Auditory Theory: Acoustics

Lecture 011 Hearing VI

Reading Assignment for Lecture 012

Before next lecture please read Sections

• 3.3 Tuning Systems

pages 144 to 151 of Acoustics and Psychoacoustics. We may have a brief quiz on these sections at the beginning of the next class.

Brain Bullets

• Hearing notes
  • Musical taste is always evolving with time; what one composer is experimenting with may well become part of the established tradition a number of years later.
• Western harmony
  • A chord consists of at least two notes sounding together and it can be described in terms of the musical intervals between the individual notes which make it up.
  • The development of harmony in Western music can be viewed in terms of the decreasing musical interval size between adjacent members of the natural harmonic series as the harmonic number is increased.
  • The musical interval between adjacent harmonics must reduce as the harmonic number is increased since it is determined in terms of the f₀ of the notes concerned by the ratio of the harmonic numbers themselves (e.g. 2:1 > 3:2 > 4:3 > 5:4 > 6:5, etc.).
  • The earliest polyphonic Western music, known as 'organum', made use of the octave, the perfect fifth and its inversion, the perfect fourth. These are the intervals between the 1st and 2nd, the 2nd and 3rd, and the 3rd and 4th members of the natural harmonic series respectively (see Figure 3.3). Later, the major and minor third began to be accepted, the intervals between the 4th and 5th, and the 5th and 6th natural harmonics, with their inversions, the minor and major sixth respectively which are the intervals between the 5th and 8th, and the 3rd and 5th harmonics respectively. The major triad, consisting of a major third and a minor third, and the minor triad, a minor third and a major third, became the building block of Western tonal harmony. The interval of the minor seventh started to be incorporated, and its inversion the major second, the intervals between the 4th and 7th harmonic and the 7th and 8th harmonics respectively. Twentieth century composers have explored music composed using whole tones, the intervals between the 8th and 9th, and between the 9th and 10th harmonics, semitones, harmonics above the 11 th are spaced by intervals close to semitones and microtones (intervals of less than a semitone), harmonics above the 16th are spaced by microtones.
• Consonance and dissonance
  • The development of Western harmony follows a pattem where the intervals central to musical development have been gradually ascending the natural harmonic series.
  • when the frequencies are equal (unison) the tones are judged to be 'perfectly consonant'
  • when their frequency difference is greater than one critical bandwidth, they are judged consonant
  • For frequency differences of between 5% and 50% of the critical bandwidth the interval is dissonant
  • Maximum dissonance occurs when the frequency difference is a quarter of a critical bandwidth.
  • For each note of the chord, each harmonic that would be resolved by the hearing system if the note were played alone, that is all harmonics up to about the seventh, contributes to the overall perception of consonance or dissonance depending on its frequency proximity to a harmonic of another note in the chord.
• Hearing musical intervals
  • Musical intervals can be ordered by decreasing consonance on this psychoacoustic basis.
  • To determine the degree of consonance of a musical interval consisting of two complex tones, each with all harmonics present, the frequencies up to the frequency of the seventh harmonic of the lower notes are found, then the critical bandwidth at each frequency mid-way between harmonics of each note that are closest in frequency is found to establish whether or not they are within 5% to 50% of a critical bandwidth and therefore adding a dissonance contribution to the overall perception when the two notes are played together.
  • The contribution to dissonance depends on where the musical interval occurs between adjacent harmonics in the natural harmonic series. The higher up the series it occurs, the greater the dissonant contribution made by harmonics of the two notes concerned.
  • This increase in dissonance of any given interval, excluding the unison and octave which are equally consonant at any pitch on this basis, manifests itself in terms of preferred chord spacings in classical harmony. As a rule when writing four-part harmony such as SA TB (soprano, alto, tenor, bass) hymns, the bass and tenor parts are usually no closer together than a fourth except when they are above the bass staff, because the result would otherwise sound 'muddy' or 'harsh'