(Road to discuss harmony) Combination of tones (Road to discuss harmony) Linear superposition If two driving forces are applied simultaneously, the response will be the sum of the responses to the driving forces individually. For instance: doubling the driving force doubles the response. In linear systems independent signals do not influence each other. Linear systems (examples): Loudspeakers, microphones and amplifiers should be linear to some extent. Is ear a linear system? Linear addition of two sound waves: Please review lecture 6 (interference)

2. Beats Slightly mismatched frequencies cause audible “beats” Question: The beat frequency between tones with frequencies f1 and f2 is 2.0 Hz. In order to increase the beat frequency, one must __. increase f1 increase f2 decrease f1 decrease f2 There is not enough information to choose Second-order beats are the beats between two tones whose frequencies are nearly but not quite in a simple ratio. They are also called beats between mistuned consonances.

Example f1 = 16 Hz f2 = 18 Hz

2a. Beats (calculations - optional)

Consonance and Dissonance Consonance - sounds that are pleasant Consonant intervals in descending order of consonance: λ2:λ1 f2:f1 examples # of half steps 1:1 1:1 unison (C,C) 0 1:2 2:1 octave (C,C) 12 2:3 3:2 perfect fifth (C,G) or (F,C) 7 3:4 4:3 perfect fourth (C,F) or (G,C) 5 3:5 5:3 major six (C,A) or (Eb,C) 9 4:5 5:4 major third (C,E) or (Ab,C) 4 5:8 8:5 minor six (C,Ab) or (E,C) 8 5:6 6:5 minor third (C,Eb) or (A,C) 3

Octave (C/C) 1 1:2 Perfect fifth (C:G) 1 2:3 Perfect fourth (C:F) 1 3:4 Octave (C’/C) 2:3

Helmholtz Theory (1877) Dissonance occurs when partials of the two tones produce 30-40 beats per second The more partials of a tone coincide with the partials of another, the less chance that beats in the range will produce roughness This explains why simply frequency windows define most of the consonant intervals

Consonance and Dissonance between two pure tones When two pure tones are sounded together, consonance or dissonance depends upon their frequency difference rather than on their frequency ratio If the frequency difference is greater than a critical band, they sound consonant If the frequency difference is less than a critical band, they sound dissonant According to Plomp and Levelt (1965) the maximum dissonance occurs at ¼ the critical bandwidth According to Kameoka and Kuriyagowa (1969) it also depends on the sound pressure level: Δf = 2.27(1 + (Lp–57)/40)f 0.447 (f is the frequency of the primary tone, and Lp is sound pressure level) The critical bandwidth changes depending on the octave of the two tones The higher the octave, the closer two notes could be and still be consonant

Consonance and Dissonance between two complex tones In this case we have to consider the roughness between the fundamental notes as well as between the harmonics This is what explains why some intervals are more consonant than others In the case of the perfect fifth the two lower harmonics coincide and the two produce frequency differences within the critical bandwidth

Musical Scales and Temperament Musical scale – a succession of notes in ascending or distending order In Western music octave is divided in 12 semitones Chromatic scale - all 12 semitones Most music makes use of 7 selected notes (major or minor scales) There are many ways to construct musical scales Different scales are different ways of dividing octave (almost always) “Standard” scales: Pythagorean scale Mean-tone temperament Scale of just intonation Equal temperament Tuning – an adjustment of pitch in any instrument so that it corresponds to an accepted norm (scale) Temperament – a system of tuning in which the intervals deviate from acoustically pure (Pythagorean) intervals Intonation – the degree of accuracy with which pitches are produced

Scales and logarithms When we go from octave to octave up, each time we multiply frequency by 2 Examples: If we go 3 octaves up, frequency is 2x2x2 = 23 = 8 times higher If we go 7 octaves up, frequency is 27 = 128 times higher On keyboards and on musical staff distance between notes is changed linearly If we go 3 octaves, it is 3 time as much as one octave If we go 7 octaves, it is 7 time as much as one octave This means that keyboard and musical staff have logarithmic scale: distance between keys and notes is proportional to the logarithm of the frequency

(All semitones are the same) Equal temperament (All semitones are the same) Octave is divided into 12 equal semitone intervals Semitone ratio: A 440 B flat 466 B 494 C 523 C sharp 554 D 587 D sharp 622 E 659 F 698 F sharp 740 G 784 A flat 831 880 (~6% up) Whole tone: Advantage: 5th and 4th are reasonably good 3d and 6th are OK Modulation from key to key is easy