How two pen pals on opposite sides of an ocean used the early internet to carve out a new field of music theory
If you have ever bent a guitar note and felt something click into place — a resonance that standard tuning never quite delivers — you have already brushed against the question that defines the life's work of Dave Keenan and Douglas Blumeyer. Why do some intervals feel locked and alive while others feel restless or dull? And is the 12-note octave really the best answer humanity could find?
These two men, separated by 25 years in age and by the width of the Pacific Ocean, have spent the better part of the last decade building the most thorough publicly available guide to a branch of music theory called Regular Temperament Theory (RTT). Their collaboration began with a simple exchange of questions in 2021 and grew into nearly 2,000 emails — by their own estimate, well over a million words — and a nine-part wiki article series that is now considered the definitive introduction to the field. This article is about who they are, what RTT actually is, and why any musician with a serious interest in harmony should know about their work.
The Two Men
Dave Keenan is an Australian music theorist, software engineer, and retired sustainable-design consultant based in Brisbane, Queensland. Born in 1959, he holds a Master of Science in Computer Science from the University of Queensland. He describes himself as having drawn diagrams of chord structures as a child — "5-limit lattices," he would later recognize — instead of practicing his piano scales. In 1976, at age 17, sitting in the back of his father's car with a programmable HP25C calculator, he conducted his first computer search for exotic tunings and independently discovered what he would later learn was the Bohlen-Pierce scale, an unusual non-octave tuning built on ratios of the number 3 rather than 2.
Keenan's most visible public contribution before his RTT work is Sagittal notation, a comprehensive microtonal notation system he co-developed with the late George Secor starting in 2001. Sagittal — named for its arrow-like accidentals, evoking the astrological archer Sagittarius — was designed as a universal system capable of notating virtually any tuning system imaginable: from just intonation to equal divisions of the octave with dozens or hundreds of notes, all on a conventional five-line staff using a single coherent family of symbols. The system is now included in the SMuFL font standard (used by MuseScore, Sibelius, and other notation software) and remains the most systematic attempt to solve microtonal notation for mainstream use. Keenan joined the online microtonality community in 1998 through the Mills College tuning email list, which eventually migrated to the Yahoo Tuning Groups and then to Facebook and Discord, and it was there that he began collaborations with figures like Paul Erlich and Graham Breed that helped shape modern RTT.
Douglas Blumeyer is a software engineer and composer based in the San Francisco Bay Area, known online as "Cmloegcmluin." He studied neither music nor mathematics formally — he went to university for film, not science. He discovered xenharmonics (the broad field of tuning systems outside of standard 12-tone equal temperament) only in the final months before graduating, and spent roughly fifteen years composing in just intonation and various equal divisions of the octave before realizing he had little deep understanding of what temperament theory actually meant. His musical work includes the interactive web application Musical Patterns, recordings on SoundCloud, and a drafted book called Fun Musical Ideas (completed in 2014 after he challenged himself to post one new musical idea per week for a year, producing 38 posts condensed into 26 chapters).
Blumeyer first caught Keenan's attention in the tuning Facebook groups by asking precise, pedagogically careful questions about terminology for his work on Metallic MOS scales — scales built on the metallic means (the golden ratio, silver ratio, bronze ratio, and so on) rather than the integer ratios of conventional tuning. Keenan later wrote that what struck him immediately was Blumeyer's care about consistent terminology: "This is a kindred spirit. This is a guy I'd like to work with."
What Regular Temperament Theory Actually Is
Before explaining what Keenan and Blumeyer built together, it helps to understand the problem RTT is trying to solve — and to understand it from first principles, the way a musician would need to.
The physics of consonance
When two notes sound together, your ear and brain are doing something very specific: detecting the ratio between the two frequencies. The ratio 2:1 is an octave. The ratio 3:2 is a perfect fifth. The ratio 5:4 is a major third. The ratio 7:4 is the "harmonic seventh" — an interval you have probably heard in blues or barbershop singing but that does not exist in standard 12-tone equal temperament. The closer a ratio is to having small whole numbers in numerator and denominator, the more the overtones of the two notes coincide, and the more the brain registers consonance or "lock." This is the foundation of just intonation (JI).
The problem with pure just intonation is that its intervals do not chain together in neat closed loops. Stack four perfect fifths (3:2) and you get a ratio of 81:16, which when reduced to a single octave gives 81:64 — a Pythagorean major third about 22 cents sharper than the pure 5:4 major third (80:64). The difference between these — 81:80 — is called the syntonic comma. It is about a fifth of a semitone, barely audible in isolation, but it creates serious problems when you try to move through many keys. In pure JI, you end up needing a slightly different pitch for nearly every harmonic function, and a fixed-pitch instrument (like a keyboard) becomes impractical.
Temperament is the ancient solution: deliberately detune certain intervals by a small amount so that the errors cancel across a chain. Meantone temperament — the tuning underlying most Western music from roughly the 16th to 19th centuries — narrows each fifth by about a quarter of the syntonic comma, so that four slightly-narrow fifths produce a pure 5:4 major third. The specific equivocation "four fifths equal one major third" is called a vanishing comma: the syntonic comma (81:80) is made to vanish within the tuning system.
12-tone equal temperament (12-TET) is a further compromise: the fifth is narrowed to exactly 700 cents (down from the pure 702), making every key sound equally in tune — and equally slightly out of tune. But the major third lands at 400 cents, which is 14 cents sharp of the pure 386-cent major third. That is a significant error, audible to any trained ear. 12-TET also makes the harmonic seventh (7:4, at 969 cents) completely unavailable: the closest approximation is the minor seventh at 1000 cents, which is 31 cents flat.
What RTT observes
The key insight of Regular Temperament Theory — articulated by Keenan, Paul Erlich, Graham Breed, and others in the early 2000s — is that all of this can be described more systematically by asking: which small prime numbers do you want to approximate, and how many stacked generators does it take to reach each of them?
In 12-TET, a single generator — one equal step of 100 cents — reaches 2 in 12 steps, 3 in 19 steps, and 5 in 28 steps. That statement is what RTT calls a map: ⟨12 19 28]. You write it as a row vector. The entire tuning system — which intervals are available, which commas vanish, how many notes per octave are needed — is completely determined by this single compact mathematical object.
Meantone temperament uses two generators: the octave (2:1) and the slightly-narrow fifth. The mapping describes how many octaves and fifths you need to reach each prime. Four fifths minus two octaves reaches the prime 5 (approximately). This is not a single row vector but a 2×3 matrix — two maps stacked — representing a rank-2 temperament. With two generators rather than one, you have a two-dimensional pitch space, and you can build music that moves freely through many keys without ever closing back into a fixed 12-note cycle.
The space between just intonation (where every prime gets its own generator, giving as many generators as primes and no errors) and equal temperament (where a single generator must approximate everything) is what RTT calls the "middle ground." Meantone is just one point in it, targeting primes 2, 3, and 5. There are infinitely many other points, targeting different prime combinations, with different commas vanishing, creating harmonic textures and possibilities that simply do not exist in 12-TET.
The Guide: Nine Articles, Three Levels
In April 2021, Blumeyer decided to teach himself RTT seriously. He was guided primarily by Keenan, and what followed was an extraordinary correspondence. By the time their main article series was complete in 2023, they had exchanged nearly 2,000 emails, none of them short. The resulting series — Dave Keenan & Douglas Blumeyer's Guide to RTT — is structured into nine articles across three levels of technical depth.
The basic level targets musicians with no particular technical background:
1. Introductions — both authors introduce themselves and provide their individual perspectives on what RTT is and why it matters. Douglas's section includes "four lessons" that build from the physics of frequency ratios to a frank cost-benefit analysis of temperament versus just intonation.
2. Mappings — the central RTT object: how to describe a temperament as a matrix that tells you how many generators reach each prime. Includes examples progressing from 12-TET to meantone to more exotic temperaments.
3. Tuning fundamentals — how to think about minimizing "damage" (tuning error) to the intervals you care about most, and why the choice of which intervals to target is itself a musical decision.
4. Exploring temperaments — the dual relationship between mappings and commas: how specifying which commas vanish determines the mapping, and vice versa.
The intermediate level, for engineers and mathematicians:
5. Units analysis — a dimensional-analysis approach to the matrices of RTT, clarifying what their entries mean and why they behave as they do. The insight is that much confusion in RTT literature comes from losing track of units.
6. Tuning computation — derivations of how to compute optimal generator tunings using matrix methods, and why the methods work. Includes Lagrange multipliers for held-interval constraints.
The advanced level, for theoreticians:
7. All-interval tuning schemes — the tuning schemes named TOP, TE, CTE, and POTE, which optimize simultaneously across all possible intervals using dual norms, avoiding the need to specify any particular target-interval set.
8. Alternative complexities — what happens when you measure interval complexity with something other than the default log-product formula, including sum-of-prime-factors, count-of-prime-factors, and their Euclidean variants.
9. Tuning in nonstandard domains — temperaments that target subgroups of primes (like 2.3.7, omitting 5) or unusual prime combinations, rather than the standard prime limits.
Plus a reference appendix: Conventions for names, variables, units, and notations — a systematic table of notation conventions designed to be internally consistent and to serve as quick reference at any level. Crucially, it specifies a variable-styling convention: whether a symbol is uppercase or lowercase, bold or roman, italic or upright, encodes whether the object is a scalar, vector, or matrix, and whether it carries "simple" units like cents or "compound" units. The appendix even includes instructions for a Windows keyboard tool (WinCompose) that lets you type RTT special characters — angle brackets, mathematical symbols, accidentals — without interrupting your workflow.
The guide was also condensed into a six-page conference paper, Regular Temperament Theory: Exploring the Landscape between JI and ETs with Linear Algebra, presented at the 8th International Conference on Mathematics and Computation in Music (MCM 2022) in Atlanta, Georgia.
What makes this guide different
Prior to this guide, RTT was documented in scattered posts across the Yahoo Tuning Groups archive (now preserved partly thanks to another of Blumeyer's projects — a viewing interface for that archive), academic papers written for specialists, and wiki pages written by and for people who mostly already knew the material. The field had brilliant practitioners but almost no systematic pedagogy.
What Keenan and Blumeyer brought was a self-aware pedagogical project. Their governing principle, stated explicitly, is "Encode, don't encrypt": every name and symbol should convey meaning rather than just assign a label. They deliberately avoided eponyms (naming concepts after people) in favor of descriptive terms. Their variable-naming convention is cross-referenced with linear-algebra standards so that a reader who has done a university course in matrix algebra can slot RTT notation into a familiar framework, while a reader who has not can still follow the rules described in the appendix. They structured the guide so that a musician can get genuine practical value from the first four articles without ever needing to read the last five.
Tuning Damage and Why It Matters to Your Ear
One of the core concepts the guide develops carefully is damage — and understanding it encodes a real musical choice.
When you tune a regular temperament, you choose an exact size in cents for each generator. Different choices produce different amounts of error across the target intervals you care about. If your music is built around pure 5:4 major thirds and 3:2 fifths, you might choose generator sizes that minimize the maximum error to either interval — a "minimax" strategy. Alternatively, you might minimize the sum of squared errors (miniRMS), which tolerates occasional large errors in exchange for smaller average errors. Or you might weight the errors by how simple the intervals are: simpler intervals (like the fifth) are heard more clearly and should be tuned more accurately, while complex intervals can absorb larger errors. This last approach is called simplicity-weighting, and it is the basis of most named tuning schemes in the literature.
These are not arbitrary aesthetic preferences. They encode fundamentally different philosophies about what makes music work. The minimax philosophy says: no single interval should be badly tuned, even if it means every interval is slightly wrong. The miniRMS philosophy says: most intervals should be nearly perfect, and the outliers are acceptable. The simplicity-weighted philosophy says: the ear hears simple ratios most acutely, so tune those most carefully.
The most famous tuning schemes — TOP (Tenney-Optimal), TE (Tenney-Euclidean), CTE (Constrained TE), POTE (Pure-Octaves TE) — are all all-interval tuning schemes: they avoid specifying any finite list of target intervals by summing damage across all possible intervals simultaneously, weighted by complexity. This is elegant and computationally convenient for wiki documentation and automated searches, but the guide is explicit that for practical music-making, you will often do better by tuning directly for the intervals you actually plan to use.
The RTT Library in Wolfram Language
Alongside the written guide, Keenan and Blumeyer developed a complete RTT code library in Wolfram Language, free on GitHub at github.com/DandDsRTT/rtt-library. Wolfram Language is the language underlying Mathematica, but the library runs entirely free in a browser through Wolfram Cloud.
The library implements the full computational pipeline of RTT:
- Tuning optimization:
optimizeGeneratorTuningMap[m, tuningSchemeSpec]— feed in a temperament mapping matrix and a named tuning scheme, get back the optimal generator sizes in cents. Supports minimax, miniRMS, miniaverage, unity-weighted, complexity-weighted, and simplicity-weighted versions. - Temperament exploration: canonical form computation, dual (finding the comma basis from a mapping, or vice versa), map-merge and comma-merge.
- Temperament addition: functions
sumanddiffthat implement the precise mathematical operation of combining two regular temperaments into one. - Nonstandard domains: changing the prime-number basis of a temperament, for work in subgroups like 2.3.7 (omitting prime 5).
- Exterior algebra: a full suite of functions for the wedge product, interior product, and related operations.
The library has 279 commits and has been the computational backbone for numerous examples and verifications throughout the written guide. One concrete illustration from the guide itself: computing the "held-octave minimax-lols-S" tuning of the porcupine temperament reduces to a single function call: optimizeGeneratorTuningMap["[⟨1 2 3] ⟨0 -3 -5]]", "held-octave minimax-lols-S"], returning {1200, 162.737} — the octave (held pure) and the generator in cents.
Their Broader Wiki Contributions
The collaboration produced not just the guide and library but a substantial body of new and improved pages on the Xenharmonic Wiki. A partial list conveys the scope:
On theory: saturation, torsion, contorsion, and defactoring; defactoring algorithms; temperament merging; temperament addition; domain basis; eigenmonzo (the unchanged interval of a temperament, a concept Blumeyer's page proposed renaming to clarify its meaning); generator form manipulation; extended bra-ket notation; optimal ET sequence; uniform map and simple map.
On the mathematics of interval representation: undirected value — a function analogous to absolute value, but which reciprocates instead of negates, expressing that a musical interval of a fifth (3:2) and a musical interval of a fourth (4:3) are the "same interval" in different directions. Just as absolute value removes the sign of a number (positive/negative), undirected value removes the direction of a ratio (superunison/subunison). The notation uses horizontal bars above and below the quantity, evoking both the absolute-value vertical bars and the fraction bar. The terminology and notation were developed jointly by Keenan and Blumeyer in 2020.
On terminology: trivial temperament, which carefully distinguishes the two edge cases of the RTT framework — just intonation (which tempers nothing, leaving all intervals pure) and single-pitch tuning (which tempers everything, making every interval vanish to zero). Acknowledging these as degenerate but valid temperaments completes the logical space of the theory; inverse-complexity-prescaled complexity, an article that is explicitly a "cautionary tale" about a mathematical dead-end Blumeyer explored and documented in full — complete with graphs — so that others would not waste time rediscovering it.
Blumeyer's Independent Theoretical Work
Beyond the collaborative guide, Blumeyer has created a significant body of original theoretical content on the Xenharmonic Wiki, much of it exploring the mathematical varieties of pitch series and scales.
Gjaeck and the Blumeyer Comma
Gjaeck is a 13-note (tridecatonic) MOS scale built from 57-EDO. Where most temperament design targets the familiar prime harmonics 3, 5, and 7 (the building blocks of conventional Western harmony), Gjaeck is designed specifically to approximate the higher prime harmonics 11, 13, 17, and 19. These are the harmonics associated with neutral intervals (the "bluesy" pitches halfway between a major and minor third), the 19-limit undertone series, and the spectral tuning of certain non-Western musical traditions.
The scale follows the step pattern ssLssLsLssLsL (eight small steps and five large steps, a 5L8s MOS), with small steps of 4 EDO-steps and large steps of 5 EDO-steps. The generator is the 35th step of 57-EDO, approximately 736.8 cents — an interval close to but distinct from a conventional fifth, associated with the 21st harmonic.
Embedded within Gjaeck is a microtonal comma called the Blumeyer comma (2432/2431, approximately 0.712 cents). A comma is any very small ratio that the tuning system treats as zero — the "loop" that allows a finite set of pitches to navigate an infinite harmonic lattice. The Blumeyer comma expresses the relationship that one step up in harmonic 11, one step up in harmonic 13, one step up in harmonic 17, and one step down in harmonic 19, collectively returns you to the unison within 57-EDO. In formula: 11 × 13 × 17 / 19 ≈ 128, which 57-EDO maps to exactly 7 octaves (399 = 57 × 7 scale steps). Discovering and naming a comma means having found a genuine structural relationship — a closed loop in harmonic space — that had not previously been identified.
Yer: a JI scale in the 2.11.13.17.19 subgroup
Yer (named after the Slavic letter Ъ) is a just-intonation scale also targeting the prime harmonics 2, 11, 13, 17, and 19 — the same unusual "subgroup" as Gjaeck, but without any tempering. Blumeyer presented Yer at Micro-Camp 2016 in Delton, Michigan. It is a demonstration that high prime harmonics can be organized into a musically coherent pitch set when you deliberately sidestep the gravitational pull of the simpler 3- and 5-limit harmonics that dominate conventional Western music theory.
Xenharmonic series: a mathematical taxonomy of pitch systems
The xenharmonic series page — which Blumeyer substantially developed — catalogs the many mathematical variations on the harmonic series that the xenharmonic community has explored. The harmonic series (f(n) = n) is the foundation of JI, but pitch systems can be generated from almost any mathematical function. The catalog includes:
Dumb Fibonacci series: only the harmonics that are Fibonacci numbers — 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. The name is Blumeyer's own, characteristic of a humor that runs throughout his writing: the series is mathematically well-defined and internally consistent, but its musical applications are, as the wiki page honestly notes, "limited, highly contrived, or as yet unknown."
Triangulharmonic series: harmonics that are triangular numbers — 1, 3, 6, 10, 15, 21. (Triangular numbers are numbers of the form n(n+1)/2: 1, 3, 6, 10...) The series forms octave-spanning modes at specific positions. The first interesting one, called the First Triangulharmonic Mode, spans from the 105th to the 210th harmonic (since 210 = 2 × 105) and produces a six-note scale. The article documents that the next such pair does not occur until 3,570 and 7,140 — producing a 35-note scale — and the one after that at 121,278 and 242,556, producing a 204-note scale that is, as Blumeyer notes with characteristic dry precision, "decreasingly likely to be of much interest."
Metallic harmonic series: built on the sequence of metallic means (φ ≈ 1.618, 1 + √2 ≈ 2.414, (3 + √13)/2 ≈ 3.303, and so on) rather than integers, where the nth metallic mean is (n + √(n² + 4))/2. Each successive metallic mean is further from 1 than the last, and the series converges toward the ordinary harmonic series as the metallic index grows. Blumeyer provides a full table comparing the metallic harmonic series pitch-by-pitch against the ordinary harmonic series, with the difference in cents at each step. The table confirms that with each successive metallic mean, you get closer and closer to an ordinary harmonic.
Logharmonic series: where f(n) = log_b(n) instead of n, producing a pitch system whose steps shrink logarithmically rather than following the constant-ratio pattern of the harmonic series. The page is dense with tables and includes a proof connecting the logharmonic series to the Riemann zeta function — the function at the heart of one of mathematics' most famous unsolved problems — when the series is interpreted as an audio waveform. Blumeyer documents that the difference between the logharmonic series and the related matharmonic series approaches the Euler-Mascheroni constant γ ≈ 0.5772, one of the fundamental constants of mathematics.
Powharmonic series: where f(n) = nᵖ, with p as a free parameter. At p = 1 you recover the harmonic series. At p = -1 you get the subharmonic series. At p > 1 you get a stretched version of the harmonic series with wider intervals. At non-integer values of p you get intermediate cases. The page also documents the closely related edharmonic series (a-edharmonic series), built by moving first by one equal division of a, then two, then three, and so on — a series whose step differences approach a limit related to the Euler-Mascheroni constant through the harmonic numbers.
Oddharmonic series: only the odd harmonics — 1, 3, 5, 7, 9, 11. The page is brief but includes Blumeyer's concept of TOH — Tritave of Odd Harmonics — which generates modes of the odd harmonics that repeat at the tritave (ratio 3:1) rather than the octave (2:1). The third TOH mode is a nine-note scale: 9, 11, 13, 15, 17, 19, 21, 23, 25 (with 27 closing the tritave). It is an unusual and acoustically rich structure with no conventional Western equivalent.
Arithmetic tunings
The arithmetic tuning page — developed by Blumeyer — provides a comprehensive taxonomy of every tuning that has equal step sizes in some measurable quantity. Equal steps of frequency gives otonal divisions (OD) and arithmetic frequency sequences (AFS). Equal steps of pitch gives equal divisions of the octave (EDO) and arithmetic pitch sequences (APS). Equal steps of string length gives utonal divisions (UD) and arithmetic length sequences (ALS). The overtone series is an equal-frequency-step tuning. Any EDO is an equal-pitch-step tuning. The undertone series is an equal-length-step tuning. By working out the full logical grid of these relationships — including the analogies between rational and irrational cases, and between sequences and divisions — Blumeyer established a clean taxonomy that had not previously been written up in one place.
N2D3P9 and harmonic complexity
Among Blumeyer's more technical contributions — developed closely with Keenan — is the complexity metric N2D3P9, presented at MCM 2022. It provides a way to measure the harmonic complexity of a JI interval in a manner that correlates well with human perception of consonance. The name encodes the formula: numerator squared (N²), divided by 3 times the denominator (3D), all to the 9th power of 2 (P9 standing for prime power). It assigns smaller numbers to simpler intervals — those that lock more clearly into tune — and larger numbers to complex ones, doing so in a way that better matches experimental data on perceived consonance than the more commonly used Tenney height (numerator × denominator) or Benedetti height (product of prime factors) measures.
Keenan's Broader Contributions
Keenan's contributions to the microtonal field are difficult to enumerate fully because he has been active for decades and his influence is pervasive. Beyond Sagittal notation and the RTT guide, he was a central figure in co-developing the regular mapping paradigm itself — the insight that all temperaments can be understood as regular linear maps from the prime-number lattice to a smaller generator lattice — alongside Paul Erlich, Graham Breed, Gene Ward Smith, and others in the Yahoo Tuning Groups era of the early 2000s.
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