~ sharp 15 ( #15 ) ~

~ augmented double octave ~

'here we theoretically 'correct' an interval sequence to create a new tonal organization for the evolving, modern guitarist ...'

Historical origins. The following discussion and theory ideas originated way back in '82' with hearing this arpeggio sounded as the closing lick for the jazz standard "Smoke Gets In Your Eyes", as performed on piano by Dr. Alan Frank. Here in 'C' major. Example 1.

Interesting sound yes? And man if that ain't a true blue Hollywood chord!
wiki ~ "Smoke Gets In Your Eyes"

In a nutshell. Turns out that there's a way to advance our 12 pitch, closed chromatic loop a bit and along the way create a new environment for exploration, improvisation and composition. We do this by 'correcting' our core diatonic arpeggio, built up from our most common diatonic, relative major / minor group of pitches, through a more symmetrical build of alternating major and minor thirds. Compare the interval formulas. Example 2.

major scale interval sequence
maj 3rd
min 3rd
maj 3rd
min 3rd
min 3rd
maj 3rd
min 3rd
# 15 symmetrical sequence
maj 3rd
min 3rd
maj 3rd
min 3rd
maj 3rd
min 3rd
maj 3rd

What results is a rather remarkable architecture that retains nearly all of the original framework of our two part, modal basis of the Ionian / Aeolian, relative major / minor tonal centering. With #15, our paired central modes are the now ancient Lydian and Dorian groups that we inherited from Pythagoras 2500 years or so ago. There's also a bit of a tritone interval shift as the Four / Seven tritone of the major scale is replaced by #11, which when capped by #15, takes on a new coloring from its traditional role in V7.

Correcting the diatonic arpeggio / Lydian major. Examine the following chart for intervals of the diatonic major scale respelled into its arpeggio. Turns out there's two 'minor 3rds' in a row to get the pitches right. In building the #15 arpeggio's, we 'fix' this with an unwavering alternating of major 3rd and minor 3rds, (a simple swap of the second minor 3rd with its next interval works the magic). Thus the idea of 'correcting' the diatonic sequence. Compare the intervals and letter names of this theory from the root pitch 'C.' Example 3.

major scale interval sequence
.
maj 3rd
min 3rd
maj 3rd
min 3rd
min 3rd
maj 3rd
min 3rd
C major arpeggio
C
E
G
B
D
F
A
C
# 15 symmetrical sequence
.
maj 3rd
min 3rd
maj 3rd
min 3rd
maj3rd
min 3rd
maj 3rd
C arpeggio #15
C
E
G
B
D
F #
A
C#

Fairly straightforward yes? How about permutating this to an interval sequence of minor 3rd / major 3rd?

Correcting the diatonic arpeggio / Dorian minor. In this next idea we follow the same build process but are thinking diatonic minor. Using the same pitches, we're now in the relative minor of 'C' major, so building up from the root pitch 'A.' Now we'll swap around the last two intervals in the arpeggio. Ex. 4.

minor scale interval sequence
.
min 3rd
maj 3rd
min 3rd
maj 3rd
min 3rd
min 3rd
maj 3rd
A minor arpeggio
A
C
E
G
B
D
F
A
# 15 symmetrical sequence
.
min 3rd
maj 3rd
min 3rd
maj 3rd
min 3rd
maj 3rd
min 3rd
A Dorian arpeggio
A
C
E
G
B
D
F#
A

Cool? For those already in the know the 'F#' in A minor implies, 'A' Dorian from the parent scale of 'G' major.

Interesting, even as we evolve to Dorian mode pitches, we attain the perfect closure of the loop. Of course if we continued along with this symmetrical sequence, our next pitch would be a up a major 3rd from 'A', so to a 'C#', so no longer in the A minor tonality. This 'transition' from minor to major, and vice versa, becomes the basis for the consistent looping, architectural structure that the #15 system provides; that one modal grouping of the pitches 'steps' by either major or minor 3rd right into the next, creating an alternating pattern of a Lydian / Dorian / Lydian or vice versa weaving and dance of these ancient musical pitches and colors.

Closing the loop. In this next idea we run the above pitch sequences out in our alternating major 3rd / minor 3rd sequence and look to close the loop. The pitches are broken down into groups of eight, so same number as generally found in a full octave, closed scale loop. Remember though we're still working with tertian arpeggios for now. Example 5.

C
E
G
B
D
F#
A
C#
C#
E
G#
B
D#
F#
A#
C#
Db
F
Ab
C
Eb
G
Bb
D
D
F
A
C
E
G
...
...

The loop is closed. From our starting point of C E G ... To C E G again in the fourth row etc. So by just counting each letter pitch once in the above tables, and no enharmonic pitches either, after 24 interval episodes of the major 3rd / minor 3rd sequence, we arrive back to our starting pitch for the same sequence of pitches to begin again. So we've perfectly closed our loop.

Arpeggios into scales. From this last group of arpeggios, let's make them into seven pitch scales. Example 6.

C
E
G
B
D
F#
A
C#

... as a closed loop becomes C Lydian ...

C
D
E
F#
G
A
B
C

Next loop.

C#
E
G#
B
D#
F#
A#
C# / Db

... as a closed loop becomes C# Dorian ...

C#
D#
E
F#
G#
A#
B
C#

Next loop with the C# now as Db.

Db
F
Ab
C
Eb
G
Bb
D

... as a closed loop becomes Db Lydian ...

Db
Eb
F
G
Ab
Bb
C
Db

Next loop.

D
F
A
C
E
G
B
D

... as a closed loop becomes D Dorian ...

D
E
F
G
A
B
C
D

The pattern ~ Lydian / Dorian double helix. Looking at the way this forms up, we see that when we extend our Lydian group it evolves into Dorian. Dorian when symmetrically continued becomes Lydian again as we ascendingly cycle chromatically through key centers. The reverse of this is also perfect; ascending Dorian arpeggio pitches continued through the #15 symmetrical cycle will morph into Lydian, then Dorian in a similar cycle of key centers. Here's a couple of measures of this evolution followed by a shorthand chart for all of the 12 Lydian and Dorian centers. Example 7.

wiki ~ double helix
C Lydian evolves into C# Dorian
C# Dorian evolves into Db Lydian
Db Lydian evolves into D Dorian
D Dorian evolves into D Lydian
D Lydian evolves into Eb Dorian
Eb Dorian evolves into Eb Lydian
Eb Lydian evolves into E Dorian
E Dorian evolves into E Lydian
E Lydian evolves into F Dorian
F Dorian evolves into Gb Lydian
Gb Lydian evolves into G Dorian
G Dorian evolves into G Lydian
G Lydian evolves into Ab Dorian
Ab Dorian evolves into Ab Lydian
Ab Lydian evolves into A Dorian
A Dorian evolves into A Lydian
A Lydian evolves into Bb Dorian
Bb Dorian evolves into Bb Lydian
Bb Lydian evolves into B Dorian
B Dorian evolves into B Lydian
B Lydian evolves into C Dorian
C Dorian evolves into C Lydian
... and so right back to where we started :)

Or vice versa. The flip side of this is also true; Dorian to Lydian creates a minor to major, perfectly sequential pattern of key centers too.

Here are the pitches. In this next charting we simply spell out the sequencing of the letter name pitches for each of the entries in the above chart by alternating our two intervals patterns; first major 3rd / minor 3rd then the minor 3rd / major 3rd pattern. Example 8.

maj / min
C
E
G
B
D
F#
A
C#
min / maj
C#
E
G#
B
D#
F#
A#
C#
maj / min
Db
F
Ab
C
Eb
G
Bb
D
min / maj
D
F
A
C
E
G
B
D
maj / min
D
F#
A
C#
E
G#
B
D#
min / maj
Eb
Gb
Bb
Db
F
Ab
C
Eb
maj / min
Eb
G
Bb
D
F
A
C
E
min / maj
E
G
B
D
F#
A
C#
E
maj / min
E
G#
B
D#
F#
A#
C#
E#
min / maj
F
Ab
C
Eb
G
Bb
D
F
maj / min
F
A
C
E
G
B
D
F#
min / maj
F#
A
C#
E
G#
B
D#
F#
maj / min
F#
A#
C#
E#
G#
B#
D#
F##
min / maj
G
Bb
D
F
A
C
E
G
maj / min
G
B
D
F#
A
C#
E
G#
min / maj
Ab
Cb
Eb
Gb
Bb
Db
F
Ab
maj / min
Ab
C
Eb
G
Bb
D
F
A
min / maj
A
C
E
G
B
D
F#
A
maj / min
A
C#
E
G#
B
D#
F#
A#
min / maj
Bb
Db
F
Ab
C
Eb
G
Bb
maj / min
Bb
D
F
A
C
E
G
B
min / maj
B
D
F#
A
C#
E
G#
B
maj / min
B
D#
F#
A#
C#
E#
G#
B#
min / maj
C
Eb
G
Bb
D
F
A
C
maj / min
C
E
G
B
D
F#
A
C#

Lots of patterns emerge and one group evolves into another. Not the easiest to sound out on a guitar or bass though. Keyboards with their sustain pedal might be a better choice to hear the coolness that lives here.

A 24 pitch loop. In this loop we get each of our 12 pitches of the chromatic scale twice (not counting the last 'C'). Thus the 24 total. Example 9.

C
E
G
B
D
F#
A
C#
E
G#
B
D#
F#
A#
Db
F
Ab
C
Eb
G
Bb
D
F
A
C
.
.
.

Easy enough yes? The symmetry of our interval loop eventually creates the perfect closure. Along the way we pick up all 12 of the major / minor key centers, now evolved from Ionian to Lydian for major and Aeolian to Dorian for minor.

So where in the music. In the fictional work "Atlas Shrugged" there's a composer who writes the most profound music the world has ever known. His name is Richard Halley and there's zero theory info about his music in the novel. The author, now perished, left no additional information of the music. No charts exist, at least that we know of. In the narrative of the story, performances of his compositions 'completely enthralled and captivated' the greatest minds of the era ...

wiki ~ Atlas Shrugged
wiki ~ Ayn Rand

In every era's music there evolves a new way forward to understand our pitches and their effect on our psyche. Mr. Halley's art might have been just such a music. Somehow the fiction of this novel and my own journey crossed paths some 30 years ago, and since then I've always linked the two. For the evolutionary effect I hoped to be achieved by anyone creating or hearing music generated in this system, would be a deeper dedication to their efforts to bring about world peace.

And hats off to George Russell. Theorist, player, educator, composer and way more, George Russell's theory book, Lydian Chromatic Concept Of Tonal Organization, sets the table for my ideas to flourish. And although I've been through his work a number of times, and actually went to Boston in 2001 to try and talk with him, the following theory and system for composing, improvising and performance is not to be found in his book. Sometimes I wish it was. So while we share the same pitches, his thing is different than mine.

So where is this #15 colortone in the music. I've never seen it or heard of it. There's a sharp one blue note, so I'd imagine up that up a couple of octaves we might call it sharp 15 but that's not what we're after here. I've been able to write it into an original composition or two but still yet to be publicly performed. So all in all, #15 and its potential artistic possibilities is still yet to be explored.

Lovely chord. Unfortunately it's not really playable on my six stringed instrument. Maybe this will be music for the seven and even eight (?) stringed guitarists? One never knows. I'll have to find a seven stringer and try it out for sure. Never thought of doing this really ... but as we often say here in Alaska ... yet another first :)

another first

Sounds polytonal. And indeed it is. Examining the first two measures of this last idea we can see the C major 7 arpeggio of measure one. Measure two has a complete D major 7th arpeggio also. The voicing of the chord to close out the line is a combination of root, major 3rd and major 7th of each of these two arpeggios.

The flip side. Might we flip the major 3rd / minor 3rd symmetry to minor 3rd / major 3rd and create an extended minor sounding arpeggio? Absolutely. We don't end up at sharp 15 though. We close back on our starting pitch. Examine the pitches and their sound from our relative minor pitch A natural of C major. Example 2.

Lovely chord. Unfortunately it's not really playable on my six stringed instrument. Maybe this will be music for the seven and even eight (?) stringed guitarists? One never knows. I'll have to find a seven stringer and try it out for sure. Never thought of doing this really ... but as we often say here in Alaska ... yet another first :)

Glimpsing a new and distant horizon. So while there's not any real call for this interval and these types of polychords in our American songbook, they do create a potentially new system of composition which harkens back to our days of polyphony, which was the rage back in Europe and probably elsewhere, during the last millennia.

wiki ~ polyphony

A new system of tonal organization. The curious aspect of this symmetrical arpeggiation is that each one organically evolves into a next series of a different hue. Which in turn evolves into the next etc., creating yet another, larger perfectly closed looping of our pitches. And while we have a similar situation today with our standard theory thinking and pitch groupings, the ability to arpeggiate and create various polyphonies, to modulate or move away from our key center we generally must borrow pitches of the new key center we want to move towards. Whereas in this new system, following along the pitches of the natural, symmetrical arpeggio can perhaps more unobtrusively move us from one tonal environment to another while getting off at any pitch along the way and exploring its own mode.

Review ... and we're out of numbers too :( In creating a perfectly symmetrical arpeggio of an alternating major 3rd / minor 3rd sequence, we can arrive at a pitch which is beyond the boundaries of our initial starting point and key center. An inverse arpeggio of minor 3rd / major 3rd interval symmetry creates a similarly identical pair of stacked chords. That our major 3rd / minor 3rd arpeggio eventually morphs into our minor 3rd / major 3rd arpeggio is the basis of the system of tonal organization for modulation, composition and improvisation.

And #15 is the last of our numers Amigos ... thus we yet again bump into the perfect closure of all things numbers in our local musical universe.

modulation
composition
improv
perfect closure
"Now ain't that something ..." :)

wiki ~ Jerry Garcia
1
#1
b2
2
b3
3
4
#4
b5
5
#5
b6
6
b7
7
8
b9
9
#9
-10
10
11
#11
12
b13
13
b14
14
15
#15
Footnotes:

(1) Isacoff, Stuart. Temperament ... The Idea That Solved Music's Greatest Riddle, p. 40-42. USA Alfred A. Knopf, New York. 2001

(1)Duffin, Ross W. How Equal Temperament Ruined Harmony, p.32. USA W.W.Norton and Company, NY, New York. 2007.
(2)Aebersold, Jamey and Slone, Ken. The Charlie Parker Omnibook. New York: Atlantic Music Corp., 1978.

Russell, George. The Lydian Chromatic Concept Of Tonal Organization. USA Concept Publishing Company, Cambridge, Mass. 1982