Every theorist from Pythagoras to Partch, from Aristoxenus to Erose.
Scroll to the end for a quick summary.
Music has never been tuned the same way twice. Across two and a half millennia, Western musicians, mathematicians, and philosophers have argued passionately about which intervals are “correct,” which compromises are acceptable, and what tuning even means. This is that story.
I. The Ancient World: Ratios, Consonance, and the Cosmos
Pythagoras (c. 570–495 BCE)
The story of Western tuning begins, as so many stories in Western thought do, with Pythagoras of Samos — though the man himself is so entangled with legend that we must be careful to speak of “the Pythagoreans” as much as the philosopher himself. The foundational myth is well known: walking past a blacksmith’s shop, Pythagoras supposedly heard consonant harmonies ringing from hammers of different weights, and realised that musical beauty was a function of numerical ratio. The story is almost certainly apocryphal — hammer weights don’t actually produce harmonic relationships in this way — but it encapsulates a genuine insight that shaped two and a half millennia of music theory: that the intervals we perceive as consonant correspond to simple whole-number ratios between vibrating string lengths.
The Pythagoreans identified the octave with the ratio 2:1, the perfect fifth with 3:2, and the perfect fourth with 4:3. These three relationships, derived from the first four integers (1, 2, 3, 4) — the tetractys, which the Pythagoreans regarded as sacred — were the building blocks of their tonal universe. From these ratios they constructed a complete scale by a process of cyclic generation: beginning on any pitch, one could ascend by successive perfect fifths (each of ratio 3:2) and descend by octaves (2:1) to keep all pitches within a single octave span. Twelve such fifths nearly — but not exactly — return to the starting pitch. The result of stacking twelve pure 3:2 fifths produces a pitch slightly sharper than the seven octaves it nominally traverses; this discrepancy, equal to approximately 23.46 cents (where 100 cents equals one equal-tempered semitone), is the Pythagorean comma. It is the first crack in the edifice of just intonation, and every tuning system that follows is, in one way or another, a response to it.
The Pythagorean scale that results from stacking fifths gives diatonic intervals with beautifully pure fifths and fourths, but produces major thirds of ratio 81:64 — significantly wider and more dissonant than the pure 5:4 major third that the ear finds most pleasing. For a culture whose polyphony was built on fourths and fifths, this was no great defect. But as Western music slowly moved toward thirds as primary consonances, Pythagorean tuning would come to feel increasingly strained.
Beyond its practical applications, the Pythagorean framework carried enormous cosmological weight. The ratios of the musical scale were understood to be the same ratios that governed the movements of the heavenly spheres — the musica universalis. To tune a monochord was, in a deep sense, to touch the mathematical fabric of the cosmos.
Aristoxenus (c. 375–335 BCE)
If Pythagoras represents one pole of the ancient tuning debate — the mathematical, rationalist, numerical — then Aristoxenus of Tarentum represents the other: the empirical, perceptual, and humanistic. A student of Aristotle and a prolific theorist (ancient sources credit him with 453 works, of which fragments on harmonics survive), Aristoxenus was deeply skeptical of the Pythagorean approach. He argued that music is fundamentally a matter of perception, and that the ear — not arithmetic — is the proper judge of musical intervals.
His Elementa Harmonica proposes a theory of melody built not on numerical ratios but on continuous, divisible space. For Aristoxenus, the tone is a unit of distance on a linear scale, and the fourth spans exactly two and a half tones. This is not a ratio-based claim but a perceptual one: he is describing how the trained musical ear hears and divides intervals, not how a string theoretically vibrates. In this framework, the whole tone can be divided into equal halves — something that is impossible in strict Pythagorean arithmetic, where the tone has ratio 9:8 and its geometric mean is irrational.
Aristoxenus thus anticipates, by more than two thousand years, the logic of equal temperament. His insistence on perceptual continuity and equal interval division runs directly counter to the Pythagorean tradition of discrete integer ratios. The tension between these two approaches — let us call them the “just” tradition and the “equal” tradition — is the fundamental dialectic of Western tuning history.
Claudius Ptolemy (c. 100–170 CE)
The astronomer and geographer Ptolemy was also a profound musical theorist. His Harmonics, written in Alexandria in the second century CE, is the most mathematically rigorous and comprehensive treatment of ancient Greek tuning theory to survive. Ptolemy was critical of both extremes: he rejected the Pythagoreans for placing mathematics above hearing, and criticised Aristoxenus for the reverse. His own method, he claimed, was a true synthesis — one begins with the ear’s judgment of consonance and then seeks the simplest rational explanation.
Ptolemy’s most important contribution was his systematic enumeration and comparison of different genera — ways of dividing the four-note unit called the tetrachord. Where Pythagoreans admitted only ratios derivable from the numbers 2 and 3 (the so-called 3-limit or Pythagorean system), Ptolemy explicitly included ratios involving the number 5. His “syntonic diatonic” tetrachord — which divides the 4:3 fourth using the ratios 9:8, 10:9, and 16:15 — yields a scale with pure major thirds of 5:4 and pure minor thirds of 6:5. This is what later theorists would call 5-limit just intonation, and it remains the mathematical ideal against which all practical tuning systems are measured.
Ptolemy’s work also includes the homalon (even) and malakon (soft) diatonic genera, and he tabulates many more possibilities with admirable mathematical clarity. His framework is essentially a theory of harmonic lattices avant la lettre — a mapping of the relationships between pitches using prime number ratios. The Harmonics would be lost to medieval Europe for centuries, preserved in the Byzantine and Islamic scholarly traditions, and would eventually prove enormously influential when rediscovered in the Renaissance.
II. The Medieval Transmission: Boethius and the Scholastic Inheritance
Boethius (c. 480–524 CE)
Anicius Manlius Severinus Boethius stands at one of the great fault lines of Western intellectual history: the collapse of the Western Roman Empire and the beginning of the Latin Middle Ages. A Roman aristocrat, statesman, and philosopher — executed on charges of treason by the Ostrogothic king Theodoric — Boethius was the principal conduit through which ancient Greek learning, including musical theory, was transmitted to medieval Europe. His De institutione musica(c. 500 CE), written partly as a paraphrase and compilation of Nicomachus, Ptolemy, and others, became the canonical text of music theory in the medieval university curriculum, where it remained authoritative for nearly a thousand years.
De institutione musica is primarily a Pythagorean document. Boethius upholds the supremacy of numerical ratio and presents a complete account of consonance based on the relationships 2:1, 3:2, 4:3, and their multiples. He divides music into three kinds: musica mundana (the cosmic music of the spheres), musica humana (the harmony of the human body and soul), and musica instrumentalis (sounding, practical music) — and he is explicit that the third kind is the least important. True understanding of music, for Boethius, is mathematical and contemplative, not performative.
For practical tuning, Boethius’ influence reinforced Pythagorean practice across the medieval period. The ecclesiastical modes — the tonal frameworks of Gregorian chant and early polyphony — were theorised in Pythagorean terms, with fifths and fourths as the primary consonances and major thirds treated as dissonances or, at best, imperfect consonances requiring resolution. This remained the theoretical orthodoxy in Europe until the fifteenth century, even as actual musical practice increasingly embraced thirds as stable, pleasing harmonies.
III. The Renaissance Revolution: Temperament and the Birth of Tonality
Bartolomé Ramos de Pareja (c. 1440–after 1491)
The first explicit challenge to the Pythagorean monopoly in European theory came from the Spanish theorist Bartolomé Ramos de Pareja, whose Musica Practica (Bologna, 1482) scandalized the scholarly establishment. Ramos proposed dividing the monochord not by Pythagorean methods but using ratios drawn from the 5-limit — including the pure major third 5:4 and pure minor third 6:5. He derived a complete diatonic scale from these simpler ratios, dismissing the traditional Pythagorean constructions as needlessly complicated and out of step with what singers actually do.
The response from established theorists was fierce. Ramos was accused of ignorance, arrogance, and musical heresy, and the controversy persisted for decades. But Ramos had put his finger on something real: the polyphonic music of the late fifteenth century, with its rich triadic textures and flowing counterpoint, demanded pure thirds, and Pythagorean thirds (81:64) were too wide and harsh for this aesthetic. The practical direction of music had simply outrun its theoretical framework, and Ramos was the first to say so in print.
Pietro Aron (c. 1480–1545)
Pietro Aron, a Florentine priest and theorist, took the practical implications of this shift a step further. His Toscanello in musica (1523) is notable for being one of the first theoretical works addressed primarily to practical musicians rather than scholastic philosophers, and for containing one of the earliest written descriptions of a specific temperament — a deliberate adjustment of pure intervals to make them work across multiple keys on a fixed-pitch keyboard instrument.
Aron described what we now call quarter-comma meantone temperament. The core problem it solves is this: if you tune a keyboard with pure 5:4 major thirds, the fifths must be slightly narrowed (tempered) to make the arithmetic work out. In quarter-comma meantone, each fifth is narrowed by exactly one quarter of the syntonic comma (the small interval of ratio 81:80 that separates a Pythagorean major third from a pure 5:4 major third). The result is that four consecutive fifths produce a perfectly pure major third. The keyboard sounds beautiful in the “home” keys — those with few sharps or flats — with pure or nearly pure thirds and only slightly compromised fifths. But the temperament is uneven: as one moves into remote keys, the discrepancies accumulate, and one interval — the notorious wolf fifth, so called for its howling dissonance — becomes intolerably out of tune. Meantone temperament thus creates a system with distinct tonal regions: a comfortable home territory and inhospitable outskirts.
Aron’s significance lies not in any mathematical novelty but in bringing the practical question of keyboard temperamentinto the theoretical literature for the first time. The meantone framework he articulated would dominate keyboard tuning practice for the next two centuries.
Gioseffo Zarlino (1517–1590)
If Ramos opened the door to 5-limit just intonation and Aron showed its practical implications for keyboard temperament, Gioseffo Zarlino — choirmaster of St. Mark’s Basilica in Venice and the dominant music theorist of the sixteenth century — provided the comprehensive theoretical synthesis. His monumental Le istitutioni harmoniche (1558) is the most important theoretical treatise of the Renaissance, and its influence persisted well into the eighteenth century.
Zarlino’s central argument was the extension of the Pythagorean tetractys to what he called the senario (the first six integers). He argued that just as the integers 1–4 yielded all the Pythagorean consonances, the integers 1–6 yield all the consonances recognised by contemporary practice — including the pure major third (5:4), minor third (6:5), major sixth (5:3), and minor sixth (8:5). This was both a mathematical and a perceptual claim: the senario explained why these intervals sound consonant. Zarlino thus gave 5-limit just intonation its definitive theoretical rationale.
He also grappled extensively with the problem of temperament, acknowledging that his pure ratios cannot all be realised simultaneously on a fixed-pitch keyboard. He discussed various compromises, favoured meantone-type solutions, and addressed the irreconcilable tension between harmonic purity and transposability. His engagement with these questions, even when he did not resolve them, defined the agenda for theorists who followed.
Zarlino also developed the first fully articulated theory of mode and harmony that closely anticipates the modern concept of major and minor keys — distinguishing between harmonies built on major thirds (which he associated with cheerfulness) and those built on minor thirds (which he associated with sadness). This is one of the earliest theoretical articulations of what would become the tonal system.
Francisco de Salinas (1513–1590)
The blind Spanish organist and theorist Francisco de Salinas represents a fascinating convergence of Renaissance humanist scholarship and musical empiricism. His De musica libri septem (1577) is a vast, erudite work drawing on newly recovered Greek sources — including Ptolemy — and on extensive practical experience as an organist and musical educator.
Salinas is notable for several reasons. He was among the first Western theorists to take seriously the full range of Ptolemy’s tuning systems, and he discussed both 5-limit and 7-limit ratios (involving the prime number 7). He described meantone temperament with mathematical precision and discussed its limitations. Perhaps most strikingly, he provided detailed transcriptions of Spanish folk and popular music, making his treatise a rare document of contemporary vernacular practice embedded within a scholarly framework.
Salinas also engaged rigorously with the question of what exactly a musician does when singing a cappella without a keyboard instrument to constrain pitch: he argued that singers naturally gravitate toward the pure ratios of just intonation, adjusting flexibly depending on harmonic context. This anticipates later discussions about the relationship between just intonation and practical performance that remain alive today.
Michael Praetorius (1571–1621)
Michael Praetorius, German composer and encyclopedist, is primarily known as a composer of Lutheran church music, but his three-volume Syntagma Musicum (1614–1620) is an invaluable document of musical practice at the turn of the seventeenth century, including detailed and practical accounts of tuning and temperament.
In Syntagma Musicum II (De Organographia), Praetorius describes the tuning practices used for organs and other keyboard instruments in contemporary Germany, providing one of the most thorough practical accounts of meantone temperament and its variants. He distinguishes carefully between different types of meantone — noting that the exact size of the tempered fifth can vary while preserving the essential character of the system — and documents the challenges of organ building and tuning in an era when instruments in different churches or towns might be tuned at different pitch levels and to different temperament schemes.
Praetorius also wrote about the tension between organists, who were locked into whatever temperament the instrument used, and singers and string players, who could adjust pitch freely. This tension — between fixed and flexible pitch, between keyboard temperament and vocal/string just intonation — is a thread running through the entire subsequent history of the art. His encyclopaedic reach and his engagement with actual instrumental practice make him an essential bridge between Renaissance theory and Baroque practice.
IV. The Baroque Crisis: Circular Temperaments and the Birth of Equal Temperament
Andreas Werckmeister (1645–1706)
Andreas Werckmeister, a German organist and theorist, is perhaps the single most important figure in the development of well temperament — the family of tuning systems that would dominate keyboard practice from the late seventeenth century through much of the eighteenth, and that made possible the music of Johann Sebastian Bach.
The central problem Werckmeister addressed was the limitation of meantone temperament: its wolf fifth rendered certain keys unusable. The music of his era was increasingly exploring remote harmonies and modulations across the full circle of keys, and a system that worked beautifully in C major but produced howling dissonance in G-sharp major was simply inadequate. The solution Werckmeister proposed was a circular temperament: one that distributed the Pythagorean comma unequally among the twelve fifths, allowing all keys to be used while preserving the character of distinct key colours.
In his Musicalische Temperatur (1691), Werckmeister presented several such systems, the most famous being “Werckmeister III.” In this temperament, four of the twelve fifths are tempered narrow (by one quarter of the Pythagorean comma each), while the remaining eight are left pure. The result is a system where all keys are playable — there is no wolf fifth — but the keys are not equal: C major sounds purer and brighter than, say, F-sharp major, which has a more tense and characteristic colour. This inequality of keys was understood not as a defect but as a positive feature: different keys have different expressive characters (Affekte), and a well temperament exploits this systematically.
Werckmeister’s work is commonly invoked in discussions of Bach’s Well-Tempered Clavier (1722), the title of which — Das Wohltemperirte Clavier — refers explicitly to this tradition. Whether Bach used Werckmeister III, some other well temperament, or equal temperament is a matter of ongoing scholarly debate, but Werckmeister’s theoretical framework is unquestionably the context within which Bach’s pedagogical and artistic aims must be understood.
Beyond temperament, Werckmeister developed a theologically inflected theory of harmony in which the consonances of music were reflections of divine order — a continuation of the Boethian tradition, now adapted to Lutheran Pietism. For Werckmeister, properly tuned music was literally a form of prayer.
Johann Georg Neidhardt (1685–1739)
Neidhardt was a German theorist and the most systematic analyst of the well-temperament tradition after Werckmeister. His Sectio Canonis Harmonici (1724) and Gäntzlich erschöpfte, mathematische Abteilungen (1732) presented a comprehensive taxonomy of temperaments organised by their suitability for different contexts — village churches, small towns, large cities, courts. He proposed a series of temperament schemes of increasing refinement, from those suited to provincial use (where few remote keys would be needed) to near-equal temperaments for sophisticated metropolitan music-making.
Neidhardt’s most significant contribution was his systematic mathematical analysis of the trade-offs involved in different distributions of the comma. He demonstrated rigorously that equal temperament — distributing the Pythagorean commaequally among all twelve fifths — is the logically limiting case of well temperament, and he was among the first theorists to describe and endorse it with mathematical precision. His work represents the intellectual moment when equal temperament ceased to be a vague ideal and became a precisely calculable system.
Jean-Philippe Rameau (1683–1764)
Rameau was the dominant figure in French music of the first half of the eighteenth century — a composer of operas and harpsichord music, and a theorist of revolutionary importance. His Traité de l’harmonie (1722) is the founding document of modern harmonic theory, and his subsequent writings developed a comprehensive account of tonal music as a system grounded in natural acoustics.
Rameau’s tuning contributions are primarily associated with his theory of the corps sonore — the “sounding body” or resonating string — and its partials. When a string vibrates, it produces not a single pitch but a series of overtones at frequencies that are integer multiples of the fundamental: 1f, 2f, 3f, 4f, 5f, 6f, and so on. This harmonic series — which includes a perfect fifth (3:2), a major third (5:4), and a minor seventh (7:4, approximately) in its lower reaches — is, Rameau argued, the natural basis of harmony. The major triad, for instance, corresponds to the 4th, 5th, and 6th partials of the harmonic series, and its consonance is thus explained by natural law rather than mathematical convention.
This argument had important implications for tuning: it grounded the preference for 5-limit ratios in physical acoustics rather than merely Pythagorean mathematics. It also raised the question of the 7th partial — whether the ratio 7:4 should be admitted as a consonance — which Rameau considered but ultimately set aside. His theoretical framework was enormously influential in France and across Europe, providing the harmonic vocabulary still used in music pedagogy today.
V. The Enlightenment Systematisers
Johann Philipp Kirnberger (1721–1783)
A student of Bach himself, Kirnberger was a conservative German theorist who spent much of his career arguing for the superiority of just intonation and theoretically pure tuning against what he saw as the corrupting influence of equal temperament. His Die Kunst des reinen Satzes in der Musik (“The Art of Pure Composition,” 1774–79) contains extensive discussion of tuning, in which he argued that pure intervals — particularly the 5:4 major third and 6:5 minor third — are not merely theoretically ideal but are the basis of actual compositional beauty.
Kirnberger proposed a specific temperament (“Kirnberger II” and “Kirnberger III”) that preserved pure fifths and pure thirds wherever possible, reserving the discrepancies for less frequently used intervals. His temperaments are notable for being among the most asymmetric — highly unequal — of the well-temperament tradition, and they were designed explicitly to honour the sound of Bach’s music as Kirnberger remembered hearing his teacher perform it.
He engaged in pointed polemics against Marpurg (see below) over the merits of equal temperament, representing a genuine intellectual divide about whether mathematical regularity or acoustic purity should take precedence.
Francesco Antonio Vallotti (1697–1780)
The Italian Franciscan friar and Paduan organist Vallotti developed a well temperament that bears his name and that has attracted renewed attention in the early music revival of the late twentieth and early twenty-first centuries. Described in his Della scienza teorica e pratica della moderna musica (written c. 1728–79, published posthumously), the Vallotti temperament narrows six consecutive fifths (from F to B) each by one-sixth of the Pythagorean comma, leaving the remaining six fifths pure. It is a gentle, elegant system that produces sweetly coloured keys without extreme contrast and that adapts well to the Italian keyboard repertoire of the seventeenth and eighteenth centuries.
Thomas Young (1773–1829)
The English polymath Thomas Young — better known for his wave theory of light and the double-slit experiment — also made contributions to music theory. His 1800 paper “Outlines of Experiments and Inquiries Respecting Sound and Light” proposed a specific well temperament (“Young I” and “Young II”) based on physical reasoning about the nature of beatingbetween mistuned intervals.
Young’s approach was empirically grounded: rather than distributing commas according to abstract mathematical schemes, he sought to distribute them in a way that minimised the dissonance of beating across all keys. His analysis of beats — the periodic amplitude fluctuations produced when two slightly mistuned pitches are sounded simultaneously — as the physical mechanism of dissonance was scientifically ahead of its time and anticipated the psychoacoustic research of Hermann von Helmholtz later in the century.
Jean le Rond d’Alembert (1717–1783)
The French philosopher and mathematician d’Alembert is best known as co-editor of the great Encyclopédie. His musical-theoretical contribution was largely as an expositor and populariser of Rameau’s work: his Élémens de musique théorique et pratique (1752) presented Rameau’s harmonic theory in a clearer, more accessible form than Rameau himself had managed, and helped spread the French harmonic system across Europe.
D’Alembert’s own contributions to tuning theory were modest but not trivial. He was among the first to treat the question of temperament with the precision of the mathematical physicist, calculating the sizes of tempered intervals in logarithmic units and discussing the conditions under which equal temperament was the uniquely rational solution to the problem of keyboard tuning. His mathematical clarity helped establish equal temperament not merely as a practical compromise but as the logically inevitable endpoint of the temperament tradition.
Friedrich Wilhelm Marpurg (1718–1795)
Marpurg was a prolific German theorist and champion of equal temperament. His polemical exchange with Kirnberger over the merits of just intonation versus equal temperament was one of the great theoretical controversies of the eighteenth century. Marpurg argued forcefully that equal temperament — in which all twelve semitones are precisely equal and all fifths are tempered by exactly the same tiny amount — was the only truly rational system, because it alone treats all keys with complete symmetry and admits free modulation without restriction.
His Versuch über die musikalische Temperatur (1776) laid out the mathematical case for equal temperament with great rigour and attacked the supposedly “pure” systems of Kirnberger and others as arbitrary and aesthetically inconsistent. The controversy was eventually resolved — or rather, dissolved — by the gradual universal adoption of equal temperament in the nineteenth century. But the Marpurg–Kirnberger debate remains a fascinating window into the period when the question of tuning still felt genuinely open.
VI. The Early Twentieth Century: Microtonality and the Expansion of the Pitch Continuum
By the late nineteenth century, equal temperament had largely conquered Western keyboard and orchestral practice. But this very consolidation provoked a reaction. The chromatic saturation of Wagnerian harmony, the non-Western influences entering European music through colonial expansion and world expositions, and the general spirit of modernist experimentation all pointed beyond the twelve tones of equal temperament toward richer and more complex tonal resources.
Ferruccio Busoni (1866–1924)
The Italian pianist and composer Ferruccio Busoni was one of the first major figures of Western art music to theorise the inadequacy of twelve-tone equal temperament and to call explicitly for its expansion. His Sketch of a New Esthetic of Music (1907) diagnosed the twelve-note chromatic scale as an artificial limitation — a “cage” that constrained musical expression — and proposed that composers explore divisions of the octave into third-tones (36 equal divisions) or even sixth-tones (72 equal divisions).
Busoni was primarily a visionary and an essayist rather than a practising microtonal composer — he did not actually build the instruments needed to realise his proposals — but his influence on subsequent generations was enormous. He posed the right question with unusual clarity and urgency: if equal temperament is a convention rather than a law of nature, what convention should replace or augment it?
Ivan Wyschnegradsky (1893–1979)
The Russian-French composer Ivan Wyschnegradsky devoted his entire adult life to the exploration of quarter-tone music — the division of the octave into 24 equal steps rather than 12. He settled in Paris after the Russian Revolution and spent decades composing for specially constructed or paired quarter-tone pianos, as well as for quarter-tone ensembles of other instruments.
Wyschnegradsky’s theory was not merely about adding extra notes but about a fundamentally different relationship with musical space. His concept of pansonority envisioned a continuous tonal world in which all divisions of the pitch continuum — quarter-tones, third-tones, arbitrary microtonal gradations — were part of a single unified musical space. His treatise Manuel d’harmonie à quarts de ton (1932, published 1980) provided a complete harmonic theory for quarter-tone music. Though his music remained outside the mainstream, it inspired later microtonal composers and theorists, and his systematic approach to non-standard divisions of the octave was genuinely pioneering.
Julián Carrillo (1875–1965)
The Mexican composer and theorist Julián Carrillo developed what he called Sonido 13 — “The Thirteenth Sound” — a system and philosophy of microtonality that went beyond quarter-tones to explore sixteenth-tones, thirty-second-tones, and even finer divisions. He coined the term Sonido 13 (the “thirteenth sound,” beyond the twelve of the chromatic scale) as a slogan for this expansion of the tonal universe.
Carrillo was unusual among early microtonal theorists in that he actually constructed instruments to realise his ideas: specially fretted guitars, pianos with extra strings, and other devices. He composed a substantial body of microtonal music and attracted significant attention — including from Leopold Stokowski, who conducted his work — but also considerable resistance from the Mexican musical establishment. His systematic cataloguing of microtonal intervals and his practical approach to instrument construction make him an important figure in the pre-electronic history of microtonality.
VII. The Mid-Twentieth Century: Just Intonation Renaissance and Alternative Tuning Systems
Harry Partch (1901–1974)
Harry Partch is perhaps the most uncompromising and radical figure in the history of Western tuning. An American composer who rejected equal temperament entirely as an impoverished, historically contingent system, Partch spent his life developing an elaborate system of 43-note-per-octave just intonation and building the extraordinary instruments needed to perform it.
Partch’s tuning system was grounded in the extended just intonation tradition: he worked with ratios not merely in the 5-limit (as Zarlino had) or even the 7-limit (as de Salinas had approached) but in the 11-limit — incorporating prime numbers up to 11. His Monophony (later called Genesis of a Music, 1949; revised 1974) is both a comprehensive theoretical treatise and a polemical manifesto. Partch argued that Western music had taken a wrong turn with equal temperament, severing the connection between music and the natural harmonic series and producing a sterile, expressively limited art form.
His 43-tone scale divides the octave into 43 unequal steps, each of which is a specific ratio from the harmonic series. His hand-built instruments — the Adapted Viola, the Chromelodeon, the Spoils of War, the Cloud Chamber Bowls, the Quadrangularis Reversum, and many others — are simultaneously sonic inventions and sculptural objects, and their visual and textural presence is inseparable from his music. Partch influenced numerous later composers and tuning theorists, and his career represents the most thorough-going attempt in the twentieth century to build a complete alternative musical culture based on just intonation.
Adriaan Fokker (1887–1972)
Adriaan Fokker was a Dutch physicist (a collaborator of Lorentz and Einstein) who, in the second half of his life, turned his scientific rigour to the study of musical tuning. His most lasting contribution was the systematic exploration of 31-tone equal temperament (31-EDO: Equal Division of the Octave into 31 steps).
31-EDO had been anticipated by the seventeenth-century theorist Christiaan Huygens, who noticed that 31 equal divisions of the octave provide an exceptionally good approximation to the pure 5:4 major third (within about 1 cent), the pure 6:5 minor third, and other 5-limit ratios. Fokker demonstrated that 31-EDO is essentially a completed and regularised form of quarter-comma meantone temperament — the same system Aron and Zarlino had used, now extended to all 31 notes of the complete chromatic scale without a wolf fifth. The 31-note octave includes not only all the notes of Western chromaticism but also additional enharmonic distinctions — D-sharp and E-flat are different pitches — that just intonation requires.
Fokker built a 31-tone organ and composed music for it, founded the Huygens-Fokker Foundation to promote research in 31-EDO and related systems, and published extensively on the theory of periodicity blocks — a concept that describes the regions of the harmonic lattice that a given equal temperament most efficiently represents. His work on periodicity blockswas a major anticipation of later developments in regular temperament theory.
VIII. The Late Twentieth Century: Computers, New Scales, and the Systematic Study of Tuning
Wendy Carlos (b. 1939)
Wendy Carlos is best known to the general public as the artist behind Switched-On Bach (1968) and the soundtracks to A Clockwork Orange and The Shining. But her contributions to tuning theory are both original and underappreciated. With the synthesiser as her instrument — a technology that, for the first time, made arbitrary pitch ratios easily realisable — Carlos explored tuning systems that had no precedent in the acoustic instrument tradition.
Her most significant tuning contribution, published in Computer Music Journal in 1987, was the development of three non-octave-based scales: Alpha (α), Beta (β), and Gamma (γ). These scales abandon the 2:1 octave — the most fundamental structural principle of virtually all Western tuning — as the basic repeating interval. Alpha repeats at roughly 15.39 semitones, Beta at 18.75 semitones, and Gamma at 34.29 semitones. All three are designed to optimise approximations of the pure fifth (3:2) and major third (5:4) at the cost of the pure octave. Carlos argued that the octave’s dominance in Western music is a cultural convention rather than an acoustic necessity, and that scales built without it can produce harmonically rich textures that are subtly but distinctly different from anything achievable in octave-repeating systems.
Carlos also wrote and spoke extensively about the history and acoustics of temperament, bringing these ideas to a broad audience through her recordings and writings.
Heinz Bohlen and John Robinson Pierce (Bohlen–Pierce Scale, 1970s–80s)
The Bohlen–Pierce scale is one of the most rigorously designed alternative tuning systems of the twentieth century, developed independently by the German acoustical engineer Heinz Bohlen and, later, by the information theorist John Robinson Pierce. The scale divides not the octave (2:1) but the tritave (3:1 — the interval of an octave plus a fifth) into 13 equal steps.
The motivation is acoustic: if one eliminates the even-numbered harmonics from a sound (as certain instruments using odd-numbered partials do, such as a clarinet to some extent), the interval of repetition is the tritave rather than the octave. The Bohlen–Pierce scale is designed to approximate the just intervals that arise from this odd-harmonic series — particularly the ratios 5:3, 7:5, 7:3, and 9:7 — with the same kind of accuracy that good equal temperaments approximate 5-limit just intonation. The scale has a strange, alien beauty to ears conditioned by octave-based tuning, and it has attracted a small but dedicated community of composers and instrument builders.
The Bohlen–Pierce scale is significant theoretically because it demonstrates that the octave is not the only possible organisational principle for a musical system, and that the logic of equal temperament — approximating just ratios with a regular equal division — can be applied to bases other than 2.
Erv Wilson (1928–2016)
Ervin Wilson is perhaps the most original and least publicly known figure in the history of tuning theory. A largely self-taught American-Mexican theorist working in Los Angeles, Wilson spent five decades developing an extraordinary body of theoretical work — in the form of dense, hand-drawn diagrams sent to a small network of correspondents — that anticipates and exceeds much of what has since been developed in academic music theory and mathematics.
Wilson’s contributions are numerous and difficult to summarise briefly. His Combination Product Sets (CPS) — including the hexany, eikosany, and dekatesserany — are elegant and highly symmetric structures within the just intonation lattice: subsets of the harmonic series that minimise the number of pitches while maximising the number of consonant intervals. The hexany, for instance, is a six-note set built from all products of pairs of four factors, and it has the remarkable property that every one of its fifteen intervals appears in consonant just-intonation relationships.
Wilson developed Moments of Symmetry (MOS) scales — scales generated by repeatedly stacking a single interval that has exactly two step sizes, producing a structure with deep mathematical properties related to the Farey sequence and continued fractions. This framework encompasses the diatonic scale (generated by fifths, with two step sizes) and many exotic alternatives. He explored horagram diagrams, mapping the entire landscape of generator-and-period scale spaces. He developed scale trees that organise all possible scales by their generator ratios.
Wilson corresponded with Partch, Carlos, and many others, but never published in the conventional sense. His work circulated in photocopied manuscripts and is now archived and studied at the Xenharmonic Wiki and Anaphoria.com. The theoretical framework developed by the early twenty-first century tuning community is inconceivable without Wilson’s foundational contributions.
IX. The Twenty-First Century: Regular Temperament Theory and Online Communities
The final chapter in this history is the most technically demanding and, in some ways, the most intellectually exciting: the development, by a loosely connected online community of theorists in the late 1990s and early 2000s, of what has come to be called regular temperament theory (RTT). This framework provides a unified mathematical language for describing, classifying, and comparing all possible tuning systems — just intonation, equal temperaments, and everything in between — in terms of the mathematical structure of the relationships they approximate.
Paul Erlich (b. 1972)
Paul Erlich is an American music theorist and guitarist whose work has been central to the development of regular temperament theory. His 1998 paper “Tuning, Tonality, and Twenty-Two Tone Equal Temperament” (published in the Journal of Music Theory) introduced the concept of periodicity blocks (building on Fokker’s earlier work) to a new audience and made a sophisticated case for the musical coherence of 22-tone equal temperament (22-EDO). In 22-EDO, the chromatic scale has 22 equal steps; it approximates 5-limit just intonation differently from 12-EDO, with notably pure major thirds and a characteristic harmonic colour.
Erlich subsequently developed the concept of harmonic entropy — a measure of the perceived dissonance or “rootedness” of a dyad (two-note combination) based on the density of rational approximations near a given frequency ratio. Harmonic entropy formalises the intuition that simple ratios are more consonant than complex ones, but in a continuously graded, psychoacoustically motivated way rather than through arbitrary cutoffs. It has become one of the standard tools of the community for discussing consonance and dissonance quantitatively.
Erlich’s 2006 paper (circulated informally as “A Middle Path Between Just Intonation and the Equal Temperaments”) introduced the Tenney-Euclidean (TE) tuning optimisation method and systematised the framework of regular temperaments — defining a temperament by which commas (small intervals) it “tempers out” (i.e., treats as zero), and deriving the optimal tuning from this algebraic specification.
Gene Ward Smith (1948–2021)
Gene Ward Smith is a mathematician (his academic work is in algebraic number theory and Galois theory) who became a central theorist of the online tuning community in the early 2000s. His contributions are primarily in the mathematical formalisation of regular temperament theory. He developed the algebraic framework in which temperaments are described as homomorphisms from the free abelian group of just intonation (the lattice of pitch ratios) to simpler groups, providing a rigorous algebraic language for the intuitions of the community.
Smith gave names to hundreds of temperaments — “meantone,” “miracle,” “magic,” “pajara,” “orwell,” “porcupine,” and many more — each defined by the specific commas it tempers out, and showed how these can be systematically enumerated and their properties (complexity, accuracy, efficiency) compared. His online contributions — on the Yahoo! tuning lists and subsequently at the Xenharmonic Wiki — form an enormous body of theory that is still being absorbed and extended.
Graham Breed (fl. 2000s)
Graham Breed, a British programmer and theorist, developed some of the first practical computational tools for regular temperament theory — most notably the “Graham complexity” measure for temperaments and an online temperament finder that, given a set of just-intonation intervals one wants to approximate well, automatically finds the equal temperament (or rank-2 temperament) that best satisfies those constraints. His website and software tools made the theoretical work of Smith, Erlich, and others accessible to a much wider community of composers and instrument builders. Breed’s practical orientation — bridging the gap between abstract algebra and usable musical tools — was essential to the community’s development.
Dave Keenan (fl. 2000s)
Dave Keenan is an Australian engineer and theorist whose contributions to the tuning community are in notation, interval naming, and the theory of sagittal notation — a comprehensive, visually systematic notation system for microtonal music that he developed jointly with George Secor. Sagittal notation uses a consistent set of arrow-like accidentals that indicate the direction and size of microtonal deviations from standard staff notation, and it is designed to work for any tuning system, from 5-limit just intonation to arbitrary equal temperaments. It remains the most comprehensive attempt to solve the notation problem for microtonal music.
Keenan also contributed extensively to the theory of temperament families and to the naming conventions used in the community. His combination of engineering precision and musical sensitivity, and his willingness to engage in extended collaborative dialogue to refine ideas, made him an important intellectual force in the development of the field.
Anders Thidell (1966–2022)
The Swedish luthier and guitarist Anders Thidell developed True Temperament — a system of fret placement for guitars and other fretted string instruments that abandons the straight, equally-spaced frets of conventional guitar lutherie in favour of curved, individually positioned frets designed to produce acoustically accurate intonation for each note on each string. Standard guitar frets are a physical embodiment of 12-tone equal temperament, and like all equal temperaments they involve trade-offs: no interval except the octave is acoustically pure. True Temperament frets — which appear as sinuous curves across the fingerboard — are positioned to realise a chosen well temperament or other tuning system with a precision impossible on a standard instrument.
Thidell founded True Temperament AB in Sweden and began producing commercial guitars and replacement necks, attracting professional players who found that the improved intonation and the distinct tonal character of well-tempered key colours enriched their playing. The True Temperament system has been adopted by a growing number of guitarists, and it represents a rare case of historical temperament theory finding a practical home in contemporary popular music instrument design.
Hidekazu Wakabayashi (fl. 2010s)
Hidekazu Wakabayashi is a Japanese pianist, composer, and microtonal theorist best known for developing Iceface tuning — a distinctive 12-note subset of 24-EDO that has attracted a devoted following in the xenharmonic community.
The idea for Iceface came to Wakabayashi in a dream in which he saw a woman playing a piano presumably tuned in the same way Iceface is tuned. The tuning itself is elegantly simple: Wakabayashi tuned a piano and harp so that the normal sharps and flats are raised 50 cents — a quarter-tone higher — which he called Iceface tuning. The result keeps all the natural (white) keys exactly as in 12-EDO while raising every black key by a quarter-tone, producing a 12-note scale with a suite of neutral intervals — seconds, thirds, sixths, and sevenths — sitting exactly halfway between the standard major and minor versions. Xenharmonic WikiXenharmonic Wiki
The term “iced” is used to describe the quality of these neutral intervals. Unlike “neutral,” “iced” sounds less ambiguous and gives the impression of a distinct quality, rather than simply placing the interval between major and minor — because iced intervals sound unique, neither major nor minor. The scale can be thought of as Mohajira[7] superimposed over a diatonic major scale
Kite Giedraitis (b. 1950s-60s)
The American guitarist and theorist Kite Giedraitis has been one of the most inventive advocates for 41-tone equal temperament (41-EDO) as a practical, performer-friendly vehicle for extended just intonation. 41-EDO is a remarkable system: it provides excellent approximations to the 7-limit and 11-limit harmonic series — including pure-sounding major thirds, minor thirds, dominant sevenths, and eleventh-partial intervals — while remaining an equal temperament that supports free transposition and familiar scale structures. It is, in a sense, a higher-resolution version of 12-EDO that unlocks harmonic resources unavailable in standard tuning.
Giedraitis developed the concept of skip fretting — a guitar fretboard design that uses only a selected subset of the 41 equal divisions per octave, laid out across the strings in a pattern that makes familiar chord shapes and scale fingerings work intuitively. His Kite Guitar (named after his own nickname) places 41-EDO frets in an arrangement that preserves much of the logic of standard guitar technique while opening up a richer harmonic palette. He also developed a systematic notation and naming system for 41-EDO intervals, and has composed and performed extensively in the tuning, providing a body of music that demonstrates its expressive potential. His work is an exemplary case of making a theoretically demanding tuning system genuinely playable and learnable.
John O’Sullivan (b. 1970)
John O’Sullivan is an Irish microtonal theorist and composer whose primary contribution is the development of the Eagle 53 system — a comprehensive framework for music in a 12-tone subset of the 53-tone equal temperament (53-EDO), one of the most historically significant and acoustically rich of all equal temperaments. 53-EDO has been admired for centuries — it was noted by Mersenne and Leibniz, and its virtues as a vehicle for 5-limit just intonation were analysed in detail by the nineteenth-century theorist R. H. M. Bosanquet — because its 53 equal steps provide extraordinarily close approximations to just fifths, thirds, and many 7-limit and 11-limit intervals simultaneously.
O’Sullivan’s Eagle 53 system goes beyond simply advocating for 53-EDO as a good approximation to just intonation. He developed a complete musical framework: notation, harmony theory, characteristic chord progressions, scale structures, and compositional practice tailored specifically to 53-EDO’s properties. The “Eagle” name evokes the system’s aspiration to soar above the limitations of 12-EDO, and the framework is designed to be a self-contained musical language rather than merely a theoretical curiosity. O’Sullivan’s work exemplifies the twenty-first-century tuning community’s characteristic combination of mathematical rigour, compositional practicality, and evangelising zeal — the conviction that a richer tonal world is not only theoretically possible but artistically necessary.
Zhea Erose (b. 1990s)
Zhea Erose is a composer, theorist, and educator whose work has developed two of the most distinctive and systematically elaborated frameworks in the contemporary microtonality community: primodality and NEJI (Near-Equal Just Intonation).
Primodality is Erose’s theory of harmony and modality grounded in the harmonic series of individual prime numbers. Rather than deriving scales from a combination of primes (as standard just intonation lattice theory does), primodality builds distinct harmonic worlds from the overtone series of each prime taken in isolation: the 2-series, the 3-series, the 5-series, the 7-series, the 11-series, and so on. Each prime generates its own characteristic collection of intervals and its own modal logic, and composition in primodal style involves moving between these distinct prime-flavoured harmonic spaces. The system makes explicit something that earlier theorists like Partch and Fokker approached more implicitly: that the prime factorisation of a ratio is not merely a mathematical property but a perceptual and compositional one, determining the qualitative character of an interval in ways that can be systematically explored.
NEJI (Near-Equal Just Intonation) addresses one of the central practical challenges of just intonation: how to design equal temperaments that closely approximate a target just intonation scale, rather than approximating all just intervals uniformly. Where conventional regular temperament theory typically seeks equal temperaments that best represent a given prime limit in the abstract, NEJI starts from a specific equal temperament scale and finds the just intervals that most faithfully realise that particular collection. This allows composers who think in just intonation terms to access the practical benefits of equal temperament — consistent intonation across transpositions, compatibility with fixed-pitch instruments, simpler notation — without sacrificing the harmonic identity of their source material. Erose has applied NEJI extensively in her own compositions and has taught the framework to a growing audience through her educational writing and online presence. Her work represents a generation for whom microtonality is not an avant-garde provocation but an everyday compositional language.
Recurring Themes
Across two and a half millennia, a few deep tensions recur in Western tuning theory.
Purity versus practicality.Just intonation offers the purest intervals but demands either a fixed tonal center or an impractically large number of pitches. Equal temperament is practical but sacrifices some consonance. Every intermediate position — meantone, well temperament, regular temperament — is a different negotiation of this tradeoff.
The ear versus mathematics. From Aristoxenus arguing against Pythagorean number-mysticism to Rameau grounding harmony in the harmonic series to Erlich formalizing harmonic entropy, the question of whether tuning is a mathematical fact or a perceptual phenomenon has never been fully resolved.
The octave and its alternatives. For most of this history, the 2:1 octave has been taken as an unquestionable invariant — the interval of equivalence around which everything else is organized. Carlos, Bohlen and Pierce, and the later Xenharmonic community showed that this assumption, too, is a choice.
Access and instruments. Almost every advance in tuning theory has been accompanied by an instrument-making problem. Meantone needed the wolf fifth worked around; well temperament needed retuning protocols; Partch needed to build everything from scratch; microtonal guitarists need refretted instruments. The history of tuning is inseparable from the history of instrument technology.
The conversation is ongoing. New equal temperaments are explored, new just intonation structures are catalogued, new instruments are built or programmed, and the ancient question — what are the right pitches? — continues to find new answers.
Summary: Key figures and their contributions
Pythagoras (500s BCE): Identified musical consonance with simple integer ratios (2:1 octave, 3:2 fifth, 4:3 fourth). Stacking pure fifths produces a scale with harsh major thirds, and twelve fifths never quite close into a circle — leaving a gap called the Pythagorean comma.
Aristoxenus (300s BCE): Rejected ratio-based theory in favour of perception. Argued the ear is the proper judge of intervals and that tones occupy continuous, equally divisible space — anticipating equal temperament by two millennia.
Ptolemy (100s CE): Synthesised both traditions. Introduced ratios involving the number 5 (pure 5:4 major third, 6:5 minor third), laying the foundation for what later became 5-limit just intonation.
Boethius (400s–500s CE): Transmitted Greek tuning theory to medieval Europe. His Pythagorean framework dominated Western music theory for nearly a thousand years.
Ramos de Pareja (late 1400s): First European theorist to publicly argue for 5-limit tuning over Pythagorean ratios, reflecting the shift toward triadic harmony in Renaissance polyphony.
Aron (early 1500s): Described quarter-comma meantone temperament — the first practical keyboard tuning system designed to produce pure major thirds, at the cost of a small narrowing of each fifth and one unusable “wolf” interval.
Zarlino (mid 1500s): Provided the definitive theoretical justification for 5-limit just intonation via the senario (first six integers). Also articulated the earliest clear theory of major and minor as distinct harmonic qualities.
de Salinas (late 1500s): Engaged with 5- and 7-limit ratios, described meantone rigorously, and argued that unaccompanied singers naturally gravitate toward just intonation.
Praetorius (early 1600s): Encyclopaedic documenter of contemporary tuning practice, including meantone variants and the tension between fixed-pitch keyboard tuning and the flexible intonation of voices and strings.
Werckmeister (late 1600s): Developed well temperament — distributing the Pythagorean comma unevenly so all keys are usable while retaining distinct harmonic characters. The theoretical context for Bach’s Well-Tempered Clavier.
Neidhardt (early 1700s): Systematically classified well temperaments by context and demonstrated mathematically that equal temperament is their logical limiting case.
Rameau (early 1700s): Grounded harmony in the overtone series (harmonic series), explaining consonance through natural acoustics and providing the theoretical basis for modern tonal harmony.
Kirnberger (late 1700s): Conservative advocate for just intonation and highly unequal well temperaments; argued that acoustic purity was compositionally essential.
Vallotti (late 1700s): Developed a gentle, elegant well temperament (six tempered fifths, six pure) well suited to Italian keyboard music.
Young (late 1700s): Physicist who designed well temperaments based on minimising the beating of mistuned intervals — an early psychoacoustic approach to tuning.
d’Alembert (late 1700s): Popularised Rameau’s harmonic theory and helped establish equal temperament as the mathematically inevitable endpoint of temperament development.
Marpurg (late 1700s): Vigorous champion of equal temperament, arguing it was the only rationally consistent system; engaged in famous polemics with Kirnberger.
Busoni (early 1900s): Called for moving beyond 12-tone equal temperament toward third-tones and sixth-tones; more visionary than practising microtonal composer, but highly influential.
Wyschnegradsky (early 1900s): Devoted his career to quarter-tone music, developing a complete harmonic theory for 24-EDO and composing extensively for specially built instruments.
Carrillo (early 1900s): Explored divisions as fine as sixteenth- and thirty-second-tones; built instruments to realise them and composed a substantial microtonal body of work.
Partch (mid 1900s): Rejected equal temperament entirely and built a 43-note just intonation system in the 11-limit, alongside the hand-built instruments needed to perform it.
Fokker (mid 1900s): Systematically developed 31-EDO as a complete realisation of meantone temperament; introduced the concept of periodicity blocks for analysing how equal temperaments approximate just intonation.
Wendy Carlos (late 1900s): Designed non-octave-repeating scales (Alpha, Beta, Gamma) optimised for pure fifths and thirds; brought tuning theory to a wide audience through recordings and writing.
Bohlen & Pierce (late 1900s): Designed a scale dividing the tritave (3:1) rather than the octave into 13 equal steps, optimised for odd-harmonic timbres — demonstrating that the octave need not be the structural basis of a musical system.
Erv Wilson (late 1900s): Developed Combination Product Sets, Moments of Symmetry scales, horagrams, and scale trees — a vast body of largely unpublished theory that became foundational for the 21st-century tuning community.
Paul Erlich (early 21st century): Central figure in regular temperament theory; developed harmonic entropy as a psychoacoustic measure of consonance and formalised the Tenney-Euclidean optimisation framework for temperaments.
Gene Ward Smith (early 21st century): Provided the algebraic formalisation of regular temperament theory and named hundreds of temperament systems defined by the commas they temper out.
Graham Breed (early 21st century): Built the first practical computational tools for regular temperament theory, making the field accessible to composers and instrument builders.
Dave Keenan (early 21st century): Co-developed Sagittal notation — the most comprehensive notation system for microtonal music, designed to work across any tuning system.
Anders Thidell (early 21st century): Developed True Temperament — curved, individually positioned guitar frets that realise well temperament or other tunings with acoustic accuracy on a commercial instrument.
Kite Giedraitis (early 21st century): Advocated for 41-EDO as a practical extended-JI system; developed skip fretting and the Kite Guitar to make 41-EDO playable with familiar technique.
John O’Sullivan (early 21st century): Developed Eagle 53 — a complete compositional framework (notation, harmony theory, scale structures) built around the historically rich 53-EDO.
Zhea Erose (early 21st century): Developed primodality (harmony derived from the overtone series of individual primes) and NEJI (Near-Equal Just Intonation — equal temperaments optimised to approximate a specific target JI scale rather than a prime limit in general).
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