Physical sound attributes directly relate to music attributes {music, hearing} {hearing, music}. Physical-sound frequency relates to music pitch. Music is mostly about frequency combinations. However, above 5000 Hz, musical pitch is lost. Physical-sound intensity relates to music loudness. Physical-sound duration relates to music rhythm. Physical=sound spectral complexity relates to music timbre.
However, frequency affects loudness. Intensity affects pitch. Tone frequency separation affects time-interval perception. Harmonic fluctuations, pitch changes, vibrato, and non-pitched-instrument starting noises {transient, sound} affect timbre. Timbre affects pitch.
emotion: chords
Chords typically convey similar feelings to people. Minor seventh is mournful. Major seventh is desire. Minor second is anguish. Humans experience tension in dissonance and repose in consonance.
emotion: pitch change
Music emotions mostly depend on relative pitch changes (not rhythm, timbre, or harmony).
emotion: key
Music keys have characteristic emotions. Composers typically repeat same keys and timbre, and composers have typical moods.
song: melody
Note sequences can rise, fall, or stay same. People can recognize melodies from several notes.
song: musical phrase
People perceive music phrase by phrase, because phrases have repeated often and because phrases take one breath. Children complete half-finished musical phrases using tones, rhythm, and harmony.
brain
No brain region is only for music. Music uses cognitive and language regions.
Musical pitch makes musical notes {tone, hearing}.
octave
Tones can be double or half other-tone frequencies. Octaves go from a note to similar higher note, such as middle-C at 256 Hz to high-C at 512 Hz. Hearing covers ten octaves: 20 Hz, 40 Hz, 80 Hz, 160 Hz, 320 Hz, 640 Hz, 1280 Hz, 2560 Hz, 5120 Hz, 10240 Hz, and 20480 Hz.
octave tones
Within one octave are 7 whole tones, 7 + 5 = 12 halftones, and 24 quartertones.
overtones
Tones two, four, eight, and so on, times fundamental frequency are fundamental-frequency overtones.
sharpness or flatness
Fully sharp tone has frequency one halftone higher than tone. Slightly sharp tone has frequency slightly higher than tone. Fully flat tone has frequency one halftone lower than tone. Slightly flat tone has frequency slightly lower than tone.
musical scales
Musical scales have tone-frequency ratios. Using ratios cancels units to make relative values that do not change when units change.
equal temperament scale
Pianos have musical tones separated by equal ratios. Octave has twelve equal-temperament halftones, with ratios from 2^(0/12) to 2^(12/12) of fundamental frequency. Frequency ratio of halftone to next-lower halftone, such as C# to C, is 2^(1/12) = 2^.08 = 1.06. Starting at middle-C, ratios of tones to middle-C are 2^0 = 1 for middle-C, 2^.08 = 1.06 for C#, 2^.17 = 1.13 for D, 2^.25 = 1.19 for D#, 2^.33 = 1.26 for E, 2^.42 = 1.34 for F, 2^.50 = 1.41 for F#, 2^.58 = 1.49 for G, 2^.67 = 1.59 for G#, 2^.75 = 1.68 for A, 2^.83 = 1.78 for A#, 2^.92 = 1.89 for B, and 2^1 = 2 for high-C. See Figure 1. F# is middle tone.
equal-temperament scale: frequencies
Using equal-temperament tuning and taking middle-C as 256 Hz, D has frequency 289 Hz. E has frequency 323 Hz. F has frequency 343 Hz. G has frequency 384 Hz. A has frequency 430 Hz. B has frequency 484 Hz. High-C has frequency 512 Hz. Low-C has frequency 128 Hz. Low-low-C has frequency 64 Hz. Lowest-C has frequency 32 Hz. High-high-C has frequency 1024 Hz. Higher Cs have frequencies 2048 Hz, 4096 Hz, 8192 Hz, and 16,384 Hz. From 32 Hz to 16,384 Hz covers nine octaves.
tone-ratio scale
Early instruments used scales with tones separated by small-integer ratios. Tones had different frequency ratios than other tones.
tone-ratio scale: all possible small-integer ratios
In one octave, the 45 possible frequency ratios with denominator less than 13 are: 3/2; 4/3, 5/3; 5/4, 7/4; 6/5, 7/5, 8/5, 9/5; 7/6, 11/6; 8/7, 9/7, 10/7, 11/7, 12/7, 13/7; 9/8, 11/8, 13/8, 15/8; 10/9, 11/9, 13/9, 14/9, 16/9, 17/9; 11/10, 13/10, 17/10, 19/10; 12/11, 13/11, 14/11, 15/11, 16/11, 17/11, 18/11, 19/11, 20/11, 21/11; 13/12, 17/12, 19/12, and 23/12.
tone-ratio scale: whole tones
In octaves, the seven whole tones are do, re, mi, fa, so, la, ti, and do, for C, D, E, F, G, A, and B. The seven tones are not evenly spaced by frequency ratio. Frequency ratios are D/C = 6/5, E/C = 5/4, F/C = 4/3, and G/C = 3/2. For example, C = 240 Hz, D = 288 Hz, E = 300 Hz, F = 320 Hz, and G = 360 Hz. C, D, E, F, and G, and G, A, B, C, and D, have same tone progression. Frequency ratios are A/G = 6/5, B/G = 5/4, C/G = 4/3, and D/G = 3/2. For example, G = 400 Hz, A = 480 Hz, B = 500 Hz, C = 532 Hz, and D = 600 Hz.
tone-ratio scale: halftones
Using C as fundamental, the twelve halftones have the following ratios, in increasing order. 1:1 = C. 17:16 = C#. 9:8 = D. 6:5 = D#. 5:4 = E. 4:3 = F. 7:5 = F#. 3:2 = G. 8:5 = G#. 5:3 = A. 7:4 or 16:9 or 9:5 = A#. 11:6 or 15:8 = B. 2:1 = C.
tone-ratio scale: quartertones
The 24 quartertones have the following ratios, in increasing order. 1:1 = 1.000. 33:32 = 1.031. 17:16 = 1.063, or 16/15 = 1.067. 13:12 = 1.083, 11:10 = 1.100, or 10/9 = 1.111. 9:8 = 1.125. 8:7 = 1.143, or 7:6 = 1.167. 6:5 = 1.200. 17:14 = 1.214, or 11/9 = 1.222. 5:4 = 1.250. 9:7 = 1.286. 4:3 = 1.333. 11:8 = 1.375. 7:5 = 1.400. 17:12 = 1.417, 10:7 = 1.429, or 13/9 = 1.444. 3:2 = 1.500. 14/9 = 1.556, 11:7 = 1.571, or 19:12 = 1.583. 8:5 = 1.600. 13:8 = 1.625. 5:3 = 1.667. 12:7 = 1.714, or 7:4 = 1.75. 16:9 = 1.778, or 9:5 = 1.800. 11:6 = 1.833, or 13:7 = 1.857. 15:8 = 1.875. 23:12 = 1.917. 2:1 = 2.000. Ratios within small percentage are not distinguishable.
tone intervals
Two tones have a number of tones between them. First interval has one tone, such as C. Minor second interval has two tones, such as C and D-flat, and covers one halftone. Major second interval has two tones, such as C and D, and covers two halftones. Minor third interval has three tones, such as C, D, and E-flat, and covers three halftones. Major third interval has three tones, such as C, D, and E, and covers four halftones. Minor fourth interval has four tones, such as C, D, E, and F, and covers five halftones. Major fourth interval has four tones, such as C, D, E, and F#, and covers six halftones. Minor fifth interval has five tones, such as C, D, E, F, and G-flat, and covers six halftones. Major fifth interval has five tones, such as C, D, E, F, and G, and covers seven halftones. Minor sixth interval has six tones, such as C, D, E, F, G, and A-flat, and covers eight halftones. Major sixth interval has six tones, such as C, D, E, F, G, and A, and covers nine halftones. Minor seventh interval has seven tones, such as C, D, E, F, G, A, and B-flat, and covers ten halftones. Major seventh interval has seven tones, such as C, D, E, F, G, A, and B, and covers eleven halftones. Eighth interval is octave, has eight tones, such as C, D, E, F, G, A, B, and high-C, and covers twelve halftones.
tone intervals: pairs
Tones have two related ratios. For example, D and middle-C, major second, have ratio 289/256 = 9/8, and D and high-C, minor seventh, have ratio 9/16, so high-C/D = 16/9. The ratios multiply to two: 9/8 * 16/9 = 2. E and middle-C, major third, have ratio 323/256 = 5/4, and E and high-C, minor sixth, have ratio 5/8, so high-C/E = 8/5. F and middle-C, minor fourth, have ratio 343/256 = 4/3, and F and high-C, major fifth, have ratio 2/3, so high-C/G = 3/2. G and middle-C, major fifth, have ratio 384/256 = 3/2, and G and high-C, minor fourth, have ratio 3/4, so high-C/G = 4/3. A and middle-C, major sixth, have ratio 430/256 = 5/3, and A and high-C, minor third, have ratio 5/6, so high-C/A = 6/5. B and middle-C, major seventh, have ratio 484/256 = 15/8, and B and high-C, minor second, have ratio 15/16, so high-C/B = 16/15.
The ratios always multiply to two. Tone-interval pairs together span one octave, twelve halftones. For example, first interval, with no halftones, and octave, with twelve halftones, fill one octave. Major fifth interval, with seven halftones, such as C to G, and minor fourth interval, with five halftones, such as G to high-C, fill one octave. Major sixth interval, with nine halftones, such as C to A, and minor third interval, with three halftones, such as A to high-C, fill one octave. Major seventh interval, with eleven halftones, such as C to B, and minor second interval, with one halftone, such as B to high-C, fill one octave. Minor fifth interval and major fourth interval fill one octave. Minor sixth interval and major third interval fill one octave. Minor seventh interval and major second interval fill one octave.
tone intervals: golden ratio
In music, ratio 2^0.67 = 1.59 ~ 1.618... is similar to major sixth to octave 1.67, octave to major fourth 1.6, and minor seventh to major second 1.59. Golden ratio and its inverse can make all music harmonics.
tone harmonics
Tones have harmonics {tone harmonics} that relate to tone-frequency ratios.
tone harmonics: consonance
Tone intervals can sound pleasingly consonant or less pleasingly dissonant. Octave tone intervals 2/1 have strongest harmonics. Octaves are most pleasing, because tones are similar. Tones separated by octaves sound similar.
Major fifth and minor fourth intervals are next most pleasing. Major-fifth 3/2 and minor-fourth 4/3 tone intervals have second strongest harmonics.
Major third 5/4 and minor sixth 8/5 intervals are halfway between consonant and dissonant. Minor third 6/5 and major sixth 5/3 intervals are halfway between consonant and dissonant.
Major fourth 7/6 and minor fifth 12/7 intervals are dissonant. Major second 8/7 and minor seventh 7/4 intervals are dissonant, or major second 9/8 and minor seventh 16/9 intervals are dissonant. Minor second 16/15 and major seventh 15/8 intervals are most dissonant.
Ratios with smallest integers in both numerator and denominator sound most pleasing to people and have consonance. Ratios with larger integers in both numerator and denominator sound less pleasing and have dissonance.
Three tones can also have consonance or dissonance, because three tones make three ratios. For example, C, E, and G have consonance, with ratios E/C = 5/4, G/E = 6/5, and G/C = 3/2.
Tone ratios in octaves higher or lower than middle octave have same consonance or dissonance as corresponding tone ratio in middle octave. For example, high-G and high-C have ratio 6/4 = 3/2, same as middle-G/middle-C.
Tone ratios between octave higher than middle octave and middle octave have similar consonance as corresponding tone ratio in middle octave. For example, high-G and middle-C have ratio 3/1. Dividing by two makes high-G one octave lower, and middle-G/middle-C has ratio 3/2.
tone harmonics: beat frequencies
Frequencies played together cause wave superposition. Wave superposition makes new beat frequencies, as second wave regularly emphasizes first-wave maxima. Therefore, beat frequency is lower than highest-frequency original wave.
If wave has frequency 1 Hz, and second wave has frequency 3 Hz, they add to make 1-Hz wave, 3-Hz wave, and 2-Hz wave, because every other 3-Hz wave receives boost from 1-Hz wave. Rising 1-Hz wave maximum coincides with first rising 3-Hz wave maximum and falling 1-Hz wave maximum coincides with third falling 3-Hz maximum, while first falling 3-Hz wave maximum, middle rising and falling 3-Hz maximum, and third rising 3-Hz maximum cancel.
If one wave has frequency 2 Hz, and second wave has frequency 3 Hz, they add to make 2-Hz wave, 3-Hz wave, and 1-Hz wave, because every third 3-Hz wave receives boost from 2-Hz wave. First rising 2-Hz wave maximum coincides with first rising 3-Hz wave maximum, while first falling 3-Hz wave maximum, middle rising and falling 3-Hz maximum, and third rising and falling 3-Hz maximum cancel.
Beat frequency is difference between wave frequencies: 3 Hz - 2 Hz = 1 Hz in previous example. Beat frequencies are real physical waves.
Small-integer frequency ratios have lower beat frequencies and reduce beat frequency interference with original frequencies. Two waves with small-integer frequency ratios superpose to have beat frequency that has small-integer ratios with original frequencies. Two waves with large-integer frequency ratios superpose to have beat frequency that has large-integer ratios with original frequencies.
Middle-C has frequency 256 Hz, and middle-G has frequency 384 Hz, with ratio G/C = 3/2. The waves add to make 384 Hz - 256 Hz = 128 Hz beat wave, with ratio C/beat = 2/1 and G/beat = 3/1.
Middle-C has frequency 256 Hz, and middle-E has frequency 323 Hz, with ratio E/C = 5/4. The waves add to make 323 Hz - 256 Hz = 67 Hz beat wave, with ratio C/beat = 4/1 and E/beat = 5/1.
Middle-C has frequency 256 Hz, and middle-D has frequency 289 Hz, with ratio D/C = 9/8. The waves add to make 289 Hz - 256 Hz = 33 Hz beat wave, with ratio C/beat = 8/1 and D/beat = 9/1.
Middle-C has frequency 256 Hz, and middle-A has frequency 430 Hz, with ratio A/C = 5/3. The waves add to make 430 Hz - 256 Hz = 174 Hz beat wave, with ratio C/beat = 3/2 and D/beat = 5/2.
Middle-C has frequency 256 Hz, and middle-B has frequency 484 Hz, with ratio B/C = 15/8. The waves add to make 484 Hz - 256 Hz = 228 Hz beat wave, with ratio C/beat = 9/8 and B/beat = 17/8.
Roger Shepard [1964] gradually increased or decreased all tones of a chord, keeping the tones separated by octaves. Pitch repeats when reaching the next octave, so tones rise or fall but do not keep rising or falling {Shepard tone} {Shepard scale}, an auditory illusion.
Brain recognizes music by rhythm or by intonation differences near main note {music, processing}. Brain analyzes auditory signals into tone sequences with pitches, durations, amplitudes, and timbres. First representation {grouping structure} segments sound sequence into motifs, phrases, and sections. Second representation {metrical structure} marks sequence with hierarchical arrangement of time points {beat}.
Brain can find phrasing symmetries {time-span reduction}, using grouping and metrics.
Brain can hierarchically arrange tension and relaxation waves {prolongational reduction}. In Western music, prolongational reduction has slowly increasing tension followed by rapid relaxation.
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225