Music
Notes
Pitches, Frequencies and Intervals
Music is related to sound caused by vibration. In equal temperament, every pair of adjacent pitches is separated by the same interval. The pitches of an equal temperament can be produced by repeating a generating interval. Classical western music theory divides frequencies into 7 pitches, C, D, E, F, G, A, B. The human ear’s perception of pitch mainly depends on the frequency ratio, not the frequency difference. The equal temperament uses the diatonic scale. Johann Sebastian Bach stipulates that there are 12 pitches in an octave, and the relationship between pitches and frequencies is shown as follow:
| Octave→ Note↓ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| C | 16.352 (-48) | 32.703 (-36) | 65.406 (-24) | 130.81 (-12) | 261.63 (+0) | 523.25 (+12) | 1046.5 (+24) | 2093.0 (+36) | 4186.0 (+48) | 8372.0 (+60) |
| C#/Db | 17.324 (-47) | 34.648 (-35) | 69.296 (-23) | 138.59 (-11) | 277.18 (+1) | 554.37 (+13) | 1108.7 (+25) | 2217.5 (+37) | 4434.9 (+49) | 8869.8 (+61) |
| D | 18.354 (-46) | 36.708 (-34) | 73.416 (-22) | 146.83 (-10) | 293.66 (+2) | 587.33 (+14) | 1174.7 (+26) | 2349.3 (+38) | 4698.6 (+50) | 9397.3 (+62) |
| D#/Eb | 19.445 (-45) | 38.891 (-33) | 77.782 (-21) | 155.56 (-9) | 311.13 (+3) | 622.25 (+15) | 1244.5 (+27) | 2489.0 (+39) | 4978.0 (+51) | 9956.1 (+63) |
| E | 20.602 (-44) | 41.203 (-32) | 82.407 (-20) | 164.81 (-8) | 329.63 (+4) | 659.26 (+16) | 1318.5 (+28) | 2637.0 (+40) | 5274.0 (+52) | 10548 (+64) |
| F | 21.827 (-43) | 43.654 (-31) | 87.307 (-19) | 174.61 (-7) | 349.23 (+5) | 698.46 (+17) | 1396.9 (+29) | 2793.8 (+41) | 5587.7 (+53) | 11175 (+65) |
| F#/Gb | 23.125 (-42) | 46.249 (-30) | 92.499 (-18) | 185.00 (-6) | 369.99 (+6) | 739.99 (+18) | 1480.0 (+30) | 2960.0 (+42) | 5919.9 (+54) | 11840 (+66) |
| G | 24.500 (-41) | 48.999 (-29) | 97.999 (-17) | 196.00 (-5) | 392.00 (+7) | 783.99 (+19) | 1568.0 (+31) | 3136.0 (+43) | 6217.9 (+55) | 12544 (+67) |
| G#/Ab | 25.957 (-40) | 51.913 (-28) | 103.83 (-16) | 207.65 (-4) | 415.30 (+8) | 830.61 (+20) | 1661.2 (+32) | 3322.4 (+44) | 6644.9 (+56) | 13290 (+68) |
| A | 27.500 (-39) | 55.000 (-27) | 110.00 (-15) | 220.00 (-3) | 440.00 (+9) | 880.00 (+21) | 1760.0 (+33) | 3520.0 (+45) | 7040.0 (+57) | 14080 (+69) |
| A#/Bb | 29.135 (-38) | 58.270 (-26) | 116.54 (-14) | 233.08 (-2) | 466.16 (+10) | 932.33 (+22) | 1864.7 (+34) | 3729.3 (+46) | 7458.6 (+58) | 14917 (+70) |
| B | 30.868 (-37) | 61.735 (-25) | 123.47 (-13) | 246.94 (-1) | 493.88 (+11) | 987.77 (+23) | 1975.5 (+35) | 3951.1 (+47) | 7902.1 (+59) | 15804 (+71) |
The relationship between frequencies and their corresponding pitches have changed a lot. Nowadays, standard pitch is defined as A4=440 Hz.
Pitch interval is the difference between two pitches perceived by humans.
In the mid-range of human hearing, human perception of intervals between notes of different pitches is roughly logarithmic; in the high-frequency range, it does not conform to the logarithmic rule. This characteristic of human hearing is determined by the physiological structure of the human auditory system. That is to say, for the three tones with frequencies of 200Hz/400Hz/800Hz, two of the intervals are roughly equal; while for the three tones with frequencies of 200Hz/400Hz/600Hz, humans will hear these two intervals different.
For two tones with a double frequency difference, such as 440Hz/880Hz, the interval between them is set to octave.
The most common temperament used today is equal temperament. The relationship of frequency ratio and pitches of the equal temperament is shown as follow:
| Intervals | Equal Temperament |
| C:C | 1.0000 |
| D:C | 1.1225 |
| E:C | 1.2599 |
| F:C | 1.3348 |
| G:C | 1.4983 |
| A:C | 1.6818 |
| B:C | 1.8877 |
| C’:C | 2.0000 |
For the 12 pitches in an octave, the name of intervals and their frequency ratio is shown as follow:
| Number of Semitones | Pitch Interval | Frequency Ratio |
| 0 | Perfect Unison | 1:1 |
| 1 | Minor Second | 16:15 |
| 2 | Major Second | 9:8 |
| 3 | Minor Third | 6:5 |
| 4 | Major Third | 5:4 |
| 5 | Perfect Fourth | 4:3 |
| 6 | Augmented Fourth | 45:32 |
| 6 | Diminished Fifth | 64:45 |
| 7 | Perfect Fifth | 3:2 |
| 8 | Minor Sixth | 8:5 |
| 9 | Major Sixth | 5:3 |
| 10 | Minor Seventh | 16:9 |
| 11 | Major Seventh | 15:8 |
| 12 | Perfect Octave | 2:1 |