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We have seen that we perceive pitch as associated with frequency, and even when the frequency is complex, we hear it as a single unit determined by the fundamental frequency or by a subjective tone (implying a missing fundamental). Even though we reduce complex sounds to a single unit (pitch), when we hear two musical tones, either simultaneously, or successively, we identify it as an interval. An interval is defined as the ratio between the frequencies of two tones. If the ratio between two pitches is 2:1 the interval is an octave.
| C1 | C2 | G2 | C3 | E3 | G3 | (Bb3) | C4 | D4 | E4 | (F#4) | G4 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 100 | 200 | 300 | 400 | 500 | 600 | 700) | 800 | 900 | 1000 | 1100 | 1200 |
| Octave | 1:2 |
|---|---|
| Perfect 5 | 2:3 |
| Perfect 4 | 3:4 |
| Major 3 | 4:5 |
| minor 3 | 5:6 |
| Major 6 | 3:5 |
| minor 6 | 5:8 |
| Major 7 | 8:15 |
| minor 7 | 5:9 |
| Major 2 | 8:9 |
| minor 2 | 15:16 |
These are called just or natural intervals -- if we made a scale out of them, we would have a just scale. In fact singers and violinists and instrumentalist who have control over their intonation will play justly -- they sometimes have to adjust to an equal tempered instrument like the piano. As long as you stay in the same key (with the same notes, their is no problem with just intonation, but as soon as you begin to modulate their are difficulties. In fact, you can see already that we have two M2 that are different ratios -- 9/8 and 10/9: if we began on D and determined the intervals, the E would be 9/8 * 900 = 1012.5. The F would be 6/5*900 = 1080 -- from C the F would be 800*4/3 = 1066.6. The result is that the further you modulate from the original key on a tuned instrument, the more unusual the intervals get. To solve this problem the idea of equal temperment was developed in the late 17th - early 18th century.
The equal tempered scale (the one we are most used to) has the octave divided into 12 equal semitones -- this means that any starting note will give you equal intervals -- all keys will sound the same. The intervals created in this system are determined by taking the 12th root of two (or 1.05946) . Thus an octave has a ratio of 2, and a tritone has a ratio of root 2.
The problem with this system is that none of the intervals are exactly in tune (except the octave). Since the semitone can be divided into 100 cents, we find, in equal temperment, the following deviations from the natural intervals:
Pythagorean: This scale is created by building fifths (and transposing by octaves. The result is a slightly altered scale -- containing two seconds M2 (204 cents) and m2 (called a Limma) (256/243 = 90 cents)
Other non-Western cultures have scales that have more divisions of the octave (Persian scale has 16 divisions -- some in between our semitones)
These are relative terms -- however, in general consonance and dissonance seem to be related to interval and harmonic series. For example if a two pitches are at unison the harmonic series will coinside -- they have the same components. If they are an octave appart, they will have a 50 % coincidence -- all the components of the lower are present in the upper, but only 1/2 of the upper are present in the lower . The next most consonant interval is P5 which has 16.7% coincidence. (Intensity of influence). Continuing in order are:P4 - 8.3, M6 at 6.7, M3 at 5.0, m3 at 3.3, m6 at 2.5. then m7,M2,M7,m2,Tritone
Another way of looking at the concept of consonance and dissonace is that the more complex the ratio (even in sine tones) the more dissonant the sound -- the tritone has a ratio of 1 to 1.414 therefore is dissonant.
Helmholtz and others have maintained that the development of harmony is closely related to the harmonic series and our perception of intervals. For example: the interval of a a fifth C-G (200,300) will generate beats (subconscious or conscious) at 100 inplying a root. The interval of a third C-F (200-266) will imply the root F (at 66) -- the third E - G (1000,1200) will imply the root 200 (C). In fact most composers find out that when they build chords on the harmonic series they tend to get a more balanced sound (piano writing)
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