…the first fold orders the paper in two parts, may they be even or uneven.
this simple fold helps to understand division, contrast, reflection, opposites or contradiction, symmetry, mirroring as well as the transition from one pole (or „polarity“ as in eastern philosophy) to the other.
the book „fun with origami“ by mitsuo okuda inspired the above cycle of simple folds: yama- ori means mountain fold, tani- ori means valley fold, kabuse ori (outside reverse fold) nagawari ori (inside reverse fold) tumami ori (rabbit- ear fold) hikiyose ori sounds like ordering a special sushi in a restaurant, but are either basic forms or starting points for more complex foldings. (https://en.wikipedia.org/wiki/Yoshizawa–Randlett_system)
here is a recent folding…it took a while, but turns out to be a rewarding result for me (->wabi sabi)
another application on ->vamp A1 is related to a much older polyphonic technique called isorhythm. (https://en.wikipedia.org/wiki/Isorhythm)
looking into ways to generate interesting textures in music a workshop held by pianist and composer harmen fraanje (https://de.wikipedia.org/wiki/Harmen_Fraanje) at conservatorium maastricht in 2019 was eye opening. the technique of isoryhmic development can be described as follows: melodic color (or melody) and bass color (bass) contain the same rhythm (->iso) of the talea (repeating rhythmic pattern) but have different phrase length. the least common multiple for all three layers defines the circulating form. algorithmic composition invented way back in 14th century in france (french motets) see montpellier codex (https://en.wikipedia.org/wiki/Montpellier_Codex)
pianist and composer christoph stiefel from switzerland (http://www.christophstiefel.ch) dedicates most of his (latest) work to the technique of isorhythm, solo piano, trio and larger ensembles. listening strongly recommended.
for vamp A1 the „code“ of the b section is: melody (6) notes, bass (4) notes, talea (3) rhythmic events within a 3/4 and 4/4 (7/4) bar.
2x(6), 3x(4), 4x(3) equals 12 resulting in 4 bars of 7/4:
adding the -> vamp A1 material using the four note voicings to harmonize the melody (top- note defining chord choice) above the given bass note sounds like this:
a little sketch that has been inspired by the last hexachord on A1. including its mirror axis as an additional note (c) the following scale occurs:
c, db, eb, f, f#, (g), a, b (the handwritten sketch is the first attempt)
adding (g) results in an octatonic scale that is closely related to symmetric scale „M6“ of composer, organist and teacher olivier messiaen, a major figure and role model in the line french composers for serious (->ernste) music (https://de.wikipedia.org/wiki/Olivier_Messiaen). his methods and findings are published under the name „the technique of my musical language“. the „modes of limited transposition“ or symmetric scales as M1 = whole tone scale (222222) or M2 = half step whole step scale (12121212) are common knowledge for (jazz)musicians. M6 consisting of 2 phrygian tetrachords or two major scale tetrachords (2211221) steered through the interval of a tritone, is a bit more exotic, yet great to use for the application of common chord/scale development. (https://en.wikipedia.org/wiki/Mode_of_limited_transposition)
messiaen described the sound and inherent symmetry of these scales as containing „the charm of impossibilities“ . (->wabi sabi)
in this example M6 applicable chords are (stacked in 3rds)
| b | c | db | eb | f | f# | g | ab(g#) |
| g | a | b | c | (c#)db | eb | f | f# |
| eb | f | f#(gb) | g | a | b | c | db(c#) |
| c | db | eb | f | f# | g | a | b |
here a link to a online accessible research of saxophonist dick de graaf, probably the most thrilling compendium i have come across last year, outlining, amongst other topics the application of messiaen modes in a jazz/improvised context. https://www.researchcatalogue.net/view/354613/376802
this site led to another finding: the band octurn, led by baritone saxophonist bo van der werf (http://www.octurn.com) based in belgium, that took the inspiration and craft olivier messiaens to another level.
here is a picture of all three note groups i derived from the origami, measuring the angles:
it is fun to go through them, to play one sound at a time and see what applications would be possible (see handwritten chord suggestions) – or if the sound stands for itself- can i use the mirrored hexachord as is or do i have to tweak it? do i want to change it at all? do i add a 7th note, hence the note c, that is its mirror axis? as a rule i tried to keep a sound as long as possible as it is, to let the structure speak for itself.
here a few words to explain the handwritten material on this paper: A1- A8 are the triangles occurring in the unfolded prime fish (->prime fish) in violin- clef, its mirror (around axis c) in bass- clef. the first trichord is the prime form without rotations/ inversions:
„A1 prime“ is constructed with its intervals to the reference note c: 2 half steps c to d/ three half steps c to eb/ eight half steps c to ab.
| A1 prime | d, eb, ab |
the following chords are already an unfolding process (intervals speaking), here c is the reference note to d, the first occurring note with 2 half steps distance; d becomes the reference note for f, the second interval with 3 half steps distance and f becomes the reference note to db, 8 half steps away…
proof: if A1 123 carries the notes d, f, c#(db), in numbers or half steps: 2, 3, 8 the measured angles of the triangle: 23, 48, 113 degrees translate to intervals: major second (2), minor third (3) , minor sixth (8). rotating the intervals (A1 123 -> A1 231 -> A1 312 etc…) will result in different chord structures, since the reference notes are changing (at this stage i did not take care of correct enharmonic spelling…)(https://en.wikipedia.org/wiki/Enharmonic)
| A1 123 | d, f, c# (db) |
| A1 321 | g#, b, c# |
| A1 231 | d#, b, c# |
| A1 132 | d, bb, c# (db) |
| A1 312 | ab, bb, db |
| A1 213 | eb, f, db |
A1 total notes: (c) db(c#), d, eb(d#), f, ab(g#), bb, b…
probably most of us have played with lego at some point. genius game, genius marketing. just proves the point how it is possible to create interest and manipulate or more friendly „to play with“ the expectations of your audience on a largest possible scale.
in case of the compositions here- smallest scale- it is a bit like this: I created my own building blocks (-> tri ads) to experiment with and to decide what sounds good to me – if it feels somewhat meaningful, I will go on building context around an idea. meaningfulness is a key to hold on to something. japanese aesthetics or eastern philosophy helps along the way to accept beginning and imperfection. wabi sabi as a concept: (https://en.wikipedia.org/wiki/Wabi-sabi)
having looked into the structure of three- note groups, there is the possibility to apply these to other musical parameters, e.g. rhythm and melody. the three- note groups visualised on a chromatic circle make it easy to apply them to a basic 12/8 rhythm (in different groupings and transpositions) as well. creating melodic fragments in combining both.
a major inspiration on visualising musical material in a simple, yet universal way, i found in miles okazaki´s (https://de.wikipedia.org/wiki/Miles_Okazaki) works. „fundamentals of guitar“, a book meant for guitar players in the first place includes great (m-based) content on organising musical material for oneself. there is a wonderful pdf available on mr. okazaki´s website called „visual reference for musicians“ (http://www.milesokazaki.com/archives/musicians-visual-reference-2014/) a highly recommendable resource for every musician/ instrumentalist.
unfolding the little children origami fish inspired this little rhythm:
a rhythmic retrograde 65432 / 23456 layered as a core idea.
…adding some changes and full band:
here is a sample of 6 note groupings ordered around the mirror axis c:
the example shows triangle A2 (-> prime fish) plus rotations and two possible ways to generate a triad:
c as starting point, each note is a reference point of the following note: c plus (2) generates d, plus (3) generates f, plus (7) generates c. the sum of angles 111+48+21= 180 degrees, as is (7)+(3)+(2)= (12).
the resulting half- steps are defined through this: each half step would equal an angle of 15 degrees- since the exact angle in a triangle does usually not align with the row of 15, the rule is plus/minus 7.5 degrees.
0-7,5 = (0) or prime interval
7,5- 22,5 = (1) minor second
22,5- 37,5 = (2) major second
37,5- 52,5 = (3) minor third
52,5- 67,5 = (4) major third
67,5- 82,5 = (5) perfect fourth
82,5- 97,5 = (6) tritone
97,5- 112,5 = (7) perfect fifth
112,5-127,5 = (8) minor sixth
127,5- 142,5 = (9) major sixth
142,5- 157,5 = (10) minor seventh
157,5- 172,5 = (11) major seventh
172,5- 180 = (12) octave
the obvious geometry of this folding inspired the translation of angles into notes.
If 180 degrees is the sum of angles in a triangle (https://en.wikipedia.org/wiki/Sum_of_angles_of_a_triangle) 12 equal parts are needed to create 12 notes of the chromatic scale (https://en.wikipedia.org/wiki/Chromatic_scale), hence 15 degrees equal a half- step in the (western) tempered system. following through the complete folding:
here step 3:
step 4 (both sides from now on):
step 5:
step 6:
and 7:
unfold again, name each occurring triangle A0-8 (diagonal mirror B0-8), attaching notes derived from inherent angles to mirror axis (note C) you get this picture:
in the following post I will slow down the process and make it digestible.
the picture above is the template for a composition called „prime fish“ an attempt to include every occurring sound, rhythm and sequence, a first rehearsal with full band (flute, clarinet, trombone, guitar, fender rhodes, drums and doublebass) reveals this:
music and geometry is fascinating. the wonderful blog of roel hollander gives more than a glimpse into this world, it is full of inspiring information about this topic: https://roelsworld.eu/en/blog-music/music-geometry/
