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