Mathematics and History of Microtuning

 

Introduction.

When we consider the subject of tuning in general two questions seem to pop up almost immediately.

1) Why did it take centuries for musicians to arrive at a scale which is simply 12 equal divisions of the octave?

2) Having now established the Equal Tempered scale as the universal norm why are people still interested in other tunings?

Here are the short answers.

1) Why did it take so long?
There is such a thing as a natural pure tone scale which can only be truly enjoyed by someone playing solo on a monophonic instrument which has 1) limited range and 2) no keyboard. This scale is wonderfully consonant and pleasant since it is based on small prime number ratios. It can also be arrived at intuitively just by tuning to beats.

This pure tone scale has two problems.

i) Scales produced by repeatedly tuning pure intervals lead to octaves that do not repeat (are not in tune). This is a pity because the octave is the most compelling consonance of all.

ii) The actual pitches for the same notes used in different keys are different.
Violinists using just intonation will play the note C4 (an octave above middle C) at a different pitch if they are in the key of C than if they are in the key of Db.

This is all very well for the violinist. He has no frets and simply slides his finger sharp on the leading note but to ask the harpsichordist to stop and retune for every modulation is to invite a strike... possibly against the violin section.

The reason it took centuries to sort out the equal tempered scale is because everyone is right. The violinist is correct when he says that his instrument sounds better in just intonation and the harpsichordist is correct when he says that he cannot modulate in just intonation.

The gradual change to equal temperament was driven largely by the tremendous improvements in Church organ design in the 16th 17th and 18th centuries and the growing dominence of instruments with keyboards. Because Church Organs embraced tremendous ranges and had keyboards it became necessary to make them a) in tune with themselves over wide ranges and b) able to modulate to accommodate the new theories of harmony that led up to Johann Sebastian Bach and Mozart.

2) Why are people still interested in other tunings?
The advent of samplers has suddenly brought world instruments into sharp focus. It is no longer necessary to spend several freezing winters in South America in order to play an Andean Flute. If you want to sell sampled sounds these days I have two words for you - ethnic instruments.

Because memory is cheap it is now common to sample every note of an ethnic instrument with the result that whatever tuning was native to that instrument is faithfully captured in the sampler. As a sound designer I can testify to the fact that the moment you reach that point in the development of a sampled ethnic instrument where you put the samples into equal temperament you suddenly lose all sense of ethnicity. Worse still, you actually change the timbre by retuning it.

Equal temperament is a hallmark of Western music made necessary by Western obsessions with polyphonic ensemble harmony. It is meaningless in other cultures which do not share the West's love of orchestras and keyboards.

Generally I would caution you against deciding what will and will not be important in the development of future music. If you had told me in 1986 that a style of music would dominate the American market in the latter part of the 20th century which would be based on 8 and 12 bit samples of 78 rpm records and the sound of an obsolete Roland Analog Drum Machine I would have laughed and laughed and laughed,

One hip hop composer once told me "All we are trying to do is something different than what everybody has being doing for the last forty years" and that defining priciple of new music remains. Ironically, just as the world has given up on pure intervals and settled for the bland universal "out of tunedness" of equal temperament computers have entered the picture which are able to retune an entire instrument in a millisecond thus solving at least one of the most compelling objections to pure tone intonation - the ability to modulate.

Frequency

Sound is transmitted through the air by the action of molecules vibrating and colliding with each other. The source of the sound sets the molecules into vibration that corresponds to the physical vibration of the sound source itself. For example, imagine that a note is played on a synthesizer that is connected to an amplifier and a speaker. The speaker cone vibrates causing the air molecules nearby to vibrate in a corresponding fashion.

These nearby molecules collide with neighboring molecules which in turn collide with their neighbors and so on until the air next to our eardrums are set into motion. We perceive the sound when our eardrums are stimulated to vibrate in response to the motion of the nearby molecules.

The notes used to create most of the music we hear today are sounds for which the molecule vibrations are very regular. The sound source, the air and our eardrums all vibrate at a regular rate called the frequency that is expressed in cycles per second. In honor of the contributions made to the study of acoustics by the German physicist Heinrich Hertz, cycles per second are also called Hertz (abbreviated Hz).

Humans are capable of perceiving frequencies roughly between 20 Hz and 20,000 Hz. If our eardrums are stimulated to vibrate at a rate less than 20 Hz or greater than 20,000 Hz we would not perceive the experience as sound. In fact depending on the amplitude (or volume) of the stimulation we might no perceive it at all. This range of perception varies from person to person. The upper limit tends to decrease with advancing age and exposure to long periods of loud sounds.

The frequencies associated with specific musical notes depend mainly on historical factors. Before the advent of technological innovations such as tuning forks and electronic tuners musical pitch varied with geography and history. During the Renaissance the frequency associated with A above middle C was generally around 460 Hz. By the baroque period this A had dropped to roughly 415 Hz, almost a whole step lower. Since then the frequency associated with this a has risen slowly until it has reached its current value of 440.

In order to specify the frequency being associated with a particular note the following convention will be used throughout this paper. The name of the note will be followed immediately with the frequency of that note. For example: an A with a frequency of 440 will be written A440.

Unison tuning

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.

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 un acceptable for polyphonic instruments or for monophonic instruments playing ensemble.

Octaves and intervals

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.

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.

audio demo (perfect 5ths beating)

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.

Basics

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.

Consonance

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,

Pure Diatonic Intervals

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.

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.

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").

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. 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.

Individual notes.

Ratios can be used to represent individual notes within the context of a key. For example the note G could be represented by the ratio 3/2 in the key of C. This idea can be generalized to represent scale degrees with ratios in any key. For example the third major scale degree would be represented by the ratio 5/4 while the third minor scale degree would be represented by 6/5. This notation is used to specify various scales without regard to a starting note.

The ratio of a note forming a specific interval with another note is calculated by multiplying the ratio of the known note by the ratio representing the interval. For example the second degree of a major scale is represented by the ratio 9/8. To find the ratio of the note a Perfect 5th above it multiply the ratio by 4/3. Mathematically 9/8 x 4/3 = 36/24 = 3/2. The note found a Perfect 4th above the second degree of a major scale is the fifth degree. In the key of C the note found a perfect fourth above D is G.

Similarly the ratio describing the interval between any two notes can be calculated by inverting the ratio of the lower note and multiplying. For example the second and fifth degrees of the major scale are represented by the ratios 9/8 and 3/2 respectively. After inverting the ratio of the lower note the interval between them is calculate by multiplying the ratios together. Mathematically, 8/9 x 3/2 = 24/18 = 4/3. In other words the interval formed by the second and fifth degrees of the major scale is a perfect 4th.

The normal musician response to this kind of insight is something equivalent to "Duuh" however we beg you to stay with us as we explore the more complex relationships that this mathematical method allows.

Octaves

As with other frequencies, ratios describing individual notes or intervals that extend beyond one octave can be brought within the scope of the octave by dividing any such ratio by two. For example, two perfect fifths up from C leads to D one octave and a whole step higher (C to G + G to D). Mathematically, 3/2 x 3/2 = 9/4. To reduce this ratio by one octave divide by two. This is equivalent to multiplying the ratio by 1/2. Mathematically, 9/4 x 1/2 = 9/8. This is the ratio that describes the major second or whole step. Returning to the previous example lowering the D by one octave would place in a major second above the original C. This technique of multiplying ratios by 1/2 in order to reduce them by one octave will be used throughout this paper.

Many musical intervals.

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.

audio demo(tuning P5 upwards)

Equal Temperament.

So far only ratios of small whole numbers have been discussed. Ironically the tuning system used exclusively today consists of no whole numbers except the octave (2/1). The foundation of equal temperament lies in the division of the scale into 12 equal intervals called semitones that correspond to 1/2 steps. The process by which this division is accomplished involves exponents and their alter egos known as roots.

Suppose for a moment that the octave was to be divided into two exactly equal intervals. The decimal equivalent of the ratio describing these intervals will be represented by the letter "r" (for ratio). If two intervals of this ratio were added together the resulting interval would be one octave. Mathematically, r^2 = r x r = 2 (recall that the decimal equivalent of the ratio 2/1 is 2). The number that satisfies this equation cannot be represented by a whole number ratio. It cannot even be written as an exact decimal equivalent. This number is the square root of two. Its decimal equivalent is 1.414213562. If you multiply this number by itself you will find that the result is very nearly equal to 2.

Of course the equal tempered scale divides the octave into twelve equal intervals called semitones. By adding 12 of these semitones together the resulting interval will be one octave. If the letter "r" is used to represent the decimal equivalent of the ratio describing these semitones the following formula illustrates the process.

r^12 = r x r x r x r x r x r x r x r x r x r x r x r = 2

Once again the number that satisfies this equation cannot be expressed as a whole number nor can it be expressed as an exact decimal equivalent. It is called the twelfth root of two and is written 12 √ 2. Its decimal equivalent is approximately 1.059463094. If you multiply this number by itself 12 times the result is very nearly two.

The difference between pure minor seconds and equal tempered seconds can be demonstrated using decimal equivalents. Recall that a pure minor second is described by the ratio 16/15. Its decimal equivalent is approximately 1.0666666667, As you have seen above, the ratio describing the equal tempered semitone is approximately 1.059463094.
The nearest whole number ratio for this is 89/84. This indicates that the pure minor second is slightly wider than an equal tempered semitone.

audio demo(pure second and equal second)

The corollary also holds: a scale constructed of pure minor seconds will have octaves that are no longer in the ratio of 2/1.

Cents

In order to easily compare various intervals, each of the equally spaced semitones are further divided into 100 equal intervals called cents. This exceedingly small interval cannot be described as a whole number ratio. Its decimal equivalent is approximately 1.00057779. The nearest whole number ratio is 1731/1730. Using this method of measurement the intervals found in the equal tempered scale are easy to derive. They are listed in the following table along with their pure interval counterparts.

Interval Equal tempered Cents Pure Cents
Unison 0 0
Minor 2nd 100 111.73
Major 2nd 200 203.73
Minor 3rd 300 315.64
Major 3rd 400 386.31
Perfect 4th 500 498.04
Tritone 600 590.22 or 609.78
Perfect 5th 700 701.85
Minor 6th 800 813.69
Major 6th 900 884.36
Minor 7th 1000 996.09
Major 7th 1100 1088.27
Octave 1200 1200

Formula.

There is a formula for converting any ratio into its equivalent measure in cents. The formula involves the use of logarithms. While logarithms are related to exponents it is not important that you fully understand them in order to use the formula. It merely requires that you have a calculator which calculates logarithms (abbreviated as log). In the following formula the letter "r" represents the decimal equivalent of the ratio you wish to convert and the letter "c’ represents the number of cents into which the ratio will be converted.

c= 3986.313714 x log r

Here is the procedure for using this formula.

1. Determine the decimal equivalent of the ratio you wish to convert by dividing the upper number of the ratio by the lower number.
2. Calculate the log of this decimal equivalent using an appropriate calculator.
3. Multiply the result obtained in step 2 by 3986.313714
4. The result of 3 will be the number of cents in the selected ratio.

For example this procedure will be used to find the number of cents in a perfect 5th.

1. A perfect 5th is represented by the ratio 3/2. The decimal equivalent of this is 1.5
2. The log of 1.5 is approximately 0.176091259. Mathematically, this is written log 1.5 = 0.176091259
3. Multiplying this number by 3986.313714 reveals the number of cents in a pure 5th.
4. Mathematically 0.176091259 x 3986.313714 = 701.9550008 cents.

This indicates that the equal tempered perfect 5th (which is exactly 700 cents) is almost two cents flatter than a pure perfect 5th. The other diatonic intervals can be similarly compared to reveal that except for the octave none of the intervals in the equal tempered scale are perfectly in tune. The reasons for this temperaments universal acceptance will be discussed in the next section of this paper.

Anomalies

In the study of tunings and temperaments one encounters small anomalies or errors which are inherent in the nature of intervals. Most of them were identified early in the history of musical theory. In general an anomaly is the interval found between the two notes found at the beginning and the end of a certain series of pure intervals. This interval is usually quite small.

These anomalies illustrate a curious fact about pure intervals, If pure intervals are repeatedly tuned upwards or downwards from a starting note there will never be an exact recurrence (regarding octave displacements) of any note in the sequence. Some of the notes so generated will be within one cent or less of other notes in the sequence but there will no exact duplications. The anomalies described below are the most common such differences.

Comma

The commas are the most common anomaly in the study of tuning and temperaments. They are primarily encountered when tuning perfect 5ths. The specific intervals that give rise to each of the commas is described below.

Syntonic.

The Syntonic comma, also known as the comma of Didymus after its discoverer, becomes evident by tuning four perfect 5ths upwards followed by one major third downwards. For example tuning four perfect 5ths upwards from C results in the notes C-G-D-A-E. A major third down from E returns to C.
On a piano keyboard this C is exactly two octaves above the starting note of the sequence. However if these intervals are in their pure form as described by the ratios listed earlier the two C’s do not form perfect octaves. This can be demonstrated using the interval addition and subtraction techniques described earlier.

3/2 x 3/2 x 3/2 x 3/2 x 4/5 = 324/80

On the piano keyboard this process results in a ratio of 4/1 which describes two notes exactly two octaves apart. Using pure intervals this odd ratio of 324/80 is the result.
You will recall that multiplying a ratio by 1/2 reduces the interval it describes by one octave. This particular ratio must be lowered by two octaves to see the difference. This accomplished by multiplying the ratio by 1/4.

324/80 x 1/4 = 324/320 = 81/80

As you can see the two C’s differ by a rather small but decidedly noticeable amount. This interval is the Syntonic comma and it indicates that the note arrived at by tuning four perfect 5ths upwards and one major third downwards is 21.506 cents sharper than the original note.

audio demo (the syntonic comma)

Pythogorean.

The Pythagorean comma becomes evident while generating a scale in the manner described by Pythagoras. By tuning twelve subsequent perfect fifths upwards all of the notes in the twelve tone scale will be generated. At the end of this process the note normally considered to be the enharmonic equivalent of the starting note is reached. For example tuning twelve pure perfect 5ths upwards from C will result in the note B# (C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B#). In equal temperament C and B# are enharmonic names for the same note. However if this is lowered by seven octaves it becomes clear they are not equivalent.

(3/2)^12 x (1/2)^7 = 531441/4096 x 1/128 = 531441/524228

This rather cumbersome ratio demonstrates that B# is about 21.460 cents sharper than C in this system of tuning.

audio demo (the pythagorean comma)

NOTE: The equal tempered scale can be obtained by subtracting one twelfth of the Pythagorean comma from each of the fifths as they are tuned upwards. The ratio of the pure 5th encompasses 701.955 cents. One twelfth of the Pythagorean comma is 1.955 (23.460/1/12 = 1.955). Subtracting 1.955 cents from 701.955 cents results in exactly 700 cents, the width of an equal tempered perfect fifth. Mathematically, 701.955 – 1.955 = 700. Tuning upwards by fifths which are one twelfth of the Pythagorean comma narrower than pure fifths will result in the equal tempered twelve tone scale. This is the famous "circle of fifths" described below.

The circle of fifths.

Most musicians learn about the "circle of fifths" at some time in their musical education. In equal temperament twelve consecutive perfect 5ths close the circle at the starting note (enharmonically since B# = C). Using pure fifths the Pythagorean comma demonstrates that the circle of fifths is in fact a spiral. By continuing to add pure fifths from the thirteenth note the spiral continues to expand but never exactly closes. By the time 41 pure 5ths have been tuned the resulting note is 19.8 cents below the starting note (disregarding octave displacements). The note at the end of 53 pure 5ths is only 3.6 cents above the starting note. 306 pure 5ths end up being 1.8 cents below the starting note.

These and other "cycles" have been proposed at various times as alternative foundations upon which scales should be built. The next section of this paper will place some of these proposals in historical perspective.

Great Diesis

On the equal tempered piano keyboard three major thirds form exactly one octave. However by tuning three pure major thirds upwards and lowering the result one octave another anomaly appears.

(5/4)^3 x 1/2 = 125/64 x 1/2 = 125/128

This anomaly is know as the Great Diesis (pronounced di-a-sis) and indicates that three pure major thirds from an interval which is 41.09 cents flatter than a pure octave.

audio demo (the great diesis)

Schisma

The schisma (pronounced siz-ma or skiz-ma)_ is a very small anomaly which becomes apparent by tuning eight perfect 5ths upwards followed by one major third upward. The resulting note should be exactly five octaves above the starting note. The following formula illustrates this process and lowers the resulting note by five octaves.

(3/2)^8 x 5/4 x (1/2)^5 = 6561/256 x 5/4 x 1/32 = 32805/32768

The schisma indicates that the note resulting from this tuning procedure is 1.954 cents sharper than the starting note.

audio demo (shisma)

NOTE: It is very interesting that a Pythagorean comma is equal to the Syntonic comma plus a the schisma (21.506 cents + 1.954 cents = 23.460 cents)

Diaschisma

Tuning four perfect 5ths downward followed by two major thirds downward will result in the enharmonic equivalent of the starting note three octaves lower. (C-F-A#-D#- G#-E-C). Raising this note by three octaves reveals yet another anomaly known as the Diaschisma.

(2/3)^4 x (4/5)^2 x (2/1)^3 = 16/81 x 16/25 x 8/1 = 2048/2025

The diaschisma indicates that the note resulting from this tuning procedure is 19.553 cents sharper than the starting note.

audio demo (Diaschisma)

History of Microtuning

Ling Lun

The mathematical derivation of the pentatonic or five note scale is attributed to Ling Lun who was purported to be a musician in the court of Emperor Huang Ti in the 27th Century BC (although many scholars believe this antiquity to be exaggerated). He started with a length of bamboo called a "lu" which was closed at one end and open at the other. A tone is produced by blowing across the open end in a manner similar to that used to play a tone with a bottle today.

The bamboo tube was measured into 81 equal parts. Another tube was cut to a length of 54 equal parts which is two thirds of the original tube’s length. Still another tube was cut to a length of 72 parts which is the length of the second tube plus one third of that length. A fourth tube was cut to 48 parts which is two thirds of the previous tube’s length. Finally a fifth tube was cut to a length of 64 parts which is derived by increasing the length of the fourth tube by one third again. This process results in a set of tubes which produce a series of pitches in the following ratios with respect to the frequency of the longest tube.

1/1 9/8 81/64 3/2 27/16 2/1
9/8 9/8 32/27 9/8 32/27

The sixth tube was exactly half the length of the longest tube
The ratios appearing below those of tubes themselves represent the intervals between the consecutive note produces by the tubes.

As you will recall the ratios 81/64 and 27/16 do not appear in the table of generally accepted pure ratios found in part one of this lecture. This is due to the fact that this scale, produced by altering the lengths of bamboo tubes by one third, is based on the "3-limit" (notice that none of the ratios contain numbers which are multiples of any prime number higher than 3. It would be many centuries before these ratios would be replaced with smaller number ratios from the "5-limit" used in the table of pure diatonic intervals found in the previous section.

NOTE The intervals between consecutive notes in this scale can be preserved using pitches from the table of pure intervals by starting on the note which forms a minor seventh with the root of the scale.

16/9 1/1 9/8 4/3 3/2 16/9
9/8 9/8 32/27 9/8 32/27

Ling Lun (cont.)
By continuing the process described above, Ling Lun produced 12 lun that formed the first known 12 tone scale. It is believed that this scale was not used musically. The twelve lun were divided into two groups of six. The first group produced the following scale. A lu one half the length of the longest was added to provide the octave.

Group 1
1/1 9/8 81/64 729/512 6561/4096 59049/32768 2/1

Intervals
9/8 9/8 9/8 9/8 9/8 65539/59049

The second group produced essentially the same scale offset from the first group by roughly one half step.

Group 2
2187/2048 19683/16384 177147/131072 3/2 27/16 243/128

Intervals
9/8 9/8 65539/59049 9/8 9/8

Notice that each set forms a whole tone scale (consecutive intervals of 9/8 or 203.9 cents) with the exception of one interval with the ratio 65539/59049 (180.4 cents). This smaller interval occurs naturally in the process of constructing the lu. If the interval between 59049/32768 and 2/1 were adjusted to be 9/8 the octave would be sharp by a Pythagorean comma. Consecutive pitches between the two groups are separated by one of two intervals, 256/243 (90.2 cents) or 2187/2048 (113.7 cents).

Pythogoras

Pythagoras of Samos lived in Greece during the sixth century B.C. His prodigious studies in so many areas of science have had a profound influence on Western thought to this day. This influence is so great that his name has become synonymous with many fundamental concepts.

Musically, Pythagoras took a similar approach to Ling Lun. Instead of bamboo tubes however he used a single string stretched between two bridges and held with a certain tension. This simple instrument was called a monochord. Pythagoras determined that the frequency of a pitch produced when a whole string was vibrating could be doubled by stopping the string at its midpoint, Of course, this produced a note one octave above the fundamental pitch. Successive octaves could be obtained by dividing the string into halves, quarters, eighths and so on.

A scale consisting entirely of octaves is not very musically useful, so Pythagoras began dividing the string of the monochord into thirds. He found that setting two thirds of the string into vibration a pitch was produced that formed an interval of 3/2 with the fundamental pitch of the whole string. During this process Pythagoras also noticed that the interval formed by this pitch and the octave above the fundamental pitch of the mono chord was 4/3. This process of forming intervals with the frequencies obtained form different proportional lengths of a single string is known as the Harmonic Proportion.

That Pythagoras did not proceed by dividing the string of the monochord into fifths, sevenths, and so on is a strange twist of fate that was to have an impact on musical theory to this day. Many historians feel that it was the perception of the number "3" as perfect or divine that prevented explorations into higher number ratios. Pythagoras and his followers established a brotherhood dedicated to a pure life and pure fifths based on the ratio 3/2. This idea spread throughout Greece and later to the rest of the known world. As it so often happens in the history of music practice preceded theory. The scales already in use could not be described using consecutive intervals of 3/2. For example the note produced by the ancient eight stringed Lyre tuned in the Dorian mode could be expressed with the following ratios (arranged in descending order)

2/1 16/9 128/81 3/2 4/3 32/27 256/243 1/1

As consecutive intervals of 3/2 these pitches can be expressed in the following descending order

3/2 1/1 4/3 16/9 32/27 128/81 256/243

The ratios used above to represent descending 5ths have been adjusted to express the relative pitch of each note within its own octave. The scale thus derived remained the basis for tuning throughout the Middle Ages.

The sequence described above can be approximated on a modern keyboard by the notes B,E,A,D,G,C,F. Placed within a single ascending octave this sequence becomes E,F,G,A,B,C,D. Oddly enough this sequence is what we now call the Phrygian mode. The names of the modes were confused in the Middle Ages. Of course these notes tuned in Equal Temperament are not those describe by Pythagoras and used by the ancient Greeks.

audio demo (pythagorean scale)

Archytas

While Pythagoras had made great progress quantifying the musical resources of his time the scales he described were virtually unsingable unless the singers were accompanies by instruments so tuned. Archytas (c. 400 B.C.) a native of Tarentum, Italy and a friend of Plato substituted the ratio 5/4 for the Pythagorean 81/64 first used by Ling Lun. He also substitued the ratio 8/7 for 9/8. These actions opened the door for the eventual acceptance of ratios with the 5-limit and 7-limit as valid musical intervals.

Aristoxenus

A school of musical theorists know as the Harmonists developed between the time of Pythagorus and Archytas. In a reaction against the mathematical foundation of this school Aristoxenus (c 330 B.C.) a student of Aristotle wrote as many as 453 works. Among them "Elements of Harmony" is said to be the earliest extant treatise on Greek Music. He believed tthat the ear, not mathematical calculation, should be the final judge of musical merit. Although he did not realize it this is consistent with the theory that small whole number ratios are inherently more consonant than larger number ratios.

Erastosthenes

Erastosthenes (276-196 B.C.) a native of Cyrene (Africa) was the director of the great Library at Alexandria. It was he who substituted the ratio 6/5 for the Pythagorean 32/27 which affirmed Archytas' use of ratios within the 5-limit. In addition he was the first proponent of the Arithmetical Proportion (although its discovery is generally attributed to Pythagoras.) In this process the string of the mono chord is divided into a number of equal parts. The notes that result as the string is stopped at the various divisions are used to form a scale. Different scales can be formed by dividing the string into a different number of parts. The arithmetic Proportion would continue to be used by many musical theorists through out history.

King Fang

During the third century B.C King Fang made a remarkable discovery. He calculated the lengths and the resulting frequencies for sixty lu that would result in a scale based on 59 consecutive intervals of 3/2. He noticed that the frequency of the fifty fourth lu was almost identical to the first (3.6 cents higher) after the octaves were taken into account. This anticipates the discovery of the fifty three cycle of perfect 5ths in the West by 18 centuries.

Ptolemy

Ptolemy (139 A.D. ->?) a native of Alexandria he was a mathematician, astronomer geographer and musical theorist. His "Harmonics" may the first complete exposition of just intonation in which he transforms Greek scales into ratios of the smallest numbers compatible with the nature of each. In doing so Ptolemy defined the scale which was to become the major scale Europe.

Ratios

1/1 9/8 5/4 4/3 3/2 5/31 5/8 2/1

Intervals
9/8 10/9 16/15 9/8 10/9 9/8 16/15

There is evidence that just intonation was being used in China as early as the third century B.C. A bronze kin, known as the "scholar’s lute’ was tuned to the following scale

Ratios
1/1 8/7 6/5 5/4 4/3 3/2 5/3 2/1

Intervals
8/7 1/20 25/24 16/15 9/8 10/9 6/5

Ho Cheng-Tien

Once again the Chinese found themselves far ahead of the West in discoveries of a musical nature. Ho Cheng-Tien (ca 370-447) gave the string lengths for the twelve tone Equal tempered scale thirteen centuries before such a scale would be considered in Europe. Apparently, however those string lengths were arrived at by ear more than by calculation. The formulation of Equal Temperament would be achieved virtually simultaneously in China and Europe thirteen hundred years hence.

Walter Odington

During the Medieval period and English monk by the name of Walter Odington (c. 1240-1280) noticed that it had become popular to sing intervals which were closer to 2/1 or 3/2. He wrote that some of the intervals in this new popular art (faux bourdon) were "imperfect consonances." In particular Odington identified the thirds 5/4 and 6/5 as such imperfect consonances and stated that singers used them intuitively rather than the Pythagorean ratios 81/64 and 32/27. He also mentioned the major Chord possibly for the first time in recorded musical history.

Nicholas Faber and the Halberstadt Organ.

On February 23 1361 Nicholas Faber completed the construction of an organ for the cathedral in the Saxon city of Halberstadt. The organ had three manuals, the third of which consifted of nine front keys and five raised rear keys in groups of two and three. Excluding the outer front keys this was the first appearance of what was to become the modern keyboard. Its practical application to the musical developments of the time would soon follow.

The introduction of the now familiar keyboard was the result of the growing acceptance of thirds and fifths as simultaneous consonances. The beauty of Ptolemy’s just intonation was effectively illustrated by the newly emerging triad. Ironically the appearance of the keyboard was also a portent of of the eventual rejection of just intonation as music became more harmonic and chromatic. The pure intervals required tempering in order to render this new music playable on keyboard instruments. This is due in a large part to the fact that each note was now playing several different musical roles. The freedom of intonation inherent in a voice does not exist for the keyboard.

Francisco De Salinas

Among the first theorists to temper the pure intervals for the sake of the keyboard was a blind Spanish organist and professor by the name of Francisco De Salinas. (1513-1590) He is generally credited with devising the meantone temperament in which the perfect 5ths are tuned slightly flat so that the major thirds can be be tuned closer to pure. This also resulted in a major second or whole step which fell between the two whole step ratios 9/8 and 10/9. This average or mean whole tone is the source of the temperament’s name.

Due to the nature of mean tone temperament the twenty four possible major and minor triads fell into two groups "good" and "bad". The sixteen good triads sounded much more pure than they do in equal temperament but the remaining eight triads sounded very much worse than they do today. Mean tone temperament had not solved the problem of playing in any key.

Don Nicola Vincentino

It was becoming clear that strictly pure intervals were not compatible with the keyboards then being developed. One solution was to redesign the keyboard so that many more than twelve notes per octave were available. One such instrument, called the Archicembalo was built by Don Nicola Vincentino (c. 1550) This harpsichord like instrument included thirty one notes per octave arranged in six banks of keys. This is one of the first examples of alternative solutions to the problem of keyboards and their inherent limitations. Unfortunately for Vincentino he enjoyed no support from his contemporaries.

Chu Tsai-Yu

Although China had not developed harmonic music with anything like the vigor found in Europe at this time there was a significant theoretical development made by Prince Chu Tsai-Yu in 1596. He published a work in which he very accurately calculated the string lengths for the twelve tone Equal Tempered Scale.

The discovery was the result of Prince Chu’s puzzlement over the discrepancy between the just intonation of the "scholars lute" and the Pythagorean tuning of the twelve lu. He resolved the discrepancy by devising a formula for equal temperament. As you’ll recall this formula divides the octave into 12 equal intervals. The ratio formed by consecultive notes in this system is 12√2/1 or approximately 1.059463094/1

Marin Mersenne

Marin Mersenne (1588 – 1648) was a French Monk mathematician and physicist. He was one of the first theorists to advocate the use of ratios within the seven limit. He noticed that the natural overtone series of the trumpet produced a major triad and other more complex harmonies. He decided that since the natural harmonic series went beyond the the major tonality so should the musical resources of the time go beyond the arbitrarily imposed 5-limit. The was the first to declare that the interval 7/6 was consonant and designed many keyboards with greater resources than the already common 7 white 5 black keyboard.

Andreas Werkmeister.

The now familiar keyboard had become widely accepted in the Baroque period. The virtues of equal temperament were being extolled by some and eschewed by others. In an attempt to reach a compromise between pure sounding intervals and harmonic capability many musicians and theorists devised various temperaments with which music could be played in any key but which also closely approximated as many pure intervals as possible. These are often referred to as well temperaments because they worked well in any key.

It was these well temperaments which inspired Back to write the "Well Tempered Clavier". Contrary to popular belief this set of keyboard pieces written in each of the major and minor keys was not intended for performance in equal temperament. It was wellknow at the time that well temperaments afforded each key its own color or unique character. This was due to the fact that semitones were not identical as they are in equal temperament. It was Bach’s intention to illustrate these key color in his "Well Tempered Clavier".

Andreas Werkmeister (1645-1706) was an organist composer and theorist highly respected by Handel, Buxtehude and many other musicians. He was one of the first to clearly state the principles of well temperament. Among these principles was the premise that well temperaments should favor the primary intervals found in keys with few sharps and or flats at the expense of tonalities with many sharps and flats, This was a very practical idea since most of the music being composed at the time was written in these in these simple key signatures due to the long established use of key sensitive tunings such as the mean tone temperament. Later even though composers such as Mozart and Haydn had developed the compositional facility to modulate into any key they tended to use key signatures few sharps or flats because of the widespread use of well temperaments. Music performed on keyboards so tuned sounded better in C and other simple keys.

Johann Philipp Kirnberger

Johann Philipp Kirnberger (1721-1783) was a student of J.S.Bach. As a composer conductor and theorist he developed many well temperaments.

Francescantonio Valotti and Thomas Young.

Many of the well temperaments were based on adjusting some or all of the fifths in a Pythagorean tuning of consecutive perfect 5ths. You will recall that by tuning upwards from a given note by twelve intervals of 3/2 the final note will form an interval with the starting note which is sharper than the octave by 23.46 cents (the Pythagorean comma). Man well temperaments are based on tuning the octave pure and placing the comma elsewhere (hopefully where it will not be played). One solution was to divide the comma into two three four six or even twelve parts and temper certain intervals by the amount represented by the partial comma.

Francescantonio Valotti and Thomas Young independently devised a well temperament in which the first six Pythagorean fifths were lowered by one sixth of a comma while the second six fifths went untempered. The only difference between the temperaments was that Valotti started on F and Young started on C.

Hermann Helmholtz and Alexander Ellis.

As seemingly impossible to stem as the advancing tide equal temperament finally achieved universal acceptance by the late 19th century. As the Romantic and later periods in music history saw more chromatic modulations and extended chords, total harmonic flexibility was required of keyboard instruments.

Hermann Helmholtz (1821-1894) compiled what has since been considered the definitive exposition of all major acoustical, theoretical and pertinet physiological knowledge gathered to his time. "On the Sensations of Tone" is still considered one of the essential texts on the subject to this day. He and his English translator, Alexander Ellis (1814-1890) have been instrumental; in recording and preserving the various fundamental music concepts which have developed throughout history.

It is interesting to note that Helmholtz was actively opposed to equal temperament. He felt that it was not appropriate to sacrifice pure intervals for the convenience of keyboard instruments.

As for Ellis, his major contribution to the study of musical theory was the invention of cents. As you will recall there are 1200 cents in an octave or 100 cents per equal tempered semitone. This single development has allowed all theorists since that time to compare the size of intervals with relative ease.

Harry Partch

The pioneer of just intonation in this century was Harry Partch (1901-1974). His development of a scale consisting of forty three tones per octave is based on small number just ratios stands as a milestone in musical history. He designed and built and entire orchestra of acoustic instruments to play his music. There are several recordings of his work currently available on the Columbia label. The music world is indeed fortunate that Partch also recorded his theories in a voluminous work entitled "Genesis of a Music".