Microtonality allows for the use of intervals outside the standard Western tuning – and/or, depending on your perspective, better representations of familiar intervals – but it can be difficult to explore both conceptually and practically. This document starts with some basic just-intonation-centric microtonal concepts and works toward a particular approach with high utility and manageable complexity. Particularly, we are interested in a highly accurate system (thus excluding things like 19edo) that is highly relatable to our existing concepts of musical intervals and notation (thus excluding most non-diatonic linear temperaments). This document seeks to explain concepts in a way that will be comprehensible to someone who has a solid understanding of conventional Western music theory but little experience with microtonality.
But first, some definitions.
A "temperament" is a way of thinking about the system of intervals created by a stacking and combining a (more or less) specific set of "generating" intervals. Commonly used temperaments usually have 1, 2, or 3 generators, one of which is typically either an octave or an octave fraction (like the 1/12th-octave semitone used by 12edo).
A "planar temperament" is a temperament that has 3 generators, one of which is an octave. These temperaments can be displayed as a 2-dimensional grid or lattice structure by assuming octave equivalence between intervals, meaning we're not particularly interested in differentiating between a fifth and a twelfth. Having 2 dimensions also means that planar temperaments can be notated using 2 sets of accidentals (♯ and ♭ would be one set).
Just intonation means using intervals based on "exact" integer ratios (to the extent possible on the instrument in question). The motivation for this typically stems from the fact that the simplest integer ratios, 2/1 and 3/2, are cross-culturally recognized as exceptionally stable consonances, and justification is often provided in terms of the harmonic series, the sequence of integer-multiple overtones of the fundamental frequency of a vibrating string, air column, or similar medium. Chords composed of integer ratios have a particular quality because the component sound waves stay in phase with each other, although if the integers are large the effect is much less noticeable because the relationship between the waves' periods becomes complex.
Just intonation chords also tend to avoid dissonance between the harmonics of their component tones. For example, in a just 5/4 dyad, the fifth partial of the lower tone is the same pitch as the fourth partial of the higher tone, forming a unison. In 12edo's closest approximation of this dyad, the partials are a noticeable 14 cents apart. Of course, the specific effects of harmonic interactions depend on the specific intervals and timbres (i.e. harmonic spectra) in use, and some instruments also generate prominent inharmonic (i.e. non-integer) partials.
But back to writing about planar temperaments – let's start with a diagram of the interval space generated by a just perfect fifth (3/2) and just major third (5/4):
Each tile displays an interval's integer ratio, cents value, and "error" – its deviation in cents from just intonation. Because our generators are just intervals, all our errors are zero. One step right from any tile corresponds to our perfect fifth, and one step up corresponds to our major third.
Each tile's darkness is proportional to the sum of the prime factors of the integers in its ratio. This is a rough way to estimate how "complex" an interval will sound to the ear, according to the tenets of just intonation, and therefore (inversely) how useful it will be in forming musical consonances. A tile's color is determined by which primes are involved: ratios involving only primes 2 and 3 are gray, while ratios involving 5 are tinted red.
It's also worth noting that this is just a subset of the actual interval space, which extends infinitely without repeating. This is true for all axes generated by intervals that are not octave fractions.
Near the origin of the diagram, our 1/1 unison, are the simplest intervals. The ratios 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, and 15/8 form a just major scale. As we get farther from the origin, the intervals become more complex and less useful for constructing consonances, so music in 5-limit just intonation (that is, just intonation where the highest prime used is 5) generally sticks to intervals near the center of the diagram.
Of course, we can still recognize intervals if they're a few cents out of tune. Performing music in just intonation would be quite difficult otherwise, given the complexities involved in the intonation of acoustic instruments. In fact, the threshold of audible indistinguishability is usually cited as 5 or 6 cents. If we account for that in our diagram and consider prime factors up to 11, we get the following result:
The new colors represent new primes: blue tiles are 7-based intervals, green tiles are 11-based intervals, and intermediate colors are formed by intervals that use multiple primes above 3. The practice of using certain just intervals to approximate other just intervals in this way is called fudging. Of course, the pitches themselves haven't changed – we're just using some of them in a different conceptual framework.
There is a fairly straightforward approach to notating this interval space. We can start by notating the center row using the symbols we're used to: sharps, flats, and a chain of 7 fifths (F C G D A E B). In this system, a sharp or flat means raising or lowering a pitch by five fifths minus three octaves, about 114 cents. We can use another pair of accidentals to represent raising or lowering by the difference between 81/64 and 5/4, which is about 22 cents. Ups (^) and downs (v) are one established choice for this interval. The result looks like this:
A just major triad with pitches 1/1, 5/4, and 3/2 is notated as C vE G ("C down-E G") in this system, and its chord symbol is Cv ("C down-major"). The 4:5:6:7 "harmonic tetrad" formed by adding 7/4 to this chord is notated as C vE G vB♭.
Note that there are no enharmonic equivalents in this space like there are in 12edo; F♯ and G♭, for example, are two different pitches.
Strictly speaking, nothing we've discussed so far is a temperament, since a temperament implies that the cents values used for some intervals systematically deviate from their just equivalents. One actual temperament we could use that would give the same approximate grid layout is called marvel temperament. Marvel is usually defined as the temperament that "tempers out" the ratio 225/224, meaning that it canonically uses the same interval to represent any pair of intervals that differ only by that ratio – for example, 16/15 and 15/14. Its non-octave generators approximate the pure 3/2 and 5/4 generators that we have used so far, but a specific value for these generators isn't implied by the temperament alone.
One criterion for determining the optimal tuning for a temperament is POTE, or Pure Octaves Tenney Euclidian. This means that our octaves remain at exactly 1200 cents, but our other generators are adjusted in a way that minimizes error for the Tenney Euclidian metric (which is beyond the scope of this document), assuming certain mappings of generators onto primes. The 7-limit POTE tuning for marvel yields the following result:
Our simple 3- and 5-limit interval mappings have stayed the same, but many of the more complex ones have been replaced with simpler ratios of higher primes. All the properties discussed for the just intonation lattice, including principles of notation, also hold for this one.
This doesn't mean that the same notation corresponds to the same ratios, however, since some of our pitches have drifted. A C harmonic tetrad now uses a vvA♯ ("double-down A sharp") instead of a vB♭.
The 700.4-cent generator used above is strikingly close to the 700-cent fifth of 12edo. Additionally, an up or down is now about 18 cents, which is close to 1/6th of a 12edo semitone and suggests using 72edo as a tuning (12 * 6 = 72). If we "quantize" both marvel generators to 72edo intervals, this is the result:
Compared to the POTE lattice, the error values are slightly different and a few ratios have changed. Most notably, the tile for 16/15 is now displayed as 15/14, since the cents value is closer to 15/14 then before – but remember that by the definition of marvel temperament, these ratios are represented by the same interval.
There is one very important difference between this diagram and the previous ones: this diagram shows all the pitches that are possible in our tuning! The leftmost and rightmost columns are identical, and every interval on the top row has an exact match on the bottom row. This has implications for our notation:
Now a double sharp or flat is equal to a 12edo whole tone, meaning that at most one sharp or flat is needed to represent any interval, enharmonically speaking. At most 3 ups or downs may be needed, and 3 ups are now equivalent to 3 downs plus one semitone.
Our familiar pitch-agnostic interval terminology can also be adapted to this system. 6/5 is an ^m3 ("up minor third"), and 7/4 is a vvA6 ("double-down augmented sixth").
One reason I'm interested in this marvel-72edo system is that it's fairly accessible on a regular MIDI keyboard with my microtonal music sequencer Faunatone. The keyboard layout handles the 12edo-equivalent chain of fifths, and ups and downs can be handled via support for accidentals and key signatures. 41edo and 53edo achieve similar levels of accuracy in the same domain with fewer notes, but they are harder to organize both conceptually and physically unless you have a large generalized keyboard (and these tend to be quite expensive).
This system is versatile, but it's certainly not the only practical approach to microtonality. It's also not really suitable for instruments like pianos that use a discrete, linearly-arranged set of pitches, unless you have 6 pianos (this has been done). It's technically possible to play 72edo music on a guitar using a tuning like E vA vvD vvvG ^^B ^E, but playing a fifth dyad is impossible in this tuning without bending a string.
72edo's approximations of intervals beyond the 11-limit are generally not as accurate, so musicians interested in exploring higher primes in a near-just tuning are encouraged to look elsewhere.