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Easy Scales by Interpolating between Harmonic Series: Difference between revisions

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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.
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.


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Take, for example, the diatonic major scale in 12EDO, where notes are approximately equal to
Take, for example, the diatonic major scale in 12EDO, where notes are approximately equal to
<u>'''C major diatonic in 12EDO &lt;&lt;8-9-12&gt;&gt; scale'''</u>


{| class="wikitable"
{| class="wikitable"
|-
|-
| | C
|+ C major diatonic in 12EDO &lt;&lt;8-9-12&gt;&gt; scale
| | D
|-
| | E
| C
| | F
| D
| | G
| E
| | A
| F
| | B
| G
| A
| B
|-
|-
| | 1/1
| 1/1
| | 9/8 or 10/9
| 9/8 or 10/9
| | 5/4
| 5/4
| | 4/3
| 4/3
| | 3/2
| 3/2
| | 5/3
| 5/3
| | 15/8 or 17/9
| 15/8 or 17/9
|}
|}
This can be derived from the following harmonic series
This can be derived from the following harmonic series
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The '''x/12 and x/9''' harmonic series become particularly stressed in the '''(Maqam) Rast''', also known as the '''"Blues" scale''', of
The '''x/12 and x/9''' harmonic series become particularly stressed in the '''(Maqam) Rast''', also known as the '''"Blues" scale''', of
<u>'''Maqam Rast &lt;&lt;9-12&gt;&gt; scale'''</u>


{| class="wikitable"
{| class="wikitable"
|-
|-
| | C
|+ Maqam Rast &lt;&lt;9-12&gt;&gt; scale
| | D
|-
| | D#-E
| C
| | F
| D
| | G
| D#-E
| | A
| F
| | A#-B
| G
| A
| A#-B
|-
|-
| | 1/1
| 1/1
| | 9/8 or 10/9
| 9/8 or 10/9
| | '''11/9'''
| '''11/9'''
| | 4/3
| 4/3
| | 3/2
| 3/2
| | 5/3 or 27/16
| 5/3 or 27/16
| | '''11/6'''
| '''11/6'''
|}
|}
Here the x/9 series uses the "blue tone" of 11/9 and grows into
Here the x/9 series uses the "blue tone" of 11/9 and grows into
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Doing such gives us the scale
Doing such gives us the scale
<u>'''10 note &lt;&lt;7-8-9-12&gt;&gt; "extended color diatonic" Harmonic Segment Scale'''</u>


{| class="wikitable"
{| class="wikitable"
|-
|-
| | 1/1
|+ 10 note &lt;&lt;7-8-9-12&gt;&gt; "extended color diatonic" Harmonic Segment Scale
| | 11181/10000
|-
| | 5/4
| 1/1
| | 9/7
| 11181/10000
| | 4/3
| 5/4
| | 3/2
| 9/7
| | 156341/100000
| 4/3
| | 5/3
| 3/2
| | 26/15
| 156341/100000
| | 28/15
| 5/3
| 26/15
| 28/15
|-
|-
| |  
|
| | between 10/9 and 9/8
| between 10/9 and 9/8
| |  
|
| |  
|
| |  
|
| |  
|
| | between 14/9 and 11/7
| between 14/9 and 11/7
| |  
|
| | between 12/7 and 7/4
| between 12/7 and 7/4
| | between 13/7 and 15/8
| between 13/7 and 15/8
|}
|}


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-----
-----
'''Appendix-'''
'''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 http://x31eq.com/temper/uv.html] to reveal temperaments and ultimately scales likely to contain the above harmonic series segments.
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 http://x31eq.com/temper/uv.html] to reveal temperaments and ultimately scales likely to contain the above harmonic series segments.
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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.
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.
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