Two tones heard together can sound pleasing {consonance, music} {harmonics, music} or sound harsh {dissonance, music}. Humans experience tension in dissonance and repose in consonance.
musical intervals
Two musical notes have musical notes between them. Perfect first means one note with zero notes between. Perfect octave means notes are separate by eight notes. Second means notes are separate by two notes. Third means notes are separate by three notes, and so on for fourth, fifth, sixth, and seventh.
frequency ratio
Tones have vibration frequencies, and two tones have frequency ratio.
ratio pairs
Frequency ratio can be higher frequency to lower frequency or lower to higher. Example is 200/100 or 100/200. Because tones are the same, both ratios have same consonance or dissonance. Musical intervals therefore are ratio pairs, each other's inverse.
Perfect first has ratio 1/1, whose inverse is also 1/1.
Perfect octave has ratios 2/1 and 1/2. Interval 2:1 is the same as interval 1/2, with octave and fundamental exchanged. Fundamental has octave 2/1 and subfundamental 1/2. Octave is most pleasing {perfect consonance}.
Perfect fifth has ratio 3/2, and perfect fourth has ratio 4/3 = 1/((3/2) * (1/2)). They are very pleasing.
Major third has ratio 5/4, and minor sixth has ratio 8/5 = 1/((5/4) * (1/2)). Minor third has ratio 6/5, and major sixth has ratio 5/3 = 1/((6/5) * (1/2)). They are pleasing {imperfect consonance}.
Other ratio pairs are 7/4 and 8/7, 7/5 and 10/7, 9/5 and 10/9, 7/6 and 12/7, 11/6 and 12/11, 9/7 and 14/9, 11/7 and 14/11, 13/7 and 14/13, 9/8 and 16/9, 11/8 and 16/11, 13/8 and 16/13, and 15/8 and 16/15. They are inharmonious. Dissonance increases with distance from octave.
musical interval pairs
Musical-interval pairs add to nine notes: perfect first and perfect octave, perfect fifth and perfect fourth, major sixth and minor third, major third and minor sixth, major seventh and minor second, and major second and minor seventh.
overtones
Fundamental tones have overtones. First overtone has frequency two times fundamental frequency. First overtone is the octave.
Second overtone has frequency three times fundamental frequency. Because 3/1 * 1/2 = 3/2, second overtone is same tone as perfect fifth and perfect fourth but over two octaves.
Third overtone has frequency four times fundamental frequency. Because 4/1 * 1/2 = 2/1, third overtone is same tone as octave, but over two octaves. First three overtones sound harmonious, stable, and pleasing.
Fourth overtone has frequency five times fundamental frequency. Because 5/1 * 1/4 = 5/4, fourth overtone is same tone as major third and major sixth, but over three octaves. It is somewhat harmonious.
Arts>Music>Theory>Tone>Interval
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Date Modified: 2022.0224