INDEX

College of Santa Fe Auditory Theory

Lecture 011 Hearing VI

INSTRUCTOR CHARLES FEILDING

  1. Hearing notes
  2. Harmonics and the development of Western harmony
  3. Consonance and dissonance
  4. Hearing musical intervals
  5. Brain Bullets

Note Data from Note Name

Input note name in the form "A#3" (no b)
Note Name:
MIDI Note Number: A440=A4
Cents Detune: cents
Frequency: Hz

 

3.3 Hearing notes 136

Fig 3.15 All two note musical intervals ocurring up to one octave

The music of different cultures can vary considerably in many aspects including, for example, pitch, rhythm, instrumentation, available dynamic range, and the basic melodic and harmonic usage in the music. 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. The perception of chords and the development of different tuning systems is discussed in this section from a psychoacoustic perspective to complement the acoustic discussion earlier in this chapter in consideration of the development of melody and harmony in Western music.

3.3.1 Harmonics and the development of Western harmony

Hearing harmony is basic to music appreciation, and in its basic form harmony is sustained by means of chords. 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.

A basis for understanding the psychoacoustics of a chord is given by considering the perception of any two notes sounding together. The full set of commonly considered two note intervals and their names are shown in Figure 3.15 relative to middle C. Each of the augmented and diminished intervals sound the same as another interval shown if played on a modern keyboard, for example the augmented unison and minor second, the augmented fourth and diminished fifth, the augmented fifth and minor sixth, and the major seventh and diminished octave, but they are notated differently on the stave and depending on the tuning system in use, these 'enharmonics' would sound different also.

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. Figure 3.3 shows the musical intervals between the first ten harmonics of the natural harmonic series. 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.

3.3.2 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. These changes have occurred partly as a function of increasing acceptance of intervals which are deemed to be musically 'consonant', or pleasing to listen to. as opposed to 'dissonant', or unpleasant to the listener. The psychoacoustic basis behind consonance and dissonance relates to critical bandwidth, which provides a means for determining the degree of consonance or dissonance) of musical intervals

Fig 3.16 The perceived consonance and dissonance of two pure tones

Figure 2.6 illustrates the perceived effect of two sine waves heard together when the difference between their frequencies was increased from 0 to above one critical bandwidth. Listeners perceive a change from 'rough' to 'smooth' when the frequency difference crosses the critical bandwidth. In addition, a change occurred between 'rough fused' to 'rough separate' as the frequency difference is increased within the critical bandwidth. Figure 3.16 shows the result of an experiment by Plomp and Levelt (1965) to determine to what extent two sine waves played together sound consonant or dissonant as their frequency difference is altered. Listeners with no musical training were asked to indicate the consonance or pleasantness of two sine waves played together. (Musicians were not used in the experiment since they would have preconceived ideas about musical intervals which are consonant.) The result is the continuous pattern of response shown in the figure, with no particular musical interval being prominent in its degree of perceived consonance. Intervals greater than a minor third were judged to be consonant for all frequency ratios. The following can be conduded:


Few musical instruments ever produce a sinusoidal acoustic waveform, and the results relating consonance and dissonance to pure tones can be extended to the perception of musical intervals heard when instruments which produce complex periodic tones play together. 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. This contribution can be assessed based on the conclusions from Figure 3.16. The overall consonance (dissonance) of a chord is based on the total consonance (dissonance) contribution from each of these harmonics.

3.3.3 Hearing musical intervals

Musical intervals can be ordered by decreasing consonance on this psychoacoustic basis.

Table 3.6 The degree of consonance and dissonance of a two note chord in which all harmonics are present for both notes a perfect fifth apart, the f0 for the lower note being 220Hz.


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. If the harmonic of the upper note is mid-way between harmonics of the lower note, the test is carried out with the higher frequency pair since the critical bandwidth will be larger and the positions of table entries indicates this. (This exercise is similar to that carried out using the entries in Table 3.5.)

For example, Table 3.6 shows this calculation for two notes whose f0 values are a perfect fifth apart (f0 frequency ratio is 3:2), the lower note having an f0 of 220 Hz. The frequency difference between each harmonic of each note and its closest neighbour harmonic in the other note is calculated (the higher of the two is used in the case of a tied distance), to give the entries in column 3, the frequency mid-way between these harmonic pairs is found (column 4), the critical bandwidth for these midfrequencies is calculated (column 5). The contribution to dissonance of each of the harmonic pairs is given in the right-hand column as follows:

(i) if they are in unison (equal frequencies) they are 'perfectly consonant', shown as 'C' (note that their frequency difference is less than 5% of the critical bandwidth).


(ii) if their frequency difference is greater than the critical bandwidth of the frequency mid-way between them (i.e. the entry in column 3 is greater than that in column 5) they are 'consonant', shown as 'c'.

(iii) if their frequency difference is less than half the critical bandwidth of the frequency mid-way between them (i.e. the entry in column 3 is less than that in column 6) they are 'highly dissonant', shown as 'D'.

iv) if their frequency difference is less than the critical bandwidth of the frequency mid-way between them but greater than half that critical bandwidth (i.e. the entry in column 3 is less than that in column 5 and greater than that in column 6) they are 'dissonant', shown as 'd'.

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. The case of a two-note unison is trivial in that all harmonics are in unison with each other and all contribute as 'c'. For the octave, all harmonics of the upper note are in unison with harmonics of the lower note contributions as 'c'. Tables 3.6 to 3.10 show the contribution to dissonance and consonance for the intervals perfect fifth (3:2), perfect fourth (4:3), major third (5:4), minor third (6:5) and major whole tone (9:8) respectively. The dissonance of the chord in each case is related to the entries in the final column which indicate increased dissonance in the order C, c, d and D, and it can be seen that the dissonance increases as the harmonic number increases and the musical interval decreases.

Table 3.7 The degree of consonance and dissonance of a two note chord in which all harmonics are present for both notes a perfect fourth apart the f0 of the lower note being 220 Hz

Table 3.8 The degree of consonance and dissonance of a two note chord in which all harmonics are present for both notes a major third apart the f0 of the lower note being 220 Hz

Table 3.9 The degree of consonance and dissonance of a two note chord in which all harmonics are present for both notes a minor third apart the f0 of the lower note being 220 Hz

 

The harmonics which are in unison with each other can be predicted from the harmonic number. For example, in the case of the perfect fourth the fourth harmonic of the lower note is in unison with the third of the upper note because their to values are in the ratio (4:3). For the major whole tone (9:8), the unison will occur between harmonics (the eighth of the upper note and the ninth of the lower) which are not resolved by the auditory system for each individual note.

As a final point, the degree of dissonance of a given musical interval will vary depending on the f0 value of the lower note, due to the nature of the critical bandwidth with centre frequency

Table 3.10 The degree of consonance and dissonance of a two note chord in which all harmonics are present for both notes a whole tone apart the f0 of the lower note being 220 Hz

Table 3.11 The degree of consonance and dissonance of a two note chord in which all harmonics are present for both notes a major third apart the f0 of the lower note being 110 Hz


(e.g. see Figure 3.8). Tables 3.11 and 3.12 illustrate this effect for the major third where the to of the lower note is one octave and two octaves below that used in Table 3.8 at 110 Hz and 55 Hz respectively. The number of 'D' entries increases in each case as the f0 values of the two notes are lowered.

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'.

Table 3.12 The degree of consonance and dissonance of a two note chord in which all harmonics are present for both notes a major third apart the f0 of the lower note being 55 Hz

Fig 3.17 Different spacings of the chord of C major. Play each and listen to the degree of muddiness or harshness each produces.

Figure 3.17 shows a chord of C major in a variety of four-part spacings and inversions which illustrate this effect when the chords are played, preferably on an instrument producing a continuous steady sound for each note such as a pipe organ, instrumental group or suitable synthesiser sound. To realise the importance of this point, it is most essential to lislell to the effect. The psychoacoustics of music is after all, about how music is perceived, not what it looks like on paper!

Reading Assignment

Before next lecture please read Sections

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

You Need To Know