Skip-fifteen scale: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
There are actually two '''skip-fifteen scales''', a utonal and an otonal version. Both are heptatonic scales (seven tones per octave).
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-01 23:02:02 UTC</tt>.<br>
: The original revision id was <tt>564704053</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>
<h4>Original Wikitext content:</h4>
<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">There are actually two **skip-fifteen scales**, a utonal and an otonal version. Both are heptatonic scales (seven tones per octave).


I (Mason Green) am not sure if anyone else has written about this type of scale before. It might be a re-discovery, but I find it interesting.
I (Mason Green) am not sure if anyone else has written about this type of scale before. It might be a re-discovery, but I find it interesting. [In fact, the otonal skip15 is the [[Dwarves|dwarf scale]] for the 13-limit 7edo patent val. It is epimorphic and [[constant structure]]. It is also the 13-limit otonal heptad, and has been listed here as [[oheptad]], and the utonal version as [[uheptad]].]


A skip-fifteen scale is the octave-repeating scale in just intonation consisting of all the harmonics between 8 and 16, but skipping the 15th (hence the name). The utonal version instead uses the subharmonics between 1/16 and 1/8, excluding 1/15.
A skip-fifteen scale is the octave-repeating scale in just intonation consisting of all the harmonics between 8 and 16, but skipping the 15th (hence the name). The utonal version instead uses the subharmonics between 1/16 and 1/8, excluding 1/15.


==Otonal skip15 scale==  
==Otonal skip15 scale==


|| Corresponding harmonic || Cents || Distance above last note ||
{| class="wikitable"
|| 8 (or 16) || 0 || (231.174) ||
|-
|| 9 || 203.910 || 203.910 ||
| | Corresponding harmonic
|| 10 || 386.314 || 182.404 ||
| | Cents
|| 11 || 551.318 || 165.004 ||
| | Distance above last note
|| 12 || 701.955 || 150.637 ||
|-
|| 13 || 840.528 || 138.573 ||
| | 8 (or 16)
|| 14 || 968.826 || 128.298 ||
| | 0
== ==
| | (231.174)
==Utonal skip15 scale==
|-
| | 9
| | 203.910
| | 203.910
|-
| | 10
| | 386.314
| | 182.404
|-
| | 11
| | 551.318
| | 165.004
|-
| | 12
| | 701.955
| | 150.637
|-
| | 13
| | 840.528
| | 138.573
|-
| | 14
| | 968.826
| | 128.298
|}


|| Corresponding subharmonic || Cents || Distance above last note ||
==Utonal skip15 scale==
|| 1/16 (or 1/8) || 0 || (203.174) ||
|| 1/14 || 231.174 || 231.174 ||
|| 1/13 || 359.472 || 128.298 ||
|| 1/12 || 498.045 || 138.573 ||
|| 1/11 || 648.682 || 150.637 ||
|| 1/10 || 813.686 || 165.004 ||
|| 1/9 || 996.090 || 182.404 ||


==Why skip the 15?==  
{| class="wikitable"
|-
| | Corresponding subharmonic
| | Cents
| | Distance above last note
|-
| | 1/16 (or 1/8)
| | 0
| | (203.174)
|-
| | 1/14
| | 231.174
| | 231.174
|-
| | 1/13
| | 359.472
| | 128.298
|-
| | 1/12
| | 498.045
| | 138.573
|-
| | 1/11
| | 648.682
| | 150.637
|-
| | 1/10
| | 813.686
| | 165.004
|-
| | 1/9
| | 996.090
| | 182.404
|}
 
==Why skip the 15?==


Although it's also possible to include the fifteenth (sub)harmonic, thus generating an octatonic scale, I believe it's better to omit it. Specifically, this helps ensure that:
Although it's also possible to include the fifteenth (sub)harmonic, thus generating an octatonic scale, I believe it's better to omit it. Specifically, this helps ensure that:
Line 41: Line 86:
* Every interval two steps wide is perceived as a third, and
* Every interval two steps wide is perceived as a third, and
* The 15:13 interval (which is tonally ambiguous in that it is almost halfway between a third and a whole tone) is omitted. Omitting this interval ensures a clear separation between tones and thirds.
* The 15:13 interval (which is tonally ambiguous in that it is almost halfway between a third and a whole tone) is omitted. Omitting this interval ensures a clear separation between tones and thirds.
* It is possible to cycle through all the pitch classes by taking the steps two at a time. In combination with the last property, this makes a circle of (unequal) thirds. This circle of thirds is 4:5:6:7:9:11:13:16 (or its utonal counterpart), a [[https://en.wikipedia.org/wiki/Fifteenth|double octave]]-repeating scale in which every interval is a third. There are seven different types of thirds used in the circle, each type occurring exactly once in the circle: supermajor (9:7), wide major (13:11), major (5:4), wide neutral (16:13), narrow neutral (11:9), minor (6:5), and subminor (7:6).
* It is possible to cycle through all the pitch classes by taking the steps two at a time. In combination with the last property, this makes a circle of (unequal) thirds. This circle of thirds is 4:5:6:7:9:11:13:16 (or its utonal counterpart), a [https://en.wikipedia.org/wiki/Fifteenth double octave]-repeating scale in which every interval is a third. There are seven different types of thirds used in the circle, each type occurring exactly once in the circle: supermajor (9:7), wide major (13:11), major (5:4), wide neutral (16:13), narrow neutral (11:9), minor (6:5), and subminor (7:6).


There is also a "phantom third" (14:11) which is not part of the circle of thirds, but appears in the skip15 scales as an interval three (rather than two) steps wide. This phantom third means that neither skip15 scale is [[Rothenberg propriety|proper]] (since it's narrower than the supermajor third, which occurs as a two-step interval in the scales).
There is also a "phantom third" (14:11) which is not part of the circle of thirds, but appears in the skip15 scales as an interval three (rather than two) steps wide. This phantom third means that neither skip15 scale is [[Rothenberg_propriety|proper]] (since it's narrower than the supermajor third, which occurs as a two-step interval in the scales).


The "Barbados third" or "fird" (13:10), which is 454.217 cents, also occurs in the skip15 scale but not in the circle. However, this interval is also sometimes considered a narrowed fourth rather than as a third.
The "Barbados third" or "fird" (13:10), which is 454.217 cents, also occurs in the skip15 scale but not in the circle. However, this interval is also sometimes considered a narrowed fourth rather than as a third.


==Approximations==  
==Approximations==
 
The skip15 scales (and the circle of thirds) can be approximated in many high edos, which are capable of distinguishing between all eight different thirds (the seven that are part of the circle, plus the "phantom" 11:14) third. Note that in order to ensure that every third has a unique approximation, it may be necessary to use a non-patent val in some cases.</pre></div>
<h4>Original HTML content:</h4>
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Skip-fifteen scale&lt;/title&gt;&lt;/head&gt;&lt;body&gt;There are actually two &lt;strong&gt;skip-fifteen scales&lt;/strong&gt;, a utonal and an otonal version. Both are heptatonic scales (seven tones per octave).&lt;br /&gt;
&lt;br /&gt;
I (Mason Green) am not sure if anyone else has written about this type of scale before. It might be a re-discovery, but I find it interesting.&lt;br /&gt;
&lt;br /&gt;
A skip-fifteen scale is the octave-repeating scale in just intonation consisting of all the harmonics between 8 and 16, but skipping the 15th (hence the name). The utonal version instead uses the subharmonics between 1/16 and 1/8, excluding 1/15.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Otonal skip15 scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Otonal skip15 scale&lt;/h2&gt;
&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;Corresponding harmonic&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Distance above last note&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8 (or 16)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;(231.174)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;203.910&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;203.910&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;386.314&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;182.404&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;551.318&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;165.004&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;701.955&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;150.637&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;840.528&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;138.573&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;968.826&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;128.298&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt; &lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Utonal skip15 scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Utonal skip15 scale&lt;/h2&gt;
&lt;br /&gt;
 


&lt;table class="wiki_table"&gt;
The skip15 scales (and the circle of thirds) can be approximated in many high edos, which are capable of distinguishing between all eight different thirds (the seven that are part of the circle, plus the "phantom" 11:14) third. Note that in order to ensure that every third has a unique approximation, it may be necessary to use a non-patent val in some cases.
    &lt;tr&gt;
        &lt;td&gt;Corresponding subharmonic&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Distance above last note&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1/16 (or 1/8)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;(203.174)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;231.174&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;231.174&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;359.472&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;128.298&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1/12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;498.045&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;138.573&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;648.682&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;150.637&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;813.686&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;165.004&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;996.090&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;182.404&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
[[Category:heptatonic]]
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-Why skip the 15?"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Why skip the 15?&lt;/h2&gt;
[[Category:just_intonation]]
&lt;br /&gt;
[[Category:Dwarves]]
Although it's also possible to include the fifteenth (sub)harmonic, thus generating an octatonic scale, I believe it's better to omit it. Specifically, this helps ensure that:&lt;br /&gt;
[[Category:7-tone scales]]
&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;Every interval one step wide is perceived as a (major or neutral) tone, and&lt;/li&gt;&lt;li&gt;Every interval two steps wide is perceived as a third, and&lt;/li&gt;&lt;li&gt;The 15:13 interval (which is tonally ambiguous in that it is almost halfway between a third and a whole tone) is omitted. Omitting this interval ensures a clear separation between tones and thirds.&lt;/li&gt;&lt;li&gt;It is possible to cycle through all the pitch classes by taking the steps two at a time. In combination with the last property, this makes a circle of (unequal) thirds. This circle of thirds is 4:5:6:7:9:11:13:16 (or its utonal counterpart), a &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Fifteenth" rel="nofollow"&gt;double octave&lt;/a&gt;-repeating scale in which every interval is a third. There are seven different types of thirds used in the circle, each type occurring exactly once in the circle: supermajor (9:7), wide major (13:11), major (5:4), wide neutral (16:13), narrow neutral (11:9), minor (6:5), and subminor (7:6).&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
There is also a &amp;quot;phantom third&amp;quot; (14:11) which is not part of the circle of thirds, but appears in the skip15 scales as an interval three (rather than two) steps wide. This phantom third means that neither skip15 scale is &lt;a class="wiki_link" href="/Rothenberg%20propriety"&gt;proper&lt;/a&gt; (since it's narrower than the supermajor third, which occurs as a two-step interval in the scales).&lt;br /&gt;
&lt;br /&gt;
The &amp;quot;Barbados third&amp;quot; or &amp;quot;fird&amp;quot; (13:10), which is 454.217 cents, also occurs in the skip15 scale but not in the circle. However, this interval is also sometimes considered a narrowed fourth rather than as a third.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-Approximations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Approximations&lt;/h2&gt;
&lt;br /&gt;
The skip15 scales (and the circle of thirds) can be approximated in many high edos, which are capable of distinguishing between all eight different thirds (the seven that are part of the circle, plus the &amp;quot;phantom&amp;quot; 11:14) third. Note that in order to ensure that every third has a unique approximation, it may be necessary to use a non-patent val in some cases.&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 20:30, 23 April 2023

There are actually two skip-fifteen scales, a utonal and an otonal version. Both are heptatonic scales (seven tones per octave).

I (Mason Green) am not sure if anyone else has written about this type of scale before. It might be a re-discovery, but I find it interesting. [In fact, the otonal skip15 is the dwarf scale for the 13-limit 7edo patent val. It is epimorphic and constant structure. It is also the 13-limit otonal heptad, and has been listed here as oheptad, and the utonal version as uheptad.]

A skip-fifteen scale is the octave-repeating scale in just intonation consisting of all the harmonics between 8 and 16, but skipping the 15th (hence the name). The utonal version instead uses the subharmonics between 1/16 and 1/8, excluding 1/15.

Otonal skip15 scale

Corresponding harmonic Cents Distance above last note
8 (or 16) 0 (231.174)
9 203.910 203.910
10 386.314 182.404
11 551.318 165.004
12 701.955 150.637
13 840.528 138.573
14 968.826 128.298

Utonal skip15 scale

Corresponding subharmonic Cents Distance above last note
1/16 (or 1/8) 0 (203.174)
1/14 231.174 231.174
1/13 359.472 128.298
1/12 498.045 138.573
1/11 648.682 150.637
1/10 813.686 165.004
1/9 996.090 182.404

Why skip the 15?

Although it's also possible to include the fifteenth (sub)harmonic, thus generating an octatonic scale, I believe it's better to omit it. Specifically, this helps ensure that:

  • Every interval one step wide is perceived as a (major or neutral) tone, and
  • Every interval two steps wide is perceived as a third, and
  • The 15:13 interval (which is tonally ambiguous in that it is almost halfway between a third and a whole tone) is omitted. Omitting this interval ensures a clear separation between tones and thirds.
  • It is possible to cycle through all the pitch classes by taking the steps two at a time. In combination with the last property, this makes a circle of (unequal) thirds. This circle of thirds is 4:5:6:7:9:11:13:16 (or its utonal counterpart), a double octave-repeating scale in which every interval is a third. There are seven different types of thirds used in the circle, each type occurring exactly once in the circle: supermajor (9:7), wide major (13:11), major (5:4), wide neutral (16:13), narrow neutral (11:9), minor (6:5), and subminor (7:6).

There is also a "phantom third" (14:11) which is not part of the circle of thirds, but appears in the skip15 scales as an interval three (rather than two) steps wide. This phantom third means that neither skip15 scale is proper (since it's narrower than the supermajor third, which occurs as a two-step interval in the scales).

The "Barbados third" or "fird" (13:10), which is 454.217 cents, also occurs in the skip15 scale but not in the circle. However, this interval is also sometimes considered a narrowed fourth rather than as a third.

Approximations

The skip15 scales (and the circle of thirds) can be approximated in many high edos, which are capable of distinguishing between all eight different thirds (the seven that are part of the circle, plus the "phantom" 11:14) third. Note that in order to ensure that every third has a unique approximation, it may be necessary to use a non-patent val in some cases.

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