AH 36 Tone PCS • Pebber Brown Page 1
Allan Holdsworth - 36 tone system
Allan Holdsworth was born in Bradford, England in 1946. He has remained one of the worlds
most intriguing guitarists and is well known for his blend of tonal and atonal melodic legato soloing.
He started playing guitar at the age of 18 and was primarily self taught with the exception of his
relationship with his father, jazz pianist Sam Holdsworth. He never asked his father for formal music
lessons but did obtain knowledge of scales and chords in great detail from him and further explored
them on his own. He developed a highly unorthodox system of scales and chords based on notes
derived from a three-octave chromatic scale.1
In the early 1970's, Holdsworth worked with many jazz-rock and jazz-fusion groups in England
including: Ian Carr’s group, Nucleus (1972), Jon Hiseman’s group, Colosseum (1972 -1973). He
became recognized worldwide during his tenure with the group Soft Machine (1973-1975) later in
1975, he joined world class drummer Tony Williams group Lifetime and recorded two monumental
albums. Between 1976 and 1979 Holdsworth toured and recorded with fusion-rock groups Gong, UK,
and was a member of Bill Bruford's group (former drummer of Yes, UK and King Crimson). His
recording and session work has included albums from fusion violinist Jean Luc Ponty, Jeff Clyne,
Spontaneous Music Ensemble and several guest solos on various compilation jazz-fusion albums from
the 1970's-1990's.2
According to Ferguson and Kernfield, "His single-line improvisations are often reminiscent of
John Coltrane’s sheets of sound. He is the author, with C. Hoard, of Reaching for the Uncommon
Chord (Wayne, NJ, c1985, and London, 1988) [incl. transcrs. by F. Amendola]."3
He formed his own group in 1980 called "I.O.U" with British drummer/keyboardist Gary
Husband, bassist Paul Carmichael and recorded one album and toured with the British lineup. He later
moved to California in 1982 and re-formed his group with American drummer Chad Wackerman and
bassist Jeff Berlin. His trio format continued on for two decades, with the lineup swapping drummers
and bass players around with many different high caliber top-notch jazz-fusion musicians over the
years. He also has featured the addition of several different jazz and fusion keyboard players including
Kei Akagi, Gordon Beck and Steve Hunt in the 1980's and 1990's. He is the author of 3 books and 1
instructional video and has recorded 11 albums.
His sound is a very harmonically advanced and sophisticated blend of jazz and rock with more
modern and atonal/disjunct chromatic interval patterns. He has not reached massive public acclaim
due to the advanced harmony and tonal sound of his music. Around 1984, he was approached by
Eddie Van Halen, and Van Halen/Warner Bros high-profile producer Ted Templeman, to produce an
album of commercial rock tunes with both Holdsworth and Van Halen soloing on it, but Allan backed
away and had a minor falling out with both of them due to the forced commercial nature of the songs
that were scheduled to be on that record and his dislike of putting out anthing that wouldnt be a regular
part of his jazz-fusion sound. His solo guitar tone is more of a medium-gain over-driven sustained
rock sound instead of a more regular traditional clean jazz guitar tone. His chordal tone is a very clean
sound blended with slight echo and chorus and on occasion he mixes it with chordal synthesizer
sounds. He is one of the pioneers of chordal "volume swells" with multiple echo and sustain creating a
very ethereal chordal effect. Holdworth has also experimented with many synthesizers and was one of
the few guitarists who made use of the revolutionary guitar synthesizer called the Synthaxe (now
1 Jim Ferguson and Barry Kernfeld. "Holdsworth, Allan." The New Grove Dictionary of Jazz, 2nd ed.. Grove Music
Online. Oxford Music Online. Oxford University Press, accessed May 2, 2015,
http://www.oxfordmusiconline.com.ccl.idm.oclc.org/subscriber/article/grove/music/J205200.
2 Ibid.
3 Ibid.
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discontinued). His sound is based on the use of his formidable legato technique and harmonic
knowledge of thousands of unusual/synthetic scales and chords based on his 36 note chromatic scale
permutation system.
Holdsworth's 36 note scale/chord system
What Allan does to generate his library of synthetic scales and chords is to take three chromatic scales
in a row (three octaves):
C-C#-D-D#-E-F-F#-G-G#-A-A#B-C-C#-D-D#-E-F-F#-G-G#-A-A#B-C-C#-D-D#-E-F-F#-G-G#-A-A#B-C.
Using the integer 0 as the starting point, the 36 note system would be numerically notated as:
0-1-2-3-4-5-6-7-8-9-10-11-12-13-14-15-16-17-18-19-20-21-22-23-24-35-26-27-28-29-30-31-32-33-34-35.
First octave: Second Octave: Third Octave:
• C =0 C = 12 C = 24
• C# = 1 C# = 13 C# = 25
• D =2 D = 14 D = 26
• D# = 3 D# = 15 D# = 27
• E =4 E = 16 E = 28
• F =5 F = 17 F = 29
• F# = 6 F# = 18 F# = 30
• G= 7 G = 19 G = 31
• G# =8 G# = 20 G# = 32
• A =9 A = 21 A = 33
• A# = 10 A# = 22 A# = 34
• B = 11 B = 23 B = 35
For this paper, I will be focusing on my own interpretation of the 36 tone system combined with my
own analysis of it based on the ideas of Schillinger's permutation theories and using the concepts of
Tymoczko's 12 note PCS representation. The 36 tone chromatic system could then be represented as an
extension of Tymoczko's PCS into a new 36 note PCS:
Holdsworth derives scales by
selecting all the possible independent 5
note scales, 6 note scales, 7 note scales
and 8, 9 and 10 note scales created by
mathematically factoring the notes. As
an introduction into this system, the
first 5 note scales produce the following
(factoring within ONE octave only)
1st group = C-C#-D-D#-E (All
adjacent chromatic tones, 0-1-2-3-4).
He then derives all the possible
combinations mathematically by
displacement of the notes.
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Here is an example using only ONE octave:
0-1-2-3-4, 0-1-2-3-5, 0-1-2-3-6, 0-1-2-3-7, 0-1-2-3-8, 0-1-2-3-9, 0-1-2-3-10, 0-1-2-3-11, 0-1-2-3-12.
In this example, the first four tones remain consistent with the displacement only occurring on
the last tone of the series. The first 5 note displacement pattern (0-1-2-3-4-5) produces a scale of 5
adjacent semitones, which initiates the entire method and sequence, but it is not very usable as a
practical scale or chord. Holdsworth states that as a practical measure, he eliminates any scales that
have more than three half-steps in a row.
One continues displacing 5 notes at a time until sequences of notes are reached that can be
applied as sources for melodic or chordal ideas. The 5 note displacement sequence 0-2-4-7-9 thus
produces the notes C-D-E-G-A, which is commonly known as a major pentatonic scale.
The next step is to map out the scale and derive a set of harmonies and chords from it by
stacking the restricted group of available tones. Two note harmonies or dyads can be created by
selecting any two of the five tones and harmonizing them in sequence up the scale. The first pair of
two harmonized notes, C(0) and D(2), would be the basis for the next pair of notes D(2) and E(4). The
next two would be E(4) and G(7) and then would be followed by G(7) and A(9) and then finally A(9)
and C(0).
• The first pair of stacked dyad harmonies would be:
◦ D(2) E(4) G(7) A(9) C(0)
◦ C(0) D(2) E(4) G(7) A(9)
Dyad stacks from this 5-note scale could be represented in Tymoczkos 12 note PCS as a simultaneous
harmonic dyad pairs with no movement:
• Triad harmonies from this scale would
logically be created using the same
methodology. The first triad stack would
be C(0)-D(2)-E(4), harmonized up the
scale following with D(2)-E(4)-G(7),
E(4)-G(7)-A(9), G(7)-A(9)-C(0) and A(9)-
C(0)-D(2).
◦ E(4) G(7) A(9) C(0) D(2)
◦ D(2) E(4) G(7) A(9) C(0)
◦ C(0) D(2) E(4) G(7) A(9)
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Three note triad stacks from this 5-note scale could then be represented in Tymoczkos 12 note
PCS as a simultaneous harmonic triad groups with no movement:
• Tetrad harmonic stacks from this scale displacement pattern would be mapped out as C(0)-
D(2)-E(4)-G(7), D(2)-E(4)-G(7)-A(9), E(4)-G(7)-A(9)-C(0), G(7)-A(9)-C(0)-D(2), and A(9)-
C(0)-D(2)-E(4).
◦ G(7) A(9) C(0) D(2) E(4)
◦ E(4) G(7) A(9) C(0) D(2)
◦ D(2) E(4) G(7) A(9) C(0)
◦ C(0) D(2) E(4) G(7) A(9)
•
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• Pentad harmonic stacks from this scale displacement pattern would be mapped out as C(0)-
D(2)-E(4)-G(7)-A(9), D(2)-E(4)-G(7)-A(9)C(0), E(4)-G(7)-A(9)-C(0)-D(2), G(7)-A(9)-C(0)-
D(2)-E(4), A(9)-C(0)-D(2)-E(4)-G(7). This reaches the maximum available notes of a Pentad
stack without repeating any notes.
◦ A(9) C(0) D(2) E(4) G(7)
◦ G(7) A(9) C(0) D(2) E(4)
◦ E(4) G(7) A(9) C(0) D(2)
◦ D(2) E(4) G(7) A(9) C(0)
◦ C(0) D(2) E(4) G(7) A(9)
This can be the source for generating vertical harmony using this method, and also as a source for
creating horizontal scale patterns. A two note melodic sequence starting as C(0)-D(2) as the dyad pair
would produce the following horizontal scale/melodic sequence: C(0)-D(2), D(2)-E(4), E(4)-G(7),
G(7)-A(9), A(9)-C(0).
Using Tymoczko's 12 note PCS, the dyad pair melodic sequence can be notated with motion indicated
by using dotted lines with arrows:
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Using permutations, the first two notes can be reversed so that C(0)-D(2) becomes D(2)-C(0). The
sequence using the permutation of D(2)-C(0) would become D(2)-C(0), E(4)-D(2), G(7)-E(4), A(9)-
G(7), C(0)-A(9).
This is only using the adjacent notes (C-D) in this particular scale. The next sequence can use non-
adjacent scale tones to create a vertical melodic sequence. C-D as a dyad pair can be changed so the
second note of the pair uses the next available scaletone so that D would be replaced with E. The dyad
pair becomes the next interval of C-E and the scale sequence of dyad pairs becomes C-E, D-G, E-A,
G-C and A-D (or 0-4, 2-7, 3-9, 7-0, 9-2). The permutation of this dyad sequence becomes E-C, G-D,
A-E, C-G, and D-A (or 4-0, 7-2, 9-4, 0-7, 2-9).
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This displacement method can continue with the next dyad pair containing the notes C-G as its basis so
that the scale sequence would be C-G, D-A, E-C, G-D, and A-E (or 0-7, 2-9, 4-0, 7-4). The
permutation of the C(0)-G(7) sequence would be G-C, A-D, C-E, D-G and E-A 7-0, 9-2, 0-4, 7-2, 9-4).
These scale sequences/chord stacks and permutations are only a small sample of the available
sequences and chords that one can create with this 36 tone chromatic system. Instead of using a
simple C-D-E-G-A major pentatonic scale, one can systematically go through each set of 5 notes,
using both adjacent and non-adjacent tones and derive thousands of scaletone patterns, sequences, and
chords. This can all be categorized and added to the forest full of trees/silva as new ideas for musical
melodic structures and scale based chordal sources. For example one can jump forward and go past 5
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and 6 note sequences and jump ahead into 7 note sequences, which can form the basis of more
commonly know diatonic scales. As an example, using only one octave of the 3 octave system, the
scales and chord stacks can be derived in the same exact manner, by first starting with a displacement
of the last note in the scale/chord stack by using 0-1-2-3-4-5-6-7, and displacing the last note as
follows: 0-1-2-3-4-5-6-8, 0-1-2-3-4-5-6-9, 0-1-2-3-4-5-6-10, 0-1-2-3-4-5-6-11, and 0-1-2-3-4-5-6-12.
One then could displace the second to last note in the 0-1-2-3-4-5-6-7 scale sequence/chord stack so
that it would start out as 0-1-2-3-4-5-7-8, skipping 6.
This pattern would continue on as: 0-1-2-3-4-5-7-9, 0-1-2-3-4-5-7-10, 0-1-2-3-4-5-7-11. The scale
created by the pattern would have the notes C-C#-D-D#-E-F-F#-G, then C-C#-D-D#-E-F-F#-A,
followed by C-C#-D-D#-E-F-F#-A#, C-C#-D-D#-E-F-F#-B, C-C#-D-D#-E-F-F#-C. Since this
closely resembles a chromatic scale with one non-adjacent tone in it, one can further explore the
iterations of 7 tones until a satisfactory and tonal set of halfsteps and wholesteps are generated which
can create a wide variety of common and synthetically based scales.
For example, if one uses the scale iteration C-D-D#-F#-G-A-B (0-2-3-6-7-9-11), this produces what is
commonly known as a the Lydian Diminished scale, also known as the jazz melodic minor + 4 scale
(with the diatonic formula of 1-2-b3-#4-5-6-7).
Lydian Diminished Scale:
• C=0 D=2 Eb(D#)=3 F#=6 G=7 A=9 B=11
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For the Lydian diminished scale, the first grouping of adjacent dyad stacks would extend into all 7
tones as a horizontal representation:
• D Eb F# G A B C
• C D Eb F# G A B
The horizontal/melodic representation of the C Lydian Diminshed scale would be a 2 note dyad
sequential pair: C-D, D-D#, F#-G, G-A, A-B, B-C (0-2, 2-3, 3-6, 6-7, 7-9, 9-11)
Using Schillinger's idea of permutations, this dyad permutation of the melodic sequence would reverse
the notes and be represented as: D-C, D#-D, G-F#, A-G, B-A, C-B (2-0, 3-2, 6-3, 7-6, 9-7, 11-9)
A continuation of the technique would select the next non-adjacent tone for a dyad stack:
• D# F# G A B D D#
• C D D# F# G A B
The horizontal representation of this tertiary dyad stack would be represented as a melodic sequence:
C-D#, D-F#, D#-G, F#-A, G-B, B-C (0-3, 2-6, 3-7, 6-9, 7-11, 11-0)
The melodic permutation of the tertiary dyad pairs would be represented as:
D#-C, F#-D, G-D#, A-F#, B-G, C-B (3-0, 6-2, 7-3, 9-6, 11-7, 0-11)
These constitute working tools to use to add to the repertory of sources for melodic ideas and this
system is capable of continuing on to use every non adjacent interval and permutation in the scale for
two-note dyadic harmonies and two-note dyadic melodic sequences.
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Three note Triadic notes can also be expressed as three-note chords or three-note melodic sequences.
The three note triads created by using adjacent scale tones would represented as:
• D# F# G A B C D
• D D# F# G A B C
• C D D# F# G A B
Horizontal three-note melodic sequences would be represented as:
C-D-D#, D-D#-F#, D#-F#-G, F#-G-A, G-A-B, A-B-C
The horizontal permutations can follow Schillingers displacement rule so that C-D-D# can be
permutated with displacement so it is expressed as:
C-D-D#, C-D#-D, D-D#-C, D-C-D#, D#-C-D, D#-D-D.
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