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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.


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.
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.
Line 8: Line 6:


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 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
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
|}
|}


Line 147: Line 145:
'''(x/10)''' = 10:12:15:16:18
'''(x/10)''' = 10:12:15:16:18


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.
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...'''
'''More scales to come later...'''


-----
-----
'''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.
. However, most of the more advanced scales on my method would require tempering out several different [[comma]]s 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]].


. 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 scale]]s 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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