Harmonic Configurations

Introduction

I've been delving for some time into a concept of music theory I chose to call "harmonic configuration".
I'm creating this page to gather resources about it or related topics.
For now, I’ll do this in a messy way and think about how to reorganize it later and add more factual resources.

What is this?

The idea of harmonic configurations provides a way of analyzing harmony by looking at which notes are present or not, without assuming a root or tonal center.
Traditionally, we think of harmony in terms of modes anchored to a tonic.
This view can become computationally complex when analyzing music automatically, because it involves many potential roots and modal variants.

Harmonic configurations bypass that by treating pitch collections as absolute sets, essentially binary patterns of included and excluded notes across the 12-tone chromatic scale.
If we only consider complete scales exempt from consecutive semitones, there are 57 of those "unrooted" configurations (while there are 33 "rooted" scales satisfying those conditions).
In some contexts we might see the term "scale" used to refer to a "pitch-class set without root" but since terms like "modes" and "scales" are somehow flexible I choose the term "configuration" to avoid this ambiguity.
Those can be a useful tool when programming music-related tasks, I also tend to think they have an interest regarding the practice of an actual instrument.

Terminology

I'll try to define some terms in order to avoid confusion, some of these terms are rooted in academia, others are neologisms or practical constructs and I had to assign them to a definition arbitrarily. I hope I don't break the terminology myself too much in this page.

Pitch class
A note identity within the octave (e.g. C, C♯, D...)
Pitch-class set
An unordered collection of pitch classes.
Mode
A rotation of a configuration that assigns a root and changes interval perception (e.g. Dorian, Phrygian).
Scale
Ordered pitch-class set with a tonic and functional relationships (e.g. C Dorian, D Phrygian).
Harmonic configuration
A pitch-class set analyzed without assigning a tonic/root.
Anhemitonic
A set with no semitones (e.g. major pentatonic).
Hemitonic
A set that includes at least one semitone.
Tonal condensation
Presence of three adjacent chromatic notes (e.g. B–C–C♯).
Ancohemitonic
A scale with no tonal condensation (consecutive semitones).
Complete scale
A scale without “gaps”; no interval between adjacent notes exceeds a minor third, nothing can be added without creating a condensation.
Moment of symmetry
Scales built with two step sizes arranged symmetrically (e.g. diatonic).
Mode of limited transposition
A symmetrical scale that repeats under few transpositions (e.g. whole-tone).
Cardinality
Number of pitch classes in a scale (e.g. 5 for pentatonic).

Data

Here are some tables.
I did them manually and might have done some typos, I hope not.
About the names of the scales: I know some people disagree about the names some of these scales should have. In some cases I had to pick one, sorry if it's not the one you're used to.
X shows the presence of notes in a scale.
The "index" is calculated in a binary way (little-endian, starting at C).
The first part of the table shows the 33 "usual" ancohemitonic scales in a relative way.
Overall here I choose to focus on ancohemitonic scales since I consider them to be the main building blocks for conventionnal chords.
The next parts shows the data organized as "configurations" in an absolute way.
Here is the table in ods format. I plan to include more data someday, including:
- tables for any combination of notes in the chromatic scale.
- more names for presented scales.
- lists of chords corresponding to a given scale.
- random comments about particular scales (eg symmetry, typical uses, etc).
etc.

Ancohemitonic modes














relative note position in mode
1 #1 2 #2 3 4 #4 5 #5 6 #6 7 intervals index
(assuming base of C)
notes
(assuming base of C)
C C# D D# E F F# G G# A A# B

Diatonic group Ionian X
X
X X
X
X
X 2,2,1,2,2,2,1 2773

Dorian X
X X
X
X
X X
2,1,2,2,2,1,2 2902

Phrygian X X
X
X
X X
X
1,2,2,2,1,2,2 3418

Lydian X
X
X
X X
X
X 2,2,2,1,2,2,1 2741

Mixolydian X
X
X X
X
X X
2,2,1,2,2,1,2 2774

Aeolian X
X X
X
X X
X
2,1,2,2,1,2,2 2906

Locrian X X
X
X X
X
X
1,2,2,1,2,2,2 3434
Melodic group Melodic Major X
X
X X
X X
X
2,2,1,2,1,2,2 2778

Half-diminished X
X X
X X
X
X
2,1,2,1,2,2,2 2922

Superlocrian X X
X X
X
X
X
1,2,1,2,2,2,2 3498

Melodic minor X
X X
X
X
X
X 2,1,2,2,2,2,1 2901


X X
X
X
X
X X
1,2,2,2,2,1,2 3414

Lydian augmented X
X
X
X
X X
X 2,2,2,2,1,2,1 2733

Bartok scale X
X
X
X X
X X
2,2,2,1,2,1,2 2742
Harmonic minor group Harmonic minor X
X X
X
X X

X 2,1,2,2,1,3,1 2905


X X
X
X X

X X
1,2,2,1,3,1,2 3430


X
X
X X

X X
X 2,2,1,3,1,2,1 2765


X
X X

X X
X X
2,1,3,1,2,1,2 2870


X X

X X
X X
X
1,3,1,2,1,2,2 3290


X

X X
X X
X
X 3,1,2,1,2,2,1 2485


X X
X X
X
X X

1,2,1,2,2,1,3 3500
Harmonic major group Harmonic major X
X
X X
X X

X 2,2,1,2,1,3,1 2777


X
X X
X X

X X
2,1,2,1,3,1,2 2918

Mixolydian altered X X
X X

X X
X
1,2,1,3,1,2,2 3482


X
X X

X X
X
X 2,1,3,1,2,2,1 2869


X X

X X
X
X X
1,3,1,2,2,1,2 3286


X

X X
X
X X
X 3,1,2,2,1,2,1 2477


X X
X
X X
X X

1,2,2,1,2,1,3 3436
Whole tone group Whole tone X
X
X
X
X
X
2,2,2,2,2,2 2730
Augmented group Augmented X

X X

X X

X 3,1,3,1,3,1 2457


X X

X X

X X

1,3,1,3,1,3 3276
Diminished group Half whole X X
X X
X X
X X
1,2,1,2,1,2,1,2 3510

Whole half X
X X
X X
X X
X 2,1,2,1,2,1,2,1 2925

Ancohemitonic configurations














funky 2-letters notation
Do Dé Ré Ri Mi Fa Fo So Sa La Li Si intervals
(starting with first note >=C)		 index
absolute pitch class
C C# D D# E F F# G G# A A# B

Diatonic group
X
X
X X
X
X
X 2,2,1,2,2,2,1 2773



X
X X
X
X
X X 2,1,2,2,2,1,2 1451


X
X X
X
X
X X
2,1,2,2,2,1,2 2902



X X
X
X
X X
X 1,2,2,2,1,2,2 1709


X X
X
X
X X
X
1,2,2,2,1,2,2 3418


X
X
X
X X
X
X 2,2,2,1,2,2,1 2741



X
X
X X
X
X X 2,2,1,2,2,1,2 1387


X
X
X X
X
X X
2,2,1,2,2,1,2 2774



X
X X
X
X X
X 2,1,2,2,1,2,2 1453


X
X X
X
X X
X
2,1,2,2,1,2,2 2906



X X
X
X X
X
X 1,2,2,1,2,2,2 1717


X X
X
X X
X
X
1,2,2,1,2,2,2 3434
Melodic group
X
X
X X
X X
X
2,2,1,2,1,2,2 2778



X
X X
X X
X
X 2,1,2,1,2,2,2 1461


X
X X
X X
X
X
2,1,2,1,2,2,2 2922



X X
X X
X
X
X 1,2,1,2,2,2,2 1749


X X
X X
X
X
X
1,2,1,2,2,2,2 3498


X
X X
X
X
X
X 2,1,2,2,2,2,1 2901



X X
X
X
X
X X 1,2,2,2,2,1,2 1707


X X
X
X
X
X X
1,2,2,2,2,1,2 3414


X
X
X
X
X X
X 2,2,2,2,1,2,1 2733



X
X
X
X X
X X 2,2,2,1,2,1,2 1371


X
X
X
X X
X X
2,2,2,1,2,1,2 2742



X
X
X X
X X
X 2,2,1,2,1,2,2 1389
Harmonic minor group
X
X X
X
X X

X 2,1,2,2,1,3,1 2905



X X
X
X X

X X 1,2,2,1,3,1,2 1715


X X
X
X X

X X
1,2,2,1,3,1,2 3430


X
X
X X

X X
X 2,2,1,3,1,2,1 2765



X
X X

X X
X X 2,1,3,1,2,1,2 1435


X
X X

X X
X X
2,1,3,1,2,1,2 2870



X X

X X
X X
X 1,3,1,2,1,2,2 1645


X X

X X
X X
X
1,3,1,2,1,2,2 3290


X

X X
X X
X
X 3,1,2,1,2,2,1 2485




X X
X X
X
X X 1,2,1,2,2,1,3 875



X X
X X
X
X X
1,2,1,2,2,1,3 1750


X X
X X
X
X X

1,2,1,2,2,1,3 3500
Harmonic major group
X
X
X X
X X

X 2,2,1,2,1,3,1 2777



X
X X
X X

X X 2,1,2,1,3,1,2 1459


X
X X
X X

X X
2,1,2,1,3,1,2 2918



X X
X X

X X
X 1,2,1,3,1,2,2 1741


X X
X X

X X
X
1,2,1,3,1,2,2 3482


X
X X

X X
X
X 2,1,3,1,2,2,1 2869



X X

X X
X
X X 1,3,1,2,2,1,2 1643


X X

X X
X
X X
1,3,1,2,2,1,2 3286


X

X X
X
X X
X 3,1,2,2,1,2,1 2477




X X
X
X X
X X 1,2,2,1,2,1,3 859



X X
X
X X
X X
1,2,2,1,2,1,3 1718


X X
X
X X
X X

1,2,2,1,2,1,3 3436
Whole tone group
X
X
X
X
X
X
2,2,2,2,2,2 2730



X
X
X
X
X
X 2,2,2,2,2,2 1365


X

X X

X X

X 3,1,3,1,3,1 2457
Augmented group


X X

X X

X X 1,3,1,3,1,3 819



X X

X X

X X
1,3,1,3,1,3 1638


X X

X X

X X

1,3,1,3,1,3 3276
Diminished group
X X
X X
X X
X X
1,2,1,2,1,2,1,2 3510


X
X X
X X
X X
X 2,1,2,1,2,1,2,1 2925



X X
X X
X X
X X 1,2,1,2,1,2,1,2 1755

Videos

I'm speaking about this in French in a couple of videos:

This one speaks about musical modes in general, not configurations, but approaches music generation and introduces the concept of musical "biomes".

This one is the first that I made specifically about configurations, where you can see me struggle to explain and demonstrate the concept.

In this one, I explain how one can optimize calculations when coding programs requiring some understanding of harmony.

Here I am running a large set of classical MIDIs through a configuration analysis pass.

Later, I discovered some oddities in the distributions and talk about them a bit.

Resources

Here are some resources somehow linked to the present topic. Again, this is a draft and I hope to be able to order them in a more useful way someday.

Emanuele Di Mauro's théorie de la gravitation tonale (also in French) explores the same idea in detail, sometimes using different words than mine:
https://gravitation-tonale.fr/
YouTube account

I think this video explains the "traditional" 33 complete rooted scales without consecutive semitones pretty well.

An interactive tool.

A large table of chords.
A large table of scales.
A collection of yaml data.
A collection of csv files.

Other unsorted links and references that were given to me or that might possibly have something to do with the topic:
Gammes, rythmes et maths
Allan Holdsworth lessons
The Forte number
Xenharmonic music
some pdf with intriguing diagrams
The Thesaurus of Scales
Cascade of modes from light to dark

Work

Some of my somewhat related work I sometimes talk about:

An old contextual harmonizer
The MIDI analyzer
A Max for Live device that makes use of this concept