tuning, scales and temperament harmonic and inharmonic spectra · 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 differentfrequencies. 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 asaxophone. In both cases there are different frequencies present above the common fundamental notesounded. These different frequencies are part of what enables you to distinguish between differentinstruments. (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 wholenumber multiples of the lowest, and most often loudest, frequency. The lowest tone is calledthe fundamental and the higher (i.e. in frequency) spectral components, the tones over thefundamental, are called overtones, or harmonics.

(2) Inharmonic, in which the criteria in (1) are not met i.e in which the spectral componentsare mostly NOT whole number multiples of the lowest frequency, which is often NOT theloudest 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. Forexample, taking the tone Middle C whose fundamental frequency is approximately 261 Hz, the harmonicseries 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 varyingdistances from the frequencies of these notes depending on the tuning system used. In particular the 7thpartial (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 ratioof 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

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scales http://www.anu.edu.au/ITA/ACAT/drw/PPofM/scales/scales1.html

interval.

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

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Scales - An Introduction

Throughout history there have been numerous ways devised to divide the octave (frequency ratio of 2/1sometimes written explicitly as 2:1) into smaller intervals. All the scales[1] thus produced have theirown characteristics and are thus more or less suited for the playing of a particular type of music (orparticular musics).

Most recent Western music has used a scale with 12 steps - semitone intervals - called the chromaticscale (from the Greek khroma, meaning colour). Other cultures use scales with a different number ofsteps, 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 yielda 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 thehigher tone is increased by a factor of 9/8 of the lower, or the lower is decreased by a factor of 8/9 of theupper.

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

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

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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 beC4 G4 D5 A5 E6 B6

Step 2: Reduce the ratios obtained in step 1 into the range of a single octave by descending from thesenotes 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 perfect4th from the beginning point. This can be obtained by descending from the beginning point by a perfectfifth (2/3) and then moving up one octave (2/3) x 2 = 4/3 = 1.3333.... Placing it in the ascendingsequence 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 1/2step is 256/243).

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Discussion

Both the perfect 4th and 5th have exact ratios, 3/2 and 4/3 as expected (The scale was constructed usingthese intervals.)The Major 3rd however, is too wide, or sharp. The ratio should be 5/4 = 1.250 but the Pythagorean scalehas 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 istoo narrow, or flat.

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Construction of the Pythagorean Chromatic Scale

Step 1: As with the Pythagorean diatonic scale, begin at some arbitrary point, but this time ascend anddescend 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:

DiscussionThere 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

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Construction of the Just Diatonic Scale

The Just ScalesThe 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 soattractive 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,

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

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The Just Chromatic Scale

A chromatic Just scale can be constructed by a procedure analogous to that used for the Pythagoreanchromatic scale. In this case, the ambiguities are more numerous for the Just scale than for thePythagorean. For example the tritone has two different values as does the raised 1st degree ( or lowered2nd 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 forexample, there results notes which have two quite different ratios. Take for example a comparison of Cmajor with D major.

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

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The Tempered Scale

If you start from a particular note and ascend or descend in either perfect fifths or thirds, one neverarrives at a note that is a whole number of octaves from the starting note. It is partly this reason that boththe Pythagorean and Just scales have posed problems for recent Western musicians (in the last fewhundred years). Adjustments have been made to these scales - in the spirit of acoustical compromise - tosatisfy three incompatible requirements:

1. True intonation - that is exact thirds, fourths, fifths, etc2. 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 instrumentswhich 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 tworequirements. In the equally tempered scale all semitone intervals are exactly the same ratio, irrespectiveof frequency location. This implies that all whole-tone intervals are likewise equal. In order toaccomplish this the octave is divided into twelve equal semitone intervals.

The two criteria then, are1. 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:

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Furthermore, since f(note13) = a f(note 1),we have a = 2or 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 givesactual approximate frequencies that result if we begin with Concert A, A4 = 440 Hz (the standard ofpitch).

DiscussionThe following table allows a comparison between some 12 tone equal temperament and theoreticalratios:

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

IntonationA competent violinist or vocalist (that rarest of beasts!) plays or sings in an intonation different fromequal temperament when not accompanied by a keyboard instrument. In a string quartet, for example, the

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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" certainnotes and in doing so depart from equal temperament. The note B in a C major passage is often playedsharp in anticipation of a move to the tonic C, whilst the same note B played as the mediant of G majoris often played flat relative to the demands of equal temperament.

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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 causesmajor difficulties with intonation. Some of these difficulties could be eliminated if the octave wasdivided 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 anoctave. This could be so because it is particularly appropriate for the human hand, although to date noextensive work has been done to assert this. In subdividing the octave into 12, perhaps we have been ledby 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 inthe same manner as we did for 12 tone equal temperament except with the different base. For example, ifwe 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:

The actual frequencies can then easily be calculated.

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