The term “isomorphic” is defined as “being of identical or similar form, shape, or structure.”
The idea of “isomorphism” as applied to music notation is that any given musical pattern (like a scale, chord, interval, or particular melody) should always have the same basic appearance regardless of which pitch it starts on. There should be consistency in the relationship between what you see and what you hear.
Each aspect of music — notation, nomenclature, and instruments — can be simplified through isomorphism. Using an isomorphic notation system with an isomorphic nomenclature and/or an isomorphic instrument so that these different aspects match each other, would likely result in added benefits.
There are several basic isomorphic patterns, different ways of distinguishing the twelve notes of the chromatic scale. (Numbers are used in the table below just to illustrate the different patterns in a formal, abstract way.)
| Basic Isomorphic Patterns | A | A# Bb | B | C | C# Db | D | D# Eb | E | F | F# Gb | G | G# Ab |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 6-6 pattern (binary), cycles every 2 semitones (major second). | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 |
| 4-4-4 pattern (tertiary), cycles every 3 semitones (minor third). | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 |
| 3-3-3-3 pattern (quaternary), cycles every 4 semitones (major third). | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 |
| 2-2-2-2-2-2 pattern (senary), cycles every 6 semitones (tritone). | 1 | 2 | 3 | 4 | 5 | 6 | 1 | 2 | 3 | 4 | 5 | 6 |
See also: 6-6 and 7-5 Pitch Patterns Tutorial and 4-4-4 Pitch Pattern.
Notation Systems
Isomorphic notation systems are pitch-proportional and have regularly repeating patterns. This typically takes the form of a regular line pattern with a consistent interval distance between each line. For example, notations with lines a major second apart have a 6-6 line pattern, notations with lines a minor third apart have a 4-4-4 line pattern, and notations with lines a major third apart have a 3-3-3-3 line pattern. Isomorphism can also be achieved through the use of alternating hollow and solid notes (a 6-6 pattern) or different patterns of notehead shapes.
Nomenclatures
The traditional musical nomenclature (note names, interval names, etc) is tied to the same complicated framework as traditional notation (sharps and flats, enharmonic equivalents, etc). So there is good reason to consider new nomenclatures that do not have these complications. There are also benefits to using a nomenclature that correlates well with a given alternative, isomorphic notation system.
(Of course, there are also benefits to having a standard nomenclature for easy communication between musicians. For that reason is probably easier for an individual to adopt an alternative notation system or instrument, than it would be to adopt an alternative nomenclature.)
Instruments
Instruments are isomorphic when the same musical pattern can be played in the same basic way regardless of the starting pitch. Common examples of isomorphic instruments are stringed instruments like the violin, viola, cello, string bass, bass guitar, and mandolin. (The guitar is mostly isomorphic, but in its usual tuning it has a slightly irregular pattern, because the interval between the G and B strings is a major third while other neighboring strings are all separated by a perfect fourth. Some advocate tuning the guitar completely in either perfect fourths or major thirds for this reason.) The traditional piano keyboard is not isomorphic (it has a 7-5 pattern), so it is necessary to learn many different fingerings to play the same scale or chord when starting on different pitches. A number of new keyboard layouts and instrument designs have been introduced to provide the benefits of isomorphism. See Isomorphic Instruments.
There are benefits to using instruments that correlate well with a given notation and nomenclature. This is why tablature is popular: the notation matches the instrument. It is also why isomorphic instruments are a logical companion to isomorphic notation systems. That is why many people interested in isomorphic notation systems are also interested in isomorphic instruments.