Lesson 3

- Lesson 3 -
Musical Scales and Temperament

Musicians today are familiar with what are referred to as the major and minor scales. For a given base note, these scales contain different subsets of eight notes out of 13 total that ascend in pitch frequency from that of the base note to its first octave. But, this system of notes, the equal temperament scale, is not the only one that's been around or is even around today. Let's compare it to some other scales.

The word "scale" is derived from a Latin word (scala) meaning "ladder" or "staircase". A musical scale is a succession of notes arranged in ascending or descending order. Most musical composition is based on scales, the most common ones being those with five notes (pentatonic), twelve notes (chromatic), or seven notes (major and minor diatonic, Dorian and Lydian modes, etc.). Western music divides the octave into 12 steps called semitones. All the semitones in an octave constitute a chromatic scale or 12 tone scale. However, most music makes use of a scale of seven selectred notes, designated as either a major scale or a minor scale and carrying the note name of the lowest note. For example, the C-major scale is played on the piano by beginning with any C and playing white keys until another C is reached.

Listen to the C-major scale.

Other musical cultures use different scales. The pentatonic or five-tone scale, for example, is basic to Chinese music as well as Celtic and Native American music. It can be played on the piano by beginning with C# and playing on the black keys, or by playing the notes C D E G A C.

Listen to the five-tone scale.

But, how did different cultures come up with the system of notes that they use today? And why is there some similarity between them. They are not the only system of notes that could be used or that have been used over the years. The construction of scales has appealed to mathematicians as well as to musicians since the time of the great Greek philosophers of the B.C.'s. We will look and listen to three important systems of notes and see how they compare. They are the just scale, Pythagorean scale and the scale of equal temperament.

Scale of Just Intonation

The scale of just intonation (or just diatonic scale) is based on the major triad, a group of three notes that sound particularly harmonious (for example, C : E : G ).

Listen to the scale of just intonation (C : E : G).

The spacing of these notes consists of a major third (C : E) and a minor third (E : G). When these intervals are made as consonant as possible, the notes in the major triad are found to have frequencies in the ratio 4 : 5 : 6. Refer to Lesson 2 for a review of the most consonant musical intervals.

The three major triads in a major scale are the tonic, subdominant, and dominant chords (also called the I, IV, and V chords, since they are built on the first, fourth and fifth notes of the scale). The frequencies of all the seven notes in the just diatonic scale can be determined by letting these three triads consist of notes with the frequency ratios 4 : 5 : 6. For example, let's start at a middle C (C4) in the scale used today, the tempered scale. This has a fundamental frequency of 261.63 Hz.

Tonic chord: 261.63 Hz : 5/4x261.63 Hz : 6/4x261.63 Hz = 261.63 Hz : : . This is in the ratio of 4 : 5 : 6or (divide by 4) 1 : 5/4 : 3/2. Although their frequencies are not exactly the same as the tempered scale we are familiar with, the three notes that best approximate this triad are: C : E : G.

Listen to the tonic chord (C:E:G, scale of just ) .

Listen to the tempered chord (C:E:G).

Dominant chord: Let's start with this approximate G of 3/2x261.63 Hz and build a 4 : 5 : 6 triad off of it. 5/4x3/2x261.63 Hz : 6/4x3/2x261.63 Hz = Hz : : . Again the frequencies do not exactly match the tempered scale we are familiar with, but the three notes that best approximate this triad are: G : B : D. Drop D down to same octave to get the following ratios referenced to C4 (261.63 Hz): 3/2 : 15/8 : 9/8.

Listen to the dominant chord (G:B:D).

Listen to the tempered chord (G:B:D).

Subdominant chord: Let's start with the C5 (2x261.63 Hz) one octave up from middle C and make it the third component of our triad. Thus, we'll have: 4/3x261.63 Hz : 5/3x261.63 Hz : 2x261.63 Hz. Referenced to C4, the ratios are: 4/3 : 5/3 : 2. Again the frequencies do not exactly match the tempered scale we are familiar with, but the three notes that best approximate this triad are: F : A : C.

Listen to the subdominant chord (F:A:C).

Listen to the tempered chord (F:A:C).

Advantages of just scale => optimal consonance ... can you hear it in the examples above? 1) Has 2 minor triads (ratio 10 : 12 : 15) E : G : B and A : C : E.

Disadvantages of just scale:
1) Five perfect fifths, but D to A is imperfect (not 3 : 2 ratio)
2) Five perfect fourths, but A to D is imperfect (not 4 : 3 ratio)
3) Because of the two different whole tone intervals, musical instruments would have to be retuned for each change of key ... this is not practical.

The Pythagorean scale

This scale is based on creating the largest possible number of perfect fourths and fifths. In order to do this, the other intervals, such as major and minor thirds and sixths, must vary from the just intonation scale. It is noted that: 1) An octave is a fourth plus a fifth; therefore, going up a fourth leads to the same letter as going down a fifth and vice versa. (4/3 x 3/2 = 2) 2) All notes on the scale (sharps and flats included) can be reached by going up or down 12 successive fifths or 12 successive fourths ... almost. For example, if 3/2 is multiplied by itself 12 times, one obtains 129.7, which means that going up 12 perfect fifths takes one up seven octaves (128 = 27) plus one fourth of a semitone extra (1.7). This leads to the circle of fifths.

Starting from, for example, C4 (261.63 Hz), we can determine the major scale by multiplying (going up) by a fourth: 1 : 4/3 and then going up by five successive fifths: 1 : 3/2 : 9/4 ( x 1/2) : 27/8 ( x 1/2) : 81/16 (x 1/4) : 243/32 ( x 1/4). Moving these notes to the same key (multiply by 1/2 or 1/4 if their value is greater than 2 or 4, respectively) and placing them in ascending order yields the following: 1 : 9/8 : 81/64 : 4/3 : 3/2 : 27/16 : 243/128 : 2. Compare this key to that of just intonation and the well-tempered scale with which you are familiar.

Listen to the tonic chord (C:E:G, pythagorean scale).

Listen to the tempered chord (C:E:G).

Listen to the dominant chord (G:B:D, pythagorean scale).

Listen to the tempered chord (G:B:D).

Listen to the dominant chord (G:B:D, pythagorean scale).

Listen to the tempered chord (G:B:D).

If the last two fifths obtained are omitted, 81/64 (E) and 243/128 (B), then we end up with the pentatonic scale: C : D : F : G : A : C with appropriate intervals.

Listen to the pentatonic scale.

Advantages: 1) All the fifths and fourths are perfect. 2) Only two types of intervals ... whole tones and semitones. (Whole tone is the same as the major whole tone of the just scale, but the semitone is about 20 percent smaller and 10 percent less than half of a whole tone.)

Disadvantages: 1) Continuing around the circle of fifths to obtain the frequencies of the flats and sharps leads to another interval. The ratio of F# to F turns out to be 2187/2048 = 1.068, which is called a chromatic semitone. It is larger than the diatonic semitone (256/243 = 1.053) previously determined. So this scale has one size whole tone, but two sizes of half tones. 2) Poor tuning of thirds. E / C = A / F = B / G = 81/64 = 1.266 ? 1.25 = 5/4.

Nonetheless, studies have shown that many concert violinists tend to favor Pythagorean tuning in their performance, which points out the great importance of fifths and fourths in music.

What about tuning for different keys? Try to construct the major scale based on fourths and fifths but in the D4-major key. Take D4 = 261.63 Hz x 9/8 = 294.33 Hz. Find the notes of the scale up to D5. How do they compare in frequency and sound to the notes found above in the C4-major key? Is retuning of instruments needed each time the key is changed?

Because Pythagorean thirds sound out of tune, numerous alterations of the Pythagorean scale have been developed. Nearly all of them flat the third (E in C-major scale) so that the major third (C to E) and minor third (E to G) are close to the corresponding just intervals. Similar adjustments are made in other notes. Such compromises form the bases of various meantone temperaments.

Scale of Equal Temperament

The most convenient temperament of all is the scale equal temperament (tempered scale). This is the scale most commonly used today. It consists of five equal whole tones and two semitones; the whole tones are twice the size of the semitones. Twelve equal semitones make up an octave.

Advantage. The tempered scale does not require retuning to play different scales and strikes a balance between consonance of fifths, fourths, major and minor thirds. None are perfect like on the just or Pythagorean scale but all are close to perfect.

Our preferences

To many listeners an over-stretched intonation is acceptable, whereas a compressed intonation is not.

Acoustics & Vib. Lab.