Scope & “Why?”s

Scope & “Why?”s


Harmonious exists to help answer some questions about harmony and music theory. Learn about the scope of the material and the motivating “why” questions.


If you are a casual musician, do you ever wonder, “How are songs put together? Why do certain chord progressions sound good? How do I write my own songs and chord progressions?

“How can I, as a songwriter, take advantage of the expertise of my audience members, when they listen to my music, to hack their brains and delight their ears?” After all, the average music listener has a lifetime of experience with thousands of melodies and songs, and they already know what sounds good.

The short answer is harmony.


Harmony is the study of what notes sound good together and how to embellish melodies with more notes to sound more full. Melody and harmony are related, so the study of harmony helps composers write melodies from chord progressions and chord progressions from melodies.

And of course, what “good together” means depends on context and style of music, but useful conventions (mapped out in the whole of Harmonious) can help us analyze and synthesize harmony across highly diverse styles and tastes.


The aim is for Harmonious to be a complete, authoritative, and practical reference on everything harmony-related in twelve-tone equal temperament (and nearby tunings).

Harmonious combines elements of musical set theory and atonal theory and jazz theory with harmony and music theory in the extended common practice, as outlined by Professor Dmitri Tymoczko in the 2011 textbook, A Geometry of Music. (See the About page for more details and bibliographic references.)

Everything in Harmonious is interconnected, but layers of content include:

While mapping out all of these connections, Harmonious also attempts to map out the richness and ambiguity inherent in harmony as precisely as possible.

An Exhaustive Approach

Instead of presenting new material in every-increasing levels of specialization, Harmonious presents a general view and sets bounds to what will be covered. Harmonious focuses on three major questions that may be of interest to both the casual and serious musician:

Q1. How many possible chords and scales are there?

  • What are all the possible chords that can be played?
  • What are all the ways to play a given chord (on the keyboard and the guitar)?
  • (Should I just try to find lists of chords and scales and hope smart people have found them all? or is there a better approach?)

Q2. Why are some chords and scales so common?

  • Which chords and scales from this exhaustive list are the most useful or versatile?
  • Is there some sort of simple mathematical measurement that can predict roughly how useful or versatile a chord or scale might be, regardless of cultural or stylistic considerations?

Q3. What can mathematics tell us about how all of these chords and scales are related?

  • If we have an exhaustive list of chords and scales, we can easily enumerate all of the possible relationships, but which relationships are meaningful?
  • Is there some sort of measurement that can predict how closely related (in a musical sense) a certain pair of chords or scales are?
  • How would understanding those relationships help us learn to analyze and compose the music we are interested in pursuing?

The majority of the remaining tutorials in this outline will try to show how atonal theory or musical set theory can help us extend our understanding of tonality beyond the common practice, oriented around major–minor keys of the diatonic scale.

We will start from square one, with a fresh, twenty-first-century music theory perspective, answering Q1 as a setup for answering the next questions. Interestingly Q2 and Q3 have a shared explanation! (How a chord or scale relates musically to other similar chords and scales determines how useful that chord or scale is, and generally how common it may be, across styles of music.)