Interval Arithmetic and Harmony

Intervals and scales

Knowing something about intervals and scales is important if you want to understand music more completely, including the origins of the scales we use.

When we want to talk about the relationship between two notes we use the term 'interval'. Intervals are defined mathematically in terms of frequencies. The frequency of a note is how fast it vibrates. The frequency of A above middle C is 440 vibrations per second, for instance. An interval is defined in terms of the ratio of frequencies of the two notes. The term 'interval' technically is a misnomer because it is a frequency ratio, not a frequency difference.

The most basic interval, the octave, is the range between a note and the next higher instance of that note, such as middle A and high A. An octave represents a 1 to 2 (written 1:2) frequency ratio, or 2:1 from the perspective of the higher note. For instance high A has a frequency of 440 x 2 = 880 vibrations per second. Low A (A below middle C) has a frequency of 220.

After the octave (2:1 ratio), the next most natural interval is the ratio 3:2. This ratio is called a perfect fifth. The next most natural interval is 4:3, the perfect fourth. The fourth plus the fifth make an octave

When you combine two intervals, the resulting interval's frequency ratio is the first ratio times the second ratio. As an example a perfect fourth plus a fifth produces the ratio 4/3 x 3/2 = 2, which is the ratio for an octave. To "subtract" one interval from another, you divide the larger interval's ratio by the second ratio.

The difference between a fourth and a fifth, as an interval, is called a pure whole step or whole tone, and has a frequency ratio of 8:9 (3/2 divided by 4/3). This interval is also called a major second.

The frequency ratio 4:5 is called a major third, and 5:6 is a minor third.

A minor sixth is the interval which together with a major third, makes an octave. Its ratio is 5:8. A major sixth together with a minor third also make an octave. The major sixth's ratio is 3:5.

A major sixth plus a whole tone is called a major seventh, and has a ratio of 8:15.

Suppose you put together a series of notes that represent the following intervals from the first (tonic) note: major second, major third, fourth, fifth, major sixth, and major seventh. This will be a series of seven notes that are very convenient for constructing music, because there are no large jumps between successive notes, and certain combinations of these notes form pleasing harmonies. This series of notes is the major scale. The jump or 'difference' between the major third and the fourth is called a half step or half tone. This is also the 'difference' between the major seventh and the octave. The sequence of intervals, with note 1 repeated an octave higher as note 8, is arranged in this pattern:

1 - whole – 2 - whole – 3 - half - 4 – whole – 5 - whole – 6 - whole – 7 - half - 8

Tuning issues every musician should know about

If you decided to tune a piano and chose C as the

tonic note, and then tuned the other notes to the ratios given above, all the intervals relative to C would have their correct ratios and you could play in the key of C. You could then tune the F sharp and the B flat and play in the key of G or F. But it would be impossible to tune all the notes so that the intervals would be correct in all of the keys (you can prove this by working through the arithmetic).

If you divide the octave into twelve equal steps, the ratio between steps would be the twelfth root of 2, or 1.05946. If you tuned all the keys of a piano such that the interval between successive keys was this ratio, it would be very close to the correct tuning in any key. This is called equal temperament tuning and is the standard way of tuning keyboard instruments. You can then play in any key, but the intervals are no longer 'pure'. For example all fifths are slightly flat (ratio 1.4983 instead of 1.5000). You can hear this flatness if you listen carefully.

We hear harmonies, not intervals

A single note by itself of course has no harmonic meaning. But the same is also true of a two-note interval. There must be a third note to define the harmony and therefore the quality of the sound. Notes 1 and 3 of the scale for example sound different depending on whether they are part of the 1 harmony or the 6 harmony. So it's rather meaningless to talk about the affective quality of the major third for example, because its feel will depend on what triad it is part of. A major second (whole note) by itself sounds dissonant, but in the context of a seventh chord the seventh and the tonic are not heard as a major second, they are heard as a minor seventh.

The major triad, consisting of a major third and perfect fifth, does have a certain quality: bright and joyful. But it takes all three notes to establish that quality. Likewise the minor triad (minor third plus fifth) has an introspective quality, and all three notes are required to define that attribute.

Therefore it makes little sense to spend a lot of time studying intervals, except in the sense of learning how to place each note of the scale in relation to the tonic. That's important if you are a singer.

Music theory myth: it's been said that a seventh chord (major triad plus minor seventh) feels like it wants to resolve to the tonic because it contains a tritone, or augmented fourth, which is thought to be unstable. This is a myth because our ears (brain) do not hear intervals, they hear harmonies. The simplest example would be the major triad, which contains a major third, a minor third, and a fifth. You might hear the major third and the fifth, but you don't hear the minor third, even unconsciously. You really just hear the notes in relation to the tonic. By the way, this myth about seventh chords is not the only lie that is perpetrated in the name of music theory.

The major triad has a consonant or pleasing sound because the frequencies all blend -- they are in the proportion 4:5:6. The minor triad frequencies have the proportion 10:12:15. A diminished triad is 25:30:36.

But why, for example, the multiple proportion 4:5:6 would be heard universally as bright and lively is still unknown.

To summarize: we hear harmonies, not intervals, and harmonies are defined as a set of notes in relation to the tonic. Said another way, scale degree (what note of the scale it is) is more important than intervals, in understanding harmonic relationships.

Minor and modal scales

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