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.