Easy Scales by Interpolating between Harmonic Series
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=Easy Scales by Interpolating between Harmonic Series= A very easy way to construct a scale that's instantly recognizable, even without repeated listening/priming in the absence of listening the music in 12EDO, is to interpolate between harmonic series. I'm using the notation <<harmonic series numbers from the root>> to denote what harmonic series certain scales contain. If this seems unclear or conflicts with an existing notation, please let me know. **Some of the most prominent scales in existence can be very quickly derived from just a few interlocked/interpolated 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 || || 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 **(x/9)** - 1/1, 10/9, 12/9 (4/3), 15/9 (5/3), 17/9 which is the same as the notes C D F A B and contains the **subdominant major chord F A C** **(x/8)** - 1/1,9/8,10/8 (5/4), 12/8 (3/2), 15/8 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 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 <<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/compromised 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 segments from a root tone** 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 || In addition, adding 16/15 to the above scale can yield an additional 5 note harmonic series, making the above a **larger <<7-8-9-12-15>> scale** **(x/15)** = 15:16:20:26:28 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...** ---- **Appendix-** Above calculations such as the interpolation of (11/7)/(14/9) can also be expressed as commas e.g. 99/98, which can be plugged into Graham Breed's Temperament Finder on [[http://x31eq.com/temper/uv.html]] to reveal temperaments and ultimately scales likely to contain the above harmonic series segments. . However, most of the more advanced scales on my method would require tempering out several different commas and, presumably, also countless related commas indirectly. I am afraid this would often result in huge and much more challenging to play in (think: well over 15 notes) scales needed to contain, say, the x/7,x/8,x/9,and x/12 harmonic series from the root tone with reasonable accuracy in regular temperament theory. If any **experts on Xenharmonic math**, including related lists, can find a way to related the input of harmonic series segments to, say, MOS scales guaranteed to have them I would really appreciate it.
Original HTML content:
<html><head><title>Easy Scales by Interpolating between Harmonic Series</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><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>
<br />
A very easy way to construct a scale that's instantly recognizable, even without repeated listening/priming in the absence of listening the music in 12EDO, is to interpolate between harmonic series.<br />
<br />
I'm using the notation <<harmonic series numbers from the root>> to denote what harmonic series certain scales contain. If this seems unclear or conflicts with an existing notation, please let me know.<br />
<strong>Some of the most prominent scales in existence can be very quickly derived from just a few interlocked/interpolated harmonic series.</strong><br />
Take, for example, the diatonic major scale in 12EDO, where notes are approximately equal to<br />
<u><strong>C major diatonic in 12EDO <<8-9-12>> scale</strong></u><br />
<table class="wiki_table">
<tr>
<td>C<br />
</td>
<td>D<br />
</td>
<td>E<br />
</td>
<td>F<br />
</td>
<td>G<br />
</td>
<td>A<br />
</td>
<td>B<br />
</td>
</tr>
<tr>
<td>1/1<br />
</td>
<td>9/8 or 10/9<br />
</td>
<td>5/4<br />
</td>
<td>4/3<br />
</td>
<td>3/2<br />
</td>
<td>5/3<br />
</td>
<td>15/8 or 17/9<br />
</td>
</tr>
</table>
This can be derived from the following harmonic series<br />
<br />
<strong>(x/9)</strong> - 1/1, 10/9, 12/9 (4/3), 15/9 (5/3), 17/9<br />
which is the same as the notes C D F A B and contains the <strong>subdominant major chord F A C</strong><br />
<strong>(x/8)</strong> - 1/1,9/8,10/8 (5/4), 12/8 (3/2), 15/8<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 />
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>"Blues" scale</strong>, of<br />
<u><strong>Maqam Rast <<9-12>> 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 "blue tone" 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 "blue tone" 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/compromised 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 />
<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 segments from a root tone</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 />
<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 />
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 />
<u><strong>10 note <<7-8-9-12>> "extended color diatonic" 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 />
In addition, adding 16/15 to the above scale can yield an additional 5 note harmonic series, making the above a <strong>larger <<7-8-9-12-15>> scale</strong><br />
<strong>(x/15)</strong> = 15:16:20:26:28<br />
<br />
Feel free to add comments or find/punch loopholes in this theory, but this is what I'd consider an idea <strong>"extended diatonic"</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 />
<strong>More scales to come later...</strong><br />
<hr />
<strong>Appendix-</strong><br />
Above calculations such as the interpolation of (11/7)/(14/9) can also be expressed as commas e.g. 99/98, which can be plugged into Graham Breed's Temperament Finder on <a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow">http://x31eq.com/temper/uv.html</a> to reveal temperaments and ultimately scales likely to contain the above harmonic series segments.<br />
. However, most of the more advanced scales on my method would require tempering out several different commas and, presumably, also countless related commas indirectly. I am afraid this would often result in huge and much more challenging to play in (think: well over 15 notes) scales needed to contain, say, the x/7,x/8,x/9,and x/12 harmonic series from the root tone with reasonable accuracy in regular temperament theory.<br />
<br />
If any <strong>experts on Xenharmonic math</strong>, including related lists, can find a way to related the input of harmonic series segments to, say, MOS scales guaranteed to have them I would really appreciate it.</body></html>