Easy Scales by Interpolating between Harmonic Series: Difference between revisions
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Some of the most prominent scales in existence can be very quickly derived from harmonic series. Take, for example, the diatonic major scale in 12EDO, where notes are approximately equal to | Some of the most prominent scales in existence can be very quickly derived from harmonic series. Take, for example, the diatonic major scale in 12EDO, where notes are approximately equal to | ||
__**C major diatonic in 12EDO <<8-9-12>> scale**__ | |||
|| C || D || E || F || G || A || B || | || C || D || E || F || G || A || B || | ||
|| 1/1 || 9/8 or 10/9 || 5/4 || 4/3 || 3/2 || 5/3 | || 1/1 || 9/8 or 10/9 || 5/4 || 4/3 || 3/2 || 5/3 || 15/8 or 17/9 || | ||
This can be derived from the following harmonic series | This can be derived from the following harmonic series | ||
| Line 20: | Line 22: | ||
which is the same as the notes C D E G B and contains the **tonic major chord C E G** along with the **dominant major chord G B D** | which is the same as the notes C D E G B and contains the **tonic major chord C E G** along with the **dominant major chord G B D** | ||
**(x/12) -** 1/1 5/4 4/3 3/2 5/3 | **(x/12) -** 1/1 5/4 4/3 3/2 5/3 | ||
the same as the notes C E F G A. | the same as the notes C E F G A. | ||
The **x/12 and x/9** harmonic series become particularly stressed in the **(Maqam) Rast**, also known as the **"Blues" scale**, of | |||
__**Maqam Rast <<8-9-12>> scale**__ | |||
|| C || D || D#-E || F || G || A || A#-B || | |||
|| 1/1 || 9/8 or 10/9 || **11/9** || 4/3 || 3/2 || 5/3 or 27/16 || **11/6** || | |||
Here the x/9 series uses the "blue tone" of 11/9 and grows into | |||
**(x/9)** - 1/1 10/9 **11/9** 12/9 15/9 17/9 | |||
Meanwhile the addition of the "blue tone" of 11/6 and removal of 5/4 changes the x/12 series to | |||
**(x/12)** - 1/1 4/3 3/2 5/3 **11/6** | |||
And the removal of the 5/4 and 15/8 shrinks the x/8 series into | |||
**(x/8) -** 1/1 9/8 3/2 | |||
If you don't like the small x/8 series, you can modify that scale by deleting the 11/9 and adding back the 5/4 and removing the 11/9 to make the scale more balanced between the size of the different series e.g. | |||
(x/8) - 1/1 9/8 5/4 3/2 | |||
and | |||
(x/9) - 1/1 10/9 4/3 5/3 17/9 | |||
**But why limit yourself to/by x/8, x/9, and x/12 harmonic extensions of the root?** | |||
---- | |||
**Part 2- Attempting to create scales from scratch given harmonic series** | |||
A common pattern in the above scales is use of and interpolation between the x/9 and x/12 harmonic series | |||
It turns out the x/7 harmonic series has many notes in common with or close to the x/8, x/9, and x/12 harmonic series. | |||
We can take advantage of this to construct a more advanced scale in order the create the series | |||
**(x/7)** = 7:9:11:12:13 (AKA 1/1 9/7 11/7 12/7 13/7) | |||
**(x/8)** = 8:9:10:12:14:15 | |||
**(x/9)** = 9:10:12:14:15 | |||
**(x/12)** = 12:15:16:18:20 | |||
Some of these ratios are near each other, but not equal, such as 9/8 in the (x/8) and 10/9 in the (x/9) series or 14/9 in the (x/9) series and 11/7 in the x/7 series. | |||
One way to evenly distribute error is to take the harmonic/geometric median between such fractions e.g. square-root-of ((11/7)/(14/9)) * 14/9 = about 1.56. | |||
Doing such gives us the scale | |||
__**10 note <<7-8-9-12>> "extended color diatonic" Harmonic Segment Scale**__ | |||
|| 1/1 || 11181/10000 || 5/4 || 9/7 || 4/3 || 3/2 || 156341/100000 || 5/3 || 26/15 || 28/15 || | |||
|| || between 10/9 and 9/8 || || || || || between 14/9 and 11/7 || || between 12/7 and 7/4 || between 13/7 and 15/8 || | |||
Feel free to add comments or find/punch loopholes in this theory, but this is what I'd consider an idea **"extended diatonic"** scale in that it stresses likely the most important harmonic series in the diatonic scale while adding an extra (x/7) into the mix, giving the scale a wider range of tonal color. | |||
**More scales to come later...**</pre></div> | |||
<h4>Original HTML content:</h4> | <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"><html><head><title>Easy Scales by Interpolating between Harmonic Series</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Easy Scales by Interpolating between Harmonic Series"></a><!-- ws:end:WikiTextHeadingRule:0 -->Easy Scales by Interpolating between Harmonic Series</h1> | <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>Easy Scales by Interpolating between Harmonic Series</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Easy Scales by Interpolating between Harmonic Series"></a><!-- ws:end:WikiTextHeadingRule:0 -->Easy Scales by Interpolating between Harmonic Series</h1> | ||
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<br /> | <br /> | ||
Some of the most prominent scales in existence can be very quickly derived from harmonic series. Take, for example, the diatonic major scale in 12EDO, where notes are approximately equal to<br /> | Some of the most prominent scales in existence can be very quickly derived from harmonic series. Take, for example, the diatonic major scale in 12EDO, where notes are approximately equal to<br /> | ||
<br /> | |||
<u><strong>C major diatonic in 12EDO &lt;&lt;8-9-12&gt;&gt; scale</strong></u><br /> | |||
| Line 59: | Line 104: | ||
<td>3/2<br /> | <td>3/2<br /> | ||
</td> | </td> | ||
<td>5/3 | <td>5/3<br /> | ||
</td> | </td> | ||
<td>15/8 or 17/9<br /> | <td>15/8 or 17/9<br /> | ||
| Line 73: | Line 118: | ||
which is the same as the notes C D E G B and contains the <strong>tonic major chord C E G</strong> along with the <strong>dominant major chord G B D</strong><br /> | which is the same as the notes C D E G B and contains the <strong>tonic major chord C E G</strong> along with the <strong>dominant major chord G B D</strong><br /> | ||
<strong>(x/12) -</strong> 1/1 5/4 4/3 3/2 5/3<br /> | <strong>(x/12) -</strong> 1/1 5/4 4/3 3/2 5/3<br /> | ||
the same as the notes C E F G A. <br /> | the same as the notes C E F G A.<br /> | ||
<br /> | |||
The <strong>x/12 and x/9</strong> harmonic series become particularly stressed in the <strong>(Maqam) Rast</strong>, also known as the <strong>&quot;Blues&quot; scale</strong>, of<br /> | |||
<br /> | |||
<u><strong>Maqam Rast &lt;&lt;8-9-12&gt;&gt; scale</strong></u><br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td>C<br /> | |||
</td> | |||
<td>D<br /> | |||
</td> | |||
<td>D#-E<br /> | |||
</td> | |||
<td>F<br /> | |||
</td> | |||
<td>G<br /> | |||
</td> | |||
<td>A<br /> | |||
</td> | |||
<td>A#-B<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>1/1<br /> | |||
</td> | |||
<td>9/8 or 10/9<br /> | |||
</td> | |||
<td><strong>11/9</strong><br /> | |||
</td> | |||
<td>4/3<br /> | |||
</td> | |||
<td>3/2<br /> | |||
</td> | |||
<td>5/3 or 27/16<br /> | |||
</td> | |||
<td><strong>11/6</strong><br /> | |||
</td> | |||
</tr> | |||
</table> | |||
Here the x/9 series uses the &quot;blue tone&quot; of 11/9 and grows into<br /> | |||
<strong>(x/9)</strong> - 1/1 10/9 <strong>11/9</strong> 12/9 15/9 17/9<br /> | |||
Meanwhile the addition of the &quot;blue tone&quot; of 11/6 and removal of 5/4 changes the x/12 series to<br /> | |||
<strong>(x/12)</strong> - 1/1 4/3 3/2 5/3 <strong>11/6</strong><br /> | |||
And the removal of the 5/4 and 15/8 shrinks the x/8 series into <br /> | |||
<strong>(x/8) -</strong> 1/1 9/8 3/2<br /> | |||
<br /> | |||
If you don't like the small x/8 series, you can modify that scale by deleting the 11/9 and adding back the 5/4 and removing the 11/9 to make the scale more balanced between the size of the different series e.g. <br /> | |||
(x/8) - 1/1 9/8 5/4 3/2<br /> | |||
and<br /> | |||
(x/9) - 1/1 10/9 4/3 5/3 17/9<br /> | |||
<br /> | |||
<strong>But why limit yourself to/by x/8, x/9, and x/12 harmonic extensions of the root?</strong><br /> | |||
<hr /> | |||
<strong>Part 2- Attempting to create scales from scratch given harmonic series</strong><br /> | |||
<br /> | |||
A common pattern in the above scales is use of and interpolation between the x/9 and x/12 harmonic series<br /> | |||
It turns out the x/7 harmonic series has many notes in common with or close to the x/8, x/9, and x/12 harmonic series.<br /> | |||
We can take advantage of this to construct a more advanced scale in order the create the series<br /> | |||
<br /> | |||
<strong>(x/7)</strong> = 7:9:11:12:13 (AKA 1/1 9/7 11/7 12/7 13/7)<br /> | |||
<strong>(x/8)</strong> = 8:9:10:12:14:15<br /> | |||
<strong>(x/9)</strong> = 9:10:12:14:15<br /> | |||
<strong>(x/12)</strong> = 12:15:16:18:20<br /> | |||
<br /> | |||
Some of these ratios are near each other, but not equal, such as 9/8 in the (x/8) and 10/9 in the (x/9) series or 14/9 in the (x/9) series and 11/7 in the x/7 series. <br /> | |||
One way to evenly distribute error is to take the harmonic/geometric median between such fractions e.g. square-root-of ((11/7)/(14/9)) * 14/9 = about 1.56. <br /> | |||
Doing such gives us the scale<br /> | |||
<br /> | |||
<u><strong>10 note &lt;&lt;7-8-9-12&gt;&gt; &quot;extended color diatonic&quot; Harmonic Segment Scale</strong></u><br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td>1/1<br /> | |||
</td> | |||
<td>11181/10000<br /> | |||
</td> | |||
<td>5/4<br /> | |||
</td> | |||
<td>9/7<br /> | |||
</td> | |||
<td>4/3<br /> | |||
</td> | |||
<td>3/2<br /> | |||
</td> | |||
<td>156341/100000<br /> | |||
</td> | |||
<td>5/3<br /> | |||
</td> | |||
<td>26/15<br /> | |||
</td> | |||
<td>28/15<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td><br /> | |||
</td> | |||
<td>between 10/9 and 9/8<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td>between 14/9 and 11/7<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td>between 12/7 and 7/4<br /> | |||
</td> | |||
<td>between 13/7 and 15/8<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
Feel free to add comments or find/punch loopholes in this theory, but this is what I'd consider an idea <strong>&quot;extended diatonic&quot;</strong> scale in that it stresses likely the most important harmonic series in the diatonic scale while adding an extra (x/7) into the mix, giving the scale a wider range of tonal color.<br /> | |||
<br /> | |||
<br /> | <br /> | ||
<strong>More scales to come later...</strong></body></html></pre></div> | |||