by Billy Gard
Introduction to the Harmonic Series
Discussion of Tuning
Intruduction to Just Intervals
Just Tuning of Chords
Pythagorean vs Just Tuning
Relationship Between the Pythagorean and Just scales
Perfect vs Imperfect
Facts and Figures
When you first heard just intonation mentioned, maybe you thought it meant something like “a quality of vocalization with a beauty that will hold up under court of law.” It could come close, if by that you are referring to the laws of harmony and tonality. It is true that just intonation adds beauty. But let us do our arithmetic to show why this is so.
When you hear a single tone, it is typically accompanied by a series of higher frequencies, called harmonics, which are multiples of the main tone's frequency. The main tone itself is called the fundamental, or the 1st harmonic. The 2nd, 3rd and 4th harmonics would have frequencies 2, 3 and 4 times that of the fundamental, respectively. The ear doesn't percieve these frequencies as a chord, however, but as an added richness to the basic tone. The quality, or timbre, of the tone is determined by which harmonics are actually present, or predominant.
The following shows the first 20 tones of the harmonic series for a fundamental on C according to their approximate location on the keyboard. 1 represents the fundamental, and the number of each harmonic shows what multiple it is of the fundamental frequency. Notice how every odd number introduces a new note, and how every even number is an octave higher than the number half its value. You may want to note how harmonics 4, 5, 6 and 7 form a dominant 7th chord. The major 7th chord can be found an octave higher in the series, formed by harmonics 8, 10, 12 and 15. In fact, just by studying these harmonics alone, you can glean much of what you will learn here about just intonation.
Move your mouse over the blue button to hear the first 19 harmonics. Note how the combined sound is similar to the fundamental note played on an oboe, whose conic air column generates all the harmonics. Note also how when the fundamental first comes in by itself, it is barely audible. But as the harmonics are added, the fundamental can be heard loud and clear. You would actually hear the fundamental note even if it were missing altogether. When a male voice is heard over a telephone, which is poor at transmitting low frequencies, the fundamental pitch of the voice is pretty much absent, but is reconstructed by the ear from the presence of the higher harmonics alone. This is because a group of multiples of a fundamental frequency, even without the fundamental itself, collectively form a wave pattern that repeats itself at the fundamental frequency. And this wave pattern is audible to the ear as the “periodicity” tone. Thus the ear can hear this repeating pattern as the fundamental tone. Thus the presence of the fundamental note itself contributes very little to the sound except to add a little depth to the timbre.
Mouse here to hear just the odd harmonics. Note how the sound is similar to a clarinet, whose tubular air column generates odd harmonics.
If you have a piano, it may come as shocking news, but it is out of tune.
“Wait a minute! Not my piano. I keep it perfectly tuned.” I'm sure you do. But let me be more specific. Sure your piano is quite tuned. In fact I'll bet it sounds just as in tune no matter what key you play in, true? But for that to be the case, your piano has to be actually out of tune.
Let us listen to the chord commonly used by barbershop quartets as a “tuning chord”. In C the notes would be C-G-C-E. Why does this particular combination of notes produce such a consonant sound? Why, for instance, does it sound “dissonant” if you raise the bass note one half-step? Let's try to get a clue by looking at the ratio of the chord frequencies as played on a “perfectly tuned” piano, given a frequency of 2 for the bass note:
2, 2.996614, 4, 5.039684
If you look really closely at these four values you may notice that they come really close to the integer values 2:3:4:5. Therefore, let us fine tune the notes so that the frequency ratios really are exact integers:
2, 3, 4, 5
Now move the mouse over the red button to hear the equal temperament tuning. Then slide over to the blue button to hear the just tuning:
Now can you hear the difference? You may notice for the first time the existance of a quiver or “buzz” in the first tuning. Yet this ubiquitous buzz is present in a properly tuned piano. You may have never noticed its presence until you heard what a chord sounds like without it, simply by making the frequencies exact integers.
This method of tuning is called just intonation. It is based on the fact that the ear hears intervals with simple frequency ratios as fully consonant and in tune. We see here then that the chords that sound consonant when played on a piano do so because their frequency ratios come really close to a ratio of small integers - but not quite on. What just intonation seeks to do is adjust the notes so that they come out exact integers. Then the buzz is gone and you have a clean, locked-in chord with the “expanded sound” that barbershop quartets talk so much about.
The piano, as we know it, is tuned according to what is called “equal temperament” (or ET), which divides the octave into 12 equal “semitones”, for the purpose of producing equal intonation in all keys. As a result of this, music in ET is actually somewhat out of tune. An octave is the interval with a frequency ratio of 2/1. If any notes are an octave apart, the higher one is twice the frequency of the lower. Here you see the logarithmic nature of pitch: that the size of an interval is based on the ratio between frequencies, not the difference. The ET semitone's frequency ratio would have to be that value by which you multiply a frequency 12 times to double it, namely the 12th root of 2, or ~1.059463. This being the case, the octave is the one interval on your piano that is perfectly tuned. It has a nice clean ratio of 2/1. All other piano intervals are out of tune, because all powers of 1.059463 below the 12th are irrational numbers.
Whether it's obvious or not, when a person sings without accompaniment (a cappella), the ear automatically seeks simple frequency ratios between notes, whether consecutive notes in the melody or simultaneously sounding notes in a chord. This is the main reason why singing is different in a fundamental, though maybe unobvious way, when the song is accompanied with an equally tempered instrument. If there is an instrument present, your voice will tune to the equal-tempered tonality of that instrument. But singing alone, the ear will try to adjust the intervals to integral ratios, making for the possibility of a cleaner tonality. This is less obvious in a single line of melody, since the different notes aren't sounded simultaneously. But when singing in parts, the clean sound from tuning simultaneously sounding notes to simple integral ratios is strikingly clear. The buzz is smoothed out, and you hear what they mean by a “ringing” chord. This doesn't mean you have to know the math to sing in tune. I'm just describing what the ear does instinctively, whether or not the particular person is good at getting his voice to reproduce the tuning accurately!
In just-intonation each note in the scale is indicated by a fraction showing the frequency of that note divided by that of the tonic. The fraction can indicate either the scale note itself or the interval that the note makes with the tonic. Now the tonic itself would be called 1, which is the simplest ratio there is. The next simplest ratio, 2, would be the frequency ratio of the perfect octave, or the tonic note repeated at the top of the scale. Let us look at additional intervals in the order of their ratio values. 3/2, the next simplest ratio after the octave, would be the frequency ratio of a perfect 5th, or degree 5 (the dominant) of the scale. An equally tempered 5th, which is heard on the piano, is actually about 2 cents flat (a cent is 100th of an ET half-step, or 21/12/100), which actually isn't that bad. Next is 4/3, which is a perfect 4th, or degree 4 (the subdominant) of the scale.
In just intonation, we observe the concept of octave equivalence, which considers only the position of the note in the scale without regard to the octave. This means that doubling and halving the value of the ratio is insignificant, but the order of the numerator and denominator is significant. So the intervals 3/2, 3/1 and 3/4 would all be considered equivalent, since they all refer to the 5th degree of the scale, differing only in the octave. 2/3 would be considered a different interval because it stands for the note a 5th downward from the tonic, which is really the 4th degree. With this in mind, we like to present the intervals in “normalized” form, which always expresses the note's ratio to the nearest tonic below it. This is done by doubling or halving the fraction until it has a value between 1 and 2, and reducing it to lowest terms. The inverse of the 3/2 (a 5th above the tonic), which is 2/3 (a 5th below the tonic), would be written as 4/3 (i.e. a 4th above the tonic). So we can consider the 4th to be the inverse of the 5th.
Going on to the higher ratios: Frequencies with the ratio 5/3 form a just major 6th, or scale note 6 (the submediant). A piano 6th is slightly sharp. The ratio 5/4 forms a just major 3rd, or scale note 3 (the mediant). A piano 3rd is slightly sharp. Note how the ratios using the smallest numbers result in the most familiar intervals in music. That's why they sound so musical!
Now let us stop here and introduce the adding together of intervals. If you go up a major 3rd and then up again a minor 3rd, you end up at a 5th. Because of the logarithmic nature of tuning, this would be done by multiplying the interval ratios: 5/4 * 6/5 = 30/20 = 3/2. To subtract intervals, you divide one interval by the other. If you go up a 5th and then down a major 3rd, you leave a minor 3rd (or the flatted mediant scale note): 3/2 / 5/4 = 3/2 * 4/5 = 12/10 = 6/5. A piano minor 3rd is flat. If you go up a minor 3rd and down a 5th, you end up on the minor 6th (or the flatted sub-mediant scale note): 6/5 / 3/2 = 6/5 * 2/3 = 12/15 = 4/5 = 8/5. The piano minor 6th is flat.
Now as you move to higher numbers in the frequency ratio, the dissonance that begins to form can be of two main types. Those ratio values that are multiples of the smaller numbers result in a sound that is tight, although shallow and horizontal. A major 7th (called the leading tone in the scale) is of this variety, being in effect a major 3rd upon a perfect 5th: 3/2 * 5/4 = 15/8. When you use the higher prime numbers in ratios, you get into the extended intervals that have a sound that is deep and vertical. The harmonic 7th: (7/4) is of this variety. 9/8 (an acute major 2nd, and supertonic scale note) would be of the former variety, being factorable down to 2 and 3. The largest prime factor of the ratio values is called the “limit” of the ratio. For instance, 4/3 would be a 3-limit interval because 4 can be factored down to 2. 5/3 would be a 5-limit interval. 7/4 is called a septimal interval, which in just-intonation parlance refers to a 7-limit value.
Let's construct a sequence of intervals above a tonic frequency of 24. Multiplied by 2/1, you'd get 48. 3/2 gives you 36. 4/3 gives you 32. 5/4 gives you 30. 5/3 gives you 40. 9/8 Gives you 27, and 15/8 gives you 45. So using ratios with an limit of 5 or less, we get 24:27:30:32:36:40:45:48, Ptolemy's 5-limit just major scale!
Some of the just-tuned chords can be found in the just major scale, while others cannot. We show the tuning of a chord in terms of the ratio between the notes. For instance, 4:5:6 would be a just major triad. You may notice that this is the ratio of the numbers 24, 30 and 36 in the above major scale. In fact, the I, IV and V chords in the major scale are all perfectly tuned major triads, and collectively cover all the notes in the scale. The iii and vi are both a just minor triad (10:12:15). Given the values in the ratio of any chord, the note which would have the value of 1 is the fundamental, which can be several octaves below the chord, and can often be heard below a well-tuned chord if you listen for it, even though it isn't really present in the chord. For the 4:5:6 major triad, the fundamental (1) is two octaves below the chord root (4). For the 10:12:15 minor triad, the fundamental is three octaves and a major 3rd below the chord root! This means that strictly speaking, the minor triad can be thought of as a major 7th chord without its root!
You may be asking by now, “isn't the ii chord a minor triad?” Yes, but it's out of tune. Believe it or not, there are two triads in the just major scale that are out of tune. One is the minor ii triad, which has the ratio 27:32:40 (woah, Nellie). The 40/27 interval of its 5th is slightly flat, giving it a harsh sound that has it affectionately named the “wolf” 5th. By the way, that tiny interval by which a wolf 5th comes short of a just 5th is an important one and occurs often. It is called a syntonic comma and bears the ratio 81/80. The other out-of-tune chord is is the vii diminished triad (45:54:64). The just diminished triad, by the way, is 5:6:7, and is not found in the just major scale due to the septimal factor.
In conventional music it is a well-known practice to inflect the pitch of scale tones with the # or b which, in just intonation terms, moves he note up or down by a 5-limit chroma (25/24). In just intonation, another much smaller inflection is applied to notes using + or -, which moves the note up or down by a syntonic comma. This adjustment is used to bridge the tuning discrepancy between 3-limit and 5-limit intervals.
If you grab the first four different notes in the harmonic series (i.e. no octaves), you will get the harmonic 7th chord, also known as a “barbershop 7th”. Its 4:5:6:7 tuning gives it the lovely ring that makes it unique in the 4-part genre. It is commonly called the “dominant 7th” chord, because the 7th chord built on the dominant note of the scale (36:45:54:64) is close to it. But its 7th is noticeably sharper than that of the harmonic 7th chord, making it sound harsh. You may note that if you knock the last number down a point, you would have 36:45:54:63, which would all be divisible by 9, thus reduceable to 4:5:6:7! This reveals the ratio (64/63) of the tiny interval by which the 7th of the dominant chord is lowered to make a harmonic 7th, called the septimal comma.
The Pythagorean scale is an older tuning based on 3-limit intervals, i.e. the scale is constructed by notes that are separated by perfect 5th's. If you start on an F and jump up in leaps of a 5th you will get F, C, G, D, A, E and B. These notes form the Pythagorean C major scale. All the notes in this tuning are 3-limit intervals from the tonic, since they are all powers of the 3/2 interval. While some of the ratios, such as the pythagorean B, are quite large ratios, they are still 3-limit because they factor down to 2 and 3. In the just major scale mentioned above, C, D, F and G are based on the pythagorean tuning. But E, A and B are 5-limit tunings, all flatted by one syntonic comma from their Pythagorean equivalents.
The existance of two scale tunings raises debate as to which is the best tuning to use for a melody line. The answer is determined by the way the notes are used, or the harmony that is put to, or implied by, the melody. If you run up the scale harmonizing the tones to the just-tuned I, IV and V triads, the notes are tuned to the just scale. But I find that the scale tones, played in sequence without harmony, actually sound more in tune using the Pythagorean tuning. The 3rd, 6th and 7th degrees sound slightly flat in a run up the just scale. However, when you attempt to construct chords with the Pythagorean tunings, they will sound harsh because 3-limit 3rds, 6ths and 7ths are out of tune.
In the strict root-based model, the roots of chords (and consequently the 5ths as well) would be tuned to the Pythagorean scale. The 3rds and 6ths of chords and the 7th of major and minor 7th chords would be tuned to 5-limit intervals above the chord root (by inflecting them a syntonic comma from the Pythagorean equivalents). The 7th of the dominant 7th chord is handled differently because it is a septimal interval.
In the strict fundmental-based model, the fundamentals of the chords are tuned to the Pythagorean scale. There arises in this case a difference of treatment between major and minor chords, since a major triad is rooted on its fundamental while a minor triad is rooted a just 3rd above it. A major III chord would be rooted on the Pythagorean 3rd of the scale. But a minor iii chord would be rooted on the just 3rd degree, as a result of tuning its fundamental to the tonic. A septimal minor iii chord is another situation, since it's root is a 5th above its actual fundamental. So it would properly be placed on the pythagorean 3rd, thereby putting the fundamental on the Pythagorean 6th below.
A more situational approach would be to base the tuning on the “implied root” of a chord, which varies according to the way the chord is used. For instance, the implied root of a B half-diminished 7th chord would be the B itself if it is being used as a true B 7th chord, like when resolving it to the III chord. It would be the G if it is used as a rootless G 9th chord, like when resolving it to the tonic. It would be the D, however, if it is used as a D minor 6th chord, which would probably tend to resolve to a VI or vi chord.
When you go 5 5ths up from the tonic, you get G, D, A, E and B, all scale tones. But if you go 5 5ths down instead, you get first the remaining scale tone F, and then Bb, Eb, Ab and Db, all flatted notes. When the root of a chord falls on a black note, the ear hears it as a flatted note (e.g. as a Ab instead of G#) because it is closer to the tonic in the circle of 5ths than the sharpened enharmonic. The tritone is ambiquous, because both the F# and the Gb are 6 5th's from the tonic in either direction. But using the just 5th as the generator, the F# and the Gb are actually slightly different, being separated by a tiny interval called the Pythagorean comma.
So how do I come up with the tunings for the full just chromatic scale? Well, let us extend the circle of 5ths to a 25-note Pythagorean chromatic scale centered around C: Dbb, Abb, Ebb, Bbb, Fb, Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#. Based on these, Let F, C, G and D retain their Pythagorean tuning. Going to the right (toward the sharps) every 4 notes are flatted by a syntonic comma. Conversely, to the left (toward the flats), every 4 notes are raised one comma. The 25-notes extend in both directions to B# and Dbb, the two enharmonics of C, which in this tuning are adjusted by 3 syntonic commas in each direction.
3 Dbb Abb Ebb 2 Bbb Fb Cb Gb 1 Db Ab Eb Bb 0 Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# -1 A E B F# -2 C# G# D# A# -3 E# B#
Cyan=just scale; Red=Pythagorean scale. Commatic inflection is shown vertically.
From this we get the just chromatic scale as shown in green on my Index of Intervals. This pattern of commatic inflection provides the maximum number of just 5ths and 3rds. If instead of adding a full comma every fourth note, you are to smooth the transition by adding a quarter comma every single note, the result would be something called meantone temperament, an alternate keyboard tuning that favors perfectly tuned major 3rds at the expense of slightly flatted 5ths - popular back in the Baroque period.
In closing, let's now look at what is often called the “turned chord” by barbershop quartets who use it before a song with a dominant chord pickup. Compare this chord DGBF in equal temperament:
2.996614, 4, 5.039684, 7.12719
with the same chord in just intonation:
3, 4, 5, 7
Now compare the two by moving the mouse over the glass buttons:
Intervals such as the octave, 5th and 4th are considered perfect because they have no “major” and “minor” variants. When considering why the 2nd degree of the scale seems to be the only “major” interval that isn't flatted in the minor scale, I came to think that this is directly related to it not being a true “just” major 2nd (10/9), but rather a Pythagorean wholetone (9/8) The only actual intervals that are described as having major and minor variants are 2nds, 3rds, 6ths and 7ths. These are known as imperfect intervals. In the 5-limit tuning the major and minor variant differ by the chromatic semitone, or “chroma” (25/24). For just intonation purposes, intervals that are composites of strictly perfect intervals are also considered perfect. This would include the so called acute “major” 2nd (9/8), which is really a perfect 5th upon a perfect 5th. In fact, so would any Pythagorean intervals, which are entirely composed out of perfect 5ths. Other observations I made are: a perfect and an imperfect interval combine to form another imperfect interval (like a 5th and a major 3rd form a major 7th); two major intervals form an augmented interval; two minor intervals form a diminished interval; a major and a minor interval form a perfect interval.
From all this a pattern seems to emerge: Only the simple 2- and 3-limit intervals are “perfect”, while the simple 5-limit intervals end up having major and minor variants. I say “simple” because you can extend either limit with large enough numbers to create rather clunky 3rds and 6ths in 3-limit and howling 4ths and 5ths in the 5-limit. While going through my interval table, I found what could be a new definition of a “major interval”: one whose numerator is divisible by 5. And a minor interval is one whose denominator is divisible by 5. I can also go on to define an augmented interval as one whose numerator is divisible by 25 (two major intervals combined), while a diminished interval is one whose denominator is divisible by 25 (two minor intervals combined). By the time you are looking at the “exceptions” to these rules, such as some of the grave and acute varients of these intervals, you are getting into some higher harmonics where major and minor start to lose meaning anyway.
A MIDI pitch bend of 4096 raises a note 100 cents (an equal-tempered half-step).
For any ratio frequency ratio n/p, the number of cents in the
log (n/p) * 1200/log 2 (Any log base is allowed).
The pitch bend interval size is:
log (n/p) * 49152/log 2
The following is a two-dimensional lattice showing various 5-limit intervals. The extensible 5-limit chromatic scale is shown in white, with additional note tunings in blue for the sake of completeness. The notes are separated vertically by perfect 5ths (3/2) and horizontally by major 3rds (5/4), with C (1/1) at the origin. The lattice is infinitely extensible in all directions. E.g. a vertical column of 25 notes centered around C would form the Pythagoren chromatic scale. Using this diagram allows you to analyze 5-limit chords and intervals visually, making analysis much easier. You will notice, for instance, a one-to-one relationship between the relative position of two notes in the lattice and the interval they form: a note is always a just minor 2nd above the note to the upper-right, and a just minor 3rd above the note to the lower-right (Note how D and F don't do this). You can also see that two notes forming a just major or minor interval are always in neighboring columns, major going right and minor going left. Notes forming a perfect interval are always in the same column. Also notice the column headers which indicate what kind of 5-limit interval each column has.
doubly- doubly- diminished diminished minor perfect major augmented augmented Fb+ (162/125) Ab+ (81/50) C+ (81/80) E+ (81/64) G#+(405/256) B#+(2025/1024) D##+(10125/8192) Bbb (216/125) Db (27/25) F+ (27/20) A+ (27/16) C#+(135/128) E#+(675/512) G##+(3375/2048) Ebb (144/125) Gb (36/25) Bb (9/5) D (9/8) F#+(45/32) A#+(225/128) C##+(1125/1024) Abb (192/125) Cb (48/25) Eb (6/5) G (3/2) B (15/8) D# (75/64) F##+(375/256) Dbb-(128/125) Fb (32/25) Ab (8/5) C (1/1) E (5/4) G# (25/16) B# (125/64) Gbb-(512/375) Bbb-(128/75) Db-(16/15) F (4/3) A (5/3) C# (25/24) E# (125/96) Cbb-(2048/1125)Ebb-(256/225) Gb-(64/45) Bb-(16/9) D- (10/9) F# (25/18) A# (125/72) Fbb-(4096/3375)Abb-(1024/675) Cb-(256/135) Eb-(32/27) G- (40/27) B- (50/27) D#- (125/108)
The following vectors show intervals and chords as they would be laid out in the lattice. Notice how the first two vectors show that a note is always a perfect 5th above the note below it and a major 3rd above the note to its left. From these two alone the other vectors can be constructed. Using these you can quickly locate a chord containing a particular note (e.g. look at CEG in the lattice and how it looks like the Major Triad pattern), find which notes complete the chord, and determine if the chord falls entirely within the 25-note chromatic scale (white region), or may require an alternate tuning (using notes from the blue region). Notice that the triad DFA doesn't fit the minor triad pattern, and is out of tune. A tuned D minor triad would require at least one note from the blue section. Either D-FA or DF+A+ would be in tune. If you look at the Major 9th chord pattern, you will notice that the first 4 chords are all subsets of it, being all composed of alternating major and minor 3rds.
Index of Intervals
Glossary of Just Intonation
Tables of Pitch Bends
Fun with Vowel Formants
Billy's home page