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.
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.
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!
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!
Not only are 3 and 5 sometimes excluded, some people love to exclude 2.
ReplyDeleteMadness! ;-)
ReplyDeleteJust intonation is old as the world and 13-limit scales models were known by the persians, greeks, and others for thousands of years. There is no way to escape the 13/8, 13/12, 16/13 ratios, etc. in arabian, turkish or persian music.
ReplyDelete7th harmonic in combination with 3th harmonic proposes the most basic models for Slendro tunings, known as the oldest scales we know on Earth, as well as many Aka/Baka tunings.
17th and 19th harmonics ratios are completely pertinent in many european music cultures, indian and gypsy music.
Extended JI "as passed 5-limit" is less and less recognized as a standard in the JI world today. For many extended JI starts passed 13-limit and for more and more it starts passed 19-limit, at least.
Excellent information, Jacques; thank you!
ReplyDeleteI'm aware this sort of math was known to theoreticians such as Ptolemy, Farabi, Avicenna, etc., but especially with regard to the Middle Eastern traditions, I'm under the impression that such theory was rarely put into practice (see my previous post on Persian dastgah). I'm less familiar with Hindustani/Carnatic traditions beyond a basic concept of srutis, but what I do know is there has been much contention on the actual nature of their tuning, and there is not likely to be a consensus any time soon. I likewise have only a basic knowledge of slendro and pelog, though I would dispute that they are known with certainty as being the oldest on the planet; from what I have read on the Internet (so it must be true!), a set of ancient Chinese flutes are the oldest confirmed musical artifact, and their scales are straight diatonic, and likewise a possible but unconfirmed section of a prehistoric bone flute has holes that strongly suggest a diatonic scale.
While of course I don't consider any one source definitive, I have found David B. Doty's book "The Just Intonation Primer" exceedingly helpful. Doty details usages of primes through and beyond 19, though from 19 onward he questions their usefulness. I would not be surprised that primes through 13 are pretty standard within the JI world, but 5-limit seems like the most common (and ideal) introduction to these concepts for a musician born and raised in 12-TET.