fourth<br />
</td>
        <td>13<br />
</td>
        <td>8<br />
</td>
        <td>22<br />
</td>
        <td>4/3<br />
</td>
    </tr>
    <tr>
        <td>Wolf fourth<br />
</td>
        <td>6<br />
</td>
        <td>8<br />
</td>
        <td>23<br />
</td>
        <td>27/20<br />
</td>
    </tr>
    <tr>
        <td>Tridecimal augmented fourth, tempered lesser septimal tritone<br />
</td>
        <td>17<br />
</td>
        <td>9<br />
</td>
        <td>25<br />
</td>
        <td>7/5*, 18/13<br />
</td>
    </tr>
    <tr>
        <td>Patent lesser septimal tritone<br />
</td>
        <td>2<br />
</td>
        <td>9<br />
</td>
        <td>26<br />
</td>
        <td>7/5<br />
</td>
    </tr>
    <tr>
        <td>Patent greater septimal tritone<br />
</td>
        <td>2<br />
</td>
        <td>10<br />
</td>
        <td>27<br />
</td>
        <td>10/7<br />
</td>
    </tr>
    <tr>
        <td>Tridecimal diminished fifth, tempered greater septimal tritone<br />
</td>
        <td>17<br />
</td>
        <td>10<br />
</td>
        <td>28<br />
</td>
        <td>10/7*, 13/9<br />
</td>
    </tr>
    <tr>
        <td>Wolf fifth<br />
</td>
        <td>6<br />
</td>
        <td>11<br />
</td>
        <td>30<br />
</td>
        <td>40/27<br />
</td>
    </tr>
    <tr>
        <td>Perfect fifth<br />
</td>
        <td>13<br />
</td>
        <td>11<br />
</td>
        <td>31<br />
</td>
        <td>3/2<br />
</td>
    </tr>
</table>

Approximations using the alternate (non-patent) val for 7 are shown with an asterisk (*).<br />
<br />
The 1953 scale, like the diatonic scale, possesses <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Myhill%27s_property" rel="nofollow">Myhill's property</a>. It is also strictly proper (whereas the 12-equal-tempered diatonic scale is proper, but not strictly proper).</body></html>