Appendix A — Music Theory Primer

(Modified on 2026-08-11)
765 words

This book is aimed at nascent researchers with interest and background in music. I frequently use terminology from music theory and performance practice, and sometimes reference significant figures/events from musical history—e.g., Beethoven and Metallica. If you have little musical background, you will quite understandably not understand all these terms and references; Even if you are an experienced and/or educated musician, you may not understand everything—this is normal and nothing to be ashamed of. Even experienced music theorists and musicologists don’t always use the same terms, and will disagree about the exact meaning of terms they do use.

In this appendix, I attempt to clarify the meaning of specific terms used in the rest of the book. Note that the perspectives and terminologies here are generally Western. However, I make efforts to refer to non-Western musics when relevant. More importantly, I am not committed specifically to what might be called the “Standard Western Music Theory,” as widely taught in Western nations in the last century. For example, most music theorists and psychologists recognize that Western “time signatures” are not great representations of how Western music is really structured. The “real” structure music theorists and psychologists favor is meter, which is related to, but not the same as, time signatures.

Pitch

One theoretical model of the diatonic scale is as seven stacked “perfect 5ths”—traditionally corresponding to a 3-to-1 frequency ratio. Start from “F” and stack up perfect fifths and you get F-C-G-D-A-E-B, with five whole-steps and two half-steps in the diatonic pattern. If you extend this line-of-fifths in either direction, new pitches are introduced in between the diatonic pitches. These pitches are represented/conceptualized in the Western traditions using accidentals attached to the “natural” scale steps. Add five ascending fifths to get F#-C#-G#-D#-A#, or five descending fifths to get Bb-Eb-Ab-Eb-Gb. Note that these “accidental” notes are perceived differently than the main diatonic pitches, and that, to some extent, notes with the same scale degree but different accidentals are considered similar: Bb and B can be used by musicians in (somewhat) similar ways.

If we tune our scales based on “pure” 3-to-1 perfect fifths, the next notes on the line-of-fifth (B# and Cb) are very close to, but not exactly the same, as C and B respectively. (They differ by the “Pythagorean comma,” a ratio of \(\frac{531,441}{524,288}\), which is about 0.234 semitones.) Because these “new” notes happen to be so close, Westerners settled on a 12-note “meta-scale”—seven-diatonic notes plus five accidentals. However, because accidentals can be added either above (sharp) or below (flat) the diatonic scale, Westerners long differentiated between, for example, C# and Db. C# and Db would traditionally be seen as slightly different pitches, differing by that same Pythagorean comma. Unfortunately, if you tune a line of 12-fifths—like Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#—the first and last notes will be badly out of tune. Western musicians eventually “fixed” this problem by tempering their fifths, tuning them all a little (\(\approx 2\) cents) flat. The result is that the “line-of-fifths” can be wrapped back on itself every 12 steps, resulting in the “circle-of-fifths.” This modern system, equal temperament, was fully adopted by Western musicians in the early to middle 19th century.

This history lesson serves a purpose. It illustrates that, though modern equal temperament results in (ideal) pitches that correspond to semitones, Western tonal notation actually contains a lot of information beyond “just” semitones. Tonal notes, for one, indicate what notes are part of the diatonic key and which are not, which is extremely relevant to human perception of music. What’s more, they can indicate other tonal relationships. For example, C# and D are closer together on the line-of-fifths than Db and D; the former pair can occur together in the same key, the latter pair cannot. These sorts of tonal details are relevant to music composition and music perception. If we assume, as many researchers often do, that C# and Db are the same, we may miss important relationships.

Rhythm

Beats and Tempo

The English word “beat” has many different musical meanings. In music theory, the term refers to an abstract, isochronous pulse.

“Isochronous” means “equal time,” referring to the idea that each beat is separated by an identical time span. In reality, when humans perform naturally, the time-spans between beats can and do vary, both accidentally and intentionally (rubato).
A beat is “abstract” because it may or not actually correspond with sound events. We can conceptualize, anticipate, and feel beats without actually hearing or playing them.

Psychologically, humans only perceive beats between about \(50_{bpm}\) and \(180_{bpm}\).

Meter

Around the world, musicians “count time” in various ways. In Western scores, this counting is (partially) enumerated in “time signatures.” However, the underlying principle—which seems to have (nearly) universal musical currency, though with many variations—is meter.