Microtonal Theory

These pages teach everything that I, @preinfarction, know about microtonal music theory, including my original discoveries. Posts further down the list are more advanced and generally rely on concepts presented earlier in the series.

:: Tuning Theory

Pythagorean Tuning: How natural harmonics can be used to generate a 3-limit just-intonation tuning-system and what we can do to fix it.
Intervals And Tuning Systems: Everyone uses the word "interval" incorrectly. Now you can get angry about it too.
EDO And Non-EDO Generators: Pretty easy math, but I discovered some of it and I'm quite proud.
Five-Limit Just Intonation: The One True Way? This post also introduces rank-3 interval space.
Pitches And Pitch Classes: Come learn to spell with me. There's a spiral in a heptagon. You'll love it.
Prime Harmonic Bases And Comma Bases: How to name intervals and how to find tempered intervals of an EDO.
Unimodular Matrices, Isomorphic Keyboards, and Unequal Temperaments: You and I can't afford 2D keyboards, but let's learn math about them anyway. This post introduces my favorite schismatic temperament, which is good for showing 5-limit just intonation music projected down to a 2D lattice or for performing slightly detuned 5-limit JI music that fits on a 2D lattice.
Chromatic Orders Of Rank-2 Intervals : Psst. Hey, Kid. You want some Hasse diagrams explaining the partial order over augmented and diminished intervals that is logically enforced if we assume the usual ordering of natural intervals in a rank-2 tuning system? Check it out.
Seven-Limit Just Intonation: Our first foray into otherworldly beauty.
The Bohlen-Pierce Scales: My original intervallic interpretation for a kind of music that no one ever really makes.
Higher Rank-EDO Generators: High-rank intervallic interpretations and definitions of EDOs up to 100 divisions. Discussion of the relationship between EDO definitions in terms of tuned prime harmonics versus definitions in terms of tempered commas.
Rank-3 Chromaticity And Diatonicity: Characterizing the region of small mistunings of rank-3 intervals, relative to just intonation, that keep the chromatic scale in its natural order
Intervallic Chords: Tertian Chord Names, Cadential Progressions, and Modal Mixture Grammars
High Prime Limit Just Intonation: Factors of 11, 13, 17, and 19.
Frequency-Space Just Intonation: Otonality, Utonality, and Near-Equal Just Intonation
Consonance And Timbre: Dissonance Curves, Inharmonic Spectra, Gamelan Scales, Custom Timbres, and Harmonic Entropy

:: Microtonal Ethnography

Makams And Maqamat I: A first look at using 24-EDO and rank-4 interval analysis for representing neutral tones in basic Arabic maqamat, as well as some discussion of finer grained Arabic intonation of ajnas.
Just Tuning Of Arabic Ajnas: More precise tuning of Arabic ajnas using data from Margo Schulter, Navid Goldrick, Mohamed A. Alsiadi, and especially Sami Abu Shumays.
Makams and Maqamat II: A brief diversion into the Pythagorean modes of Safi al-Din, and then a long course in using 53-EDO for understanding Turkish makams and ajnas
Makams and Maqamat III: Using 24-EDO and 53-EDO together for a detempering analysis of Arabic maqamat, with 53-EDO data about Arabic/Syrian intonation from Mohamed A. Alsiadi.
Makams And Maqamat IV: Yalcin Tura's Turkish Baglama Scale compared to 24-EDO
Makams and Maqamat V: A deep look at tetrachords. We'll start with justly intoned tetrachords from Mansur Zalzal, Ibn Sina, and al-Farabi. We'll represent tetrachords in 87-EDO , the smallest Zalzalian EDO. We'll examine the fine-grained intonation of some Turkish makams and tetrachords as presented by Ozan Yarman in 159-EDO and then try to spell them intervallically. We'll review the cultural and historical placement of some tetrachords as reported by Margo Schulter. And we'll close by looking at the lute fingering geometry of Navid Goldrick, just because I don't know where else to put it.
Makams and Maqamat VI: A look at Persian Dastgah.
Makams and Maqamat VII: Syrian Jewish liturgical maqamat per Mauro Braunstein.
Chopi Mbila Music: Analyzing the Mbila xylophone music made by the Chopi people in Mozambique
The Polish Duda: A septimal analysis of the microtonal tuning of the chanter on a Polish bagpipe
Georgian Folk Music: Three possible just tunings for the basic tetrachord of Georgian folk chant music
Indonesian Gamelan: The pelog and slendro scales of Javanese and Balinese musical ensembles centered around inharmonic metallophones
Byzantine Chant: ...
An Ethnographic Smattering: I hope to also include sections on other microtonal bagpipes, Indian Srutis, Wagogo music of Tanzania, Ancient Chinese Shí-èr-lǜ tuning, and Bunun pasibutbut singing of Taiwan.

:: Microtonal Composition

Consonance, Fluidity, and Temporal Stress: The foundations of composition - sounding good spectrally, sounding good temporally, and linking these together.
Melodic Space Identification: Mode Intersection, Harmonic Chromaticism, and Tonal Continuation: Techniques for pairing a scale with a chord to help you write a melody
The Meantone Constraint For Rank-2 Fuxian Counterpoint: ...
Five-Limit Fuxian Counterpoint: ...
Five-Limit Romantic Harmony: Functional Chord Progressions, Passing Chord Voice Leading, Upper Chord Extensions In SATB Voicing
Melodic models: The Fluid, The Accumulated Diatonic, The Scaled And Interpolated Diatonic

What is Microtonal Music?

Microtonal music is any music that isn't tuned in 12 tone equal temperament. It includes:

The ancient Greek foundations of western music, like Pythagorean tuning, the seven-limit genera of Archytas, and Ptolemy's diatonic scale.
All western music of the Renaissance and most of the Baroque period, which used tuning systems like well temperaments and meantone temperaments. Split-key keyboard instruments that distinguished between microtones have been invented, built, and used by composers, musicians and music theorists without pause from the Renaissance to today, from Vicentino's split key archicembalo harpsichord (first described in 1555) in the Renaissance, to a range of cimbali cromatico in the 1600s associated with (Trasuntino, Fabbri, Zarlino/Pesaro, Vido, ...) to a golden age of new keyboard designs in the late 1800s from people like Hermann von Helmholtz, Robert Bosanquet, Adriaan Fokker, and Thaddeus Cahill, to a spate of new weirdos in the modern era.
Modern and historic maqams of the middle east - the Arabic, the Turkish, and the Persian. Sometimes analyzed with quarter tones or with 53-EDO, or more exotic systems, but always microtonal.
The Shruti microtones of classical Hindustani music.
Gamelan, the ensemble music of Indonesia, with tuning systems like Pelog and Slendro.
I don't know a ton about Chinese music, but I hear the ancient sanfen sunyi tuning system is basically equivalent to Pythagorean tuning and was used to generate a chromatic scale called shi'er lü. I imagine there have been some developments since then.
Modern twelve tone equal temperament even, kind of; the staff notation of pitches still waggles its eyebrows to suggest an abstraction of intervallic music beneath the frequencies, an abstraction which is agnostic to the tuning system.

Microtonal music is just music. It's music as performed in all cultures in all history. It's music as analyzed with intervals and pitches - abstract representations agnostic to the tuning the performer selects. It's just music. Let's learn about music. In particular, let's learn about intervals, tuning systems, and just intonation.

My name is Ethan L. You can find me at @preinfarction on twitter. I don't have a Patreon page, but let me know if you want to give me money and I'll make one real quick.