How intervals are counted and named, why the numbers refuse to add up, and the physical account of consonance that turns a memorised list into something you can hear a reason for — and use on a mix.
An interval is the distance between two pitches, and it is the unit everything else in music is assembled from. Scales are patterns of intervals. Chords are stacks of them. Tuning is an argument about them. And, more usefully for anyone working in a control room, the reason two parts fight each other in a mix turns out to be an interval fact rather than a mixing fact.
This lesson does two things: it gets you naming intervals accurately, and it replaces the usual memorised list of which ones are consonant with a physical account of why — which is the version that stays with you and the version that is actually useful at the desk.
Interval numbers come from counting letter names, including both ends. C up to G counts C, D, E, F, G — five names, so it is a fifth. C to E counts three, so it is a third. C to the C above counts eight, so it is an octave, and a note with itself counts one, called a unison.
Counting both ends means the shared note gets counted twice whenever intervals are stacked, so the numbers do not add normally. A third stacked on a third is a fifth, not a sixth: 3 plus 3 minus 1. A fifth plus a fourth is an octave: 5 plus 4 minus 1 is 8. This trips up nearly everyone at some point, and once you know the reason it stops being confusing.
The number alone is not enough — C to E and C to E-flat are both thirds and they sound nothing alike. Each interval also carries a quality, and intervals fall into two groups that behave differently.
Unisons, fourths, fifths and octaves are called perfect. They are the same size whether the surrounding key is major or minor, and they have the simplest frequency ratios of any interval, which is where the name came from.
Seconds, thirds, sixths and sevenths come in major and minor forms, a semitone apart. These are the intervals that carry the difference between a major and a minor key.
Either group can also be stretched or squeezed by a semitone. A major interval made one semitone larger is augmented; a minor one made a semitone smaller is diminished. A perfect interval has no minor form, so it goes straight from perfect to diminished going down, and perfect to augmented going up.
Counted in semitones, the full set inside one octave runs:
The six-semitone interval in the middle is the tritone, so called because it is three whole tones. It is the one interval with no plain name — it is spelled as an augmented fourth or a diminished fifth depending on where it came from, and it sits exactly halfway through the octave, which is why it is its own inversion.
There is a method that removes the need for the table above. Ask whether the upper note appears in the major scale of the lower note. If it does, the interval is major (for 2nds, 3rds, 6ths and 7ths) or perfect (for unisons, 4ths, 5ths and octaves).
Then adjust. If the upper note is a semitone lower than the scale would give, the interval is minor — or diminished, if it was a perfect one. A semitone higher makes it augmented.
Worked: what is D up to B-flat? Count the letters — D E F G A B — six, so it is a sixth. Is B-flat in the D major scale? No; D major has a B natural. B-flat is a semitone below that, so this is a minor sixth. No table, and it works in any key.
Move the lower note of an interval up an octave, or the upper one down, and you invert it. C up to G is a fifth; G up to C is a fourth. Two rules cover every case:
So a major sixth inverts to a minor third, and a minor seventh inverts to a major second. This is not a curiosity: it is the mechanism behind chord inversions in the next course, and it explains why a chord can be voiced several different ways and remain recognisably the same chord.
Most theory teaching gives you a list — fifths and octaves consonant, seconds and sevenths dissonant — and leaves it there. There is a physical reason underneath it, and it comes straight from the harmonic series.
Any pitched sound is made of a fundamental plus harmonics at whole-number multiples of it. A note at 200 Hz has energy at 400, 600, 800, 1,000 and so on. Now sound two notes together, and ask how many of their harmonics land on exactly the same frequency.
Take a perfect fifth, ratio 3:2. A 200 Hz note against a 300 Hz note: the lower one's harmonics are 400, 600, 800, 1,000, 1,200; the upper one's are 600, 900, 1,200, 1,500. They share 600 and 1,200 and more above that — the third harmonic of the lower is the second harmonic of the upper, and the pattern repeats. Coinciding partials do not interfere with each other, so the pair sounds smooth.
The simplest ratios have the most coincidences, and they are exactly the intervals we call consonant:
Notice the order. The intervals everyone calls consonant sit at the top with small whole numbers; the ones called dissonant need much larger numbers to describe, which is another way of saying their harmonics almost never coincide.
When two partials do not coincide but sit close together, they interfere, and the result is a slow rise and fall in level at a rate equal to the difference between the two frequencies. 440 Hz against 442 Hz gives two beats a second. This is a subtraction you can hear, and it is the oldest tuning tool there is: adjust until the beating stops.
As the two frequencies separate, the beating speeds up and stops sounding like pulsing. Somewhere in the region of fifteen to twenty beats a second it turns into a distinct grating quality — roughness — and that sensation is what dissonance actually is at the level of the ear. Separate them further still and the two are heard as two clean tones with no interaction at all.
Where that boundary sits depends on the ear's frequency resolution, described in terms of critical bands. Two tones inside the same critical band interfere; two tones in different bands largely do not. And critical bands are proportionally wider at low frequencies, which produces the most practically useful consequence in this entire lesson.
The same interval is rougher played low than played high. A close-voiced major third at the bottom of a piano is muddy, and the identical third two octaves up is clean and bright. Nothing about the interval changed — what changed is whether the two notes and their lower harmonics fall inside one critical band. This is why arrangers keep low voicings open, with wide spacing at the bottom and closer stacking above, and it is also why a bass part and a low keyboard pad playing a third apart will sound like a problem no EQ move quite fixes. It is not a mix problem. It is an arrangement problem, and it gets solved by moving a note.
Those clean ratios are just intonation. Equal temperament, which is what your instruments are tuned to, divides the octave into twelve identical steps instead — which lets music change key freely and puts every interval slightly out of its pure ratio.
The tempered fifth is 700 cents against a just 702: two cents flat, and effectively inaudible. That is why equal temperament works at all. The tempered major third is 400 cents against a just 386 — 14 cents sharp, which is enough to hear as a slow shimmer between the harmonics of a sustained chord. The tempered minor third is about 16 cents narrow.
So the fifth got away with the compromise and the thirds paid for it. That is the honest reason a sustained equal-tempered major triad has a faint restlessness that a barbershop quartet or a brass section does not — ensembles without fixed pitch drift toward the pure ratios automatically, because they are tuning by ear to the point where the beating stops.
Here is where the theory turns into desk work. Two notes are consonant because many of their harmonics land in the same place. Two parts mask each other because many of their harmonics land in the same place. Those are the same sentence.
Which means the interval between two parts predicts how they will behave in a mix:
This also explains a common frustration. When two instruments will not separate no matter what you do with EQ, check what interval they are playing. If they are moving in parallel thirds or sixths in the same octave, no filter is going to give you two things, because acoustically there is close to one thing there. The fix is in the arrangement — an octave apart, or a different inversion — and it takes about ten seconds compared with the twenty minutes of EQ that will not work.
An augmented fourth and a diminished fifth are the same six semitones and the same sound. They are not the same interval, because they came from different places and are going to different places: an augmented fourth is a fourth that has been stretched and tends to resolve outward, while a diminished fifth is a fifth that has been squeezed and tends to resolve inward.
This is the same principle as enharmonic spelling for single notes, applied to pairs. The name records function — where the interval sits in the harmony and what it is likely to do next — and the sound alone cannot tell you that. It is why theory keeps two names for one sound instead of simplifying.
The tritone is the interval that carries the most of this history. It was avoided as a melodic leap in early sacred writing, on grounds that had as much to do with how hard it is to sing as with anything doctrinal, and it later became the engine of tonal harmony rather than an outcast: the tritone between the third and seventh of a dominant seventh chord is exactly the tension that makes the resolution home feel inevitable. The next course takes that apart properly.
One more distinction in naming. Two notes sounded together form a harmonic interval; two notes sounded one after the other form a melodic interval. The name and size are identical either way, but only the harmonic case produces beating and masking, because only in that case are both sets of harmonics in the air at once.
The roughness argument has a practical form that arrangers have used for a century, sometimes called the low interval limit: below a certain pitch, each interval stops being usable in close voicing.
The rough guidance is that a major or minor third becomes muddy below roughly the E or E-flat just under middle C, while a perfect fifth stays reasonably clear an octave or so lower, and an octave stays clear lower still. Those are approximations, not laws — they shift with timbre, level and the room — but the ordering is reliable, and it comes straight from the critical-band explanation above: the wider the interval, the further apart the two sets of harmonics, and the lower it can go before they collide.
At a mixing desk this is a diagnostic. When the low end is congested and no filter fixes it, look at the notes before the EQ: a keyboard voicing with a third at the bottom, or a bass and a synth an interval apart down low, will sound congested in any room and through any monitor, because the congestion is happening in the listener's ear rather than in the mix.
Intervals larger than an octave are compound intervals, and they are named by continuing the count: a ninth is an octave plus a second, an eleventh an octave plus a fourth, a thirteenth an octave plus a sixth. Those three names turn up constantly in chord symbols, and the next course explains what they are doing there.
For hearing intervals rather than counting them, the reliable method is to attach each one to a song you already know by heart and can recall instantly — a rising fifth, a falling minor third, a major seventh. The specific songs do not matter and yours will work better than anyone else's, because the point is the recall, not the reference.
Consonance is not a matter of taste. It is how much of two notes' harmonic content lands in the same place — which is also, word for word, the definition of masking.
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