Two pitches are said to be in tune when they can sound together with a minimum of beating. Beating is one of the most important phenomena found in the study of microtuning. When two pitches sound of slightly different frequencies are sounded simultaneously you will perceive them as a single pitch being modulated to grow alternately louder and softer., The cycling of this amplitude modulation is called "beating" or "beats". 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.
audio demo(beating unison)
For example, suppose that two tones with frequencies of 440 Hz and 442 Hz are sounded together. These tones are separated by an interval of only 7.85 cents. Under these conditions you would perceive a single tone with a frequency of 441 Hz ( mathematically: 440 Hz +442 Hz = 882 Hz and 882 Hz /2 = 441 Hz). The amplitude of this single perceived tone would be modulated at the rate of 2 Hz. (442 Hz – 440 Hz = 2 Hz). This modulation is called the beat frequency
Since the beat frequency is the difference between the two frequencies it stands to reason that as two similar tones are pulled apart in pitch the beats will speed up and as the two tones are moved closer together in pitch they beats will slow down and ultimately stop. This is indeed the case. Two pitches are said to be in tune when the beats stop altogether.
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.
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.
When two notes are played simultaneously they are said to form an interval. An interval can also be identified as the relationship between the frequencies of any two notes. Intervals form the foundation from which scales and micro tuning are generally studied.
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. For example the note which forms an octave above A440 is A880 (440 Hz x2 = 880 Hz)
audio demo (octaves beating)
All intervals exhibit a subjective quality that manifests itself as the degree of consonance or dissonance with which they are perceived. Consonance is the degree to which the interval sounds pleasant or restful. A consonant interval has little or no musical tension or tendency to change. Such intervals are often found at the end of musical phrases or pieces. Dissonance is the degree to which an interval sounds unpleasant or rough. Dissonant intervals generally feel quite tense and unresolved. These intervals often precede consonant intervals to convey musical direction or movement. These perceptions are purely subjective and depend on the musical context in which they are found, but most people find general agreement about the consonance or dissonance of most intervals.
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.
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.
audio demo (perfect 5ths beating)
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.
Consider the frequencies 100 Hz and 200 Hz. Dividing 200 by 100 equals 2. In mathematical terms 200/100 = 2. The frequencies 200 Hz and 100 Hz are said to be in the ratio of 2 to 1 (written 2:1 or 2/1). For example the ratios 10/5, 48/24 1024/512, and 880/440 are all equivalent to the ratio 2/1 since the first number is twice the second number. This particular ratio describes the interval of an octave.
The advantage of using ratios to describe intervals is found in the fact that the specific frequencies that form an interval have no impact on the ratio which describes it. For example consider the frequencies 200 Hz, 400 Hz, 500 Hz, and 1000 Hz. These frequencies represent two different octaves. The various ways to combine these frequencies are found in the following table.
Addition 400+200 = 600 1000+500 = 1500
Subtraction 400-200 = 200 1000-500 = 500
Multiplication 400 x 200 = 80,000 1000 x 500 = 500,000
Division (Ratio) 400/200 = 2/1 1000/500 = 2/1
As you can see the result obtained by adding subtracting or multiplying the two frequencies together will depend on the frequencies themselves even though both of the intervals are octaves. Only the ratio (division) provides the same result in both cases. This particular example illustrates that any pair of frequencies which form an octave will be in the ratio 2/1. Of course, other intervals are not described by the ratio 2/1. The ratios associated with intervals other than the octave have been derived using a variety of means throughout history. Much of this process is described in the next section of this paper.
One of the fundamental guiding principles which is evident throughout the development of musical mathematics is based on the study of psychoacoustics. This principle contends that 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. This is the objective mathematical concept that supports the subjective perception of consonance and dissonance described above,
With this in mind, here is a list of the pure diatonic intervals and the pure ratios that are generally accepted to describe them. This list also includes the decimal equivalent obtained by dividing the smaller number into the large number of each ratio. This decimal equivalent will become important when equal temperament is is considered in a mathematical context. Notice that the list is ordered roughly in order from the most consonant to the most dissonant.
(Click on an interval name to hear it.)
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… |
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.
Musical theorists have limited the primes with which intervallic ratios are specified for various reasons throughout history. These reasons will be examined in the next section. For now it is only important to realize that the pure intervals found in the traditional twelve tone diatonic scale are represented by ratios of numbers which are multiples of primes no higher than five.
This limitation excluding ratios of numbers that are multiples of primes larger than five is known as the "5 limit", (a term coined by micro tonal composer Harry Partch in "Genesis of a Music"). It was adopted about 400 B.C. and has remained a foundation of scale development to this day. For example, Partch developed a forty three tone scale using intervals whose ratios consist of numbers which are multiples of primes no higher than eleven (the "11 limit").
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.
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. It will be used to illustrate various concepts throughout this papers.
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.
Adding or subtracting a large number of intervals can quickly become quite unwieldy. Fortunately there is a shorthand method to express the addition or subtraction of a large number of if identical intervals. This method involves the use of exponents. You may recall from high school math that multiplying a single number by itself several times can be expressed with exponents. For example 2 x 2 x 2 = 2ˆ3 = 8.
Now watch this carefully because it is the central mystery of microtuning.
Exponents can also be applied to ratios. For example adding five perfect 5ths can be expressed by the following formula
3/2 x 3/2 x 3/2 x 3/2 x 3/2 = (3/2)^{5} = 243/32
As this interval is larger than 2 octaves multiply it by 1/2 twice in order to lower it two octaves.
(3/2)^{5} x (1/2)^{2} = 243/32 x 1/4 = 243/128
This ratio is not listed in the table of diatonic intervals above. It is close to the ratio for a major 7th (15/8).
Tuning perfect fifth upwards five times and lowering by two octaves should result in exactly a major seventh. That it does not is one of the most puzzling aspects of micro tuning. This puzzle is discussed below as the anomalies of micro tuning are explained.
Musical scales are basic to most Western music. Modern keyboard instruments have 12 notes per octave with a musical interval of one semitone between adjacent notes. All common Western scales incorporate octaves whose frequency ratios are (2:1). Therefore it is only necessary to consider notes in a scale over a range of one octave, since the frequencies of notes in other octaves can be found from them. Early scales were based on one or more of the musical intervals found between members of the natural harmonic series (e.g. see Figure 3.3). 3.4.1 Pythagorean tuning
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': C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, c. The final note after 12 steps around the circle of fifths, shown as c has a frequency ratio to the starting note, C, of the frequency ratio of the perfect fifth (3:2) multiplied by itself 12 times, or:
C/c = (3/2)^{12} = 129.746 An interval of twelve fifths is equivalent to 7 octaves, and the frequency ratio for the note (c') which is 7 octaves above C is:
c'/C = 2^{7} = 128.0
Thus 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:
c/c' 129.746/128 = 1.01364
If the circle of fifths were established by descending by perfect fifths instead of ascending, the resulting note 12 fifths below the starting notes would be flatter than 7 octaves by 1.0136433, and every note of the descending circle would be slightly different to the members of the ascending circle. Figure 3.18 shows this effect
Fig 3.18 The Pythagorean scale is based on the circle of fifths formed either by ascending by 12 perfect fifths (outer) or descending by 12 perfect 4ths (inner).
and the manner in which the notes can be notated. For example, notes such as 0# and Eb, A# and Bb, Bbb and A are not the same and are known as 'enharmonics', giving rise to the pairs of intervals such as major third and diminished fourth, and major seventh and diminished octave shown in Figure 3.15. The Pythagorean scale can be built up on the starting note C by making F and G an exact perfect fourth and perfect fifth respectively (maintained a perfect relationship for the sub-dominant and dominant respectively):
F/C=4/3
G/C=3/2
The frequency ratios for the other notes of the scale are found by ascending in perfect fifths from G and when necessary, bringing the result down to be within an octave of the starting note. The resulting frequency ratios relative to the starting note C are:
D/C = 3/2 x G/C x 1/2 = 3/2 x 3/2 x 1/2 = 9/8
A/C = 3/2 x D/C = 3/2 x 9/8 = 27/16
E/C = 3/2 x A/C x 1/2 = 3/2 x 27/16 x 1/2 = 81/64
B/C = = 3/2 x E/C = 3/2 x 81/64 = 243/128
The frequency ratios of the members of the Pythagorean major scale are shown in Figure 3.19 relative to C for convenience. The frequency ratios between adjacent notes can be calculated by dividing the frequency ratios of the upper note of the pair to C by that of the lower. For example:
Frequency ratio between A and B = (243/128) / (27/16) = 243/128 x 16/27 = 9/8
Frequency ratio between E and F = (4/3) / (81/64) = (4/3) x (64/81) = 256/243s
Figure 3.18 shows the frequency ratios between adjacent notes of the Pythagorean major scale. A major scale consists of the following intervals: tone, tone, semitone, tone, tone, tone, semitone, and it can be seen that:
Figure 3.19 Frequency ratios between the notes of a C major Pythagorean scale
Frequency ratio of the Pythagorean semitone = 256/243
Frequency ratio of the Pythagorean tone = 9/8
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. The dominant and sub-dominant keynotes are a perfect fifth above and below the key note respectively. This produces all the notes of the major scale (any of which can be harmonised using one of these three chords). Taking the note C being used as a starting reference for convenience, the major scale is built as follows. The notes E and G are a major third (5:4) and a perfect fifth (3:2) respectively above the tonic, C:
E/C=5/4
G/C= 3/2
The frequency ratios of B and D are a major third (5:4) and a perfect fifth (3:2) respectively above the dominant, G and they are related to C as:
B/C = 5/4 x G/C = 5/4 x 3/2 = 15/8
D/C = 3/2 x G/C x 1/2 = 3/2 x 3/2 x 1/2 = 9/8
(The result for the D is brought down one octave to keep it within an octave of the C.)
Fig 3.20 Frequency ratios bwteen the notes of a C major just diatonic scale
The frequency ratios of A and C are a major third (5:4) and a perfect fifth (3:2) respectively above the sub-dominant, F (the F is therefore a perfect fourth (4:3) above the C (perfect fourth plus a perfect fifth is an octave):
F/C = 4/3
A/C = 5/4 x F/C = 5/4 x 4/3 = 20/12 = 5/3
The frequency ratios of the members of the just diatonic major scale are shown in Figure 3.20 relative to C for convenience, along with the frequency ratios between adjacent notes (calculated by dividing the frequency ratio of the upper note of each pair to C by that of the lower). The figure shows that the just diatonic major scale (tone, tone, semitone, tone, tone, tone, semitone) has equal semitone intervals, but two different tone intervals, the larger of which is known as a 'major whole tone' and the smaller as a 'minor whole tone':
Frequency ratio of the just diatonic semitone = 16/15
and frequency ratio of the just diatonic major whole tone = 9/8
and frequency ratio of the just diatonic minor whole tone = 10/9
The two whole tone and semitone intervals appear as members of the musical intervals between adjacent members of the natural harmonic series (see Fig 3.3) which means that the notes of the scale are as consonant with each other as possible for both melodic and harmonic musical phrases. However, the presence of two whole tone intervals means that this scale can only be used in one key since each key requires its own tuning. This means, for example, that the interval between D and A is:
A/D = (5/3) / (9/8) = (5/3) x (8/9) = 40/27
which is a musically flatter fifth than the perfect fifth (3:2).
In order to tune a musical instrument for practical purposes to enable it to be played in a number of different keys, the Pythagorean comma has to be distributed amongst some of the fifths in the circle of fifths such that the note reached after twelve fifths is exactly seven octaves above the starting note (see Figure 3.18). This can be achieved by flattening some of the fifths, possibly by different amounts, while leaving some perfect, or flattening all of the fifths by varying amounts, or even by additionally sharpening some and flattening others to compensate. There are therefore an infinite varieties of possibilities, but none will result in just tuning in all keys. Many tuning systems were experimented with which provided tuning of thirds and fifths which were close to just in some keys at the expense of other keys whose tuning could end up being so out of tune as to be unusable musically. Padgham (1986) gives a fuller discussion of hming systems. A number of keyboard instruments have been experimented with which had split black notes (in either direction) to provide access to their enharmonics, giving C# and Db, D# and Eb, F# and Gb, G# and Ab, and A# and Bb-for example, the McClure pipe organ in the Faculty of Music of the University of Edinburgh discussed by Padgham (1986)-but these have never become popular with keyboard players.
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 instmments. All semitones are equal to one twelfth of an octave. Therefore the frequency ratio (r) for an equal tempered semi tone is therefore a number which when multiplied by itself 12 times is equal to 2, or:
r = 12th Root 2= 1.0595
The equal tempered semi tone is subdivided into 'cents', where one cent is one hundredth of an equal tempered semi tone.
The frequency ratio for one cent (c) is therefore: c = 100th root of 1.0595 = 1.000578
Fig 3.21 Fundamental frequency values to four significant figures for eight octaves of notes, four either side of middle C, tuned in equal temperament with a tuning reference of A440 (Middle C is marked with a black spot)
Cents are widely used in discussions of pitch intervals and the results of psychoacoustic experiments involving pitch. Appendix 2 gives an equation for converting frequency ratios to cents and vice versa. Music can be played in all keys when equal tempered tuning is used as all semi tones and tones have identical frequency ratios. However, no interval is in tune in relation to the intervals between adjacent members of the natural harmonic series (see Figure 3.3), therefore none is perfectly consonant. However, intervals of the equal tempered scale can still be considered in terms of their consonance and dissonance, because although harmonics of pairs of notes that are in unison for pure intervals (e.g. see Tables 3.6 to 3.12) are not identical in equal temperament, the difference is within the 5% critical bandwidth criterion for consonance. Beats (see Figure 2.6) will exist between some harmonics in equal tempered chords which are not present in their pure counterparts.
Figure 3.21 shows the to values and the note naming convention used in this book for eight octaves, four either side of middle C, tuned in equal temperament with a tuning reference of 440 Hz for A4; the A above middle C. The equal tempered system is found on modern keyboard instruments, but there is increasing interest amongst performing musicians and listeners alike in the use of unequal temperament. This may involve the use of original instruments or electronic synthesisers which incorporate various tuning systems. Padgham (1986) lists approximately 100 pipe organs in Britain which are tuned to an unequal temperament in addition to the McClure organ.
Before next lecture please read Sections
pages 152 to 166 of Acoustics and Psychoacoustics. We may have a brief quiz on these sections at the beginning of the next class.
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… |