what music theory do i need to start tuning?

First posted 22/09/2018, edited 7/11/2018

You don’t really need to know anything to start, many have gone into the field with only their ears and instrument and just does what feels the most natural. Sometimes what turns out to be the most natural is similar to what we’ve grown up with, or similar to the simple fractions our ears are so good at identifying, so called “just intonation”. Sometimes a xenharmonic explorer tunes up something totally unexpected and full of the unfamiliar. Some musicians (even some working at a very high level) are averse to theory, but I think it helps to know at least a little bit of what we are doing here, in order to learn from each other, share ideas, and grow all kinds of new music that will enrich future generations.

I thought I’d start with some basics. In order to keep them applicable to microtonal music only a couple of definitions need to be broadened, so the musical alphabet, notes, accidentals, scales and keys is a good place to start.

Most of the world shares musical traditions based around 5-note (pentatonic) and 7-note (heptatonic) scales, and an octave framework. Even in most modern music, these are the two most popular numbers for notes per octave in melodies and harmonies in a huge number of musical styles. You may even ask “why octaves?”, but I will just say for now that humans on average have an amazingly accute understanding of the octave as a reference, and the concept of octave equivalence is pretty well ingrained in us. Some xenharmonic musicians have gone against that, but that is a rabbit hole for another day.

Although pentatonic systems have been huge for melody for thousands of years, most musical cultures have developed a heptatonic system for talking about, counting, singing, and reading, musical notes. As the concept of zero wasn’t huge in many parts until quite a bit later, musicians tended to label the notes in their scale starting with the first one and proceeding through to the last one in each octave, where the scale pattern might be able to be seen repeated up and down the range of the instrument, or where certain scale degrees were flexible to permit subtle changes necessary for particular nuances. Many cultures used some kind of alphabet or syllable system to count notes in each octave. The most popular systems around the world are presently {A, B, C, D, E, F, G, A}, the “letter system” and {Do, Re, Mi, Fa, Sol, La, Ti/Si, Do}, “solfege”, with {Sa, Re, Ga, Ma, Pa, Dha, Ni, Sa}, “sargam” also another notable example particularly across the Indian subcontinent. We could also write {1, 2, 3, 4, 5, 6, 7, 8}, which is the basis for interval naming systems in most languages, where notes a step apart are said to form a 2nd, e.g. A-B or Do-Re or Sa-Re, and distances increase with each scale degree, e.g. C-G, Fa-Do, and Ga-Ni all being 5ths (if you miscount, make sure you’ve counted both endpoints – this is the same system of counting places in a race, but not the same counting system we use to work out, say, a difference in age).

Let’s relate these systems to each other so readers from slightly different backgrounds can still make use of the same logic. The letter system {A, B, C, …} is fixed – that is, the letters are not thought to change pitch depending on the music. However, the other three systems are usually thought of as moveable, so that the first note name lines up with the tonic or home note or pitch centre in the music. Where those systems are tied down to a fixed pitch, e.g. fixed solfege, Do is usually made equivalent to C.

Do    =   Sa     =  1
Re    =   Re/Ri =  2
Mi    =   Ga     =  3
Fa    =   Ma     =  4
Sol   =   Pa      =  5
La    =   Dha   =  6
Ti/Si =   Ni      =  7
[Do   =   Sa      = 8, although considering the octave the start of the next set is also just fine]

The exact pitch of these musical notes can depend on the instrument, technique and style, but even more important than that is the scale or key where they’re located. In most contemporary music which uses the letter system, letters are combined with sharp (♯) and flat (♭) to show these differences, and while some solfege systems use change in vowel quality to reflect these changes, they are generally not reflected in sargam, so I’d like to stick to the letter system as the quickest and arguable most universal way to refer to musical notes (at least in English).

We can make two of the most common musical scales from around the world if we take the interval of a 5th, for example between D and A, and continue to chain 5ths from either side. Taking our chain to five notes we might have C-G-D-A-E, an example of a pentatonic set, which we could rearrange in pitch order starting from any of the notes, i.e. C-D-E-G-A, D-E-G-A-C, E-G-A-C-D, G-A-C-D-E, or A-C-D-E-G. Notice how we get 5 modes out of a 5-note scale. This is a great set of notes in most tunings, as long as there is something like a 5th (otherwise we will end up with rather odd things happening, and the pitch order of our notes won’t be preserved, e.g. A might be higher than C or lower than G). The tuning range for 5ths where this works, i.e. the tuning forms a constant structure is rather large, the 5ths can be as small as half an octave, or as large as two-thirds of an octave. The convention is to write these octave fractions using a backslash, i.e. 1\2 and 2\3, as a regular slash is reserved for the ratios of just intonation, where for example 2/1 would be a perfect octave above the tonic 1/1, and 3/2 would be a perfect fifth above the tonic, with the occasional convention that inverse ratios represent pitches below the tonic.

[I’m probably using way too much jargon. Some of these things are easily searched online, but usually it’s just easier to ask people directly, as the usage isn’t totally standardised, but we are trying to create a new musical language that adequately deals with the intricacies of tuning. You can find me here, on facebook, or you can ask the Xenharmonic Alliance facebook group if you have any questions]

If we continue our chain of fifths out to 7 notes, F-C-G-D-A-E-B, we end up with a heptatonic scale with 7 modes, each starting on a different letter, most commonly called Ionian/Major (C), Dorian (D), Phrygian (E), Lydian (F), Mixolydian/Dominant (G), Aeolian/Natural minor (A) and Locrian (B).

Now where tuning comes into this is, these seven letters need not be equally spaced, and the fifths may not all be the same size. If they are, then we end up with a circle of 7 fifths, and our tuning is 7-tone equal temperament (7TET). Here, taking us up or down 7 fifths returns us to the same place, so there is no need of sharp or flat symbols, and there are only 7 possible intervals within the span of an octave. However, most of the world doesn’t use 7TET.

Let’s expand our chain of fifths from 7 notes to 12. Using our usual method of going down a fifth from F we get B, but we already have a note called B (7 fifths above), so we will call this one “B-flat”, or B♭. This 8 note set was used for a while in medieval music, thought of as 6 fixed scale degrees with the B♭/B flexible depending on the direction of the melody, and what fitted best, and giving rise to a few more options in terms of harmony. Another fifth down from B♭ and we get E♭. Taking our chain further in the other direction, and we encounter a similar problem, a fifth up from B seems to be F, so we call our new F, “F-sharp”, or F♯, and we continue with C♯ and G♯. Now we have 12 notes: E♭-B♭-F-C-G-D-A-E-B-F♯-C♯-G♯. This has been one of the most popular sets of notes for western music involving melody and harmony, in a variety of tunings over the last 650 years or so. It was also popularised as a good layout on keyboards, and the layout on 12-note-per-octave keyboards everywhere is called Halberstadt after an organ in Halberstadt from 1361 using the layout, with the 7 letters as larger keys towards the front, and the 5 accidentals (notes involving a sharp or flat) as smaller raised keys towards the back. Many of us now know the keyboard with each octave laid out as 7 white and 5 black keys.

With 12 notes we have many more than 7 intervals, but instead, even if each of the eleven fifths in the chain is the same size, we will have 23 sizes of interval, each generatable by stacking fifths (our system’s generating interval or generator) along with reducing by octaves (our system’s period). The concept of a chain of pitches generated by any two unique intervals is sometimes called a linear tuning or a rank-2 or 2D tuning, as it takes two dimensions to lay out all the possible pitches of such a sytem, although the octave component is often ignored and the pitches displayed on a line (like I have already been doing in this post). Here are our intervals.

augmented unison (A1) from C, Eb, F, G, Bb ——————– +7 fifths
minor second (m2) from C#, D, E, F#, G#, A, B —————– -5 fifths
major second (M2) from C, D, Eb, E, F, F#, G, A, Bb, B —— +2 fifths
augmented second (A2) from Eb, F, Bb ————————— +9 fifths
diminished third (d3) from C#, G# ———————————- -10 fifths
minor third (m3) from C, C#, D, E, F#, G, G#, A, B ———— -3 fifths
major third (M3) from C, D, Eb, E, F, G, A, Bb —————— +4 fifths
augmented third (A3) from Eb ————————————— +11 fifths
diminished fourth (d4) from C#, F#, G#, B ———————- -8 fifths
perfect fourth (P4), from C, C#, D, E, F, F#, G, G#, A, Bb, B -1 fifth
augmented fourth (A4) from C, D, Eb, F, G, Bb ————— +6 fifths
diminished fifth (d5) from C#, E, F#, G#, A, B —————– -6 fifths
perfect fifth (P5) from C, C#, D, Eb, E, F, F#, G, A, Bb, B — +1 fifth
augmented fifth (A5) from C, Eb, F, G, Bb ———————- +8 fifths
diminished sixth (d6) from G# ————————————– -11 fifths
minor sixth (m6) from C#, D, E, F#, G, G#, A, B ————— -4 fifths
major sixth (M6) from C, D, Eb, E, F, G, A, Bb, B ————- +3 fifths
augmented sixth (A6) from Eb, Bb ——————————– +10 fifths
diminished seventh (d7) from C#, F#, G# ———————– -9 fifths
minor seventh (m7) from C, C#, D, E, F, F#, G, G#, A, B — -2 fifths
major seventh (M7) from C, D, Eb, F, G, A, Bb —————- +5 fifths
diminished octave (d8) from C#, E, F#, G#, B —————— -7 fifths
perfect octave (P8) everywhere ————————————- 0 fifths

Some of these might look unusual to a musician just getting into more advanced harmony, but all of these, and sometimes a few more, pop up all over the place in the corpus of music written down up to the present time, especially those taking less than 7 fifths to generate.

How are our 12 notes and 23 resultant intervals to be tuned?

There are huge number of possible solutions, but I shall take us through three of the most important tuning ranges when dealing with 12-note chains of fifths like this.

If one tuned the fifths to be as harmonious as possible, one would probably find that each perfect fifth had its lower and upper notes’ frequencies in a ratio of 2:3. A 2:3 fifth is often called a just or pure fifth, and a chain of these 2:3 ratios (with a bit of 1:2 octave reduction when necessary) gets us a famous tuning we now know as Pythagorean, after Pythagoras, who was famous for discovering or rediscovering the rational proportions involving the prime numbers 2 and 3 that made harmonious sounds, for example the 1:1 perfect unison, 1:2 perfect octave, 1:3 perfect twelfth, 2:3 perfect fifth, 1:4 double octave, 3:4 perfect fourth, 3:8 perfect eleventh, etc. When two 2:3 perfect fifths are stacked one ends up with (2*2):(3*3) = 4:9, a rather sonorous major ninth.When four 2:3 fifths are stacked one gets 16:81, to most ears a rather large and active major third plus two octaves. Bringing these ratios within the octave for easy comparison with each other we have an 8:9 major second, a 64:81 major third, and a 3:4 perfect fourth. One can tell just looking at the numbers that the major third is going to be acoustically more complex and therefore less stable, but it lends itself very nicely to resolutions to a simpler 2:3 fifth or 3:4 fourth, as in much early European harmony. Continuing, one finds a minor second (-5 fifths) with a ratio of 243:256, slightly smaller than the standard half-step or semitone on today’s guitars and pianos, but rather lovely as a melodic movement or for a brief moment of tension before release. If we go a bit further the other way, we find a different type of semitone, the chromatic semitone or augmented unison, with a ratio of 2048:2187 (+7 2:3, fifths, reduced by several octaves). We find the size of this semitone is almost exactly in the ratio of 5:4 with the minor second, so we could, like others have done before, measure intervals using these small parts, called commas, or kommas, with 4 parts to the minor second, 5 parts to the augmented unison, 9 parts to the major second, 13 parts to the minor third, etc, and realise that an octave would be 53 parts, meaning even if we continued our chain endlessly to generate new Pythagorean ratios, the 53-tone equal temperament would still be a very good approximation, and for most purposes is the best simple model for Pythagorean intonation.

Another option would be to aim for sweeter, simpler major thirds. The simplest ratio in the major third region is 4:5, one comma narrower* than the Pythagorean 64:81 major third. Say we like these thirds a lot. And we want to make them our default major third. Since we have 4:5 one comma narrower than 64:81, and both span 4 perfect fifths, if we narrow each fifth by 1/4 of that comma, then our originally 64:81 major third becomes 4:5. This tuning is known as (1/4-comma) meantone, and it ruled most of the western world of music between the 15th and 18th centuries, with some usage stretching even further. It was the default tuning for most keyboard instruments, as its thirds and sixths were sweet and close to simple ratios (4:5, 5:6, 3:5, 5:8) it gave sonorous harmonies, although bare fifths were less pleasant than in Pythagorean. if we look at the semitones in meantone, their sizes are the other way around, with the augmented unison smaller, around 2/5-tone, and the minor second around 3/5-tone, their difference being the -12 fifth diminished second, here being equivalent to the ratio 125:128, which we can think of as an octave less three 4:5 major thirds. In ratio arithmetic, this is (1:2)x(5:4)x(5:4)x(5:4)=125:128.

*actually a syntonic, as opposed to Pythagorean, comma, but their difference is less than two cents, almost imperceptible.

So why doesn’t everybody still tune their keyboard to Pythagorean, or meantone? The main reason is the intervals formed by bridging the distant ends of the chain of fifths – since we don’t necessarily have a circle, but rather a 12-note chain, as spelled out earlier: Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#, and so the interval from G#-Eb, looking to many modern musicians like a perfect fifth spelled differently, is actually a different interval with a different sound, a diminished sixth. In Pythagorean, this interval is a comma (or roughly 1/53 of an octave) narrower than a perfect fifth, so is mostly unsuitable for stable harmonies, though one might be able to slip one in very briefly in passing without too many musicians noticing. In meantone, the diminished sixth is sharp of a perfect fifth by almost twice that amount, and the interval from G#-Eb in meantone was often called a “wolf” fifth – perhaps you can imagine why.

Creating a circle of twelve fifths and making sure that all intervals and keys were usable was one of the major developments in tuning theory in the 18th century. A 12-tone circulating system with sufficiently acceptable intervals in all twelve tonal centres came to be known as a well-temperament, and most well-temperaments have a range of key colours that lie somewhere between meantone and Pythagorean – that is, some with slightly narrow fifths and rich, warm major thirds, and others with strong pure fifths and more active, energetic major thirds. Even if an interval is spelled as a diminished sixth, it will still function well as a perfect fifth, and so we have an enharmonic equivalence, e.g. between C#-Db, D#-Eb, etc, or more generally, +12 fifths = a perfect unison (octave reduced).

Knowing about these three historically popular 12-note tunings teaches us about a few concepts which are really helpful for exploring tuning theory:
1) ratios (and the harmonic series, which I didn’t go into, but with which all ratios can be derived)
2) regular temperament
3) irregular temperament
4) mapping, e.g. of abstract intervals to ratios, M3=4:5 in meantone, or physical mapping of pitches on a keyboard
5) enharmonic equivalence, e.g. P5=d6 in a well-temperament.
6) different interval sizes within a sort of fuzzy “bucket”, e.g. d6<P5 in Pythagorean.

If any of these approaches are carried beyond twelve notes or applied to different numbers of pitches, or to different ratios, they are amazing resources, and can be far more exciting than the now-familiar 12-tone equal temperament, while still giving the musician the ability to use most of the knowledge they’ve gained about other tuning systems or music in general.

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.