Auditory Theory: Acoustics

Lecture 012 Hearing VII

Reading Assignment for Lecture 014

Before next lecture please read Sections

  • 4.1 A 'black box' model of musical instruments 152
  • 4.2 Stringed instruments 155

pages 152 to 166 of Acoustics and Psychoacoustics. We may have a brief quiz on these sections at the beginning of the next class.

Brain Bullets

You Need to Know

  • Unison tuning
    • Two pitches are said to be in tune when they can sound together with a minimum of beating.
    • The frequency of the perceived pitch is the average of the two pitches. This average is obtained by adding the two frequencies together and dividing by two. The rate at which the beats occur is merely the difference between the two actual frequencies.
    • Fast beating is irritating to the human ear because it disables the ear’s ability to judge distance and phase relationships within a sound. Therefore sounds which beat fast are described as dissonant and unstable while sounds that beat slowly or not at all are described as being consonant or stable.
  • Pure Intervals
    • By tuning in such a way as to minimize beating it is possible to create a scale in which intervals are optimized for consonance.
    • As it turns out this is achieved by choosing intervals whose frequency relationships are expressed in small whole number ratios.
    • The problem with pure intervals is that tuning instruments using them leads to octaves which are hopelessly out of tune and never repeat.
    • While this may be a satisfactory situation for monophonic melodic instruments playing solo it is quite unacceptable for polyphonic instruments or for monophonic instruments playing ensemble.
  • Octaves and intervals
    • The simplest interval other than the unison is the octave.
  • Musically, two notes that form an octave share the same note name (for example C).
    • The note sounds almost identical, and yet one is higher than the other.
    • The octave is one of the most compelling intervals because it demonstrates the cyclic, or repeating, nature of musical sound.
    • Mathematically an octave is obtained by doubling the frequency of any note.
    • The octave is usually considered to be the most consonant interval.
  • The other generally accepted consonant intervals are the perfect fifth, major third, major sixth, minor third and minor sixth.
  • The intervals that are generally accepted to be dissonant are the major second, minor seventh, minor second, major seventh and the triton (augmented fourth or diminished fifth).
  • A mathematical basis for these subjective perceptions can be seen in the representation of intervals by ratios.
  • Perfect 5ths
    • After the octave the fifth is the next most consonant interval to tune.
    • A fifth can be achieved by dividing a string into three parts (an octave divides the string into two parts).
    • In a perfect fifth the upper note vibrates three times for every two vibrations on the lower note.
    • It was Pythagoras who first discovered a method for tuning by fifths.
    • His discovery led to the cycle of fifths which creates every note in the diatonic scale.
    • Unfortunately using his method does not exactly create a cycle of fifths..it creates a spiral of fifths which highlight the central problem of tuning...that notes tuned repeatedly upwards by intervals do not yield in tune octaves.
  • Ratios
    • The relationship between the frequencies of the two notes forming any interval can be described mathematically as a ratio.
    • Numerically ratios behave as fractions, nothing more than one number being divided by another number.
    • This means that you can determine the ratio formed by any two frequencies by simply dividing one frequency by the other.
  • Consonance
    • The interval described by ratios in small whole numbers are more consonant and "harmonious" to the human ear than intervals described by ratios of numbers other than whole numbers.
    • The smaller the numbers in the ratio the more consonant the interval.
  • Pure Interval Pure Ratio Decimal Equivalent
    Unison 1/1 1
    Octave 2/1 2
    Perfect 5th 3/2 1.5
    Perfect 4th 4/3 1.333333333333…
    Major 6th 5/3 1.666666666666…
    Major 3rd 5/4 1.25
    Minor 3rd 6/5 1.2
    Minor 6th 8/5 1.6
    Major 2nd 9/8 1.125
    Major 7th 16/9 1.777777777777…
    Minor 2nd 16/15 1.066666666666…
    Tritone 45/32 or 62/45 1.40625 or 1.42222222222…
  • Prime limits
    • If you examine the ratios listed above, you’ll notice that none of the numbers in any of the ratios are multiples of numbers higher than 5. All of the numbers in these ratios are multiples of two, three or five. These are examples of numbers known as primes. A prime number is a number that can be divide only by itself and one. Other primes include seven, eleven and thirteen.
  • Frequency
    • The ratios in the table above can be used to calculate the frequency of any note which forms a specific interval with another of known frequency. For example to calculate the frequency of the E a perfect 5th above A440 multiply the known frequency by the value of the ratio (that is by its decimal equivalent). In this case 440 Hz x 1.5 = 660 Hz.
  • Addition and subtraction
    • Intervals can be added together in order to form other intervals. For example a Perfect 5th and a Perfect 4th placed back to back form an octave (C to F+F to C = C to C). Interestingly the same result is obtained by multiplying the ratios of the intervals being added.. In this example 4/3 x 3/2 = 12/6 = 2/1. This technique is very helpful when considering the effect of tuning several intervals upward one after the other.
    • A similar technique is used to subtract intervals. The ratio representing the interval to be subtracted is inverted (flipped over) and multiplied by the ratio describing the other interval. For example subtracting a Perfect 4th from a Perfect 5th will result in a major 2nd (C to G – G to D = C to D). Using the technique described above 3/2 x 3/4 = 9/8 which is the ratio of a major 2nd. This technique is used to discern the effect of tuning intervals downwards.
  • Tuning systems
    • The Pythagorean scale is built up from the perfect fifth. Starting for example from the note C and going up in 12 steps of a perfect fifth produces the 'circle of fifths':
    • But twelve perfect fifths (C to c) is therefore slightly sharp compared with 7 octaves (C to c') by the so-called 'Pythagorean comma' which has a frequency ratio:
  • Just tuning
    • Another important scale is the 'just diatonic' scale which is made by keeping the intervals which make up the major triads pure: the octave (2:1), the perfect fifth (3:2) and the major third (5:4) for triads on the tonic, dominant and sub-dominant.
  • Equal tempered tuning
    • The spreading of the Pythagorean comma unequally amongst the fifths in the circle results in an 'unequal temperament'. Another possibility is to spread it evenly to give 'equal temperament' which makes modulation to all keys possible where each one is equally out of tune with the just scale. This is the tuning system commonly found on today's keyboard instruments. All semitones are equal to one twelfth of an octave.