discrete isomorphic completeness and a unified isomorphic layout format
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Discrete Isomorphic Completeness and a Unified Isomorphic Layout Format
Brett Park David Gerhard
Isomorphic Hexagonal Note Layouts:
• Iso = Same; Morph = Shape
• An arrangement of tones whereby musical constructs have the same shape regardless of key / root
Major Triads (Piano): 12 shapes
Major Triads (Piano): actually 6 shapes
Major Triads (isomorphic): 12 shapes
Major Triads (isomorphic): actually 1 shape
All musical constructs are a single shape regardless of key
Advantages
• Quicker learning
• Intuitive connection to harmony
• Most compact note arrangement
• A. Milne et al. (2008)
The sum of intervals along a path that return to the start location must be 0
0
1
1
-1
5-5
-5
4
What makes an isomorphism
Chose any interval for an initial direction
What makes an isomorphism
Opposite Direction always has negative interval value
What makes an isomorphism
Second Direction can be any interval
What makes an isomorphism
Third interval is the sum of the other two
+2
What makes an isomorphism
Third interval is the sum of the other two
+2
What makes an isomorphism
Many isomorphisms, depending on interval choices
Janko
+2
-1 -1
Harmonic
-4+7
-3
Wicki-Hayden
+2
-7 +5
Physical InstrumentsJanko
P. von Jankó (1885)
Physical InstrumentsThummer
G. Paine et al. (2007)
Physical InstrumentsOpal
Peter Davies
Physical InstrumentsAxis
C-Thru Music (2007)
Virtual InstrumentsHex Player
A. J. Milne et al 2011
Virtual InstrumentsMusix Pro for iPad
Gerhard and Park 2011
Rainboard: Physical Reconfigurable isomorphic instrument
Gerhard and Park 2011
3 research questions
Is a layout “complete”?
• are all notes available to play?
Are two layouts the same / similar?
• to a translation, mirroring or rotation?
Which layouts are better than others
• Note compactness
• Ease to play melodies
• Easy to play chords
Brute Force Generation
Given two interval vectors x,y that define a unidirectional isomorphism, the isomorphism is complete (contains all note intervals) if and
only if x and y are co-prime.
{GCD (x,y) = 1}
Proof in paper
Isomorphic Completeness
Complete:gcd(-5, -2) = 1gcd(9, 2) = 1
Degenerate:gcd(4, 2) = 2gcd(6, 3) = 2
Isomorphic Completeness
Isomorphic Completeness
Two categories: Complete and degenerate.
Degenerate layouts may have their own musical purposes.
Musical interestingness of degenerate microtonal hexagonal isomorphic layouts is left
for future work
Notation
The UIL (Unified Isomorphic Layout) is a proposed notation based on Hayden’s
initial GLD notation
Unified Isomorphic Layout
L,G,D;RMS;TG
-L-D
D
-G
L
G = largest intervalL = smallest intervalD = difference (G – L)
Unified Isomorphic Layout
L,G,D;RMS;TG
-L-D
D
-G
L
R = clockwise rotationM = mirroringS = shear (Prechtl 2011)
Unified Isomorphic Layout Notes:
• Base Representation
• R=0; M=0
• T = number of tones (if omitted, T=12tet)
• Format can make use of Cents, Ratios, Roman or traditional interval shorthand namings
Unified Isomorphic Layout
-L-D
D
-G
L
GPitch!Axis!Range
Mirrored!
Pitch!Range
Isotone!Axis!Range
Mirrored!Isotone!Range
Isotone!Axis!RangeMirrored!Isotone!Range
Isotone axis: a line indicating zero pitch changePitch axis: the direction in which pitches ascend
(pitch axis orthogonal to isotone axis)
A. Milne et al.
Example: 1,4,3;000;12
Pitc
h Ax
isIsotone Axis
Harmonic Table
Wicki-Hayden
Janko
Janko (base)
Musix ProIsomorphic Exploration Tool
Musix Pro
• Any layout (up to octave adjacent interval)
Musix Pro
• Hexagon or Rectangles
Musix Pro
• Note Sizes
Musix Pro
• Scale Selection
Musix Pro• Note Identification
Conclusions• A pair of coprime intervals define a complete layout
• Degenerate layouts may be interesting
• Not limited to 12-tet
• Universal Isomorphic layout for comparing layouts
• Musix Pro tool to explore all hexagonal (and rectangular) isomorphic layouts
LINKS• Musix Pro
• http://shiverware.com/musixpro/index.html
• Rainboard
• http://www.soundonsound.com/news?NewsID=16407
• http://www.therainboard.com
• TED talk on isomorphisms, musix and the rainboard
• http://youtu.be/r3kocjx69g4
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