Dave Keenan & Douglas Blumeyer's guide to RTT: introductions

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This is article 1 of 9 in Dave Keenan & Douglas Blumeyer's guide to RTT, or "D&D's guide" for short.

Hello! Dave and Douglas here. Somehow, both of us wound up independently writing our own brief "introduction to RTT" type documents, and it's a testament to the subtle complexity of this topic that we wrote rather different things that don't conflict and actually sort of complement each other. So we'd like to share both of them here, and also use this opportunity to introduce ourselves and say a bit about what drew us to regular temperament theory in the first place.


Dave's brief introduction to RTT

Dissonance is easy. Consonance is rare. The most common kind of consonance occurs when two notes have their frequencies approximating a simple ratio.[1] This is primarily due to some of their partials coinciding, or nearly coinciding.[2]

We like to have different flavors of consonance, which correspond to different simple ratios. We want to find tunings — sets of notes — that will give us enough consonances in enough flavors, without being too complex, and without having errors that are too large. The complexity of a tuning is partly about the number of notes per octave, and partly about the number of different step sizes.

Regular Temperament Theory is a powerful tool to aid us in finding such tunings. It observes that the problem of approximating many simple ratios can be reduced to one of approximating a few small prime numbers. It then generates all of its tunings by stacking (both up and down) a small number of intervals which are called the generators. In one extreme, the generators are the small prime numbers themselves, giving just intonation (JI) as a lattice, having zero errors but high complexity. In another extreme, there is a single small generator whose iterations must approximate all the desired primes, giving an equal temperament (ET), having low complexity but high errors.

These two extremes were well explored prior to RTT. What RTT did was open up a vast middle ground between JI and ET, where the number of generators is greater than one but less than the number of primes being approximated. These are called regular temperaments (RT). Only a very small region of that middle ground had been explored prior to RTT, namely the "meantone" region that approximates primes 2, 3 and 5, using two generators which are an octave (prime 2) and a slightly narrow fifth (approximate 2:3).

When the generation of tunings is formulated in this way, the tools of linear (or matrix) algebra[3] can be applied.

The defining thing about a regular temperament is the count of each generator required to approximate each prime number. This is called the temperament's mapping, and can be represented as a matrix.[4]

We can then institute computer searches to find optimum mappings, with our desired balance of error versus complexity. Many such searches have been done and many resulting temperaments named and cataloged.

Dave's personal introduction

I write to you now as a hollow husk of a man, having been drained of all knowledge by young Douglas Blumeyer. Ha ha.

OK, he's not that young, but he is 25 years my junior. I was born in 1959, and if you can't do that math, you're probably in the wrong place. Ha ha.

As a child, instead of practicing the piano, I drew diagrams of the chords. I would later learn that these diagrams were called "5-limit lattices". I eventually found the guitar more to my liking. In 1976, while sitting in the back of my father's car, I did my first computer search for exotic tunings, using a HP25C programmable calculator, and discovered something I would later learn was called the Bohlen-Pierce scale. A few years later, I refretted a guitar for ¼-comma meantone tuning. And while studying electronics, I made a "music" generator using oscillators modulating other oscillators, and built a "graphic oscillator" where you set the wave shape using a bunch of sliders. Then I did a degree in physics and computer science and worked as an electronics designer, software engineer and teacher.

In 1998 I joined the Mills College tuning email list which later became the Yahoo tuning groups and then the present day Facebook tuning groups and Discord servers. There I met clever, friendly and like-minded people, and began several fruitful collaborations, most notably for this work, with Paul Erlich and Graham Breed. Little did I know then, that I would find myself in 2022 co-authoring a textbook on this field we now call Regular Temperament Theory.

Douglas first came to my attention on the tuning Facebook groups when he asked some questions in regard to an article he was writing, about Metallic MOS scales. What struck me was his obvious understanding of, and caring about, the pedagogical issues, particularly in relation to consistent terminology. He was essentially asking us old-timers to suggest appropriate terminology for his new ideas, so it wouldn't clash or cause confusion with any existing terminology for closely related ideas. I immediately thought: this is a kindred spirit. This is a guy I'd like to work with.

Of course I was joking about being drained. One doesn't lose knowledge by sharing it. And even if one did, Douglas has given me much more than he has taken. For example: He pretty much forced us both to learn a branch of mathematics that I had not been up to as a physics undergraduate in 1982. That was tensor analysis, or multilinear algebra. This was necessary so that we could understand some of the work of the late great Gene Ward Smith. But rest assured that you don't need to know multilinear algebra to understand this work. Nor do you need to know any abstract algebra or group theory.

There is very little that is original in the content of this work. What is original is the systematic terminology and the pedagogy. We have pushed and prodded the last 20 years of writings in regular temperament theory into a form that we believe makes it far easier to grasp than ever before.

I know some of you old-timers are going to find it strange. You may be upset, initially, by some of our renamings. For example, we have eschewed eponyms in favor of descriptive terms. One of our catch-cries, when creating short names for things, is "Encode, don't encrypt". But while we won't be honoring anyone but Euclid in our terminology, we very much want to give credit where it's due, so please tell us about any place you think we could do that better.

Another catch-cry is "Don't FUSS", which is an acronym that means don't "Foul" Up the Simple "Stuff". Which means don't allow the need to more finely distinguish later advanced ideas, to "reach back" and complicate the earlier simpler ones. But I ask you old-timers to please give it a chance. Try to adopt "beginners mind" and see how the system works as a coherent whole. And thank you so much for everything that you contributed to this work.


Douglas's brief introduction to RTT

One day while traveling I realized that three days in a row I'd met up with a friend and needed to explain what I'd been working on recently, and each one of these three times I'd found myself explaining RTT in a completely different way, and that none of these three explanations had been particularly successful. So I resolved to come up with something I could lay out — even to someone whose knowledge of music was light on theory — that could get them excited about regular temperaments. It's not exactly an elevator pitch in length, as I'd originally intended, but I think it's not half bad.

Lesson one: shapes

  1. To hear harmony, your brain detects frequency ratios. For example, a major chord is just three frequencies in the ratio [math]4\hspace{-3px}:\hspace{-3px}5\hspace{-3px}:\hspace{-3px}6[/math].
  2. Frequency ratios do not need to be exact. On a piano in the standard temperament, the actual ratio of a major chord is more like [math]4\hspace{-3px}:\hspace{-3px}5.04\hspace{-3px}:\hspace{-3px}5.99[/math], and this is still close enough to hear it as a major chord.
  3. Temperaments exploit this wiggle room, finding sweet spots within it. Standard temperament tunes the ratios so that, for example, four perfect fifths reach exactly the same frequency as a major third (plus a couple octaves). This uniquely shapes our possibilities for writing and hearing melodies and chord progressions.
  4. Many other sweet spots live within this wiggle room, where we can still hear the chords that are familiar to us from standard temperament, but melodies and progressions get shaped in other special ways. One tuning adjusts ratios so that three major seconds add up to a perfect fourth. In another tuning, four perfect fifths make a minor third instead of a major one, which switches the functional roles of major and minor.
  5. If you only listen to music in the standard temperament, your ears get that one single shape beaten in. While even this one single shape has enormous harmonic possibilities which we still haven't exhausted, the possibilities beyond this shape present an even greater frontier.

Lesson two: limits

  1. The wiggle room of tuning is due to the limitations of human perception.
  2. This limitation has pros and cons. For the same reason that our ears can tolerate [math]4\hspace{-3px}:\hspace{-3px}5.04\hspace{-3px}:\hspace{-3px}5.99[/math] as a [math]4\hspace{-3px}:\hspace{-3px}5\hspace{-3px}:\hspace{-3px}6[/math] major chord, we have trouble hearing chords with bigger numbers in their ratios as what they are. For example, suppose we sounded a [math]13\hspace{-3px}:\hspace{-3px}16\hspace{-3px}:\hspace{-3px}20[/math] chord outside of any musical context; a specially trained musician could hear this as a [math]13\hspace{-3px}:\hspace{-3px}16\hspace{-3px}:\hspace{-3px}20[/math] chord, but the average person would not. And no human today could hear a [math]113\hspace{-3px}:\hspace{-3px}157\hspace{-3px}:\hspace{-3px}203[/math] chord as [math]113\hspace{-3px}:\hspace{-3px}157\hspace{-3px}:\hspace{-3px}203[/math] without special tools or, again, a musical context that clarifies things.[5] This is because any chord that is that complex will find itself inside the wiggle room of a simpler chord, which intercepts our perception, or in other words, overshadows the more complex chord.
  3. So, humans have perceptual limits on the complexity of chords.
  4. However, those perceptual limits — even for an untrained ear — are higher than the limits of the standard temperament. When standard temperament adjusts ratios, while it stays within the wiggle room for the simplest chords, like major chords, for any chords beyond that, it strays too far outside the wiggle room, leaving them unrecognizable! We can't write with them or hear them at all in this tuning. For example the medium complexity chords [math]5\hspace{-3px}:\hspace{-3px}7\hspace{-3px}:\hspace{-3px}9[/math] and [math]8\hspace{-3px}:\hspace{-3px}10\hspace{-3px}:\hspace{-3px}11[/math] simply can't be played on a piano.
  5. It's not easy to tune ratios so that all the simplest and all the medium complexity chords can all be heard. It's a game of give and take. If you restrict yourself to only 12 pitches per octave, like on a piano, then there's only so many chords at a time that you can approximate, no matter how you tune those pitches. You could always squeeze in more and more pitches, which would give you more and more opportunities to approximate different chords, but at a certain point you will have added so many different pitches that it's impractical to write, play, or understand the music. So that's a different sort of complexity limit you'd come up against.
  6. A wide world of tunings are out there. Shop around and you'll find some you like, with just the right mix of chords, melodies, progressions you like, plus a level of practicality that you're comfortable with. And this will open up an even more enormous frontier of harmonic possibilities to you.

Lesson three: buzz

  1. While various tunings of a chord may all be recognizable as that chord, they do sound different from each other.
  2. In particular, chords have a special feeling when the ratios are exact. It's often described as a "buzzing" sound when the frequencies lock in perfectly. Many people think these chords sound better.
  3. So why not just use all exact ratios? Well, using only exact ratios may seem simpler, but in reality, it can be a lot trickier.
  4. The problem is related to the problem already mentioned about squeezing in more and more pitches. When you insist on ratios always being exact, you force yourself to use a lot more pitches in order to accomplish that.
  5. Many of these pitches end up being close to each other, and this is annoying for a couple reasons.
  6. Firstly, this wastefully increases the complexity of your pitch system, making it harder to write, play, and hear the music for almost no reason. It would be better to just have one pitch in that vicinity, rather than two, three, or more.
  7. Secondly, these close-together pitches tend to sound quite disjunct or mistuned if played together, or when you need to switch from one to the other during a melody or chord progression. It would be advantageous if you could just split the difference between them, rounding them all to the same pitch, so that no chord or melody ever sounded straight-up awkward; they all just sound a wee bit off all the time.
  8. So, while using tuning to create equivocations such as four-fifths-make-a-third reduces the number of frequencies in your system, which seems like it could be limiting, when it's done right, it makes music more efficient, and even opens up more options than were there before. Think of these equivocations like bridges connecting formerly impassable harmonic relationships, granting you newfound options for navigating harmonic space. Ultimately, we could say that a trade-off exists between exactness in the moment and exactness a few chord changes away. And both types of exactness matter in music. Having an exact [math]4\hspace{-3px}:\hspace{-3px}5\hspace{-3px}:\hspace{-3px}6[/math] is great, but then when you move around by exact [math]6[/math]'s and [math]4[/math]'s, you get to [math]\frac{6×6×6×6}{4×4×4×4} = 5.06[/math], which is not exactly the same as our [math]5[/math]. But if we moved around using these particular inexact [math]6[/math]'s and [math]4[/math]'s, like [math]\frac{5.99×5.99×5.99×5.99}{4×4×4×4} = 5.04[/math], then we'd end up at the exact same inexact [math]5[/math] that we're using, i.e. [math]5.04[/math]. So if you just accept a little inaccuracy in each moment, then you gain additional freedom to move around the way you like.
  9. You may find that you are one of the folks who falls in love with the "buzz" of exact chords. And you will find that you will go to the great lengths required to make it happen in your music, dealing with these challenges. And you will find yourself in great company!
  10. But some people actually dislike the buzzing effect, and so for these people, missing out on it is not even a bad thing. In this case the choice is clear: choose temperaments!

Lesson four: warning

  1. In the most typical case, tempering means adjusting the tuning of the prime numbers that make up the frequency ratios of just intonation — the harmonic building blocks of your music — only a little bit, so that you can still sense what chords and melodies are “supposed” to be, but in just such a way that the interval math “adds up” in more practical ways than it does in pure just intonation (JI).
  2. This is also what equal divisions (EDs) do, but where EDs go "all the way", compromising more JI accuracy for more ease of use, RTT finds a "middle path": minimizing the accuracies you sacrifice, while maximizing ease of use. Understanding that much of the “what”, you can refer to the table at the end here to see basically "why":
  3. The point is that a tempered tuning manages to score high for both usability and harmonic accuracy, and therefore the case can be made that it is better overall than either a straight ED or straight JI. On this table (which reflects my opinion), RTT got six total stars while ED and JI each only got five. (And this doesn't even account for the power RTT has to create fascinating new harmonic effects, like comma pumps and essentially tempered chords, which EDs can do to a lesser extent.) (And this table doesn't account for how many don’t find the distinctive buzzing sound of perfectly accurate JI to be a desirable effect.)
  4. But, you protest: this article series is pretty long, and it contains a bunch of gnarly diagrams and advanced math concepts, so how could RTT possibly be easier to use than JI? Well, what I’ve rated above is the ease of use after you’ve chosen your particular ED, RTT, or JI tuning. It’s the ease of writing, reading, reasoning about, communicating about, teaching, performing, listening to, and analyzing the music in said tuning. This is different from how simple it is to determine a desirable tuning up front.
  5. Determining desirable tunings is a whole other beast. Perhaps contrary to popular belief, xenharmonic musicians — composers and performers alike — can mostly insulate themselves from this stuff if they like. It’s fine to nab a popular and well-reviewed tuning off the shelf, without deeply understanding how or why it’s there, and just pump, jam, or riff away. There's a good chance you could naturally pick up what's cool about a tuning without ever learning the definition of "vanishing comma" or "generator". But if you do want to be deliberate about it, to mod something, rifle through the obscure section, or even discover your own tuning, then you must prepare to delve deeper into the xenharmonic fold.
  6. As for whether determining a middle path tuning is any harder than determining an ED or JI tuning, I think it would be fair to say that in the exact same way that a middle path tuning — once attained — combines the strengths of ED and of JI, determining a middle path tuning combines the challenges of determining good ED tunings and of determining good JI tunings. You have been warned.
ED RTT (middle path) JI
ease of use ★★★★ ★★★
harmonic accuracy ★★★ ★★★★

Douglas's personal introduction

I studied neither music nor mathematics in school, at least not extensively. If someone had told me I'd need linear algebra and stats for making the music I wanted to make one day, I'd've gotten more excited about it! I didn't manage to discover xenharmonics until just a few months before I graduated. I had no background with historical temperaments, and I spent most of my early xen years obsessing over JI, EDOs, Erv Wilson's ideas such as MOS and CPS, and my own bizarre experiments. So for nearly 15 years I got by with barely any understanding of what it really meant for music to have been written with "porcupine" or "mavila", or even "meantone".

It wasn't until fairly deep into volunteering my time to help Dave with the Sagittal notation he'd co-created that I realized he was also an expert on RTT, and so I began to pick his brain for help with mappings, commas, tunings, and such. Once he'd finally helped me get my feet wet with the basics, I dove straight into the deep end. At this point I'm still obsessing over how to make it easier for others to learn how to compose with regular temperaments, but I do plan to compose with them myself one day.

And maybe I'm deluding myself, but I think my unusual background has prepared me well for this task. Nowadays I'm a software engineer so I have the programming chops to deal with the math problems that come up in this material. And I have some storytelling skills from studying film in university and spending several years as a director and screenwriter. And I've also been a teacher so I like to think I'm halfway decent at explaining things.

Anyway, this series of articles is the reference I wish I'd had when I was learning RTT, or Regular Temperament Theory. There are other great resources out there, but this is how I would have liked to have learned it myself.

See also

In February 2023, Stephen Weigel interviewed us about this guide for his Now and Xen podcast.

To continue your journey along the Dave Keenan & Douglas Blumeyer's guide to RTT tuning series, we recommend you next read:

  • 2. Mappings: to see exactly how JI intervals are tempered by a regular temperament


  1. Where "approximating" means something like "having errors less than 30 cents", and "simple" means something like "involving a numerator and denominator whose product is less than 1000".
  2. There are weaker consonance effects that don't depend on coinciding partials, such as combination tone effects, but these have no necessary relationship to ratios of integers (small or otherwise) and therefore RTT is powerless to optimise them.
  3. We are going to teach you those aspects of linear algebra that you will need for RTT, assuming only that you have high-school math, but you will already have the required knowledge if you have done a course in either "linear algebra", "matrix algebra", "matrix theory" or "matrix analysis".
  4. Strictly speaking, it is the canonical form of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament. This will all be explained.
  5. It's important to be clear about what we mean here by "hearing" a chord such as this [math]113\hspace{-3px}:\hspace{-3px}157\hspace{-3px}:\hspace{-3px}203[/math] example. Let's begin by noting that the three pitches of a [math]113\hspace{-3px}:\hspace{-3px}157\hspace{-3px}:\hspace{-3px}203[/math] chord are in a quite similar proportion to those of a [math]5\hspace{-3px}:\hspace{-3px}7\hspace{-3px}:\hspace{-3px}9[/math] chord; we can see this in the fact that the nearby chord [math]115\hspace{-3px}:\hspace{-3px}161\hspace{-3px}:\hspace{-3px}207[/math], when divided by [math]23[/math], is exactly [math]5\hspace{-3px}:\hspace{-3px}7\hspace{-3px}:\hspace{-3px}9[/math]. Now, it's certainly an important phenomenon that if a trained listener was presented with the following three chords:
    1. the relatively simple [math]5\hspace{-3px}:\hspace{-3px}7\hspace{-3px}:\hspace{-3px}9[/math] chord,
    2. this similar but way more complex [math]113\hspace{-3px}:\hspace{-3px}157\hspace{-3px}:\hspace{-3px}203[/math] chord, and
    3. another similar chord which is even more complex still,
    this listener could reliably tell which was which. Among other things, the difference in nature of the periodicity buzz would set the feeling of the chords apart. But this is not the phenomenon being discussed here. When we say hearing this chord as what it is, we mean the ability to reliably identify its ratios, as if you were a subject being tested for the correct answer. And it's certainly understandable that by characterizing "identification" in this way — that is, like passing a test by machine-like itemization of numbers accurately — it may come across as unimportant, at least when compared with the transcendent wonder of experiencing music. But it's critical to understand that this sort of test is not the end in and of itself; it is merely the proof of having truly experienced a particular type of transcendent musical experience, namely, that of feeling a chord for the exact harmonic ratios it was intended to be. And it is this particular transcendent musical experience that RTT is designed to preserve when performing its approximations, and so it is this experience which this guide is discussing here. To be clear, there are definitely musical contexts where [math]113\hspace{-3px}:\hspace{-3px}157\hspace{-3px}:\hspace{-3px}203[/math] may be made hearable for what it is, and we don't intend to place ourselves squarely on the wrong side of history, since human faculties are enhanced ever further each year and maybe one day the average person with untrained ears will have such precise perception that they'll be able to hear this level of harmonic complexity for what it is. And there are a great many factors that contribute to our sense of consonance, concordance, chord complexity, etc. etc. and this is only one model of the problem, and it's a quite limited model. But it's the one this article is here to help introduce.