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PPoM Tuning, Scales And Temperament

Tuning, Scales And Temperament

Harmonic and Inharmonic Spectra
The sounds that we hear from vibrating objects are complex in the sense that they contain many different frequencies. This is due to the complex way that objects vibrate.

A "note" (say Middle C) played on a piano sounds different from the same "note" played on a saxophone. In both cases there are different frequencies present above the common fundamental note sounded. These different frequencies are part of what enables you to distinguish between different instruments. (Their difference in timbre.)

Spectra can be classified as being of one of two types:

  1. Harmonic, in which the spectral components (different frequencies) are mostly whole number multiples of the lowest, and most often loudest, frequency. The lowest tone is called the fundamental and the higher (i.e. in frequency) spectral components, the tones over the fundamental, are called overtones, or harmonics.
  2. Inharmonic, in which the criteria in (1) are not met i.e in which the spectral components are mostly NOT whole number multiples of the lowest frequency, which is often NOT the loudest tone. Many percussion instruments fall into this category.

The Harmonic Series
The harmonic series is a series of frequencies that are whole number multiples of a fundamental. For example, taking the tone Middle C whose fundamental frequency is approximately 261 Hz, the harmonic series on this frequency is:

Hs = f , 2f , 3f , 4f , ......... nf and where f = 261 Hz.,
= 261 Hz , 522 Hz , 783 Hz , 1044 Hz , ................... n x 261 Hz.

This is expressed in musical notation thus:

Note that these notes are 'representing' the tones. The frequencies of the actual tones are at varying distances from the frequencies of these notes depending on the tuning system used. In particular the 7th partial (B in this example) is considerably flat in Pythagorean, just and equal temperaments (see later).

We can see from the above discussion of the harmonic series, that an interval can be expressed as a ratio of the frequencies of the two tones:

The higher of the two tones (f2) is placed over the lower (f1) because we are considering and upward interval.

Example: For the perfect fifth C - G in the above description,

Scales - An Introduction
Throughout history there have been numerous ways devised to divide the octave (frequency ratio of 2/1 sometimes written explicitly as 2:1) into smaller intervals. All the scales1 thus produced have their own characteristics and are thus more or less suited for the playing of a particular type of music (or particular musics).

Most recent Western music has used a scale with 12 steps - semitone intervals - called the chromatic scale (from the Greek khroma, meaning colour). Other cultures use scales with a different number of steps, for example 4, 5, 7, 11, 12, 13.

Definition of a Musical Scale
A scale is a discrete set of pitch relationships (or intervals), most often arranged in such a way as to yield a maximum possible number of consonant combinations (or minimum possible number of dissonances) when two or more of these intervals are sounded together.

The most consonant interval after the octave is the fifth (3/2) and the next most consonant is the fourth (4/3). The difference between these two intervals

is defined as the interval of a whole-step. This means that in a whole-step interval the frequency of the higher tone is increased by a factor of 9/8 of the lower, or the lower is decreased by a factor of 8/9 of the upper.

Given the restriction on consonance in developing musical scales (see above) we can immediately develop two of the oldest scales, the Pythagorean and the Just.

Construction of the Pythagorean Diatonic Scale
The Pythagorean scale is developed from the interval of a perfect 5th.

Step 1: Begin with an arbitrary tone of frequency represented by 1, and ascend in steps of perfect 5ths:

if these were based on C4 the notes would be
C4 G4 D5 A5 E6 B6

Step 2: Reduce the ratios obtained in step 1 into the range of a single octave by descending from these notes in whole octave steps:

Step 3: Arrange the notes obtained in step 2 in ascending order:

Step 4: The above sequence of notes is missing one important note the one corresponding to a perfect 4th from the beginning point. This can be obtained by descending from the beginning point by a perfect fifth (2/3) and then moving up one octave (2/3) x 2 = 4/3 = 1.3333.... Placing it in the ascending sequence as arranged in step 3, we have the following sequence:

This is known as the Pythagorean diatonic scale. Examining the intervals between each step,

We can see that the scale consists of the familiar major scale (where the whole step is 9/8 and the 1/2 step is 256/243).

Both the perfect 4th and 5th have exact ratios, 3/2 and 4/3 as expected (The scale was constructed using these intervals.)
The Major 3rd however, is too wide, or sharp. The ratio should be 5/4 = 1.250 but the Pythagorean scale has this interval as 81/64 = 1.265.
The minor 3rd between the 2nd and 4th degrees should be 6/5 = 1.20 but in fact is i.e. it is too narrow, or "flat".

Construction of the Pythagorean Chromatic Scale
Step 1: As with the Pythagorean diatonic scale, begin at some arbitrary point, but this time ascend and descend in perfect 5ths.

Step 2: Reduce the ratios obtained in step 1 into the range of a single octave:

Step 3: Arrange the notes obtained in step 2 in ascending order:

There are two half-step intervals: Firstly, 256/243 = 1.053 between degrees 1 & 2, 3 & 4, 5 & 6, 8 & 9, 10 & 11, and 12 & 1. Secondly 2187/2048 = 1.068 between degrees 2 & 3, 4 & 5, 9 & 10, 11 & 12.

There is also an ambiguity concerning the tritone in that there are two different ratios for that interval: 729/512 = 1.4238, and 1024/729 = 1.4047

Construction of the Just Diatonic Scale
The Just Scales
The Just Scale is built to maximise the number of consonant intervals that have exact frequency ratios. The major triad, ratios 4/5/6 is the starting place. One of the features that makes the Just Scale so attractive in the West is that most of Western harmony is built around this triad.

Step 1: Form a major triad:

Step 2: Form major triads from the top and bottom notes of this triad:

Step 3: Bring the ratios from step 2 within the range of a single octave:

Step 4: Arrange notes in ascending order:

This is known as the Just diatonic scale. Examining the intervals between each step,

There are two whole tones:
9/8 = 1.125 (called the major tone), and
10/9 = 1.1' (called the minor tone).

The semitone has a ratio of 16/15 = 1.06'.
The minor third between notes 2 and 4, (4/3)/(9/8) = 32/27 = 1.'185' , does not have the desired ratio of 6/5 = 1.2.

The perfect 5th between notes 2 and 6, (10/6)/(9/8) = 40/27 = 1.'481' does not have the desired ratio 3/2 = 1.5.

The Just Chromatic Scale
A chromatic Just scale can be constructed by a procedure analogous to that used for the Pythagorean chromatic scale. In this case, the ambiguities are more numerous for the Just scale than for the Pythagorean. For example the tritone has two different values as does the raised 1st degree ( or lowered 2nd degree).

The Just scale provides "cleaner" or "simpler" harmonies than does the Pythagorean scale. However, when the scale is played simultaneously with the original, as happens for a transposing instrument for example, there results notes which have two quite different ratios. Take for example a comparison of C major with D major.

All the difficulties so far mentioned were known by the 17th Century and an attempt was made to resolve them by "tempering" the scale.

The Tempered Scale
If you start from a particular note and ascend or descend in either perfect fifths or thirds, one never arrives at a note that is a whole number of octaves from the starting note. It is partly this reason that both the Pythagorean and Just scales have posed problems for recent Western musicians (in the last few hundred years). Adjustments have been made to these scales - in the spirit of acoustical compromise - to satisfy three incompatible requirements:

1. True intonation - that is exact thirds, fourths, fifths, etc
2. Complete freedom of modulation - that is from any one key to any other.
3. Convenience in the practical use of keyboard instruments - that is to make it possible for instruments which have been tuned differently to play together without retuning.

Equal Temperament
In this form of compromise, the first of these requirements is sacrificed for the good of the latter two requirements. In the equally tempered scale all semitone intervals are exactly the same ratio, irrespective of frequency location. This implies that all whole-tone intervals are likewise equal. In order to accomplish this the octave is divided into twelve equal semitone intervals.

The two criteria then, are
1. That all semitone intervals be identical; i.e.

2. That the octave maintain its integrity; i.e.

f(note 13) = 2 x f(note 1)

The first criterion means that the ratio of all the semitone intervals is the same. Call this ratio a.

The ratio of any two semitones is thus the same:

Furthermore, since f(note13) = af(note 1),
we have a= 2
or a (12th root of 2)

The twelfth root of two is approximately 1.059463094.

Using this fact, the following table shows frequency ratios of intervals within one octave, and gives actual approximate frequencies that result if we begin with Concert A, A4 = 440 Hz (the standard of pitch).

The following table allows a comparison between some 12 tone equal temperament and theoretical ratios:

It is clear from these results that equal temperament mistunes all intervals except the octave. It drastically mistunes thirds and sixths. This results in many intonation difficulties, but in return keyboard instruments can play freely together, scores can be transposed up and down for singers, and compositions using modulations to remote keys can be effected without problems.

A competent violinist or vocalist (that rarest of beasts!) plays or sings in an intonation different from equal temperament when not accompanied by a keyboard instrument. In a string quartet, for example, the thirds and fourths will be played such that they sound best, and that is not equal temperament.

An orchestra does not play in equal temperament. Brass players tend to "lip up" or "lip down" certain notes and in doing so depart from equal temperament. The note B in a C major passage is often played sharp in anticipation of a move to the tonic C, whilst the same note B played as the mediant of G major is often played flat relative to the demands of equal temperament.

Equal Divisions of the Octave into Different Numbers of Tones
In twelve tone equal temperament the octave is divided into twelve equal ratio intervals. This causes major difficulties with intonation. Some of these difficulties could be eliminated if the octave was divided into a different number of tones, giving a scale with more or less that 12 notes.

From the early stages of the piano's development, there have been five black and seven white keys to an octave. This could be so because it is particularly appropriate for the human hand, although to date no extensive work has been done to assert this. In subdividing the octave into 12, perhaps we have been led by our hands and not our ears!

In order to divide the octave into a different number of tones all we need to do is to compute the ratios in the same manner as we did for 12 tone equal temperament except with the different base. For example, if we choose 16 divisions to the octave, then the ratio unit ratio, a, is (the sixteenth root of 2), which is, approximately, 1.041616011. Each of the intervals can then be calculated as before:

Note # Ratio Note # Ratio
Note 1 a0 f 1.0000 Note 2 a1 f 1.0416
Note 3 a2 f 1.1301 Note 4 a3 f 1.1771
Note 5 a4 f 1.2261 Note 6 a5 f 1.2772
Note 7 a6 f 1.3303 Note 8 a7 f 1.3857
Note 9 a8 f 1.4433 Note 10 a9 f 1.5034
Note 11 a10 f 1.5660 Note 12 a11 f 1.6311
Note 13 a12 f 1.6990 Note 14 a13 f 1.7697
Note 15 a14 f 1.8433 Note 16 a15 f 1.9201
Note 17 a18 f 2.0000

The actual frequencies can then easily be calculated.

1 Topic for Discussion. What is the difference between a scale and a mode?


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