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.)
| number of semitones |
interval name |
generally considered |
closest harmonic |
actual ratio |
difference |
|---|---|---|---|---|---|
| 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.