Representing Intervals and Harmonic Relationships

The Equal Tempered Scale

Until well into the 17th century, musical systems tended to be based on harmonic scales, in which each note had a harmonic relationship to the others (meaning that the relationship between frequencies is a simple mathematical ratio). The most common scale used today is not harmonic. Called the equal tempered scale, it takes advantage of some mathematical serendipity and some fuzziness in the sensitivity of the ear to create a scale that is for the most part, approximately harmonic, but has some other useful features. In harmonic scales, the notes of the scale are not evenly spaced. This has several disadvantages. The primary one is that instruments such as pianos, tuned percussion, guitars, etc, which cannot freely vary their pitch are constrained to playing in one unchanging harmonic space relative to one harmonic center, in other words, only in one musical key. In the equal tempered scale each successive note has a precise and identical mathematical relationship to the previous note. Twelve such notes are stretched across an octave, the range within which frequency doubles. A majority of those notes happen to fall tolerably close to frequencies that are harmonically related. Unlike a harmonic scale, the equal tempered scale presents a situation where harmonic relationships are not dependent on where in the scale you start. The harmonic center can be anywhere in the scale. This allows a composer or musician to shift harmonic center or key freely, and this has added a dimension of musical expression. This also gives rise to the possibilities for dissonance. In most harmonic scales, all the notes blend together harmonically. The notes in the equal tempered scale that are not in a harmonic relationship with the current harmonic center stand out from the rest and sound dissonant.

(Equal Temperament is quite specific to western musical traditions. I will go into some depth in examining it because it poses some particular challenges that harmonic systems do not. The challenge of representing harmonic relationships in any musical system is a major one, since visual perception does not seem to have an analogous feature.)

The harmonic approximations of the Equal Tempered Scale

number of
0 unison Consonant 1/1=1.000 20/12=1.000 0.0%
1 semitone Dissonant 16/15=1.067 21/12=1.059 0.7%
2 whole tone Dissonant 9/8=1.125 22/12=1.122 0.2%
3 minor third Consonant 6/5=1.200 23/12=1.189 0.9%
4 major third Consonant 5/4=1.250 24/12=1.260 0.8%
5 perfect fourth Consonant 4/3=1.333 25/12=1.335 0.1%
6 tritone Dissonant 7/5=1.400 26/12=1.414 1.0%
7 perfect fifth Consonant 3/2=1.500 27/12=1.498 0.1%
8 minor sixth Consonant 8/5=1.600 28/12=1.587 0.8%
9 major sixth Consonant 5/3=1.667 29/12=1.682 0.9%
10 minor seventh Dissonant 9/5=1.800 210/12=1.782 1.0%
11 major seventh Dissonant 15/8=1.875 211/12=1.888 0.7%
12 octave Consonant 2/1=2.000 212/12=2.000 0.0%

When different frequencies mix a new quality arises which is not found in the separate frequencies. Each musical interval (combination of 2 notes) within the octave has its unique sound. Intervals are a factor in chords, where multiple notes are played simultaneously, and in melodies where notes are strung across time. The experience of an interval is not immune to context. The same interval will have a different musical feel depending on its relationship to the current key and the recent progression of keys. This complex play of implications is perhaps the greatest challenge to translating western tonal music to a visual realm.