Summery
88-Musaics explained
Logical correspondence between spatial structures and sound structures is what distinguishes regular instruments from the others, in particular the relation "translated spatial structure <--> transposed sound structure".
The relationship (morphism) between translation and transposition is not the only one. The aim of this study is to highlight, with the smallest pre-requisite, the other natural relations between spatial and sound structures.
Musical context
We place ourselves in the context of twelve equal tone temperament, musical system that divides octave into 12 equals chromatics intervals, and we focuses mainly on the qualities of the harmonic structures inscribed in the chromatic circle.
Enharmonic equivalence
Enharmonic equivalence is a pragmatic axiom that unifies close pitches such as C4, B3#, D4bb …
It is on this principle that, today, many user interfaces (UI) of musical instrument are designed (keyboard piano, guitar frets, …).
Octave equivalence
Octave equivalence is manifested by the fact we spell different pitches by the same name, whatever this frequency, on condition that are on octave quotient.
For example, there are different sounds such as C4, C5, C6, but only one equivalence class named C.
C0, C5, …, Cn are pitches
C is Pitch Class in Musical Set Theory terminology, noted most often 0 (zero). (see glossary)
It is well known by the musician for the notation of chords: CMaj7 labels all the chords, with a Cx as root, based on a structure (4, 3, 4, 1) ie 3rd maj, fifth and major seventh.
We warn the reader that the other essential components of music (rhythm, timbre, expression …) are not covered by the 88musaic and this study.
Enharmonic and Octave equivalence
(with integer notation)
Chromatic scale may be generated in 4 ways, in steps of 1, 5, 7 or 11. (these numbers are coprimes with 12).
These 4 generatorsof the group Z12 : {1, 5, 7, 11} -- or {1, 5, n-5, n-1 } with n=12 -- are supported by 2 axes and so 4 angulars sectors : 1-0-5, 7-0-1, 7-0-11 and 11-0-5, we will call "armature" a such structure of angular sector.
2 axes support to Φ(n=12) (Euler's phi function)
Chromatic scale generated by step 1 : C0, D0, D#, E0, F0, F#0, G0, G#0, A0, A#0, B0, C1, C#1, etc.
Case of linear UI Instrument as string instrument, for example, mono string instrument (Đàn bầu, ...)
or piano keyboard (hybrid interface).
keyboard-piano linear 1-step, hybride interface (regular and semi regular)
At the bottom of the keyboard, the pitches are linearly arranged in semitones.
Scale generated by step 5 : C0, F0, Bb0, Eb1, Ab1, Db2, Gb2, B2, E3, A3, D4, G4, C5, F5, etc. (rem : Bb ~ A# ~ Cbb ...)
Conjugated with the first (step 1), case of string instruments tuning in fourths (P4guitar, bass, bajosexto, ...)
Scale generated by step 7 : C0, G0, D1, A1, E2, B2, F#3, C#4, G#4, D#5, A#5, F6, C7, G7, etc.
Conjugated with the first (step 1), case of string instruments tuning in firths (mandolin, violin, cello, ...)
Scale generated by step 11 : C0, B0, A#1, A2, G#3, G4, F#5, F6, E7, D#8, D9, C#10, C11, B11, etc.
No case (?)
some traditional instruments in regular tuning, but there are also, and increasingly, modern instruments in this case.
Thereafter we will focus our attention on sector 1-0-5 (in accordance with the historical approach, 88 musaics were discovered on guitar tuned in perfect fourths) but all what will be studied is transposable, true, to the 3 other sectors (In particular string instruments tuning in fifths - 1-0-7).
Below, an example of matrix base (angular sector 1-0-5), we choose as origin 0 (upper left corner), the value C0 of pitch notation.
2D musical game space
Matrix base in armature (1, 5)
A angular sector forms a two-dimensional array, this elements are in scientific pitch notation. This space can be considered as a musical game space.
Like keyboard of piano, let us emphasize the notes of diatonic scale
2D musical game space Diatonic/ Pentaonic UI
The black and white grid on figure represents two scales of notes (diatonic and pentatonic), linked together by the complementarity relation (union of these scales forms a chromatic scale).
Without damaging our logic, let us admit the octave equivalence relationship (i.e. C0, C1,... Cn will simply be noted by C), as shown below.
C-Diatonic-Maj / D#-Pentatonic
Rather than engraving such a structural form "in the marble" (as piano keyboard), let us concern ourselves by the various possible forms (showing in a limited matrix 12x12)
D-Diatonic / F-Pentatonic
F-Diatonic / G#-Pentatonic
As can be seen in previus figure (F-Diatonic scale) the sharp notation is not always well suited, finally we prefer the neutral numerical notation 0..11.
0..11 are Pitch Class (see glossary on this page)
Below, F-Diatonic again with this notation in a matrix 13x13.
PCS : {0, 2, 4, 5, 7, 9,10}
(a set of pitch classes) F-Diatonic
By definition, all diatonic scales are same intervallic structure (2, 2, 1, 2, 2, 2, 1) or (w, w, h, w, w, w, h) that may be represented by a geometric structure, invariant by translation-transposition as illustrated by the animation below.
is-motif in translation on matrix base
Have you noticed that the diatonic scales presented reveals the same geometric structure?
As one might expect, geometric structures are not limited to grids of dimensions 12x12 or 13x13, we can even use the circular representation.
diatonic maj is-motif,
a geometric signature
There is a bijection between this geometrical structure and intervallic structure of PCS.
We will call is-motif this type of geometry structure.
Note : The circular is-motif representation makes it possible not to determine a specific PCS (absence of "fundamental", or root, placed in the upper left corner, without label).
In absence of a root (ref to a giving PC), these representations are equivalent :
size and borders are not important for is-motif representation
From 4096 Pitch Class Set elements to 352 is-motifs
How many different is-motifs exist ? To answer this question, one has to wonder how many exist of different structures (pitch-class set) within an octave. Response is 2 ^ 12 = 4096 musical structures or pitch-class sets (PCS), it is the power set of well temperated 12 pitch class set.
Below, the n=12-row of Pascal's triangle, it gives us the number of PCS per cardinality :
The 12 equal-tempered pitch classes can be represented by a chromatic circle.
A pitch class set (PCS) is defined by select pitches-class as shown in the following example C-diatonic (same representations with different labels)
chromatic circle
c-diatonic in chromatic circle
C-Diatonic is noted
{0, 2, 4, 5, 7, 9, 11}
It is a PCS in musical set therory notation.
Hypercube is a formidable way to organize the 4096 parts of 12, but such a structure is in dimension 12, and so difficult to explore in 3D real life, let alone in 2D. Here is an overview Q12 .
But for the pleasure of the eyes, here is one : a point represents a PCS (pitch class set), as the vertex of a hypercube of dimension 12.
The elements are linked together by the direct inclusion relation, also known as Hasse diagram.
Levels of hypercube are more practicable.
The purpose of this study is to show you how these 4096 PCS can be found, from a reduced set of 88 PCS classes !
Reduction 1 : Kowing that the diatonic scale have 12 PCS representatives (C-Diatonic, C # -Diatonic, ..., B-Diatonic), we would be tempted to divide 4096 by 12, but it would be without counting the musical structures with limited transposition PCS ... Indeed, some scales have only 1, 2, 3, 4 or 6 representatives PCS in their cyclic-orbit, as shown by images below.
Diatonic Intervallic Structure
and its 12 PCS in cyclic-orbit
(transposition degree = 12)
apply cyclic transposition to dim7 chord
3 transposed repeated 4 times
dim7 Intervallic Structure
and its 3 PCS in cyclic-orbit
(transposition degree = 3)
Most of the 4096 PCS can be transposed 12 times, except for 76 of them. The 76 PCSs with limited tranposition status can be grouped into 17 is-motifs (classes of intervallic structure). By example dim7 chord has 3 representative PCS ({0, 3, 6, 9}, {1, 4, 7, 10}, {2, 5, 8, 11}) whole tone scale only 2 : {0, 2, 4, 6, 8, 10} and {1, 3, 5, 7, 9,11} (second and third row in list below)
By example dim7 chord has 3 representative PCS ({0, 3, 6, 9}, {1, 4, 7, 10}, {2, 5, 8, 11}) whole tone scale only 2 : {0, 2, 4, 6, 8, 10} and {1, 3, 5, 7, 9,11} (third and second row in list below)
The 17 classes of limited transposition intervallic structure
17 examples (from 76) of Limited Transposition representive PCS
In 1954, Edmond Costère (Lois et Styles des Harmonies Musicales. Paris: Presses Universitaires de France) determined the first (?) reduction of the 4096 PCS to 351 elementary structures. In fact, there are exactly 352 equivalence classes by transposition relation (including empty set).
Demonstration : Number of Limited Transposition PCS is, according to 17 LT-PCS in table above, SUM of (6x9, 4x3, 3x2, 2x1, 1x2) = 76. So number of distincts Intervallic Structure is (4096 - 76) / 12 + 17 = 335 + 17 = 352
Here the 352 intervallic structures type, ordered by cardinal (number of PC in PCS). We will see that we can still reduce their number.
The 352 is-motifs
Diatonic is-motif is located at (13, 18) coordinate (row, col)
From 352 Intervallic Structure Type to 180 Dual Complement is-motifs
White ink on black background or black ink on white background ?
Reduction 2 : Kowing that a is-motif is the union of 2 PCS (one and his complementary), we would be tempted to divide 352 by 2, but it would be without counting the PCS self-complemented. There is 8 is-motifs in this case.
The 8 self-complemented is-motifs, accompanied by an example of a representative PCS.
So, there are (352-8) / 2 + 8 = 172 + 8 = 180 distinct is-motifs (dual complemented is-motifs) , which allow us, through a game of transposition and complementarity, to recover (regenerate) the 4096 PCS of our powerset. However, we will see that we can further reduce this number.
Which double complementary part should be emphasized?
What color ?
The arbitrary choice of color
For simplicity, we will remain in black and white and use black to represent the smallest part (also an arbitrary choice, but logic throughout this study - the order relation will be presented in another document). Here are the 180 reductions.
180 dual complemented is-motifs
Pentatonic-Diatonic is located at (row, col) : (8, 10)
From 180 dual complemented is-motifs to 122 dual cplt rotation is-motifs
During a more thorough inspection of the 180 dual is-motifs, redundant models can be observed. For example :
two dual complemented is-motifs
Each can be derived from the other by rotation around a point.
There is therefore redundancy of patterns.
transformation of a half turn around a point
These cases of PCS, where one can be deduced from the other by rotation around a point (half turn) (the top becomes the bottom). Musically, it is an inversion.
Here is a well-known example, highlighting the relationship between a major chord and a minor chord. Given example below, in clock representation : {0, 3, 7} become {0, 5, 9}, and vice-versa. The structural relationship between major and minor quality is emphasized by geometry.
The rotation operation emphasize relationship between major and minor quality
Reduction 3 : Kowing that a is-motif is deductible frome other by rotation, we would be tempted to divide 180 by 2, but it would be without counting the invariant is-motif by half rotation. There is 64 structures type in this case !
Example of is-motif invariant by half rotation around a point :
inv. rot
It is one of this 64 other elements :
64 dual cplt is-motifs (from 180) invariant by half rotation (around a point)
So, number of dual complement rotation is-motifs is : (180-64) / 2 + 64 = 58 + 64 = 122.
122 is-motifs in complement and rotation relationship
pentatonic-diatonic is located at (row, col) : (7, 8)
From 122 dual complement rotation is-motifs to 88 musaics
If you observe well, you will still see redundant is-motifs in our last 122 : some are deductible by turning around a diagonal. We need to use 3D transformation.
For example, these 2 is-motifs below are deductible from each other by a transformation operation (left-diagonal rotation of 180°)
Here is the left-diagonal rotation of 180°
left-diagonal rotation
The same goes for another operation : right- diagonal rotation of 180°
Here is the right-diagonal rotation of 180°
right-diagonal rotation
Here another example : dominant seventh is-motif presented in a square pattern to emphasize the diagonal transformation.
C7 <=> CM7 sus2 5+ transformation
Left diagonal rotation
Transformation from one is-motif to an other can be realized also by rotation on their right diagonal.
C7 <=> C7 b9 (no 5) transformation
Right diagonal rotation
Reduction 4 : As you can expect, some is-motifs are invariant by left and / or right diagonal rotation. Finally, musaic is name we gave to is-motif in smallest reduce. There are 19 musaics variants by all transformations (rotations and complement) and 69 invariant by at least one operation (diagonals rotation, half rotation around a point, complement in complex relationship, will be presented in a next page). We get a total of 19+69 = 88 musaics
19 musaics that vary by complement and rotations (around a point, diagonals)
The majority of musaics (78%) turn out to be invariant as a result of some transformations we have just presented.
The following is an example of a left diagonal rotation invariant (so, not in the previous 19)
{0,1,2,4,7}
invariant by left (and right) diag. rot.
Finally, the expected 88 musaics!
(armature 1-5)
From a single PCS of each of these 88 musaics, we can generate all the subsets (4096) of the tempered set of the pitch classes, by transformation operations (complement, 1/2 turn rotation around of point, rotation by left and/or right diagonal, translation). So we may conclude that the cardinal of the smallest set of generators of the 4096 configurations is 88 !
Transformation operations viewed from other angles.
The first persons known to have written something about left and right diagonal transformations are Rahn et Morris (andreatta-2003), but in the form of multiplication operations (numerical notation of PC allows this type of operation) : m5 is left diagonal and m7 is right diagonal transformation, m11 is half-rotation around a point and m1 is neutral operation. These operations can also be represented as permutations. Find them here, and many more:
https://en.wikipedia.org/wiki/Multiplication_(music)
The complementarity operation is also the object of great attention in musical set theory (Forte).
Different versions of same transformation
Try by yourself simple basic transformations on a musaic (7th chord)
(a good application of Klein-Group properties for this animation !)
Oh, it's interesting, but what use?
Heuristic and multidisciplinary
The relationship between symmetries of sound structures and musical interfaces, such as musaic, is an open exploratory field (structural correspondence). This can also be a source of inspiration, regardless of any musical style and hope to be suprised by its use.
Open to any musical style
88 musaics are not synonymous with atonality, tonality can perfectly fit in.
In the case of the guitar, the tuning in IV is not an "open tuning" (the opposite is to be considered).
Pedagogical dimension
The logical framework of the instrument makes it possible to focus learning on understanding, construction and invention.
Alternatively, it is a good practical case study of group theory, that can interest mathematician and teacher (on this site, a web page is (will be) dedicated to this approach).
Global/Local
We have shown on this page the only global characteristics of musaics, which include local characteristics very useful to the musician for the understanding of music and instrumental practice (the study of local characteristics will find its place in a specialized website).
From chord on guitar tuning in fourths to musaic fragment
F7 analysis with musaic n°26
These 88 musaics are in fact canvas patterns on which musician perform finger choreographies on their instrument (in regular tuning).
Musaics (as "harmonic tablatures") helps the musician to identify hamonic colors. As noted above in the text, identification pattern (geometry and musical structure) is not limited to all fourths tuning : all regular user interfaces are concerned.
More generally, a matrix regular instrument is instance of a dual interval space (DIS), as two-dimentional array of pitches where “rows” are separated by the same interval and the “columns” by an other but also same (non-zero) interval (regular interface).
Matrix instrument are noted as follows: DIS(y,x) where y is row interval and x is culumn interval. Voilin is DIS(1,7), guitar in IV is DIS(1,5), etc.
DIS(1,7) violin
DIS(2,3)
accordion (right hand)
DIS(1,2)
DIS(1,5) LinnStrument
expressive MIDI controller in P4 regular tuning
It would seem, but this remains to be demonstrated, that only regular interfaces built on a framework based on value axes ("prime root") of Euler function are affected by this reduction. In this hypothesis, only Fifths and Fourths tuning benefit from 88musaics.
88 musaics for instruments tuning in fifths
(experimental) 88 musaics for accordion (right hand)
Afterword
We have succeeded in reducing by a path:
4096 -> 352 (transposition equivalence) - > 180 (complement) -> 122 (half center rotation) -> 88 (diagonals),
This is not the only possible way, but all converge to the 88 musaics . For example:
4096 -> 352 (transposition equivalence) -> 224 (+ half center rotation) -> 158 (+ diagonal) -> 88 (+ complement).
This last way is the one most often found, particularly in the musical set theory and mathematics (group theory) :
4096 (trivial group), -> 352 (cyclic group), -> 224 (dihedral group), -> 158 (affine group) and -> 88 ("musaic group", affine + complement)
References
(under construction)
Edmond Costère. 1954. Lois et Styles des Harmonies Musicales, Paris: Presses Universitaires de France, (hard to find !)
Musical Set Theory, Allen Forte List of pitch-class sets
Rahn, John. 1980. Basic Atonal Theory. New York: Schirmer Books; London and Toronto: Prentice Hall International. ISBN 0-02-873160-3.
Morris, Robert, 1987. Composition with Pitch-Classes, New Heaven, Yale University Press.
Andreatta, Moreno. Musical set theory http://repmus.ircam.fr/_media/mamux/papers/andreatta-2003-settheorycomplet.pd
Guerino Mazzola. 2002. The Topos of Music. Birkhäuser,. (presentation of the 88 equivalence classes, accessible via a good mathematical prerequisite)
Stephen C.Brown. Dual Interval Space in Twentieth-Century Music, . Musaic in armature 1-5 can be see as DIS(1,5).
Friend and Other Links
(under construction)
P4-Tuning group on FB : Guitar tuning in 4ths
Absolute regular notation vs tonal non-regular notation : http://tenor2015.tenor-conference.org/papers/02-PerezLopez-Bigram.pdf
new keyboard (symmetrical piano) : http://www.le-nouveau-clavier.fr/english/
[...]
Glossary
Pitch : refers to height of sound, frequency (example : A4)
Pitch Class (PC) : (abstract) set of pitches in octave equivalence ( including enharmonic equivalent) (A)
Pitch Class Set (PCS) : set of PC, use in Musical Set Theory
PC Interval : a distance between two pitches, measured in semitones. It may be null (unison), ascending or descending. But this traditional perception can not be applied to PCs, because of the octave equivalence which cancels orientation. Indeed, {0, 2} can be the expression of both a second and a seventh, therefore of an interval and its inversion.
Enharmonic and Octave equivalence
(with integer notation)
{0, 4, 7}
Intervallic Structure (is) : function that, when applied to a PCS, returns, in an ordered collection of intervals that constitute it. The sum x of these intervals always verify that x is multiple of 12 (0, 12, 24 …) .
Example : Intervallic structure of min7 (Minor seventh structure chord) is : (3,4,3,2)
root to b3 = 3 semitones
b3 to 5 = 4 semitones
5 to b7 = 3 semitones
b7 to root = 2 semitones
Examples :
is({0,2}) return (2,10) because 0-2 is major second interval (2 semitones) and 2-12 interval is de Minor 7 (10 semitones). 10 + 2 = 12, ok.
is({0, 4, 7}) return (4,3,5) (major third, minor third, fourth)
is({1, 5, 8}) return (4,3,5) (major third, minor third, fourth)
is({4}) return (0) * rem: 12 is equivalent to 0 with octave identification *
is({6}) return (0)
is({}) return ()
is({0,1,2,3,4,5,6,7,8,9,10,11}) return (1,1,1,1,1,1,1,1,1,1,1,1)
Note :
Intervallic structure in cyclic shift - different modes : (3,4,5), (4,5,3), (5,3,4), refer to the same is-motif.
minor 7 structure
is : (3,4,3,2)
IS : (3,4,3,2)
IS : (4,3,2,3)
IS : (3,2,3,4)
IS : (2,3,4,3)
is-motif : (88musaics) a geometric representation of an intervallic structure and this complement.
Musaic : (88musaics) a geometric object composed of at most 8 intervallic structures, so 4 distincts is-motifs obtained by transformation operations (rotations and complement). Origin of the word musaic commes from a neologism, condensed from “mosaic”, “music” and “muse”. It testifies to the strong interest we have in the natural relationships between forms (visual, palpable …) and musical structures (is, chord, scale, …), with the hope that musaicis could be an inspiration to revisit old musical repertoires and discover other ways. Historically, the term “harmonic mosaic” had been used.
Armature : (88musaics) An integer pair representing the intervals from which the musaic matrix frame is builted. For example, (1, 7) or (1, 5) for game on regular instruments, respectively tuned in fifths or fourths.
Diatonic/Pentatonic is-motif
whole tone scale is-motif
prime_roots : (defined for this etude) prime roots is minimal numbers to determine value of Euler Phi Function. Example, with n=12, prime-roots(n) are {1, 5} and numbers prime with 12 are {1, 5, n-5, n-1}, so {1, 5, 7, 11} (This property has been identified as an application of the Bezout theorem), therefore, Euler function can be defined by : 2 * card(prime_roots(n)).
Hypothesis : Prime_roots give dimension of geometry representation element ? (prime_roots(12) = 2 => 2D ?)
(under construction)
About author
My name is Olivier Capuozzo, I live in France, can be contacted by mail (olivier.capuozzo) on Google messaging. I would appreciate receiving feedback and testimonials from creative and educational exploitations of the 88 musaics. Apologize for my approximate English, hoping that it translates correctly concepts and thinkings in French...
This study, carried out without any financial or institutional support, was initially undertaken by Jean-Yves Fusil [1941-2011] (scientist, musician, humanist...). A first publication was made in 1984, available here. I joined Jean-Yves in this research work in the 80s , I brought an algebraic vision to this work (Musical Set Theory and group theory) and programming a software to explore the 88 musaics and their relationships which made it possible to discover more (new) concepts. This work is far from finished!
Hoping to have aroused your curiosity, thank you !
Tools
Some softwares uses
OpenScad : OpenSCAD for transformation by coding with 3D primitive functions
Gimp : for construct GIF animation and more
Ubuntu and GNU/Linux OS and tools
MusaicBox : New version JS online : https://github.com/ocapuozzo/musaicbox-app (on line : https://musaicbox.org/ )