Just Intonation

Often the first step a musician might take away from the familiar environs of the standard Western twelve-tone equal temperament is to explore a method of tuning known as just intonation.

In an equal temperament, the distance between every pair of neighboring pitches is exactly the same. This is very useful for practices such as transposition and modulation, since the relationships between notes don't change as they are moved up or down as a group.

However, today's standard is the result of numerous compromises made throughout history. For thousands of years, until only about three centuries ago in Western culture, tuning was done with pitches being related to one another by specific ratios. The smaller the numbers in the ratio, the more consonant (smooth, pleasant, at rest) the interval sounds, as it is physically easier for the human ear to identify its harmonic properties.

The mathematical procedure to apply these ratios is relatively straightforward: a basic pitch is chosen, and the frequency of that pitch is multiplied by the desired ratio to arrive at the second pitch. For example, standard concert tuning is known as A440, because the A above Middle C is set at 440 Hz (Hertz, a unit that measures vibrations per second). If we take this as our basic pitch and want to find the pitch at a ratio of 3:2 above it, we'd have 440*(3/2) = 660 Hz.

RATIO	INTERVAL
1:1	Unison
9:8	Tonus (Whole Tone)
4:3	Diatessaron (Perfect Fourth)
3:2	Diapente (Perfect Fifth)
2:1	Octave

The simplest ratio, 1:1, is two instances of the same pitch, and is therefore appropriately termed a unison. The next-simplest ratio, 2:1, is one pitch exactly double the frequency of the other, which for historical reasons is called an octave ("oct-" being a prefix meaning "eight", the octave is the eighth note at the end of a heptatonic scale, a return to the functional position of the tonic; in other words, the distance between the first and last "do" of "do re mi…").

The next-simplest ratio, 3:2, was named the diapente ("dia-" translated variously as "by/from/through/of", "pente" meaning "five") and is the fifth step of "do re mi…", namely "sol". Its inversion is the diatessaron ("by/from/through/of four"), ratio 4:3, the "fa" in "do re mi…". The distance between the diapente and the diatessaron was called the tonus, ratio 9:8, what is today called a whole tone; a whole tone up from "do" is "re".

These intervals all exist in what is known as Pythagorean tuning, an ancient Greek system in which the ratios involved exhibit no prime factors greater than 3; for example, 9:8 factors to (3*3):(2*2*2). This system can also be called 3-limit just intonation, and it's the most fundamental of all musical tunings beyond the basic unison and octave. It's easy to tune because the octave and pure perfect fifth are simple to hear, and a Pythagorean scale is achieved by stacking pure fifths and collapsing the results into the span of an octave.

However, limiting eligible ratios to the prime factor of 3 can yield some relatively complicated intervals: for example, a Pythagorean major third (more properly called a ditone, a stack of two whole tones) is the ratio 81:64, and the minor second (or semitone) could be either 256:243 (called the Pythagorean limma, or diatonic semitone) or 2187:2048 (the Pythagorean apotome, or chromatic semitone). Such complicated intervals tend to sound dissonant (harsh), as they are more difficult for the human ear to process.

A far simpler system can be constructed by raising the prime limit to 5, thus allowing ratios like 5:4 to serve as the major third, 6:5 the minor third, and 16:15 the minor second.

RATIO	CENTS	INTERVAL
1:1	0	Unison
16:15	112	Minor Second
9:8	204	Major Second
6:5	316	Minor Third
5:4	386	Major Third
4:3	498	Perfect Fourth
3:2	702	Perfect Fifth
8:5	814	Minor Sixth
5:3	884	Major Sixth
9:5	1018	Minor Seventh
15:8	1088	Major Seventh
2:1	1200	Octave

Notice that the most basic Pythagorean intervals are retained in this tuning, as they are simpler than ratios involving the prime factor 5; for example, 9:8 is simpler than 10:9, which would otherwise be the simplest ratio for a major second.

Notice also the measurements given in the unit of cents. This common measurement scheme is an extremely convenient means to compare the sizes of any intervals, and is based on 1200 cents per octave. Dividing the octave into twelve equal tones, as does the common Western standard tuning, yields one tone every 100 cents: 100 is the minor second, 200 is the major second, 300 is the minor third, and so on. Understanding this, and given the chart above, it becomes clear how impure the intervals of standard tuning are; for example, a minor third in twelve-tone equal temperament is 16 cents (about a sixth of a semitone) flat from the pure version of the same interval, while a major third is 14 cents sharp, and the same is also true (but reversing the sharp and the flat) for their inversions, the sixths.

The pure intervals have been tempered to equal widths for the sake of conveniences like unhindered transposition and modulation. However, the cost of this compromise is the loss of the distinctive characters of each key and their modes, and the harmonic purity of chords.

5-limit just intonation is perhaps the most welcoming first step out of equal temperament, as a great deal of existing traditional and contemporary popular music can be performed in this tuning and (due to the purity of the intervals) end up sounding sweeter, literally more harmonic, than it would in standard twelve-tone tuning.

On the other hand, there are serious dangers inherent in just intonation: beware, for outside the comfortable conveniences of equal temperament, there be wolves!

A wolf interval is so named because it is so dissonant (far from pure) that it "howls". The size of a fifth built on most degrees of a just-intoned scale will be pure (there are 702 cents between the just minor second and just minor sixth, for example), but certain fifths are monstrously wrong: the distance between a just major second and just major sixth is only about 680 cents, almost a quarter of a semitone flat from pure!

Taming these wolves was another cause for the compromise of equal temperament. Without tempering, the second chord of both the major and minor scales would be practically unusable for the vast majority of purposes, and having access to this chord is certainly very useful. This is one excellent reason that equal temperament started to find favor about three centuries ago, and has since become the de facto standard tuning of practically all Western music.

However, for music which cleverly evades those rare wolf intervals, and which does not have a great need for complete freedom of transposition or modulation, just intonation offers a sweetness of sound unparalleled by standard tuning. In fact, transposition and modulation are still perfectly useful techniques within just intonation, but certain modes work better together than others, as they each have a unique arrangement of intervals rather than the uniform smear of equal temperament.

For the exceptionally adventurous musician, just intonation can even be extended beyond the 5-limit into higher primes. Septimal just intonation incorporates ratios involving the prime factor of 7, most readily recognized in the "barbershop seventh" of 7:4 (so called due to its ubiquity in the purely harmonized performances of barbershop quartets, more formally known as the "harmonic seventh", "subminor seventh", or "septimal minor seventh", and sometimes as the "perfect seventh" or "pure seventh"). Even higher primes have seen occasional use by the arguably insane (such as the good folks of the Alternate Tunings group on Yahoo! and the Xenharmonic Alliance on Facebook), though a highly trained ear is required to discern such intervals. Sometimes, to achieve a particularly unusual tuning, certain low primes (even 3 and 5) are actually excluded!

With modern technology, just intonation has become widely accessible. Even an inexpensive synthesizer may have a retuning capability, often with a list of presets including basic just intonation. Many acoustic instruments support just intonation natively, though fretted instruments like guitars may require modification (the frets must be bent into carefully calculated squiggles). In most professional multitracking software, MIDI tracks can be retuned instantly, lending the purity of just intonation to a song that was originally composed in equal temperament and giving it a little extra shine. Give it a try!

The Persian Dastgāh System

An obsession of mine over the past several months has been the family of music traditions of the Middle East, most especially the Persian dastgāh system. A dastgāh is a collection of related "modes" and associated melodic rules. In theory, there are more than fifty such collections in existence, though twelve standard dastgāhs were codified in the 19th Century and are widely practiced today.

Each "mode" in a dastgāh consists of seven core pitches, to which further pitches are added to accommodate modulation or expression. These are not exactly modes in the Western sense, because the widths of certain intervals differ from one "mode" to another. As a result, a given instrument is limited only to certain sets of possible "modes" without being retuned (unless it uses a compromised tuning such as those detailed below).

A dastgāh also entails established practices regarding the role or function of each pitch in a "mode" over the course of musical development; these practices are known as sayr. Additionally, many short melodic fragments known as gushehs are incorporated into the larger context of each performance; how these gushehs are handled is part of the artistry of an individual performer. The full set of dastgāhs, with their associated sayr and gushehs, is called the radif.

Music historians consider the dastgāh system to have its roots in the court of Xosrov II, ruler of the Persian Empire from AD 590 to 628. He was patron to the musician Bārbod, who organized a musical system containing seven Royal Modes (Xosrovāni), thirty related modes (Lahn), and 360 melodies (Dastān). When the Persians were conquered in AD 642, Bārbod's system became the basis for the Arabic system of maqām (and thereby the Turkish makam); many of the original Persian names survive in the present-day maqāmat (the plural form).

The matured dastgāh system of modes and melodies grouped together into collections emerged by the end of the 19th Century, though there is little surviving evidence of the specific developments up to that time. Like the Arabic and Turkish systems, the dastgāh tradition has been passed down aurally through history by its practitioners, very few of whom left any written records. As such, tunings for the same dastgāh can vary from region to region, or even between cities or individuals.

Several attempts have been made to standardize a tuning system that could be used in which all dastgāhs could be represented with an acceptable degree of fidelity:

• In the 13th Century, Safiaddin Ormavi devised a scale with 17 tones per octave that ostensibly became the standard of Middle Eastern music theory for several centuries; however, it is not likely this system was adhered to in practice, given the nature of the instruments, the fact that most music used only about seven tones at a time, and especially the inherent variability of the aural tradition.

• Impressed with the advanced developments of European harmonic techniques and desiring to incorporate them into their own traditions, some prominent Middle Eastern musicians in the 19th and 20th centuries (notably Mikhail Mashaqa and Ali Naqi Vaziri) advocated the adoption of an equal temperament; however, because the semitone was too large to accommodate all traditional intervals, a further subdivision to a quarter-tone was proposed. The resulting 24-tone scale has found some favor for its relative practicality, as existing 12-tone instruments can be used (a basic technique involves splitting a composition between two pianos with one tuned a quarter-tone apart from the other), though many consider this tuning a poor compromise at best and culturally inappropriate at worst.

• In the 1940s, a physicist named Mehdi Barkesli conducted studies to measure the actual intervals used in practice by several Persian singers in various modes. Influenced by the writings of respected medieval scholars such as Ormavi, he presented a 22-tone scale as a means to unite all dastgāhs into a common tuning. However, this scale is built on a 7-tone scale from which the remaining tones are derived, but the basic scale itself is arbitrary, and the additional tones are calculated by means that have little relevance to actual practice.

Each of these systems makes the assumption of octave equivalence, though in reality the interval of a perfect octave may not even be present; pitches exist individually, not as members of a pitch class that repeats an octave higher or lower. Furthermore, as each pitch is an individual entity, it is inappropriate to consider one pitch a modified (raised or lowered) form of another, as is often done under influence from the Western tradition of sharps and flats.

In 1965, Hormoz Farhat presented his theory of flexible intervals in the dastgāh system. He identified five characteristic intervals from which the Persian modes may be constructed, and recognized the inherently approximate nature of these intervals in practice. These intervals range from about 90 cents (slightly smaller than the semitone of 12-tone equal temperament, though very close to the Pythagorean interval of 256:243 on which it is historically based) to about 270 cents (called a "plus-tone" by Farhat, it is significantly smaller than a minor third and perhaps best thought of as an augmented second). With this approach, there is no assumption of octave equivalence, and individual pitches serve distinct roles.

In addition to dastgāh itself, I have been studying the closely related Turkish makam, particularly the work of Karl Signell. He also identifies five characteristic intervals of makam, though two of the intervals are each about 20 cents different from their counterparts presented for dastgāh by Farhat. The makam intervals Signell offers are all multiples of a basic comma of about 23 cents (in his glossary, he identifies it as the Pythagorean comma, which has a value of 23.46 cents, though the numbers he provides for the intervals strongly suggest he actually uses the Holdrian comma of 22.64 cents, which indeed has been used by noted Turkish composers and theorists). He explains that the number of commas stacked to produce a given interval can vary; thus, those two intervals that differ between Farhat's dastgāh and Signell's makam could easily be adjusted with the addition or removal of a comma.

FARHAT	SIGNELL
90¢	90¢
135¢	114¢
160¢	180¢
204¢	204¢
270¢	271¢

There is still much for me to learn in this area, and I'm very excited to see where it leads. I'm curious to try using those five basic intervals, perhaps with appropriate adjustments to the number of stacked commas, to build some musically useful non-traditional modes, taking advantage of the precise tuning afforded by modern electronics. I also plan to study the rhythmic aspects of Middle Eastern music, which are based on cycles called uṣūl and more recently on the meter of a form of poetry known as ghazal, though that is definitely a subject for another day.

Isomorphic Keyboards

One of the first obstacles an aspiring musician will face is trying to understand the relationships between the different notes they can make with their instrument of choice: a piano looks like it's missing every third or fourth black key; a guitar in standard tuning starts repeating pitches five frets along the next string, except for one string where it's only four; a saxophone plays different notes depending on which combinations of keys are held down; and fretless instruments like a violin or sliding instruments like a trombone offer practically no guidance to the desired positions on their own.

Chords make it all even more confusing; the same chord shape can get twisted into all sorts of arrangements depending on which note is the root. Playing a basic major chord on a piano can end up on all white keys, all black keys, or some combination of both. On a guitar, a major chord might involve all six strings or only five or four, and a player's fingers need to learn their own form of gymnastics to switch from one shape to another in time with the music.

Fortunately, there is a better way. Not only is it easier to learn, but it opens up a whole range of new possibilities unavailable on many traditional instruments.

This interface is called an isomorphic keyboard, so named because one hand shape plays the same type of chord no matter what the root pitch is. The pitches are arranged on a hexagonal grid in such a way that each direction represents movement by one certain interval. Such a layout is achieved by selecting any two intervals to define the pattern.

G		Ab		A		Bb		B
	E		F		Gb		G
C		Db		D		Eb		E
	A		Bb		B		C
F		Gb		G		Ab		A
	D		Eb		E		F
Bb		B		C		Db		D
	G		Ab		A		Bb
Eb		E		F		Gb		G
	C		Db		D		Eb
Ab		A		Bb		B		C
	F		Gb		G		Ab
Db		D		Eb		E		F

For example, in the above diagram, upward movement represents an ascending perfect fifth (C to G to D to A), so therefore downward movement is by a descending perfect fifth (C to F to Bb to Eb). Movement to the "northeast" is an ascending major third (C to E to Ab to C), so "southwest" is descending (C to Ab to E to C). Such motion is universal across the entire grid; starting on C and moving northeast to E and then moving upward from there will yield a perfect fifth above E, which is B. Given this arrangement, it is clear that motion along the "northwest"/"southeast" axis is by minor thirds.

G
	E
C

To play a basic major chord in such an arrangement, the shape is a triangle: a cluster of two hexes on the left bordering one hex on the right (such as C-major, C-E-G). This shape can be moved to any position on the grid and it will yield a major chord built on whatever pitch is in the lower-left hex; for example, look for B-Eb-Gb (B-major) or D-Gb-A (D-major). A minor chord is merely flipped around: the lone hex on the right is instead moved to the left (C-Eb-G).

Additionally, since the physical pattern between pitches does not change regardless of position, compositions can be played in a different key without any need to change the relative hand movements; exactly the same motions are made around a different tonic hex.

Again, an isomorphic arrangement is not limited to the one given in this example, but can instead be built from any two intervals, including those not available in standard Western twelve-tone tuning. An axis could be defined by the 7:4 harmonic (or "barbershop") seventh, or by the 352.9-cent neutral third of 17-EDO, or any other exotic interval your music might demand. As such, tunings that are completely unattainable on a traditional instrument can be accommodated with ease.

Is such a keyboard actually practical? Absolutely. Various isomorphic layouts have been in use for real instruments since the 1800s. Early on, these were unique, custom-built acoustic instruments, though by the late 20th century technology allowed the development of several electronic interfaces that could be automatically retuned to an entirely different layout at the whim of the performer. These specialized devices have typically been as costly as many other instruments, though there are now inexpensive apps for mobile touchscreen devices that make isomorphic keyboards easily accessible and widely available, and there are even free resources online that let visitors use their computer keyboards as isomorphic instruments!

19-EDO Guitar

Recently exhumed from my bedroom closet (along with a really nice tuner I didn't even know was in there) is an old electric guitar that has certainly seen better days. I don't believe it would be worth the cost to restore, but it's still basically functional. This makes it a prime candidate for some, shall we say, unorthodox modifications.

I want to rip out all the frets, refinish the neck, and install new frets with 19 to the octave instead of 12.

Why 19? Well, it's the next equally divided octave (EDO) up from 12 (the standard tuning of modern Western music) that works well for triadic harmony (the most basic chords). Not only does it sound good, it's actually better than normal tuning at approximating 5-limit just intonation. Any piece written for standard tuning can actually sound a little bit purer if it's played in 19-EDO. For example, in 12-EDO a major third is 13.7 cents sharp from the "sweet spot" of the just major third (5:4), but in 19-EDO it's only 7.4 cents flat. Even better, the 12-EDO minor third is 15.6 cents flat from just (6:5), but in 19-EDO it's less than 0.15 cents sharp! Also, as inversions of the thirds, these benefits apply to the major and minor sixths as well.

Of course, there's still more to gain: 19-EDO offers 7 more pitches per octave than standard tuning. The main benefit here is the introduction of some extremely useful new intervals: the traditional second, third, sixth, and seventh are all available in not only two varieties (major and minor) but four each: supermajor, major, minor, and subminor. Additionally, the tritone of 12-EDO is split into two distinct intervals, one for the augmented fourth (the real tritone, a stack of three whole tones) and the other for the diminished fifth. Utilizing these extra intervals opens up broad new expressive possibilities, both melodically and harmonically.

CENTS	INTERVAL
0.0	Unison
63.2	Subminor 2nd
126.3	Minor or Neutral 2nd
189.5	Major 2nd
252.6	Supermajor 2nd or Subminor 3rd
315.8	Minor 3rd
378.9	Major 3rd
442.1	Supermajor 3rd
505.3	Perfect 4th
568.4	Augmented 4th
631.6	Diminished 5th
694.7	Perfect 5th
757.9	Subminor 6th
821.1	Minor 6th
884.2	Major 6th
947.4	Supermajor 6th or Subminor 7th
1010.5	Minor 7th
1073.7	Neutral or Major 7th
1136.8	Supermajor 7th
1200.0	Octave

Now, given this list, it's clear that the minor second and its inversion, the major seventh, deviate from 12-EDO by 26.3 cents, which is significant enough they could be considered "neutral" (halfway) intervals. Fortunately, they both fare better as approximations of their respective just intervals (16:15 and 15:8), deviating only 14.6 cents; this is still a distinct difference, though it's slightly less terrible than the 15.6-cent deviations from just that the minor third and major sixth exhibit in 12-EDO. If those intervals are tolerated, certainly these can be as well.

With such a different system, you might think I'd need to relearn how to play every scale and every chord. In fact, the new frets that correspond to the equivalent old frets won't have moved far, so the same hand positions will sound the same scales and the same chords. Effectively, the standard frets will only have shifted up or down the neck a little to make room for the new arrivals. They will be different enough that they will sound "out of tune" if played alongside a 12-EDO instrument; however, 19-EDO is actually more "in tune" to pure intervals than the current standard!

The Equally Divided Tetrachord

I've recently been studying tetrachords as well as equally divided octaves (EDOs). This morning, I was inspired to equally divide the just perfect fourth (4:3) into various sizes, starting with the tetrachord.

CENTS	ROLE
0	Hypate (Tonic)
166	Parhypate (Second)
332	Lichanos (Third)
498	Mese (Fourth)

The basic step size in the equally divided tetrachord (EDT) works out to 166 cents. This yields a second that is -34 cents from that of the standard 12-EDO major, and -37.9 cents from the just major tone (9:8), but only -1.5 cents from the 29-EDO neutral second. Likewise, the EDT also yields a third that is +32 cents from the 12-EDO minor, and +16.4 from the 5-limit minor (6:5) or +37.9 from the Pythagorean minor (32:27), but only +1 cent from the 29-EDO neutral third. As the perfect fourth of 29-EDO is only -1.4 cents from just (4:3), clearly a subset of 29-EDO makes an excellent approximation of the EDT. 36-EDO offers an even better approximation, and if its 500-cent perfect fourth is used as the basis instead, the EDT becomes an exact subset of 36-EDO.

Regardless of whether the EDT is played straight or tempered to an EDO, it features what can best be described as a neutral second that leans toward major and a neutral third that leans toward minor. Adding a disjunct EDT from the just perfect fifth (3:2) to round out an octave (additional pitches at 702, 868, 1034, and 1200 cents), the resulting pattern of the interval "leanings" (M2, m3, M6, m7) suggests the Dorian mode. Alternatively, adding a conjunct EDT from the fourth (additional pitches at 664, 830, 996, and 1200 cents), the fifth degree turns out very flat, -36 cents from 12-EDO and -37.9 from just (3:2), though still only +1.9 from the nearest degree of 29-EDO; the sixth is neutral leaning toward minor, +16.4 cents from just (8:5) and +2.4 cents from 29-EDO, while the seventh is the Pythagorean minor (16:9), +3 cents from 29-EDO.

This study has heightened my interest in 29-EDO, and I plan to explore it in greater detail. Also, of course, the perfect fourth can be divided equally into other step sizes as well; I've done the basic calculations for several of these, and a few look very interesting!