Getting back on the band wagon. Starting a new blog.

Wow, it’s been a long time since I wrote a post on here.

Over the intervening months and years I have thought several times about writing a book, or several books, and have made some small attempts towards that goal, but thinking more and more about the age we’re living in, the way we communicate and consume media leans heavily in favour of those which are internet friendly, bite-sized, and interactive. So I’m deciding to start a blog, which will hopefully document what I would like to do as a music educator. Some of it will be focused on my piano teaching, and what I do day to day (I teach in a few schools at the primary, intermediate and high school level, but more and more of my work is as a private teacher in students’ homes), and some of it will be focused on my xenharmonic/microtonal pursuits – what I have learned about tuning, how it applies to music, how musicians of any level can use it to make interesting music and to connect the dots between music, art, science, creation and analysis.

I’d like to explore both the fundamental elements of music (for beginner and intermediate musicians) and also the more specialised area of tuning, how it affects melody, harmony, timbre, technique, interpretation, style, notation, instrument design, and general musical thinking. It is interesting going down the road of teaching music from a tuning-agnostic, or tuning-variable point of view. If we are fully agnostic, it’s pretty hard to allow the use of any exact interval, but to appeal to the largest number of people and to make most of the huge corpus of existing music available for us to relate to I will be using some structures that most practicing musicians already know, and simply expanding their meanings to include what might happen outside of 12-tone equal temperament (12TET).

I’m not into telling people who have studied for years to “wipe the slate clean”, and so most of what I do microtonally can be understood through the lens of the everyday musician, or at least, I’d like to think so. I also want to make this intuitive to grasp, and interesting enough to spark creative exploration on the part of the reader (you). Feel free to ask any questions. I’m not much of a writer, so if there’s anything I can clarify, let me know. I have read some fantastic books which deal at least a little with tuning and how it can affect other musical elements. I’d like to try and inspire in a reader a fraction of what Helmholtz’ On the Sensations of Tone or Partch’s Genesis of a Music or Mathieu’s Harmonic Experience inspired in me, and try to stay true to my own direction at the same time. This is probably not a how-to guide on tuning of every kind, but hopefully it will be enough of an introduction to ideas which can be interwoven with common practice music theory.

Before I start my first post proper, maybe I should try to answer the question, why bother with microtonality? Many practicing musicians hear words like microtonal or alternative tunings and think something along the lines of I’m doing pretty well with the tuning I’ve got, why would I want anything different? It already sounds fine, and I wouldn’t know what to do with any extra notes

Thinking about tuning is just like thinking about any other element of music, it opens up a whole lot of doors into expression, the sorts of sounds and structures one can use, and I think it deserves as much respect as other variables which are taken for granted – style, form, timbre, texture, rhythm, etc. Not many musicians would submit themselves to an entire career playing music of one rhythm, or of one texture, or using one timbral colour, but so many musicians never have the doors opened to tunings outside 12TET, to the universe of other possibilities. It’s like the 12TET system is a planet, supporting all sorts of landscapes and forms of life, but it is still just one equal temperament (among the infinite, let alone the families of linear and higher dimensional regular temperaments, and all their irregular cousins, and of course, just intonation and its countless possibilities for pitch). Who says other planets out there don’t support life?

So many interesting things can be done only in particular tunings, and some of that has been very well explored in 12TET. Most of these features are not taught to musicians as features of the tuning system, but as musical facts. Here are a couple of them:

  • a chain of 12 perfect fifths is equivalent to 7 perfect octaves, so a “circle” of twelve equally sized perfect fifths can be made, and modulations can be made around the circle. F# and Gb major for example, become the same pitch, and therefore the same key, and the difference between them is lost, except in the notation.
  • a chain of 3 major thirds lands us at a perfect octave. This means we can have symmetrical augmented harmonies and melodies.
  • a chain of 4 minor thirds lands us at a perfect octave. This means we can have symmetrical diminished harmonies and melodies.
  • stacking two identical major seconds lands us on the sweetest major third, close to the simple ratio 4:5.
  • in a dominant seventh chord, the interval between the third and seventh is equivalent to its inversion, the interval between the seventh and third. This allows progressions where tritones hold and the rest of the chord moves around, and allows tritone substitution without modulation.

Following these four harmonic paths in just intonation leads to four distinct pitches, and none of those are the same as where we started (for those interested, the small resultant intervals have names too – respectively, the Pythagorean comma, the diesis, “major” diesisDydimic or syntonic comma, and the jubilisma, though I had to look up that last one!). More importantly, there are tuning systems that share these features, and we can group them in families, e.g. those which make the above intervals into unisons, or “temper out” those intervals. The temperament families referred to above would be pythagorean (12, 24, 36, 48, 60, 72, 84, 96), augmented (3, 6, 9, 12, 15, 18, 21, 24), diminished (4, 8, 12, 16, 20, 24, 28, 32), meantone (12, 19, 26, 31, 33, 43, 45, 50), and pajara (10, 12, 22, 24, 32, 34, 44, 46). I decided to give a few equal temperaments belonging to each family so you get the idea that these features are adopted on some planets, but certainly not all. Laws governing musical life are not the same everywhere in the universe. And in fact any musical universe can be dreamed up, based on any rules, or even a lack of such rules, to give any kind of imaginable structure or chaos.

There is also the (not so small) matter of sound. This is the real point for most microtonalists. Numbers are there for those interested, and I can certainly get carried away with them (as seen above) but music really should be about sounds, and the expressive capabilities gained when tuning is a flexible parameter are not insignificant. One can hear the subtle differences between string quartets playing with near-pure fifths and wide major thirds (often sought after for expressive melodic playing) vs those playing with near-pure thirds (great for music full of rich major and minor harmonies), or how a harpsichord reacts differently in 12TET vs in a good meantone or well-temperament for period repertoire. More often than not one also achieves different kinds of timbres, different kinds of harmonies, different kinds of melodies when playing for extended periods in different tunings.

Sorry, got a bit carried away on this first post. Will try to make the next one a bit more focused on exploring one topic at a time. Might write a bit about the three keyboard layouts I use, the reasons we have them and why they’re important, at least in my musical explorations.

Thanks all for following along. Hopefully back soon.

fifths. the spiral. circles. temperament.

just for anyone skeptical that a system of thirty one tones to the octave is ridiculous or impractical, keyboard (and other) instruments were built in the 16th, 20th and 21st century, and the Fokker organ in Amsterdam continues to see the performance of new compositions and old, and stands as a symbol of the thirty one tone movement.

some other keyboard instruments to tickle your fancy

and of course guitars

thirty one tone music is really out there, you just have to know where to look…

anyway, on to today’s main topic: the spiral of fifths.

so, i’m guessing, if you’re a modern musician, you’ve been taught that fifths are arranged in a circle of twelve notes, and if you go up or down twelve fifths from any note you end up back where you started right? twelve fifths up from Ab is Ab? right?

this is a modern take on ancient theory, to suit the 12-notes-per-octave agenda, equal or otherwise. in truth fifths should be arranged in a spiral, regardless of how they are tempered.

as we saw in the thirty one tone post yesterday, the more fifths we go up, the more sharps in the key, and the more fifths we go down, the more flat. we should name the keys accordingly, and not assume that any pitch in the spiral is enharmonically equivalent to any other pitch.

so if you wanted to go up and down in fifths from C, you’d have:
…Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx…

this spiral continues infinitely in both directions, at least in theory, regardless of the size of the fifths. it might help to associate the height and length of the spiral with pitch at that point, so that if we travel up one node, we are always going a fifth higher in pitch, and down three nodes means going down three fifths, or a major thirteenth.

if fifths are equal to 3/2, we have a Pythagorean system with each 12 fifths taking us up around 23c in pitch, so that in ‘enharmonic pairs’, the ‘sharper’ of the two will be higher in pitch when both notes are taken within the same octave. e.g. B# higher than C, G# higher than Ab.

if fifths are equal to 700c, as in 12EDO, then each twelve fifths we go up will take us up 0 cents in pitch, i.e. we ain’t goin’ anywhere honey . but i said the spiral was infinite didn’t i? well, the note names should take in to account where you are in the spiral, so a fifth up from D# has to be A#, not suddenly Bb. even though 12EDO tempers the two notes to the same pitch, we still spell according to the spiral. we still spell the scale of F# major as F# G# A# B C# D# E# F#, though many musicians would much prefer seeing the more familiar Bb, Eb and F substituted for those i ain’t got no damn E# key on my piano notes.

so spelling is according to the spiral, but most of the time we’re actually hearing a circle of twelve fifths right? just like the textbook theory? well yeah, in what are called circulating systems the spiral has been cut off at some point (enough double sharps already!), and turned into a circle.

in 12 note circulating tunings, including 12EDO and any of the many well-temperaments, we cut the spiral off at 12 pitches, effectively making 12 fifths up exactly the same as 0 fifths up. which means probably flattening at least a few of them a little bit from 3/2 in order to avoid a wolf between the top and bottom of the spiral, e.g. between what would have been G# and Eb, a diminished sixth, if we take the 12 note chain Eb Bb F C G D A E B F# C# G#.

this alteration of a perfect interval, in this case 3/2, in order to merge two notes into one, is called temperament, and we see it all the time. almost all instruments in the modern world are tempered, mostly equally. we can flatten fifths slightly so that 12 fifths make seven perfect octaves (as in 12EDO), or a little bit more so that four fifths makes a double compound 5/4 major third (as in quarter-comma meantone, with 31EDO very close), or even more so that three make a compound 5/3 major sixth (as in 1/3-comma meantone, 19EDO very close). or we can widen intervals, widen our fifths rather dramatically so that four make a double compound 9/7 super third (22EDO gets close to this).

of course, temperament doesn’t just apply to fifths. we’re used to 3 major thirds adding up to an octave, but this is a particular tempered worldview. in regular theory, three major thirds makes an augmented seventh. which is only ever equal to an octave when major thirds are tempered to 400c each. in most systems three major thirds will either come out flatter (if major thirds are less than 400c) or sharper (when they’re bigger) than an octave. if they’re just 5/4s, then we end up 41 cents below an octave, just like in 1/4 comma meantone. which makes progressions like Ab-C-E-G# rather exciting!!!

four minor thirds? a diminished ninth! four 6/5 thirds gets us to (6/5)^4=1296/625, or 62.5 cents sharper than an octave. ouch. in diatonic tunings (built in chains of fifths), this means when fifths are equal to 700c, augmented sevenths and diminished ninths are equal to octaves, when fifths are less than 700c (think meantones), augmented sevenths are lower and diminished ninths are higher, and when fifths are greater than 700c (for example Pythagorean or super-Pythagorean, superpyth for short), augmented sevenths are higher and diminished ninths are lower. phew.

so why is this important?

the concept of a chain of a single generator plus the period of an octave (known as linear [rank-two] tunings) is pretty fundamental to scale building . a lot of the time this means generating scales by fifths as we’ve done here, known as syntonic tuning systems, where the fifths could be 700c or 703c or 696c or 720c or 686c (if you just love that howl).

of course we could have three generators (rank-three), for example 2/1, 3/2 and 5/4, which would give us an infinite lattice of what is known as 5-limit just intonation, all ratios containing the products and divisors of primes 2, 3 and 5. but most of the time we’d choose a set of them as our scale, like this twelve note scale i posted earlier:
1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8, or the 22 shrutis of the Indian Classical tradition:
1/1 256/243 16/15 10/9 9/8 32/27 6/5 5/4 81/64 4/3 27/20 45/32 729/512 3/2 128/81 8/5 5/3 27/16 16/9 9/5 15/8 243/128 2/1

or rank-four, with say, 2/1, 3/2, 7/4 and 11/8, to give us a rather interesting set of pitches, or rank n, but rank one should do for now. and in any case, syntonic tunings are probably the best way to explain important families of tuning suitable for almost all musics of the western tradition, and to build on what many of you already know about music theory.

if you’re wondering what some good circulating fifth systems are, you can start by looking at equal temperaments which have a useable fifth and approximate other colourful intervals along the chain. or simply take a chain of fifths of any size you like and see what you find.

5 and 7 EDO, with fifths of 720c and 686c respectively, will probably be at the limits of tolerance to what you might call a useable fifth. so they might provide an upper and lower bound on some tunings to explore.

check out this page too

from smallest to biggest, a good selection of equal temperaments [along with a good selection of musical tastings] might be:

685.7c: 7EDO [this album is one of the coolest things i’ve heard, definitely not an ideal tuning but very interesting]

690.9c: 33EDO [not really recommended, but what the hell]

692.3c: 26EDO [i’m not really familiar with this one, a little strange]

694.7c: 19EDO [this one’s become rather popular, great for guitar, close to 1/3 comma]

696.0c: 50EDO [just amazing, my favourite, close to the 2/7 comma of Zarlino]

696.8c: 31EDO [funkadelic, popular, about as many frets as you can fit on a guitar, close to 1/4 comma]

697.7c: 43EDO [decent meantone, very close to 1/5 comma]

698.2c: 55EDO [Mozart’s tuning, great Baroque meantone close to 1/6 comma]

700.0c: 12EDO [your old friend] [no links necessary]

701.9c: 53EDO [absolutely wonderful for primes, also as a master system for Indian musics]

702.4c: 41EDO [like Pythagorean, but with more attitude]

[also the basic layout for this beast of a keyboard]

704.3c: 46EDO [great fun, just a little twisted]

705.8c: 17EDO [awesome as a guitar tuning]

709.1c: 22EDO [another rather popular one, very odd at first, but a gem]

711.1c: 27EDO [interesting…]

720.0c: 5EDO [great pentatonic, just a bit bizarre]

fwoah! that should do it for now. enjoy the links. next time i might spend a bit of time talking about my favourite of these, 50EDO, and why it’s so damn cool.

a brief introduction to tuning, numbers, history

so i’ve decided to write a blog about something i’m passionate about. it’s not so obvious what i should post here, who my intended audience is and all that so i guess i’ll start with some basics.

intonation in music had been a defining feature for millenia, before music could play played one had to ask, with what pitches? before one can think about melody, or harmony, or voice leading, or voicing, one must first decide on pitch.

it seems in our modern world pitch is given too little attention, with the most obvious step being selecting one from twelve, the twelve notes found on almost all instruments and in most musicians’ heads as the only notes, the ‘right’ notes, handed down through the ages from the hands of the musical gods. this is rubbish.

those twelve notes, twelve relationships, that have come to define so much about modern music, are an invention, one simple cold calculated solution to the problem of which notes to choose, and where to stop, in defining a musical universe that is practical to play around in. twelve equal divisions of the octave: 12EDO.

all in all it’s a rather good system for its simplicity, with twelve equally spaced pitches we get only twelve unique interval classes: a minor second, a major second, a minor third, a major third, a very nearly perfect fourth, an augmented fourth, a very nearly perfect fifth, a minor sixth, major sixth, minor seventh, major seventh, and perfect octave. we are taught in modern theory that these twelve intervals are enharmonically equivalent to other intervals, that for example an augmented fourth is ‘the same’ as a diminished fifth, because they share the same key on the instrument, they are the same pitch even though we might call them different names or give them a different symbol on the page. that we should try to spell according to the classical rules of harmony and voice leading, but that in practice it’s all the same, a diminished fourth IS a major third, and an augmented unison is a minor second, and so on and so forth…

but where did these interval classes come from? how long have we had this conflation of perhaps twenty-one different categories (or more) being crammed into a grid of twelve notes?

there was a time only a few hundred years ago when a G# could never be an Ab, where no two notes were ‘equivalent’ in the modern sense.  if you wanted an augmented fourth and not a diminished fifth, that’s exactly what you got, not some middle-of-the-road approximation of both.

12EDO is a tuning that optimises the tuning of fifths, their sums, and their octave inversion, fourths. its ancestor is a tuning known as Pythagorean, though Pythagoras was evidently not the first person to come up with it. The tuning uses one ratio as its generator, the ratio 3 to 2, meaning when you go up a 3/2 the higher note’s frequency is 3/2 times that of the lower note. in musical terms, 3/2 is a perfect fifth. Really perfect. beatless. beautiful. every third harmonic of the lower tone lining up with every second of the higher note, it’s a focussed powerful sound, very familiar to most traditional musics, and our modern 12EDO version of 7 ‘semitones’ is pretty darn close, with an error of less than 0.28%. if we take a chain of fifths downwards and upwards from any arbitrary pitch, we end up with a chain like this, each step to the right going up a perfect fifth:


or in musical notation, taking the starting frequency 1/1 as C, we have:


although Db is now like six octaves below F#, so let’s bring the whole scale within one octave [doubling a frequency brings it up an octave, halving it brings it down] so that each ratio is between 1 (the initial frequency) and 2 (the octave above)

256/243-128/81-32/27-16/9-4/3-1/1-3/2-9/8-27/16-81/64-243/128-729/512, rearranging from smallest to largest (i.e. lowest to highest in pitch):



Look familiar? twelve tone chromatic scale. the original, formalised actually well before Pythagoras in Ancient China (read the intro to Harry Partch’s Genesis of a Music for a more detailed and possibly humorous run-down on the history of tuning).

here though, the steps are Not equal, the augmented unison steps are larger than the minor seconds. and we can’t change the spelling willy-nilly without changing the ratios. F# means 6 fifths up from C, or 729/512, whereas Gb would be 6 fifths down, or 1024/729. different notes, Gb being less than a quarter of a modern semitone lower than its enharmonic partner. if we want to play F# with a fifth above it, and we’ve only got the twelve keys above, we have to use Db a the top, which at almost a quarter of a semitone flatter than a perfect fifth, doesn’t sound too pretty.

if we continue this chain we run into bundles of these close pairs, which means we either need an instrument with 53 keys (where the notes almost exatcly line up – see, or we need to stick to a set of twelve notes so we don’t wind up with any pairs. the clever thing is, if we narrow each fifth just a tad, the sharps will drop and the flats raise, so that if we set our fifths to the twelfth root of 2, as they are in 12EDO, all of the pairings merge to a single note

the perfect fifth is what gives most modern harmony its power, the reason power chords in rock and pop and fourth harmonisations in jazz are popular today is probably because the 4/3 fourth and 3/2 fifth are very well represented in 12EDO, just as the fifth was given extreme importance prior to the renaissance period. 3/2 is a beautiful stable consonance.

but it is not the only beautiful stable interval. ratios of the magic numbers 2 and 3 reigned supreme throughout a lot of early musical history, but 5 soon came to be seen as warm, emotive, and important in expressing human drama in a way ratios of 3 and 2 couldn’t do so easily. what’s special about 2, 3, and 5? they’re all primes, and though there are differing opinions on their fundamentality, primes are important building blocks for rational harmony.

if you’ve ever heard overtone singing in a tonal context, or tried tuning a fretless instrument playing solo, or heard any good a capella quartets or choirs or small string groups, you probably already know the sound of 5. 5/4 is the major third, the Perfect major third, the one most people will sing or play instinctively because in most contexts it sounds miles better than the 81/64 major third you get by going up four perfect fifths, or even the major third in 12EDO, which people keep telling us is ‘right’ and ‘in tune’…

so how about we substitute all those big ugly numbers out of our old scale with some simpler warmer harmonies? let’s draw up a little lattice with 3/2 fifths going horizontally and 5/4 major thirds vertically:

5/3   5/4  15/8 45/32
A-       E-    B-      F#-
4/3    1/1  3/2    9/8
F         C      G       D
16/15 8/5 6/5  9/5
Db+   Ab+ Eb+ Bb+        or, arranged by pitch:

C       Db+    D    Eb+ E-    F      F#-     G    Ab+  A-   Bb+  B-

still a 12-tone chromatic scale, but less even, with 34 different intervals…! a whole bunch of flavours to try!

i feel like i should stop here, didn’t quite expect to write that long of a first post intro, but hopefully it gives you a little taste of what numbers might mean in music, and perhaps the inkling that 12EDO might not be the ideal musical universe after all