Upon the two Series of Seven, the major key and the minor key, the whole art of music has been established; one limitation brings on the other.
To each of these a definite character has been attributed; we have learned and have taught that they should be heard as contrasts, and they have gradually acquired the significance of symbols:—Major and Minor—Maggiore e Minore—Contentment and Discontent—Joy and Sorrow—Light and Shade. The harmonic symbols have fenced in the expression of music, from Bach to Wagner, and yet further on until to-day and the day after to-morrow. Minor is employed with the same intention, and has the same effect upon us now, as two hundred years ago. Nowadays it is no longer possible to “compose” a funeral march, for it already exists, once for all. Even the least informed non-professional knows what to expect when a funeral march—whichever you please—is to be played. Even such an one can anticipate the difference between a symphony in major and one in minor. We are tyrannized by Major and Minor—by the bifurcated garment.
– Ferruccio Busoni, Sketch of a New Esthetic of Music (1907), tr. Theodore Baker
Pitch is a continuum. Harmony accomodates not only triads built of major and minor thirds, but infinitely many triads, tetrads, pentads, and even sextads and septads. It’s in a composer’s interest to learn how to use the entire continuum of pitch and harmony for musical effect. However, a tuning system that comes within the just-noticeable difference of any desired pitch would have to have, at the lowest estimate, over a hundred notes per octave; we don’t have the appropriate tools to jump into the deep end. Instead, we can ease ourselves into the water by exploring successively denser systems, each introducing a few new intervals and possibilities while preserving some familiar ones.
We can proceed by generalizing the 12edo standard based on certain properties of 12edo which we value. These might include octave period, transpositional invariance, near-just perfect fifths, consonant major and minor triads, and composite note count. I will address each of these properties.
Nearly all tuning systems repeat at a certain period interval, such that if a pitch is in the tuning, the pitches one period higher or lower are also in the tuning. Due to the phenomenon of octave equivalence, it’s intuitive to make the octave the period of a tuning, and in practice most tuning systems do indeed have octave periods. It’s therefore reasonable to restrict our search to tunings with octave periods.
In 12edo, if we transpose a set of pitches by a certain number of steps, the intervals between the resulting set of pitches will be identical to the original intervals. For example, all 12 major keys in 12edo have the same tunings for the major third, perfect fifth, et cetera. This property is called transpositional invariance. An example of a tuning without transpositional invariance is the diatonic scale, where each of its 7 modes is different and must be learned to understand the whole scale. From this example, we can see that to learn a larger tuning system without transpositional invariance would be a daunting prospect! It’s therefore reasonable to restrict our search to tunings with transpositional invariance.
All tunings with octave periods and transpositional invariance consist of divisions of the octave into a certain number of equally sized steps; that is, EDOs.
The just perfect fifth is extremely important to music, being (along with its relative under octave equivalence, the tritave) the simplest just interval not to be perceived as an equivalence, and thus a powerful source of consonance. By adding to an empty scale successive pitches a perfect fifth apart under octave equivalence, we can generate the pentatonic scale, the diatonic scale, and the 12-note Pythagorean tuning, each broadly used in human music across time and space. Moreover, the perfect fifth and octave are sufficient to generate consonant quintal and quartal chords, providing a minimal framework for consonant harmony. While no finite EDO contains a just perfect fifth, we can identify EDOs that contain close approximations to the just perfect fifth.
Given that we can generate consonant chords with only a perfect fifth, the presence of consonant major and minor triads is arguably redundant. This is especially true if we consider that the presense of consonant major and minor triads would allow us to easily fall back on standard musical logic, which is counterproductive. After some exploration, you will most likely find that you can grow comfortable without these triads, as I have. As for compositeness: any small and composite edo is likely to share factors, and therefore intervals, with 12edo. (The smallest composite edo to share no factors with 12edo is 25edo – more than twice 12edo’s size!) Preserving intervals of 12edo would be contrary to our purpose. Therefore, I’m comfortable abandoning these two properties of 12edo.
With these properties determined, we know exactly which kind of tunings we are looking for: EDOs with close approximations to the just perfect fifth. In particular, we want to find the first such tuning after 12edo in note count. Based on listening, I find it easy to discard 13edo, 14edo, 15edo, and 16edo for our purposes based on their closest approximations of the just perfect fifth. This leaves 17edo, the tuning dividing the octave into 17 equally spaced steps. This is the tuning which we will explore today!
I'll end this section by acknowledging that it's possible to come to different conclusions on these properties. I will discuss other tuning options which fulfil different properties in the appendix.
Standard musical notation is based on the circle of fifths, and can still work well in 17edo, just with different notational equivalences observed. (For example, C# is higher than Db, and E# and Gb have the same pitch; for reasons relating to 17edo's sharper fifth, a sharp or flat alters a note by 2 steps of 17edo rather than 1 as you might expect!) I will use "#" and "b" to stand for the sharp and flat accidentals in text. With standard notation, an ascending octave of 17edo from C to C can be notated C, Db, C#, D, Eb, D#, E, F, Gb, F#, G, Ab, G#, A, Bb, A#, B, C:
However, in certain cases this can be counterintuitive - for example, the neutral third above C is notated D#. Also note that as we move linearly up in pitch, the note we write on the staff jumps up and down. For this reason, it makes sense to introduce the Stein-Zimmermann half-sharp and half-flat accidentals. In text, I will use t for half-sharp and d for half-flat. Now we can write the neutral third above C as Ed. An ascending octave of 17edo now looks like this:
For intervals, minor, neutral, major, diminished, perfect, and augmented will be abbreviated respectively m, N, M, d, P, and A. For example, P4 for the perfect fourth and N6 for the neutral sixth. A will only ever stand for "augmented", not for the pitch class – "A4" should be read as an augmented fourth, not as the pitch usually standardized to 440 Hz!
Another way of expressing intervals is directly as steps of 17edo. An interval of n steps of 17edo is denoted n\17. So, P4 corresponds to 7\17 and N6 corresponds to 12\17.
Scales will usually be written as a sequence of step sizes. For example, "3 3 3 1 3 3 1" represents the sequence of major seconds (3\17) and minor seconds (1\17) that make up the Lydian mode of the diatonic scale.
There's only so much you can learn through reading, so I want to lead with some examples of 17edo being practically used in composition. This section contains a series of short etudes in various 17edo modes, which are described more technically further down. I hope to be adding more etudes every so often — my ambition is that every scale I discuss in writing will have a musical example. For each etude I'll also include a short description of my approach to the mode in question.
The Lydian mode is constructed by six P5s above the root, which lends it a certain stability. Among the diatonic modes, Lydian is the only one to contain an A4, so this interval by itself contains lots of Lydian color. Harmonically, the M3, M7 and A4 can be tense when placed closely against the root, but when placed in higher octaves and contextualized with quintal harmony, they can be calmed. Melodically, the tense M7 and A4 can resolve to the stable fifth and octave, while the M3 can be treated as a passing tone up to the A4. For me, Lydian conveys a sense of solar brightness and majesty, like a sunrise.
Taken as a single harmonic object, the Ionian mode actually sounds quite unusual and more dissonant than some other modes. Ionian is the only diatonic mode to contain both a P4 and a M7. It's possible to place these degrees directly above the root to form a sour-sounding quartal structure. For larger voicings, one way to proceed is to cluster the P4 with notes that form consonances with it (1, P5, M2) and the M7 with notes that form consonances with it (M3, M6, M2). For me, Ionian sounds like a brighter Mixolydian, tart and sour where Mixolydian is sweet.
This table contains basic information on a wide variety of heptatonic modes in 17edo that I've subjectively deemed useful.
In this table, a mode's "distinguishing features" are the degree or co-occurring degrees that distinguish it from all other modes of the same parent scale. For example, Cilician's distinguishing features are M2 and P4, meaning that no other mode of the same parent scale (malacotonic) has both a major second and a perfect fourth.
A mode's "non-diatonic features" are any non-diatonic degrees, or sets of degrees that don't co-occur in any diatonic mode. (The latter usually comes up because the N4 and N5 appear in diatonic as respectively the d5 of Locrian and the A4 of Lydian. An N4 against the root could just be in Locrian unless it co-occurs with degrees that aren't found in Locrian.)
The Trachytonic and Malacotonic parent scales may be named in other sources as Screamapillar and Mosh, respectively. I hope you understand why I opted not to perpetuate these names.
| Mode | Parent scale | Step pattern | Degrees | Distinguishing features | Non-diatonic features |
|---|---|---|---|---|---|
| Lydian | Diatonic | 3 3 3 1 3 3 1 | M2 M3 A4 P5 M6 M7 | A4 | n/a |
| Ionian | Diatonic | 3 3 1 3 3 3 1 | M2 M3 P4 P5 M6 M7 | P4 and M7 | n/a |
| Mixolydian | Diatonic | 3 3 1 3 3 1 3 | M2 M3 P4 P5 M6 m7 | M3 and m7 | n/a |
| Dorian | Diatonic | 3 1 3 3 3 1 3 | M2 m3 P4 P5 M6 m7 | m3 and M6 | n/a |
| Aeolian | Diatonic | 3 1 3 3 1 3 3 | M2 m3 P4 P5 m6 m7 | M2 and m6 | n/a |
| Phrygian | Diatonic | 1 3 3 3 1 3 3 | m2 m3 P4 P5 m6 m7 | m2 and P5 | n/a |
| Locrian | Diatonic | 1 3 3 1 3 3 3 | m2 m3 P4 d5 m6 m7 | d5 | n/a |
| Lydian d4 | Trachytonic | 3 3 2 2 3 3 1 | M2 M3 N4 P5 M6 M7 | N4 against any other degree | N4 against any other degree |
| Ionian d7 | Trachytonic | 3 3 1 3 3 2 2 | M2 M3 P4 P5 M6 N7 | N7 against any other degree | N7 |
| Mixolydian d3 | Trachytonic | 3 2 2 3 3 1 3 | M2 N3 P4 P5 M6 m7 | N3 against any other degree | N3 |
| Dorian d6 | Trachytonic | 3 1 3 3 2 2 3 | M2 m3 P4 P5 N6 m7 | N6 against any other degree | N6 |
| Aeolian n2 | Trachytonic | 2 2 3 3 1 3 3 | N2 m3 P4 P5 m6 m7 | N2 against any other degree | N2 |
| Phrygian n5 | Trachytonic | 1 3 3 2 2 3 3 | m2 m3 P4 N5 m6 m7 | N5 against any other degree | N5 against any other degree |
| Trachylocrian | Trachytonic | 2 3 3 1 3 3 2 | N2 N3 N4 N5 N6 N7 | Any two degrees | N2, N3, N6, N7, or N4 and N5 together |
| Lissolydian | Lissotonic | 3 3 2 2 3 2 2 | M2 M3 N4 P5 M6 N7 | M3 | N7, or N4 against any other degree |
| Rastoid | Lissotonic | 3 2 2 3 3 2 2 | M2 N3 P4 P5 M6 N7 | P4 and M6 | N3 or N7 |
| Nairuzoid | Lissotonic | 3 2 2 3 2 2 3 | M2 N3 P4 P5 N6 m7 | m7 and M2 | N3 or N6 |
| Bayatioid | Lissotonic | 2 2 3 3 2 2 3 | N2 m3 P4 P5 N6 m7 | m3 and P5 | N2 or N6 |
| Lissolocrian | Lissotonic | 2 2 3 2 2 3 3 | N2 m3 P4 N5 m6 m7 | m6 | N2, or N5 against any other degree |
| Sikahoid | Lissotonic | 2 3 3 2 2 3 2 | N2 N3 N4 P5 N6 N7 | N2 and N4 | N2, N3, N6, N7, or N4 and P5 together |
| 'Iraqoid | Lissotonic | 2 3 2 2 3 3 2 | N2 N3 P4 N5 N6 N7 | N5 and N7 | N2, N3, N6, N7, or P4 and N5 together |
| Dalmatian | Malacotonic | 3 2 3 2 3 2 2 | M2 N3 N4 P5 M6 N7 | M6 | N3, N7, or N4 against any other degree |
| Cilician | Malacotonic | 3 2 2 3 2 3 2 | M2 N3 P4 P5 N6 N7 | M2 and P4 | N3, N6, or N7 |
| Pisidian | Malacotonic | 2 3 2 3 2 2 3 | N2 N3 P4 P5 N6 m7 | P5 and m7 | N2, N3, or N6 |
| Lycian | Malacotonic | 2 2 3 2 3 2 3 | N2 m3 P4 N5 N6 m7 | m3 | N2, N6, or N5 against any other degree |
| Galatian | Malacotonic | 3 2 3 2 2 3 2 | M2 N3 N4 P5 N6 N7 | N4 and N6 | N3, N6, N7, or N4 against any other degree |
| Bithynian | Malacotonic | 2 3 2 3 2 3 2 | N2 N3 P4 P5 N6 N7 | N7 and N2 | N2, N3, N6, or N7 |
| Adam Freese's Sun Scale | n/a | 3 1 4 2 3 1 3 | M2 m3 N4 P5 M6 m7 | n/a | N4 against M2, P5 or M6 |
| Sun t7 | n/a | 3 1 4 2 3 2 2 | M2 m3 N4 P5 M6 N7 | n/a | N7, or N4 against M2, P5 or M6 |
| Neutral second scale | n/a | 2 repeating | N2 m3 M3 N4 P5 N6 m7 M7 m9 M9... | n/a | N2, N4 against non-Locrian degrees, N6, lack of octave |
| Equalized BP Lambda | n/a | 3 repeating | M2 M3 A4 N6 N7 m9 m10 P11... | n/a | N6, N7, lack of octave |
For more functional-harmony-centric viewpoints on 17edo, I recommend The 17-Tone Puzzle by George Secor and the 17edo section of Modes and Chord Progressions in Equal Tunings by Easley Blackwood. I also have a Youtube playlist with some of my favorite 17edo pieces by other artists.
If you accepted my axioms in the introduction, then after becoming familiar with 17edo, either 19edo, 22edo, 24edo, or 29edo may be logical next steps. I find 19edo and 22edo's fifths offensively flat and sharp respectively, but for some ears or compositional styles this is not an issue. I also recommend against 24edo, because I find that the preservation of all 12edo intervals has the effect of discouraging more-xenharmonic gestures, making them sound more dissonant in comparison. I may address 29edo in depth in the future.
If we relax the demand for a near-just P5 and instead emphasize consonant major and minor triads, 15edo begins to look appealing, and after it 19edo. If we abandon both demands, 13edo is a clear next step.
Among transpositionally-invariant tunings with near-just P5s but without octave periods, the most obvious candidates are EDFs (equal divisions of the perfect fifth), among which 8edf and 9edf are appealing options.
If we abandon the need for transpositional invariance, we're in the relatively unstudied world of just intonation scales. I recommend Bicycle as a starting point.