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17edo neutral scale: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
== 17edo neutral scale ==
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2009-08-21 03:29:30 UTC</tt>.<br>
: The original revision id was <tt>85061777</tt>.<br>
: The revision comment was: <tt>MOS name added</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">=17edo neutral scale=  


A lovely system of Middle-Eastern flavored scales!
A lovely system of Middle-Eastern flavored scales!


We can call the [[MOSScales|Moment of Symmetry]] scale derived from a 5/17 generator &amp; an octave repeat the **17edo Neutral Scale**. We build it by stacking neutral thirds; in 17edo that means the interval of five degrees of 17.
We can call the [[MOSScales|Moment of Symmetry]] scale derived from a 5/17 generator &amp; an octave repeat the '''17edo Neutral Scale'''. It is an example of a [[neutral thirds scale]]. We build it by stacking neutral thirds, the generator of the [[neutrominant]] temperament. In [[17edo]] that means the interval of five degrees of 17.


Begin anywhere. Let's call our first pitch (&amp; its octave transposition) 0:
Begin anywhere. Let's call our first pitch (&amp; its octave transposition) 0:
Line 48: Line 41:
We have arrived again at a MOS scale, of type 3L+4s ("mosh" according to the [[MOSNamingScheme]]).
We have arrived again at a MOS scale, of type 3L+4s ("mosh" according to the [[MOSNamingScheme]]).


==7-note neutral scale:==  
==Interval chain==
Viewing 17edo as a temperament on the 2.3.7.11.13 subgroup, we get the following interpretation for the 2122122212 mode of the 10-note MOS scale:
{| class="wikitable sortable right-1 right-2"
|-
! Step# of scale<ref>In terms of the 10-note MOS scale, 1-based (unison=1)</ref>
! Steps of 17edo<ref>Amount of steps of 17edo, 0-based (often called "degree")</ref>
! Note name on C
! Harmonics approximated
! #Gens up
|-
| 9
| 14
| Bb
| '''7/4'''
| -4
|-
| 2
| 2
| Dd
|
| -3
|-
| 5
| 7
| F
|
| -2
|-
| 8
| 12
| Ad
| '''13/8'''
| -1
|-
| 11
| 17
| C
| '''2/1'''
| 0
|-
| 4
| 5
| Ed
|
| +1
|-
| 7
| 10
| G
| '''3/2'''
| +2
|-
| 10
| 15
| Bd
|
| +3
|-
| 3
| 3
| D
| '''9/8'''
| +4
|-
| 6
| 8
| F+
| '''11/8'''
| +5
|}
<references/>
 
The 6th degree can be raised by a [[chroma]] to a 23/16 (-5 generators). Some may prefer using the sharper 6th degree because it makes a 7/4 with the 8th degree.
 
== 7-note neutral scale ==


degrees from 0: 0 2 5 7 10 12 15 (0)
degrees from 0: 0 2 5 7 10 12 15 (0)
cents from 0: 0 141 353 494 706 847 1059 (1200)
cents from 0: 0 141 353 494 706 847 1059 (1200)
interval classes from P1: P1 N2 N3 P4 P5 N6 N7 (P8)
interval classes from P1: P1 N2 N3 P4 P5 N6 N7 (P8)


degrees between: 2 3 2 3 2 3 2
degrees between: 2 3 2 3 2 3 2
cents between: 141 212 141 212 141 212 141
cents between: 141 212 141 212 141 212 141
interval classes between: N2 M2 N2 M2 N2 M2 N2
interval classes between: N2 M2 N2 M2 N2 M2 N2


===modes of 7-note neutral scale===  
=== modes of 7-note neutral scale ===


Naturally, with seven notes we have seven modes, depending on which note we make the starting pitch (tonic) of the scale. I have given these modes a one-syllable name for my own use. Feel free to name (or not name) these modes as you see fit:
{{Idiosyncratic terms|The 7 proposed mode names}}
==== ====
|| mode 1 : bish || from bottom || in between ||
|| degrees || 0 2 5 7 10 12 15 (0) || 2 3 2 3 2 3 2 ||
|| cents || 0 141 353 494 706 847 1059 (1200) || 141 212 141 212 141 212 141 ||
|| interval classes || P1 N2 N3 P4 P5 N6 N7 (P8) || N2 M2 N2 M2 N2 M2 N2 ||
|| solfege || do ru mu fa sol lu tu (do) || ru re ru re ru re ru ||


|| mode 2 : dril || from bottom || in between ||
Naturally, with seven notes we have seven modes, depending on which note we make the starting pitch (tonic) of the scale. I ([[Andrew Heathwaite]]) have given these modes a one-syllable name for my own use. Feel free to name (or not name) these modes as you see fit:
|| degrees || 0 3 5 8 10 13 15 (0) || 3 2 3 2 3 2 2 ||
|| cents || 0 212 353 565 706 918 1059 (1200) || 212 141 212 141 212 141 141 ||
|| interval classes || P1 M2 N3 A4 P5 M6 N7 (P8) || M2 N2 M2 N2 M2 N2 N2 ||
|| solfege || do re mu fu sol la tu (do) || re ru re ru re ru ru ||


|| mode 3 : fish || from bottom || in between ||
{| class="wikitable"
|| degrees || 0 2 5 7 10 12 14 (0) || 2 3 2 3 2 2 3 ||
! mode 1 : bish
|| cents || 0 141 353 494 706 847 988 (1200) || 141 212 141 212 141 141 212 ||
! from bottom
|| interval classes || P1 N2 N3 P4 P5 N6 m7 (P8) || N2 M2 N2 M2 N2 N2 M2 ||
! in between
|| solfege || do ru mu fa sol lu te (do) || ru re ru re ru ru re ||
|-
! degrees
| 0 2 5 7 10 12 15 (0)
| 2 3 2 3 2 3 2
|-
! cents
| 0 141 353 494 706 847 1059 (1200)
| 141 212 141 212 141 212 141
|-
! interval classes
| P1 N2 N3 P4 P5 N6 N7 (P8)
| N2 M2 N2 M2 N2 M2 N2
|-
! solfege
| do ru mu fa sol lu tu (do)
| ru re ru re ru re ru
|}


|| mode 4 : gil || from bottom || in between ||
{| class="wikitable"
|| degrees || 0 3 5 8 10 12 15 (0) || 3 2 3 2 2 3 2 ||
! mode 2 : dril
|| cents || 0 212 353 565 706 847 1059 (1200) || 212 131 212 141 141 212 141 ||
! from bottom
|| interval classes || P1 M2 N3 A4 P5 N6 N7 (P8) || M2 N2 M2 N2 N2 M2 N2 ||
! in between
|| solfege || do re mu fu sol lu tu (do) || re ru re ru ru re ru ||
|-
! degrees
| 0 3 5 8 10 13 15 (0)
| 3 2 3 2 3 2 2
|-
! cents
| 0 212 353 565 706 918 1059 (1200)
| 212 141 212 141 212 141 141
|-
! interval classes
| P1 M2 N3 A4 P5 M6 N7 (P8)
| M2 N2 M2 N2 M2 N2 N2
|-
! solfege
| do re mu fu sol la tu (do)
| re ru re ru re ru ru
|}


|| mode 5 : jwl || from bottom || in between ||
{| class="wikitable"
|| degrees || 0 2 5 7 9 12 14 (0) || 2 3 2 2 3 2 3 ||
! mode 3 : fish
|| cents || 0 141 353 494 635 847 988 (1200) || 141 212 141 141 212 141 212 ||
! from bottom
|| interval classes || P1 N2 N3 P4 d5 N6 m7 (P8) || N2 M2 N2 N2 M2 N2 M2 ||
! in between
|| solfege || do ru mu fa su lu te (do) || ru re ru ru re ru re ||
|-
! degrees
| 0 2 5 7 10 12 14 (0)
| 2 3 2 3 2 2 3
|-
! cents
| 0 141 353 494 706 847 988 (1200)
| 141 212 141 212 141 141 212
|-
! interval classes
| P1 N2 N3 P4 P5 N6 m7 (P8)
| N2 M2 N2 M2 N2 N2 M2
|-
! solfege
| do ru mu fa sol lu te (do)
| ru re ru re ru ru re
|}


|| mode 6 : kleeth || from bottom || in between ||
{| class="wikitable"
|| degrees || 0 3 5 7 10 12 15 (0) || 3 2 2 3 2 3 2 ||
! mode 4 : gil
|| cents || 0 212 353 494 706 847 1059 (1200) || 212 141 141 212 141 212 141 ||
! from bottom
|| interval classes || P1 M2 N3 P4 P5 N6 N7 (P8) || M2 N2 N2 M2 N2 M2 N2 ||
! in between
|| solfege || do re mu fa sol lu tu (do) || re ru ru re ru re ru ||
|-
! degrees
| 0 3 5 8 10 12 15 (0)
| 3 2 3 2 2 3 2
|-
! cents
| 0 212 353 565 706 847 1059 (1200)
| 212 131 212 141 141 212 141
|-
! interval classes
| P1 M2 N3 A4 P5 N6 N7 (P8)
| M2 N2 M2 N2 N2 M2 N2
|-
! solfege
| do re mu fu sol lu tu (do)
| re ru re ru ru re ru
|}


|| mode 7 : led || from bottom || in between ||
{| class="wikitable"
|| degrees || 0 2 4 7 9 12 14 (0) || 2 2 3 2 3 2 3 ||
! mode 5 : jwl
|| cents || 0 141 282 494 635 847 988 (1200) || 141 141 212 141 212 141 212 ||
! from bottom
|| interval classes || P1 N2 m3 P4 d5 N6 m7 (P8) || N2 N2 M2 N2 M2 N2 M2 ||
! in between
|| solfege || do ru me fa su lu te (do) || ru ru re ru re ru re ||
|-
! degrees
| 0 2 5 7 9 12 14 (0)
| 2 3 2 2 3 2 3
|-
! cents
| 0 141 353 494 635 847 988 (1200)
| 141 212 141 141 212 141 212
|-
! interval classes
| P1 N2 N3 P4 d5 N6 m7 (P8)
| N2 M2 N2 N2 M2 N2 M2
|-
! solfege
| do ru mu fa su lu te (do)
| ru re ru ru re ru re
|}
 
{| class="wikitable"
! mode 6 : kleeth
! from bottom
! in between
|-
! degrees
| 0 3 5 7 10 12 15 (0)
| 3 2 2 3 2 3 2
|-
! cents
| 0 212 353 494 706 847 1059 (1200)
| 212 141 141 212 141 212 141
|-
! interval classes
| P1 M2 N3 P4 P5 N6 N7 (P8)
| M2 N2 N2 M2 N2 M2 N2
|-
! solfege
| do re mu fa sol lu tu (do)
| re ru ru re ru re ru
|}
 
{| class="wikitable"
! mode 7 : led
! from bottom
! in between
|-
! degrees
| 0 2 4 7 9 12 14 (0)
| 2 2 3 2 3 2 3
|-
! cents
| 0 141 282 494 635 847 988 (1200)
| 141 141 212 141 212 141 212
|-
! interval classes
| P1 N2 m3 P4 d5 N6 m7 (P8)
| N2 N2 M2 N2 M2 N2 M2
|-
! solfege
| do ru me fa su lu te (do)
| ru ru re ru re ru re
|}


As you can see, these modes contain many neutral 2nds &amp; 3rds, making it sound very different from the traditional major-minor Western harmonic &amp; melodic system, while having a coherent structure including ample 4ths &amp; 5ths that help ground the scale. The 17edo neutral sixths, at 847 cents, come very close to the 13th harmonic - JI interval 13/8 - 841 cents. Thus, their inversions, the 17edo neutral thirds come very close to 16/13.
As you can see, these modes contain many neutral 2nds &amp; 3rds, making it sound very different from the traditional major-minor Western harmonic &amp; melodic system, while having a coherent structure including ample 4ths &amp; 5ths that help ground the scale. The 17edo neutral sixths, at 847 cents, come very close to the 13th harmonic - JI interval 13/8 - 841 cents. Thus, their inversions, the 17edo neutral thirds come very close to 16/13.
Line 110: Line 295:
Interestingly, the 7-note neutral scale does not allow you to build any minor or major triads whatsoever. You have only one minor 3rd, which occurs with a diminished 5th, but no perfect fifth, allowing you to build a diminished triad, but no minor triad. You have no major thirds at all. In JI-terms, you might say that it contains harmonies based on 2, 3, &amp; 13, while skipping 7 &amp; 11.
Interestingly, the 7-note neutral scale does not allow you to build any minor or major triads whatsoever. You have only one minor 3rd, which occurs with a diminished 5th, but no perfect fifth, allowing you to build a diminished triad, but no minor triad. You have no major thirds at all. In JI-terms, you might say that it contains harmonies based on 2, 3, &amp; 13, while skipping 7 &amp; 11.


17-tonists may find these scales helpful for escaping the familiar. Just because you //can// play diatonic music in 17edo, doesn't mean you have to. These neutral scales give you a more xenharmonic modal system to play with.
17-tonists may find these scales helpful for escaping the familiar. Just because you ''can'' play diatonic music in 17edo, doesn't mean you have to. These neutral scales give you a more xenharmonic modal system to play with.


If you continue stacking neutral thirds, you will soon come to a rather lovely 10-note neutral scale. I (or someone) will come back to that sooner or later.
If you continue stacking neutral thirds, you will soon come to a rather lovely 10-note neutral scale. I (or someone) will come back to that sooner or later.


== Some brief note on the 3, 7 and 10 note MOS ==


(Note that you will come up with similarly structured scales by using //other neutral thirds// as generators, although some of them will sound quite different. Some equal divisions of the octave containing neutral scales: [[10edo]], [[13edo]], [[16edo]], [[19edo]], [[24edo]], [[31edo]]....)</pre></div>
You can also take call the neutral sixth the generator, which I ([[Andrew Heathwaite]]) personally favour as it is an (approximate) harmonic rather than a subharmonic. But that's because it's how I use it, you might not. If you see it this way, the 3rd harmonic is harmonically opposite to the 13th harmonic, because, (13/8)^2 ~ 4/3, the perfect fourth being an upside down perfect fifth.
<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;17edo neutral scale&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x17edo neutral scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;17edo neutral scale&lt;/h1&gt;
&lt;br /&gt;
A lovely system of Middle-Eastern flavored scales!&lt;br /&gt;
&lt;br /&gt;
We can call the &lt;a class="wiki_link" href="/MOSScales"&gt;Moment of Symmetry&lt;/a&gt; scale derived from a 5/17 generator &amp;amp; an octave repeat the &lt;strong&gt;17edo Neutral Scale&lt;/strong&gt;. We build it by stacking neutral thirds; in 17edo that means the interval of five degrees of 17.&lt;br /&gt;
&lt;br /&gt;
Begin anywhere. Let's call our first pitch (&amp;amp; its octave transposition) 0:&lt;br /&gt;
&lt;br /&gt;
0 (0)&lt;br /&gt;
&lt;br /&gt;
Add a note a neutral third (five degrees) up from 0:&lt;br /&gt;
&lt;br /&gt;
0 5 (0)&lt;br /&gt;
&lt;br /&gt;
Add a note a neutral third down from 0 (remember, in 17edo, 0=17):&lt;br /&gt;
&lt;br /&gt;
0 5 12 (0)&lt;br /&gt;
&lt;br /&gt;
Between these notes we have intervals of:&lt;br /&gt;
&lt;br /&gt;
5 7 5&lt;br /&gt;
&lt;br /&gt;
Since we have two different step sizes, we have arrived at a three-note MOS scale. But let's continue; three-note scales don't give us much to work with.&lt;br /&gt;
&lt;br /&gt;
Add an N3 up from 5:&lt;br /&gt;
&lt;br /&gt;
0 5 10 12 (0)&lt;br /&gt;
&lt;br /&gt;
Add an N3 down from 12:&lt;br /&gt;
&lt;br /&gt;
0 5 7 10 12 (0)&lt;br /&gt;
&lt;br /&gt;
Add an N3 up from 10:&lt;br /&gt;
&lt;br /&gt;
0 5 7 10 12 15 (0)&lt;br /&gt;
&lt;br /&gt;
Add an N3 down from 7:&lt;br /&gt;
&lt;br /&gt;
0 2 5 7 10 12 15 (0)&lt;br /&gt;
&lt;br /&gt;
We have arrived again at a MOS scale, of type 3L+4s (&amp;quot;mosh&amp;quot; according to the &lt;a class="wiki_link" href="/MOSNamingScheme"&gt;MOSNamingScheme&lt;/a&gt;).&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x17edo neutral scale-7-note neutral scale:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;7-note neutral scale:&lt;/h2&gt;
&lt;br /&gt;
degrees from 0: 0 2 5 7 10 12 15 (0)&lt;br /&gt;
cents from 0: 0 141 353 494 706 847 1059 (1200)&lt;br /&gt;
interval classes from P1: P1 N2 N3 P4 P5 N6 N7 (P8)&lt;br /&gt;
&lt;br /&gt;
degrees between: 2 3 2 3 2 3 2&lt;br /&gt;
cents between: 141 212 141 212 141 212 141&lt;br /&gt;
interval classes between: N2 M2 N2 M2 N2 M2 N2&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x17edo neutral scale-7-note neutral scale:-modes of 7-note neutral scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;modes of 7-note neutral scale&lt;/h3&gt;
&lt;br /&gt;
Naturally, with seven notes we have seven modes, depending on which note we make the starting pitch (tonic) of the scale. I have given these modes a one-syllable name for my own use. Feel free to name (or not name) these modes as you see fit:&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc3"&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt; &lt;/h4&gt;
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;mode 1 : bish&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;from bottom&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;in between&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;degrees&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 2 5 7 10 12 15 (0)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2 3 2 3 2 3 2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 141 353 494 706 847 1059 (1200)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;141 212 141 212 141 212 141&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;interval classes&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;P1 N2 N3 P4 P5 N6 N7 (P8)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;N2 M2 N2 M2 N2 M2 N2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;solfege&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;do ru mu fa sol lu tu (do)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;ru re ru re ru re ru&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;mode 2 : dril&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;from bottom&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;in between&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;degrees&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 3 5 8 10 13 15 (0)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3 2 3 2 3 2 2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 212 353 565 706 918 1059 (1200)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;212 141 212 141 212 141 141&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;interval classes&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;P1 M2 N3 A4 P5 M6 N7 (P8)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;M2 N2 M2 N2 M2 N2 N2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;solfege&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;do re mu fu sol la tu (do)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;re ru re ru re ru ru&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;mode 3 : fish&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;from bottom&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;in between&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;degrees&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 2 5 7 10 12 14 (0)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2 3 2 3 2 2 3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 141 353 494 706 847 988 (1200)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;141 212 141 212 141 141 212&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;interval classes&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;P1 N2 N3 P4 P5 N6 m7 (P8)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;N2 M2 N2 M2 N2 N2 M2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;solfege&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;do ru mu fa sol lu te (do)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;ru re ru re ru ru re&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;mode 4 : gil&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;from bottom&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;in between&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;degrees&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 3 5 8 10 12 15 (0)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3 2 3 2 2 3 2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 212 353 565 706 847 1059 (1200)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;212 131 212 141 141 212 141&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;interval classes&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;P1 M2 N3 A4 P5 N6 N7 (P8)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;M2 N2 M2 N2 N2 M2 N2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;solfege&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;do re mu fu sol lu tu (do)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;re ru re ru ru re ru&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;mode 5 : jwl&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;from bottom&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;in between&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;degrees&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 2 5 7 9 12 14 (0)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2 3 2 2 3 2 3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 141 353 494 635 847 988 (1200)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;141 212 141 141 212 141 212&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;interval classes&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;P1 N2 N3 P4 d5 N6 m7 (P8)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;N2 M2 N2 N2 M2 N2 M2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;solfege&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;do ru mu fa su lu te (do)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;ru re ru ru re ru re&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;mode 6 : kleeth&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;from bottom&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;in between&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;degrees&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 3 5 7 10 12 15 (0)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3 2 2 3 2 3 2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 212 353 494 706 847 1059 (1200)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;212 141 141 212 141 212 141&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;interval classes&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;P1 M2 N3 P4 P5 N6 N7 (P8)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;M2 N2 N2 M2 N2 M2 N2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;solfege&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;do re mu fa sol lu tu (do)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;re ru ru re ru re ru&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;


You might also find that the 10-note scale can be formed by two 17-tone pythagoresque pentatonic scales a neutral interval apart, implying something of a different approach. And one of the loveliest things I find about them is the ease with which one can play 8:11:13 chords, so there are some frightening blues licks in this decatonic scale. R'lyeh blues anyone?


&lt;table class="wiki_table"&gt;
(Note that you will come up with similarly structured scales by using ''other neutral thirds'' as generators, although some of them will sound quite different. A neutral sixth about sharp of the 13th harmonic leads to 7L+3s like in 17-tone, whereas going flat of the 13th harmonic can lead to 7s+3L. (This boast is possible because 10-edo sits right on it.) Some equal divisions of the octave containing neutral scales: [[10edo]], [[13edo]], [[16edo]], [[19edo]], [[24edo]], [[31edo]]....)
    &lt;tr&gt;
        &lt;td&gt;mode 7 : led&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;from bottom&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;in between&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;degrees&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 2 4 7 9 12 14 (0)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2 2 3 2 3 2 3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 141 282 494 635 847 988 (1200)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;141 141 212 141 212 141 212&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;interval classes&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;P1 N2 m3 P4 d5 N6 m7 (P8)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;N2 N2 M2 N2 M2 N2 M2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;solfege&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;do ru me fa su lu te (do)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;ru ru re ru re ru re&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
[[Category:13-limit]]
As you can see, these modes contain many neutral 2nds &amp;amp; 3rds, making it sound very different from the traditional major-minor Western harmonic &amp;amp; melodic system, while having a coherent structure including ample 4ths &amp;amp; 5ths that help ground the scale. The 17edo neutral sixths, at 847 cents, come very close to the 13th harmonic - JI interval 13/8 - 841 cents. Thus, their inversions, the 17edo neutral thirds come very close to 16/13.&lt;br /&gt;
[[Category:17edo]]
&lt;br /&gt;
[[Category:Modes]]
The 17edo neutral 2nds, at 141 cents, fall between 13/12 (139 cents) &amp;amp; 12/11 (151) cents. I've found that they generally function as 13/12, since they fall 3/2 away from 13/8. But you can discover these things for yourself, if you like, &amp;amp; feel free to think of them in different ways entirely.&lt;br /&gt;
[[Category:MOS scales]]
&lt;br /&gt;
[[Category:Neutral]]
Interestingly, the 7-note neutral scale does not allow you to build any minor or major triads whatsoever. You have only one minor 3rd, which occurs with a diminished 5th, but no perfect fifth, allowing you to build a diminished triad, but no minor triad. You have no major thirds at all. In JI-terms, you might say that it contains harmonies based on 2, 3, &amp;amp; 13, while skipping 7 &amp;amp; 11.&lt;br /&gt;
[[Category:Neutral second]]
&lt;br /&gt;
[[Category:Neutral third]]
17-tonists may find these scales helpful for escaping the familiar. Just because you &lt;em&gt;can&lt;/em&gt; play diatonic music in 17edo, doesn't mean you have to. These neutral scales give you a more xenharmonic modal system to play with.&lt;br /&gt;
&lt;br /&gt;
If you continue stacking neutral thirds, you will soon come to a rather lovely 10-note neutral scale. I (or someone) will come back to that sooner or later.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(Note that you will come up with similarly structured scales by using &lt;em&gt;other neutral thirds&lt;/em&gt; as generators, although some of them will sound quite different. Some equal divisions of the octave containing neutral scales: &lt;a class="wiki_link" href="/10edo"&gt;10edo&lt;/a&gt;, &lt;a class="wiki_link" href="/13edo"&gt;13edo&lt;/a&gt;, &lt;a class="wiki_link" href="/16edo"&gt;16edo&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="/24edo"&gt;24edo&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;....)&lt;/body&gt;&lt;/html&gt;</pre></div>
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