Welcome, gentlemen, to what must surely be the most temporally improbable gathering in the history of music theory — we have assembled here the finest minds ever to wrestle with the question of how to divide the octave.
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1. Introductions
INTERVIEWER: Gentlemen, please introduce yourselves — in one sentence each — covering when and where you lived, what you contributed to tuning, and your fundamental outlook on the subject.
PYTHAGORAS (c. 570–495 BCE, Samos/Croton): I am Pythagoras of Samos, and I demonstrated in the sixth century BCE that musical intervals arise from simple whole-number ratios — the octave from 2:1, the fifth from 3:2, the fourth from 4:3 — and I hold that tuning is nothing less than the mathematics of the cosmos made audible, the music of the spheres rendered into string lengths.
ARISTOXENUS (c. 354–300 BCE, Tarentum): I am Aristoxenus of Tarentum, pupil of Aristotle and, yes, of the Pythagoreans, and I say that my predecessors have it precisely backwards — tuning is not a matter of arithmetic ratios but of trained musical perception, for the ear, not the number, is the final judge of all intervals.
PTOLEMY (c. 100–170 CE, Alexandria): I am Claudius Ptolemy of Alexandria, and in my Harmonics I sought to reconcile the rational approach of the Pythagoreans with the perceptual approach of Aristoxenus, cataloguing multiple tetrachord divisions and insisting that reason and sensation must both be satisfied.
BOETHIUS (c. 477–524 CE, Rome): I am Anicius Manlius Boethius, Roman senator and philosopher, and through my De institutione musica I transmitted the Greek theoretical tradition to the Latin West, distinguishing musica mundana, musica humana, and musica instrumentalis, holding that true music is the contemplation of numerical ratios, not mere instrument-playing.
DE PAREJA (c. 1440–1522, Baeza/Bologna): I am Bartolomé Ramos de Pareja, and in my Musica Practica of 1482 I proposed a just intonation division of the monochord using ratios from the 5-limit, including the pure major third of 5:4, breaking from centuries of Pythagorean orthodoxy and scandalising my colleagues considerably.
ARON (c. 1480–1545, Florence): I am Pietro Aron, theorist of Florence, and in my Toscanello in musica of 1523 I described for the first time in print a practical meantone temperament — specifically quarter-comma meantone — in which the major thirds are rendered pure and the fifths tempered slightly, a workable solution for the polyphonic music of my age.
ZARLINO (c. 1517–1590, Venice): I am Gioseffo Zarlino, maestro di cappella at St. Mark’s Basilica in Venice, and I grounded the theory of counterpoint in the senario — the first six integers — arguing that just intonation with pure 5-limit consonances is the natural foundation of all harmony, though I also acknowledged practical keyboard compromises.
DE SALINAS (c. 1513–1590, Salamanca): I am Francisco de Salinas, blind organist and professor at Salamanca, and in my De musica libri septem of 1577 I provided the most systematic treatment of meantone tunings of my era, discussing not only quarter-comma but third-comma and other varieties, insisting that temperament must serve the ear and the singer above all.
PRAETORIUS (c. 1571–1621, Creuzburg/Wolfenbüttel): I am Michael Praetorius, German encyclopaedist of music, and in my Syntagma Musicum I documented the practical tuning systems of my era with extraordinary detail — meantone, various irregular temperaments, organ building — concerning myself above all with what the working musician and instrument builder actually needs.
WERCKMEISTER (c. 1645–1706, Halberstadt): I am Andreas Werckmeister, organist and theorist of central Germany, and I devised a series of well temperaments — notably Werckmeister III — in which all twenty-four major and minor keys are usable yet retain distinct characters, arguing that such musica poetica is a reflection of divine order.
NEIDHARDT (c. 1685–1739, Königsberg): I am Johann Georg Neidhardt, and I produced the most exhaustive taxonomy of well temperaments before the modern era, classifying them by social setting — for the village, the town, the city, and the court — and edging ever closer, in my later work, toward 12-tone equal temperament.
RAMEAU (c. 1683–1764, Dijon/Paris): I am Jean-Philippe Rameau, composer and theorist, and in my Traité de l’harmonie of 1722 I founded modern tonal harmonic theory upon the concept of the fundamental bass, arguing that harmonic nature is grounded in the overtone series and that temperament is a necessary, if imperfect, accommodation to the keyboard.
KIRNBERGER (c. 1721–1783, Berlin): I am Johann Philipp Kirnberger, pupil of Johann Sebastian Bach, and I advocated passionately for a well temperament close to just intonation — particularly what we now call Kirnberger III — insisting that the rich variety of key characters is artistically essential and that equal temperament is a musical impoverishment.
VALLOTTI (c. 1697–1780, Padua): I am Francesco Antonio Vallotti, Franciscan friar and maestro di cappella in Padua, and I developed a well temperament using six pure fifths and six tempered fifths that provides smooth transitions among keys while preserving more harmonic warmth than equal temperament.
YOUNG (c. 1773–1829, Milverton): I am Thomas Young, English polymath — physicist, physician, Egyptologist — and also the proposer of a well temperament in 1800 that bears my name, a practical and elegant compromise that I believe balances the purity of common keys against the acceptability of remote ones.
D’ALEMBERT (c. 1717–1783, Paris): I am Jean le Rond d’Alembert, mathematician and encyclopédiste, and while my primary work was in mechanics and mathematics, I engaged seriously with Rameau’s harmonic theory, writing an exposition of it and debating the physical and mathematical foundations of consonance in ways I believe clarify what temperament can and cannot achieve.
MARPURG (c. 1718–1795, Berlin): I am Friedrich Wilhelm Marpurg, Berlin critic and theorist, and I was among the strongest advocates in the German-speaking world for 12-tone equal temperament, engaging in pointed controversy with Kirnberger on the subject, arguing that harmonic equality across all keys is the rational and progressive position.
BUSONI (c. 1866–1924, Empoli/Berlin): I am Ferruccio Busoni, pianist and composer, and in my Sketch of a New Aesthetic of Music of 1907 I called for the division of the tone into thirds and sixths — microtonal divisions — arguing that the Western twelve-note system had exhausted itself and that music must expand its tonal resources without limit.
WYSCHNEGRADSKY (c. 1893–1979, St. Petersburg/Paris): I am Ivan Wyschnegradsky, Russian-French composer, and I devoted my life to 24-tone equal temperament and beyond — quartertones and their extensions — composing a substantial body of microtonal music and writing Manual of Quartertone Harmony, driven by a mystical belief that denser tonal space approaches the continuous and the infinite.
CARRILLO (c. 1875–1965, Ahualulco/Mexico City): I am Julián Carrillo, Mexican composer and theorist, and I developed Sonido 13 — the thirteenth sound, beyond the twelve of equal temperament — exploring 96-tone equal temperament and other fine divisions, building special instruments to realise these ideas and demonstrating that the physical universe of sound is vastly richer than tradition acknowledges.
PARTCH (c. 1901–1974, Oakland/various): I am Harry Partch, American maverick, and I rejected equal temperament wholesale, constructing my own instruments — cloud chamber bowls, chromelodeons, boos — to realise an extended just intonation system based on the 11-limit tonality diamond, because I believe that music must be corporeal, immediate, and rooted in the natural ratios of the harmonic series.
FOKKER (c. 1887–1972, Bois-le-Duc/Amsterdam): I am Adriaan Daniël Fokker, Dutch physicist and music theorist, and I developed the theory of periodicity blocks to explain how finite scale systems emerge from just intonation lattices, advocated for 31-tone equal temperament, and built the 31-tone organ in Amsterdam, believing that meantone’s natural extension provides the ideal balance of purity and practicality.
2. Just Intonation vs. Temperament
INTERVIEWER: Let us begin the debate in earnest. What are the merits of tempering intervals versus using just intonation?
PYTHAGORAS: There is no question — just ratios, specifically those derived from the primes 2 and 3 alone, are the true intervals, and any deviation from them is a corruption of mathematical perfection. The Pythagorean tuning of pure fifths stacked upon pure fifths is the only tuning with cosmological justification.
ZARLINO: With the greatest respect to our founder, Pythagorean tuning produces major thirds of 81:64 — the ditone — which is rough and unsuitable for the rich triadic harmony we have developed by the sixteenth century. The pure 5:4 major third is demonstrably sweeter, and just intonation in the 5-limit is nature’s gift to the polyphonist.
ARON: And yet, Zarlino, you know as well as I do that just intonation on a fixed-pitch keyboard is impossible without wolf intervals and awkward compromises. My quarter-comma meantone sacrifices a tiny sliver of the fifth — only about five and a half cents — to keep all usable major thirds pure. That is a sensible trade.
WERCKMEISTER: Both meantone and just intonation are too restrictive. I compose and play music that passes through many keys, and I need all twenty-four to be available. My well temperaments allow this while ensuring that the keys nearest C major remain sweetest — there is a theological virtue in this graduated harmonic world.
PARTCH: You are all, with the greatest affection, describing compromises. The human ear did not evolve to hear tempered intervals — it evolved to hear the harmonic series as produced by resonating bodies. Every cent you temper is a lie told to the ear. Just intonation is not a limitation; it is a liberation from the tyranny of the keyboard’s architecture.
RAMEAU: Monsieur Partch speaks with passion but forgets practicality. The overtone series itself is the foundation I use — but nature does not hand us a keyboard. The corps sonore gives us the principle; temperament gives us the instrument. A slight tempering of fifths is the price of harmonic civilization.
FOKKER: This debate need not be binary. 31-EDO tempers the syntonic comma — as meantone does — but with so many notes that the approximations to 5-limit just intonation are extraordinarily close. The fifth is only 0.7 cents flat. You get the modulatory freedom Werckmeister desires and the harmonic purity Zarlino demands, simultaneously.
ARISTOXENUS: All of you continue to speak as though the ear will ratify your arithmetic. It will not, necessarily. Trained singers adjust their intonation contextually, and no fixed system — just or tempered — captures what a great choir does spontaneously. The living ear exceeds all your diagrams.
KIRNBERGER: Marpurg would have us flatten all this variety into 12-equal, in which every key sounds identically mediocre. Bach — my teacher, whose authority I invoke — used the variety of key characters that only a well temperamentprovides. The Well-Tempered Clavier is not an advertisement for equal temperament; it is a tour through a world of differentiated harmonic regions.
MARPURG: Kirnberger persistently misreads Bach. “Well-tempered” means usable in all keys — it does not mandate audible differences between them. And the compositional richness of modulation is in any case better served by an equal system, where no key imposes a tax on the player.
3. If Forced to Choose an EDO
INTERVIEWER: Setting aside your preferences for just intonation or well temperament — if you were absolutely forced to choose a single equal division of the octave, which would it be?
FOKKER: 31-EDO, without the slightest hesitation. It is the natural completion of the meantone tradition, with magnificent 5-limit approximations, a usable 7-limit, and enough notes to express chromaticism, enharmony, and beyond. I built an organ for it. I have heard it. It is beautiful.
PARTCH: None of them, but if tortured into choosing, perhaps 43-EDO — it approximates enough of my 11-limit ratios reasonably well. But I want it on record that this is a capitulation, not a preference.
WYSCHNEGRADSKY: 72-EDO, because it contains 12-EDO, 24-EDO, and 36-EDO as subsets, gives extremely fine approximations to many just ratios, and represents a practical ceiling of human perceptual resolution while still being theoretically manageable.
CARRILLO: 96-EDO, the system I championed — eighth-tones — because it divides each semitone into eight equal parts, giving composers extraordinary expressive nuance while retaining a logical relationship to the familiar twelve-note framework.
BUSONI: I proposed in my Sketch a division into thirty-six tones per octave — 36-EDO — which gives third-tones and sixth-tones, and this would be my choice; it is fine enough to express the tonal gradations I craved without becoming unwieldy.
NEIDHARDT: Honestly, I would choose 12-EDO. My later work already approaches it, and for the universal circulation of all keys that I sought, it is the logical terminus. I am not sentimental about the journey if the destination is this convenient.
MARPURG: Agreed with Neidhardt, and I am not embarrassed to say so plainly: 12-EDO is the rational conclusion of centuries of Western tuning thought.
WERCKMEISTER: I would resist 12-equal and choose 55-EDO — it very nearly replicates quarter-comma meantone and preserves key variety, while still being technically an equal system.
ZARLINO: 19-EDO is close to my heart — its major thirds are purer than 12-equal, it extends meantone naturally, and singers can navigate it with less pain than many alternatives.
DE SALINAS: Also 19-EDO for me, or perhaps 31-EDO — both are meantone-family and I understand meantone to be the correct framework for vocal polyphony.
RAMEAU: I would choose 12-EDO for its harmonic clarity and modulatory freedom, while noting that the slight roughness of its thirds is a real loss that I do not minimise.
YOUNG: Perhaps 53-EDO, which offers extraordinarily accurate approximations to Pythagorean and 5-limit just intervals alike — it satisfies almost everyone in this room to some degree, which is the mark of a good compromise.
PTOLEMY: 53-EDO also appeals to me, as it can represent many of the tetrachord divisions I catalogued with remarkable fidelity. The ancient world had, in some ways, intuited this richness.
4. Which Prime Limits Matter?
INTERVIEWER: Which ratios are important to approximate — those built from primes up to 3, 5, 7, or 11? In other words, what is the appropriate prime limit?
PYTHAGORAS: The primes 2 and 3 are entirely sufficient. The 3-limit gives us the octave, fifth, and fourth — all that true music requires. The introduction of 5 was a corruption; 7 and 11 are barbarisms.
DE PAREJA: I must respectfully contradict the master. The prime 5 is essential — the 5:4 major third and the 6:5 minor third are the consonances that define Renaissance harmony. Without 5 we are singing hollow, medieval intervals.
PARTCH: And stopping at 5 is itself arbitrary. My Tonality Diamond extends to the 11-limit, and each new prime — 7, 11 — opens genuinely new emotional and harmonic territory. The 7:4 is not a mistuned minor seventh; it is a completely distinct interval with its own consonant character. The 11:8 is not a mistuned tritone; it floats in a way nothing else does.
RAMEAU: The harmonic series provides my answer empirically: the partials 1 through 6 give us the major triad and, via inversion, the minor. This is 5-limit. The 7th partial produces a ratio that does appear acoustically in resonating strings, but Western harmony has not integrated it, and I am not sure it should.
FOKKER: The 7-limit is eminently musical. In my periodicity block analysis, the prime 7 introduces the septimal minor seventh and a whole world of new comma relationships — the septimal comma, the archytas comma. 31-EDOapproximates the 7-limit excellently, and this is one of its chief merits over 12-EDO.
PTOLEMY: In my Harmonics I did consider intervals approaching what we would now call the 7-limit; my so-called equable diatonic tetrachord uses ratios that approach this region. I say the question of what is consonant cannot be settled by arithmetic alone — the ear must be consulted in each case.
WYSCHNEGRADSKY: As a composer working with continuous tonal space, I find the prime limit question somewhat theoretical. But I note that quartertone harmony approximates certain 11-limit intervals naturally, and these sounds carry genuine expressive weight — the 11th harmonic in particular has a floating, otherworldly quality I exploited extensively.
ARISTOXENUS: You are all still thinking in terms of ratios. I ask instead: which intervals does the trained musical ear recognise as stable and functional? That list will not align perfectly with any prime limit. There are intervals between your ratios that a Dorian singer uses fluently without ever having heard of the number 7.
5. How Many Notes Per Octave?
INTERVIEWER: How many notes per octave is a reasonable number — in any kind of system, equal, unequal, just, or tempered?
ARISTOXENUS: The Greek tetrachord system gives us, in the diatonic genus, seven notes per octave — and this number reflects a genuine cognitive limit of melodic memory and orientation. Seven, or perhaps the chromatic expansion to twelve, is as far as practical melody can go.
BOETHIUS: I concur that the classical system of seven modal degrees — the foundation of all Western music up to my era — is cognitively natural and theologically ordered. More would be confusion; fewer would be impoverishment.
ZARLINO: Twelve chromatic notes per octave represents the correct expansion for Renaissance polyphony — enough to express the hexachordal system in all its modulations without overwhelming the performer.
WERCKMEISTER: Twelve is sufficient for my purposes — the twenty-four major and minor keys arise naturally from twelve pitch classes, and my well temperaments prove that twelve notes can be arranged to serve all compositional needs.
FOKKER: Twelve is impoverished. 31 notes per octave is the more natural number for Western tonal music — it resolves the enharmonic confusions of meantone, allows genuine representation of 7-limit harmonics, and is practically navigable on an instrument of suitable design.
PARTCH: My 43-note scale represents what I found necessary for 11-limit just intonation within the octave. But I want to be clear that I did not choose 43 for theoretical neatness — I derived it from the ratios I needed, and 43 is what I arrived at. The number follows the musical logic, not the reverse.
CARRILLO: The question of “how many” is precisely the question I devoted my life to opening up. Once you admit that 12 is arbitrary, you must ask what the finest practically playable division is — and I worked at 96 per octave. The limit is the construction of the instruments and the training of the performers.
BUSONI: I proposed 113 in my more visionary moments — but candidly, for practical composition, something between 24 and 36 seems the right order of magnitude: enough to express the microtonal gradations that twelve lacks, not so many as to require specially built instruments for every performance.
NEIDHARDT: From a practical standpoint of keyboard instruments and notation, twelve has historically won — and it wins because human fingers, human eyes reading notation, and the cognitive organisation of pitch in tonal harmony all converge on twelve as the working number. Theory may point beyond it; practice brings us back.
WYSCHNEGRADSKY: Twenty-four — the quartertone system — is the minimum expansion that genuinely opens new harmonic territory while being teachable within one or two generations. It was my life’s work to demonstrate this practically, not merely theoretically.
6. Is the Major Third a Consonance?
INTERVIEWER: Is the major third — the interval of 5:4 — a consonance?
PYTHAGORAS: Absolutely not. The major third in Pythagorean tuning is 81:64 — a complex ratio involving four iterations of the fifth — and it is rightly classified as a dissonance requiring resolution. The only consonances are the unison, octave, fifth, and fourth.
BOETHIUS: My classification follows the Greeks: the consonantiae perfectae are the fourth, fifth, and octave. The third is an imperfect consonance at best — tolerated in practice but not theoretically primary.
DE PAREJA: This is precisely the battle I fought in 1482 and won, at some cost to my reputation. The 5:4 major third is not only a consonance — it is one of the sweetest sounds available to the human ear, and any theory that calls it a dissonance is a theory built for abstract arithmetic, not for music.
ZARLINO: The major third is a consonance — I proved this from the senario. The ratios 4:5:6 produce the major triad, the most stable harmonic configuration in nature. To call the third dissonant is to condemn the major triad itself.
ARISTOXENUS: The ear of every trained musician in any era has recognised the third as a stable resting point in appropriate contexts. This is the evidence that matters, not the ratio’s complexity. The major third is a consonance because it functions as one.
RAMEAU: The major third emerges directly from the corps sonore — it is the interval between the 4th and 5th harmonics of a vibrating string. Nature declares it consonant, and that settles the matter for me.
KIRNBERGER: In the well temperament I use, the major thirds closest to C major are nearly pure and fully consonant; those in remote keys are wider and accordingly more tense. This graduated consonance is musically useful — the third is not uniformly one thing across all keys.
7. Is the Barbershop Seventh a Consonance?
INTERVIEWER: What about the harmonic seventh — the 7:4 ratio, sometimes called the barbershop seventh? Is it a consonance?
PYTHAGORAS: The number 7 lies outside my system entirely. I have nothing further to say.
ZARLINO: The senario ends at 6. The ratio 7:4 is outside the domain of legitimate musical intervals as I define them, and I would classify it as a dissonance — or rather, as something exterior to the harmonic system altogether.
PARTCH: This response from Zarlino illustrates exactly the dogmatic limitation I rebelled against. The 7:4 is supremely consonant — more consonant in isolation than the minor seventh at 16:9, by any acoustical measure. It is the seventh partial of the harmonic series. Your ears know it even if your theory forbids it. Barbershop quartets have intuited this for generations without any theoretical permission.
FOKKER: Acoustically, the 7:4 is unambiguously more consonant than the tempered minor seventh of 12-EDO — its combination tones are clean and its beat frequency at unison is zero. 31-EDO approximates it at roughly 968.5 cents, which is close enough to be functionally consonant, and I have heard it work strikingly in harmony.
RAMEAU: The 7th partial does appear in the harmonic series, I grant — but it appears higher up, is quieter, and is not part of the fundamental triadic structure that Western harmony rests upon. I would call it a “natural dissonance” — acoustically smoother than an artificial dissonance, but functionally requiring context and resolution in a harmonic grammar.
WYSCHNEGRADSKY: In quartertone harmony, we approximate the 7:4 reasonably closely, and in sustained chords it reads clearly as a stable, dark, resonant interval — distinct from both the major sixth and the minor seventh. I treated it as a qualified consonance in my compositional practice.
ARISTOXENUS: My answer is: ask a choir that has sung through it. The answer will depend on context, tradition, and training — not on whether 7 appears in Zarlino’s senario.
DE SALINAS: I confess I find the 7:4 interval striking and somewhat outside my harmonic world — but when I hear it tuned purely, as in a natural trumpet’s seventh harmonic, it does not strike me as requiring resolution. It is alien but not unpleasant. Perhaps “consonance” needs a wider definition.
8. Whose Tuning Would You Borrow?
INTERVIEWER: If you were forced to adopt another panelist’s tuning system — which would be the least worst, and why?
PYTHAGORAS: If I must — Ptolemy. He at least respects the primacy of mathematical reasoning and treats tuning as a theoretical system, even if his concessions to the ear go further than I would wish.
PARTCH: Fokker. Of all the temperament advocates here, he comes closest to understanding what just intonation is reaching for, and 31-EDO approximates enough of my ratios to be tolerable. I would still miss my 11-limit intervals.
FOKKER: Zarlino — his theoretical grounding in the 5-limit is correct as far as it goes, and his just intonation is the foundation my temperament is designed to approximate.
WERCKMEISTER: Neidhardt. He understood, as I did, that well temperament is the answer, and his later, more equal systems would serve my sacred music without the wolves that plague meantone.
KIRNBERGER: Vallotti. His temperament is elegant, practical, and preserves the key differentiation that I prize, without being as uncompromising as pure just intonation would require on a keyboard.
RAMEAU: Young. His temperament of 1800 is rational, smooth, and grounded — a man of science rather than sentiment, which I respect, and his compromise seems musically just.
ARON: De Salinas — he understood meantone more completely than I did, and his third-comma meantone gives a slightly less flat fifth in exchange for a slightly less pure third, which for some music is the better trade.
MARPURG: Neidhardt — specifically his most equal system, which is essentially 12-EDO by another name.
BUSONI: Wyschnegradsky — his quartertone universe is close to what I imagined, and his compositional output proves it is not merely theoretical.
WYSCHNEGRADSKY: Carrillo. His system is even more ambitious than mine, and his physical construction of the instruments to realise it shows a commitment I deeply admire.
PTOLEMY: The Indian sruti system — twenty-two microtonal divisions of the octave — most closely parallels my own systematic approach to tetrachordal division. The Indians appear to have arrived, from entirely different cultural premises, at a similar conclusion: that the space of musical pitch requires fine subdivision beyond the obvious consonances. I find this compelling confirmation.
ARISTOXENUS: The maqam tradition makes the most sense to me, because it treats scale and melodic gesture as inseparable — a maqam is not merely a set of pitches but a way of moving through them, an emotional and melodic character. This is precisely what I argued about Greek modes: the ethos of a scale is carried in its melodic practice, not in its arithmetic ratios.
PARTCH: The Indian sruti system is the one that most anticipates my own work — it recognises that the chromatic scale of twelve notes is an extreme simplification, and it grounds its divisions in the natural resonances of the voice and the vina. The 22-sruti framework actually approximates many 5-limit and 7-limit ratios with remarkable accuracy.
FOKKER: Gamelan tuning fascinates me as a physicist and theorist because it operates entirely outside the just intonation paradigm — its intervals are deliberately inharmonic and differ ensemble by ensemble, producing that characteristic shimmering beating effect that is itself the aesthetic goal. It is a reminder that “consonance” is culturally constructed, not acoustically absolute.
BUSONI: The Persian dastgah system attracts me because of its modal richness and its incorporation of intervals that Western theory has no name for — neutral thirds, neutral seconds. This is the kind of tonal territory I was calling for in 1907, and here it already exists, fully cultivated, in another tradition.
WYSCHNEGRADSKY: The maqam tradition, because Arab theorists — particularly al-Farabi and Ibn Sina — developed systematic descriptions of neutral intervals centuries before Europeans admitted such things existed. A neutral third of approximately 11:9 is neither major nor minor — it is its own interval, with its own character, and the maqam tradition made this its home ground.
RAMEAU: I confess that Ancient Chinese theory — the Pythagorean chain of fifths arriving at the lülu system — is the non-Western system most transparent to me, because its logic is the same logic of stacked fifths that underlies our own historical foundations. There is something remarkable in two civilisations independently discovering the same generative principle.
10. Best and Worst of 12-EDO
INTERVIEWER: What, in your view, are the best and worst aspects of modern 12-tone equal temperament?
MARPURG: Best: perfect harmonic equality across all keys, making the full circle of modulation freely available to every composer and performer without instrument-specific constraints. Worst: I struggle to identify a worst aspect, honestly — the critics are too precious about tiny differences in third quality.
KIRNBERGER: Best: its simplicity and universality for practical instrument building and notation. Worst: the destruction of key character. Every key sounds identically coloured — or rather, identically colourless. The D-flat major of Bach’s Well-Tempered Clavier should feel different from C major; in 12-EDO it is merely transposed. This is an artistic impoverishment on a civilisational scale.
PARTCH: Best: its success in standardising pitch across ensembles worldwide, which is a genuine practical achievement. Worst: the systematic, deliberate mistuning of every interval except the octave — particularly the destruction of the 7-limitand 11-limit intervals, which cannot even be approximated in 12-EDO. It is not a tuning system; it is a tuning suppression system.
WERCKMEISTER: Best: the resolution of all wolf intervals and the clean circle of fifths. Worst: the loss of spiritual differentiation between keys — I felt that different keys carried different theological and emotional characters, and 12-EDO erases this with a uniform tempering.
RAMEAU: Best: it realises, practically, the unlimited modulation that the harmonic language I described demands. Worst: the major third at 400 cents is 14 cents sharp of 5:4 — audibly rough in sustained chords — and this roughness has, I believe, pushed Western music toward faster harmonic rhythms and away from the sustained resonance that just harmony makes possible.
FOKKER: Best: simplicity. Worst: it is a very bad approximation of 5-limit just intonation — 14 cents off the major third is genuinely terrible — and it entirely fails the 7-limit. 31-EDO is strictly superior in every harmonic dimension except simplicity, and simplicity is a poor reason to choose a tuning.
BUSONI: Best: the shared global language it created, allowing Brahms and Debussy and Mahler to communicate within a common framework. Worst: the exhaustion I diagnosed in 1907 — the sense that every possible tonal combination has been explored, and the system offers no new territory. It was this exhaustion that made microtonality not a luxury but a necessity.
NEIDHARDT: Best: its inevitability — all roads in Western keyboard tuning led here, and the destination is as clean as mathematics can make it. Worst: the thirds, yes. I always knew the thirds were the price. I paid it willingly.
11. Best and Worst of Xenharmonic/Microtonal Tuning
INTERVIEWER: And what of modern xenharmonic and microtonal tuning — what are its best and worst aspects?
CARRILLO: Best: the liberation of music from an arbitrary twelve-note cage — the recognition that sound is continuous and that composers may work at any resolution they choose. Worst: the practical difficulty of notation, instrument building, and performer training, which has kept these ideas on the margins for a century despite their validity.
WYSCHNEGRADSKY: Best: the discovery of genuinely new emotional colours — the quartertone chord has a quality that no twelve-note chord can reproduce, and this is not merely technical novelty but genuine expansion of human expressiveness. Worst: the persistent assumption among mainstream musicians that microtonal intervals are “out of tune” rather than “differently tuned” — a cultural bias that dies very slowly.
PARTCH: Best: the return to the body, to the voice, to the natural resonances of the physical world — extended just intonation reconnects music to its acoustic foundations. Worst: the tendency among microtonal composers to treat the topic as primarily theoretical, producing systems of great intellectual elegance that nobody performs or listens to. A tuning system that does not make music is a mathematical curiosity.
FOKKER: Best: the systematic exploration of harmonic space beyond 12-EDO, revealing relationships — commas, temperament families, lattice structures — that 12-EDO obscures entirely. Worst: the proliferation of incompatible systems with no shared notation or instrument ecosystem, which makes xenharmonic music extraordinarily difficult to disseminate.
BUSONI: Best: it vindicates what I wrote in 1907 — the world did listen, eventually, and composers did venture beyond the semitone. Worst: much microtonal music has pursued complexity for its own sake, mistaking density of pitch content for depth of expression. New intervals must serve musical meaning, not demonstrate theoretical ingenuity.
ARISTOXENUS: Best: the xenharmonic tradition has, at last, turned attention back to the ear and away from the tyranny of the fixed keyboard. Worst: in many cases it has simply replaced one set of rigid numbers with another — the 12 notes of equal temperament replaced by 31 or 72 or 96, equally fixed. True flexibility of intonation, as my singers knew it, remains elusive.
ZARLINO: I view some of this with concern. The expansion of consonance to include ratios beyond the senario — 7, 11, 13 — risks dissolving the harmonic framework that makes tonal music intelligible. One must ask: what is the grammar of this new music? Without a grammar, there is no communication.
12. The Future of Music Tuning
INTERVIEWER: Finally, gentlemen — what do you believe the future of music tuning will look like?
PARTCH: The future, if music is healthy, will return to the harmonic series — not as historical curiosity but as living acoustic reality. Electronic instruments and computers remove every obstacle to precise just intonation. There is no longer any excuse for the compromises that keyboard geometry once forced. The question is whether composers will have the courage to abandon the twelve-note security blanket.
FOKKER: I believe the future lies in the systematic exploration of regular temperament theory — understanding which EDOs and which rank-2 temperaments best serve which musical purposes, and building instruments accordingly. The tools of linear algebra and lattice geometry will reveal the map of tuning space, and composers will navigate it consciously.
WYSCHNEGRADSKY: The future is continuous pitch — the dissolution of discrete steps entirely into glissando and spectral continua. Equal divisions are staging posts on a journey toward a music of pure frequency. Electronic synthesis makes this possible, and the inner ear has always craved it.
RAMEAU: I suspect the future will involve a renewed engagement with acoustic science — particularly psychoacousticsand the physics of combination tones — to ground tuning choices in what the ear actually perceives rather than what theorists prefer. The corps sonore is always the final arbiter.
BUSONI: The future is pluralism — many tuning systems coexisting for different musical purposes, the way different scales and modes coexist today. Twelve-EDO will not disappear, but it will be one option among dozens, each with its own repertoire, notation, and community of practice.
ARISTOXENUS: I hope the future rediscovers what trained singers and instrumentalists have always known: that living intonation is adaptive, contextual, and relational. The finest tuning is the one that responds in real time to the acoustic environment and the musical phrase — not any fixed system at all.
KIRNBERGER: My hope is more modest: that the world rediscovers the artistic richness of key differentiation. Whether via well temperament or some more sophisticated harmonic approach, the idea that D-flat major should feel different from C major seems to me a musical truth too important to lose permanently.
CARRILLO: The future is instrument building — physical or electronic — that makes fine microtonal intervals as easy to play and as natural to hear as the twelve semitones are today. Once the physical barrier falls, the theoretical arguments will resolve themselves in practice. Children who grow up with 96 divisions will hear them as naturally as we hear twelve.
13. Closing Remarks
INTERVIEWER: Gentlemen, we are near the end of our time. In one sentence each, please give your closing remark — what should the reader take away from today’s discussion?
PYTHAGORAS: Number is the foundation of all things, and the reader should know that in the ratios 2:1, 3:2, and 4:3 they hold the mathematical keys to the universe of sound.
ARISTOXENUS: Train your ear — for without a trained ear, all the ratios and systems we have debated today are merely ink on papyrus, and music is made not in theory but in sound.
PTOLEMY: Reason and sensation must always work together: a tuning theory that satisfies the intellect but offends the ear is incomplete, and one that pleases the ear but cannot be explained is incomplete in a different way.
BOETHIUS: The contemplation of musical proportion is an elevation of the soul, and the reader should understand that tuning is not merely a technical matter but a participation in the rational order that underlies creation.
DE PAREJA: Do not be afraid to trust the sweetness of the pure major third — tradition can be wrong, and the ear that hears something beautiful is not required to apologise for it.
ARON: The practical musician deserves a practical theory: meantone temperament was the right compromise for its era, and the principle — that theory must serve the music being made, not constrain it — remains valid in every era.
ZARLINO: The harmonic series and the ratios of the senario are nature’s own provision for music, and the reader should understand that the best tuning is the one that keeps faith with these natural proportions as closely as musical practice allows.
DE SALINAS: The voice is the measure of all tuning — sing before you calculate, and let the calculation follow what the voice finds true.
PRAETORIUS: The working musician and the instrument builder deserve theoretical respect — theory that cannot be built and played is an indulgence, and I urge the reader to value practical documentation alongside abstract speculation.
WERCKMEISTER: All twenty-four keys are a gift, and the reader should know that the exploration of remote harmonic regions — the dark keys, the bright keys — is not merely technical adventure but spiritual journey.
NEIDHARDT: The history of tuning is a history of increasing freedom — from the locked fifths of Pythagoras, through meantone’s pure thirds, to the universal circulation of all keys — and this freedom, however imperfect its current expression, is worth celebrating.
RAMEAU: The harmonic series is the foundation of all music, and the reader who understands that the bass note generates its own harmony through the physics of resonance will understand both the greatness and the limitations of every tuning system discussed today.
KIRNBERGER: Key character is not an accident of imperfect technology — it is an artistic resource, and its loss to equal temperament is a genuine aesthetic tragedy that the reader should at least be aware of, even if modern practice has forgotten it.
VALLOTTI: A well-designed temperament is like a well-designed city — not perfectly regular, but organised so that the most-travelled routes are the most comfortable, and even the distant quarters have their own interest and beauty.
YOUNG: A good tuning system is like a good theory in physics: it should be as simple as possible, but not simpler — and 12-equal, for all its convenience, is somewhat simpler than the musical facts warrant.
D’ALEMBERT: The reader should understand that musical tuning is ultimately a problem in applied mathematics constrained by human perception, and that the best solutions — like the best solutions in physics — are those that find elegant structure in complex empirical data.
MARPURG: Equal temperament is not a compromise but a liberation — and the reader should not allow nostalgic arguments about key colour or just thirds to obscure the genuine achievement of a system in which all harmonic regions are equally available to all composers.
BUSONI: The twelve-note scale is not the end of music’s development but a way station, and the reader should approach the vast unmapped territory of microtonal and xenharmonic sound not with anxiety but with the excitement of an explorer who has just been handed a much larger map.
WYSCHNEGRADSKY: The quartertone — and what lies beyond it — is not an eccentricity but a necessity, and the reader who truly listens to a pure 24-EDO chord will hear something that twelve notes can never provide.
CARRILLO: Sonido 13 is the sound of everything beyond what tradition gave us — and that sound is not foreign to the human ear; it is simply waiting to be heard, if we will only stop pretending that twelve is all there is.
PARTCH: Forget the keyboard — forget the twelve semitones — and listen to the way a resonating body fills the air with its natural harmonics, because that is where music began, and it is where, if we are wise, it will return.
FOKKER: The reader should know that tuning space is vast, beautiful, and largely unexplored — that 31-EDO and the periodicity block framework give us tools to navigate it, and that the journey is well worth the cost of unfamiliarity.
The Grand Tuning Symposium adjourns. The panelists retire to debate the precise tuning of the refreshments.
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