An Exhausting Yet Necessarily Incomplete Study Of The Properties Of Musical Scales

It's true, I'm writing a book.

As if making this website, and a few thousand YouTube videos isn't enough.

I've studied music theory for about 43 years. I'm a bit of a nerd. In the past decade I've worked on creating software to calculate the many interesting properties of musical scales, and in doing that I've scoured academia and music theory literature for advanced, geeky arcane stuff about musical scales. The result of that work can be seen (in part) on the pages of this website. All of that and more is going in the book.

That's right, there is stuff in the book that you will not find on this website, or in the YouTube videos. Sneaky, huh?

I've spent the past 7 years working on it, and it is still not finished. It's massive. Here is the cover (as a fake 3D photo) of Volume 1: The Splaining

See that guy on the cover? He is reading about all the properties of musical scales, and there is so much material it's making his brain hurt.

Take note: this is not a printed book. It's a PDF document, and it's a big one. If you choose to print it out, the lives of all those trees are on your conscience, not mine.

In Volume 1: The Splaining, I explain all the concepts and ideas that you will need — actually, more than you'll need — to understand everything in Volume 2: The Scales.

This one is also not a printed book, it's another PDF document and it's even bigger than Volume 1. There is one page for every scale, and as you will read in Volume 1: The Splaining (spoiler alert!!), there are 2048 of those. So yeah. It's a 2048 page book. That seems crazy when you read about it here, but once you see the book and how it's organized, you'll understand.

Here is the Table of Contents for Volume 1: The Splaining (so far):

Preface
Contents
Introduction
1. Preamble: The Goal Of This Book
2. Typographic Conventions
3. Explaining Pitch
  1. Equivalence
  2. Pitch Classes
  3. Pitch Class Sets
  4. MIDI Pitch Numbers
4. Defining Scales
5. What Universe Is This?
  1. Octave Equivalence
  2. Twelve Tone Division
  3. Equal Temperament
6. Cents
7. Generic and Specific Intervals
  1. Genus and Species
  2. Skips and Steps, Size and Span
8. Summary
Making Scales
1. Preamble
2. Binary Representation
3. Power Set
4. Cardinality
5. Applying Limitations
6. Summary
Representing Scales
1. Preamble
2. Pitch Class Set
3. Interval Structure
4. Staff Notation
5. Bracelet Notation
6. Tonnetz Diagram
7. Instrument-specific diagrams
8. Summary
Modifying Scales
1. Preamble
2. Rotation / Transposition
3. Modes
  1. Diatonic Modes
4. Reflection / Inversion
5. Forte Set Classes
  1. Inversions
6. Transpositional/Inversional Set Classes
7. Prime Form
8. Multiplication
  1. Self M-related
9. Unified Transformation Operations
10. Mutation
  1. Levenshtein Distance
  2. A Better Method
11. Complement
  1. Hypermodes
12. Invariant Properties
13. Set Group Systems
14. Summary
Symmetries
1. Preamble
2. Reflective Symmetry
3. Palindromicity
4. Chirality
  1. Bracelets vs Necklaces
5. Ridge Tones
6. Rotational Symmetry
  1. Messaien's MOLTs
  2. Truncations
7. Summary
Observing Intervals
1. Preamble
2. Cyclic Correlation
3. Interval Structure
4. Interval Matrix
5. Steps, Skips, Leaps, and Spans
6. Interval Vector
  1. Saturation
  2. Saturation Medians
  3. Deepness
  4. Hemitonia
  5. Cohemitonia
  6. Z-Relations
  7. ZC-Relations
    1. Hexachordal Theorem
  8. All-Interval Tetrachord
7. Imperfections
8. Propriety / Coherence
9. Heteromorphic Profile
  1. Contradictions
  2. Ambiguities
  3. Differences and Samenesses
10. Coherence Quotient
11. Sameness Quotient
12. Distribution Spectra
  1. Spectra Variation
  2. Trivalence
  3. Myhill’s Property
  4. Reduced and Semireduced Scales
  5. Consecutivity Property
13. Summary
Generators
1. Preamble
2. Generators
  1. Group Theory
  2. Three Gap Theorem
  3. Chopin’s Theorem
  4. Distance Models
3. Bisectors
4. Well-Formedness
  1. Symmetry Condition
  2. Closure Condition
  3. Degenerate Well-Formed Scales
  4. Cardinality Equals Variety
  5. Structure Yields Multiplicity
  6. Pairwise Well-Formed Scales
5. Summary
Special Scales
1. Preamble
2. Some Special Scales
3. Summary
Evenness
1. Preamble
2. Maximal Evenness
  1. Distributional Evenness
  2. Cohn’s Property
    1. Hexatonic Cycles
  3. Euclidean Distance From Evenness
3. Maximally Smooth Cycles
4. Maximal Area
5. Polygon Perimeter
6. Center of Gravity
7. Self-complements
8. Lewin-Quinn FC-components
9. Summary
Fragments
1. Preamble
2. Tetrachords
3. Hierarchizability
4. Triad Inclusion
5. Bi-Triadics
  1. Generic Modality Compression
6. Summary
Appendix
1. Set Theory Notation
2. Pascal’s Triangle
3. Coherence Data
Lists
1. Non-degenerate Well-formed Scales
2. Degenerate Well-formed Scales
3. Scales with the Myhill Property
4. Prime Scales with Consecutivity Property
5. Prime Semi-Reduced Scales
6. Prime Reduced Scales
7. Prime scales that are Self-M-Related
8. Balanced Scales
9. Maximally Even Scales
10. Maximal Area Scales That Are Not Maximally Even
11. Maximally Even Scales With The Myhill Property
12. Proper Scales
13. Strictly Proper Scales
14. Deep Scales
15. Palindromic Scales
16. Scales With Rotational Symmetry
17. Trivalent Scales
18. Bi-Triadic Scales
19. Scales with Contradictions but no Ambiguities
20. Prime Scales Of Hierarchizability 3 with k=2
21. Prime Scales Of Hierarchizability 2 with k=2
22. Prime Scales Of Hierarchizability 1 with k=2
23. Prime Scales Of Hierarchizability 3 with k=4
24. Prime Scales that increase hierarchizability when k increases from 2 to 3
25. Prime Scales with Consecutivity Property
26. Tetratonic Scales With No Common Triads
27. Pentatonic Scales With No Common Triads
28. Hexatonic Scales With No Common Triads
29. ... and dozens more
Bibliography
Glossary
Alphabetical Index

Things this book will not do:

Make you a better performer in any way
Give you the ability to match chord progressions with appropriate scales
Spice up your improvization skills with jazzy modes and sick licks
Be useful in any practical way for anything that matters in making your life better

Things this book will do:

Help you to understand scales, modes, intervals, interval vectors, coherence, evenness, generators, and much more
Make you sound like a smarty-pants when you're at a music theory convention and drop five-dollar words like "trivalent"
Explain all kinds of obscure and fascinating properties of musical scales, in clear understandable language with good examples and lots of diagrams and illustrations
Be a really comprehensive reference for all kinds of facts about sets of 12 things
Impress your friends with your ability to discern the hierarchizability of any pitch class set
Make you the "life of the party" as you explain the nuances of distribution spectra
Help your kids fall asleep (read it to them at bedtime!)

OK Ian, take my money already

Keep your wallet closed, it's not fully written yet. I will accept your money once I have a book to give you.

But I have a deal for you. If you join my Patreon crew as a supporter, I will give you a copy of the book when it comes out. No matter what your pledge is. You could pledge for $1 and quit after a month, and you'll get this book for $1, no strings attached*. My patrons are currently giving enough that it keeps this web server going, and occasionally I get to buy a theory book! In gratitude for that support, you will be signed up to get the book, and you'll also be treated to occasional glimpses of chapters as they're completed. I send an update to my Patrons every couple of months to let them know how things are coming along, and with alerts to new stuff that I release on the website.

* Since you would be getting an early pre-release copy of the book, I will ask that you write a review on Amazon. Ideally a favourable one, but I know that's not a given, I have to earn that by writing an incredible book.

Go to My Patreon Page and join others like yourself who totally geek out on all this nerdy stuff.