Musical Scales and Tuning



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Musical Scales and Tuning


In general, the 3/2 ratio of fundamental frequencies results in the best possible harmonic agreement between tones with different pitches. Therefore, tone pairs separated by musical intervals of perfect fifths will sound as harmonious as different pitches ever can sound. We can therefore identify a set of pitches related to each other by 3/2 frequency ratios that provide many opportunities for harmonious tonal combinations.

So reasoned the ancient Greek mathematician and philosopher Pythagoras whose method of tuning the notes of a musical "scale" is based solely on the perfect fifth. The term scale comes from the Latin scala, meaning ladder. Starting with an arbitrary frequency, f, the frequencies of the notes in the Pythagorean tuning system are found by successive multiplication by 3/2.

That is, if note N1 has a frequency of f Hz, then note N2 has a frequency of (3/2)f Hz, note N3 has a frequency of (9/4)f. Hz, and so on. Each note in the sequence is a perfect fifth (3/2 frequency ratio) higher than the previous note. The pitches obtained by taking successive perfect fifths will be spread out over quite a large pitch range. To obtain pitches so that they all lie within the same octave we must rearrange the transposed pitches into ascending (or descending) order.



Fig. 3. Musical scale of notes from C4 to C5

Finally, we have a choice about the starting note for the scale. It is possible to begin and end the scale on any of its pitches. Only one results in a scale that is divided into two symmetric halves. If we arrange the pitches so that C is the beginning (and ending) note of the ascending scale and normalise the frequencies so that the frequency of C is 1, the diatonic major scale results.

The Pythagorean diatonic major scale is symmetric in that the intervals among the first four notes occur in the same order as the intervals among the last four notes. Note that there are only two interval types, the 9/8 whole step and the 256/243 semitone.

Observation of the diatonic major scale shows why the 3/2 frequency ratio is called a perfect fifth. If we start at any note in the scale, call it number 1, and count up five notes the second note is said to be a "fifth" higher, in the sense that it lies five notes up the scale from the starting point.


Just vs Equal Temperament


The "Just Scale" (sometimes referred to as "harmonic tuning") occurs naturally as a result of the overtone series for simple systems such as vibrating strings or air columns. All the notes in the scale are related by rational numbers. Unfortunately, with Just tuning, the tuning depends on the scale you are using - the tuning for C Major is not the same as for D Major, for example. Just tuning is often used by ensembles (such as for choral or orchestra works) as the players match pitch with each other "by ear."

The "equal tempered scale" was developed for keyboard instruments, such as the piano, so that they could be played equally well (or badly) in any key. It is a compromise tuning scheme. The equal tempered system uses a constant frequency multiple between the notes of the chromatic scale. Hence, playing in any key sounds equally good (or bad, depending on your point of view).

There are other temperaments which have been put forth over the years, such as the Pythagorean scale, the Mean-tone scale, and the Werckmeister scale. For more information on these you might consult "The Physics of Sound," by R. E. Berg and D. G. Stork (Prentice Hall, NJ, 1995).

The table below shows the frequency ratios for notes tuned in the Just and Equal temperament scales. For the equal temperament scale, the frequency of each note in the chromatic scale is related to the frequency of the notes next to it by a factor of the twelfth root of 2 (1.0594630944....). For the Just scale, the notes are related to the fundamental by rational numbers and the semitones are not equally spaced. The most pleasing sounds to the ear are usually combinations of notes related by ratios of small integers, such as the fifth (3/2) or third (5/4). The Just scale is constructed based on the octave and an attempt to have as many of these "nice" intervals as possible. In contrast, one can create scales in other ways, such as a scale based on the fifth only.



Interval

Ratio to Fundamental
Just Scale


Ratio to Fundamental
Equal Temperament


Unison

1.0000

1.0000

Minor Second

25/24 = 1.0417

1.05946

Major Second

9/8 = 1.1250

1.12246

Minor Third

6/5 = 1.2000

1.18921

Major Third

5/4 = 1.2500

1.25992

Fourth

4/3 = 1.3333

1.33483

Diminished Fifth

45/32 = 1.4063

1.41421

Fifth

3/2 = 1.5000

1.49831

Minor Sixth

8/5 = 1.6000

1.58740

Major Sixth

5/3 = 1.6667

1.68179

Minor Seventh

9/5 = 1.8000

1.78180

Major Seventh

15/8 = 1.8750

1.88775

Octave

2.0000

2.0000

You will note that the most "pleasing" musical intervals above are those which have a frequency ratio of relatively small integers. Some authors have slightly different ratios for some of these intervals, and the Just scale actually defines more notes than we usually use. For example, the "augmented fourth" and "diminished fifth," which are assumed to be the same in the table, are actually not the same.

The set of 12 notes above (plus all notes related by octaves) form the chromatic scale. The Pentatonic (5-note) scales are formed using a subset of five of these notes. The common western scales include seven of these notes, and Chords are formed using combinations of these notes.

As an example, the chart below shows the frequencies of the notes (in Hz) for C Major, starting on middle C (C4), for just and equal temperament. For the purposes of this chart, it is assumed that C4 = 261.63 Hz is used for both (this gives A4 = 440 Hz for the equal tempered scale).




Note



Just Scale


Equal
Temperament


Difference

C4

261.63

261.63

0

C4#

272.54

277.18

+4.64

D4

294.33

293.66

-0.67

E4b

313.96

311.13

-2.84

E4

327.03

329.63

+2.60

F4

348.83

349.23

+0.40

F4#

367.92

369.99

+2.07

G4

392.44

392.00

-0.44

A4b

418.60

415.30

-3.30

A4

436.05

440.00

+3.94

B4b

470.93

466.16

-4.77

B4

490.55

493.88

+3.33

C5

523.25

523.25

0

Since your ear can easily hear a difference of less than 1 Hz for sustained notes, differences of several Hz can be quite significant!

Listen to the difference:
The first second of this WAV file contains a major triad starting on F# (F# - A# - C#) using the Just scale appropriate for C Major. The last part of the file contains the same triad but using the Just scale appropriate for F# Major. (This is one of the worst case situations).
Tuning Shift WAV file.

Here's another example to test your ears. The following WAV file has two "players" playing a C major scale. One of the players is using the Just Scale, the other the Equal Tempered scale. Both start on exactly the same pitch. See if you can here the notes where the pitches are different.


Major scales in different temperaments

A major scale in musical theory is a scale in which each note rises in pitch with the order: tone, tone, semitone, tone, tone, tone, semitone....



The major scale (and most other western scales) has eight notes - an octave. The simplest major scale is C major (see figure 1). It is unique in that it is the only major scale not to feature accidentals.



Figure 1. The C major scale

When writing out major (and minor scales), every line and space on the stave has to be filled, and no note can have more than one accidental. This has the effect of forcing the key signature to feature just sharps or just flats; ordinary major scales never include both.




Constructing major scales

Analysing scales with sharps
Scales and key signatures are closely linked in music. It is necessary to construct a key signature - consisting of a number of sharps or flats - in order to know which accidentals a particular major scale will have. An easy, but time consuming, way to do this would be to use the pattern of tone/tone/semitone/etc... given above. If we choose to write the scale of D-major, we know immediately that the scale begins on a D. The next note will be a tone above - an E. The note after that will also be a tone above, however it is not simply an F as would seem obvious. Because the difference between an E and an F is actually a semitone (look on a piano keyboard, there is no 'black note' in-between them) it is necessary to raise the F to become an F-sharp to achieve a difference of a whole tone.
This could be followed to create a whole scale, with all the sharps (or with a different scale, flats) put correctly in. However a cleverer way of constructing scales arises from analysing patterns in the whole series of major scales. Starting on the scale of C-major, there exists no sharps or flats. If you start a new scale on the 5th of C-major - G-major - you will find one sharp, augmenting the F. Starting the scale on the 5th of G major (a D) it will be necessary to put 2 sharps in - an F-sharp and a C-sharp. Writing this pattern out for all the scales looks like this:

C maj - 0 sharps


G maj - 1 sharp - F# (meaning F-sharp)
D maj - 2 sharps - F#, C#
A maj - 3 sharps - F#, C#, G#
E maj - 4 sharps - F#, C#, G#, D#
B maj - 5 sharps - F#, C#, G#, D#, A#
F# maj - 6 sharps - F#, C#, G#, D#, A#, E#
7 sharps - F#, C#, G#, D#, A#, E#, B#

In this table it can be seen that for each new scale (starting on the fifth of the previous scale) it is necessary to add a new sharp. The order of sharps which need to be added follows: F#, C#, G#, D#, A#, E#, B#. This pattern of the sharps can be easily remembered through the use of the mnemonic:


F C G D A E B


Father Charles Goes Down And Ends Battle

Looking closer, the last accidental added matches the tonic (first note) of the scale two-fifths before it (in this table, two lines up.) A useful rule for use in recognising major scales with sharps is that the tonic is also always one note above the last sharp.




Analysing major scales with flats
A similar table can be constructed for major scales with flats in them. In this case each new scale starts on the 5th below the previous one:

C min - 0 flats


F min - 1 flat - Bb (meaning B-flat)
Bb min - 2 flats - Bb Eb
Eb min - 3 flats - Bb Eb Ab
Ab min - 4 flats - Bb Eb Ab Db
Db min - 5 flats - Bb Eb Ab Db Gb
Gb min - 6 flats - Bb Eb Ab Db Gb Cb
7 flats - Bb Eb Ab Db Gb Cb Fb

Here, a similar pattern can be recognised, each new scale keeps all the flats of the previous scale but adds a new one following the sequence: Bb, Eb, Ab, Db, Gb, Cb, Fb. Interestingly this is the direct inverse of the pattern of sharps given above. Luckily (!) the mnemonic can now be reversed to form the sentence:


B E A D G C F


Battle Ends And Down Goes Charles' Father.

Again there is a similar, but reversed, relationship between tonics and accidentals. The tonic matches the second to last flat added on.




The circle of fifths
The information gathered from analysing scales can be used in constructing the circle of fifths:

This is a useful way of finding key signatures of major scales. Starting clockwise from the top C each new letter represents a new scale, a fifth above the one before it. This means that each new scale (clockwise) requires an extra sharp to be added to its key signature. The exact sharps to be added are found by reading off the letters starting from the F (to the left of the C.) For example, if we needed to know how many, and which, sharps a scale of E major requires, we note that E is at position 4 - it requires 4 sharps. These sharps are (reading off from F): F#, C#, G#, D#. If you were faced with a key signature of 5 sharps, you would count off 5 from the top to arrive at B - it is the scale of B major.





Fig 2. The b-major scale

Similarly, key signatures with flats can be created. Each new letter starting from F represents a new scale, and the position of the letter indicates how many flats it has. The actual flats are read anticlockwise from the Bb on position 2. Ab is on position 2, so it has 2 flats: Bb and Eb.



Web sites.


  • http://www.phy.mtu.edu/~suits/scales.html

  • http://www.hypermaths.org/quadibloc/other/musint.htm

  • http://www.silcom.com/~aludwig/The_musical_scale.htm

  • History of scales at http://www.phy.mtu.edu/~suits/scales.html

  • http://www.todayskills.com/music/theory/scale.htm

  • Full scale at http://www.phy.mtu.edu/~suits/notefreqs.html

  • http://ptolemy.eecs.berkeley.edu/eecs20/week8/scale.html

  • Scale theory: http://home.austin.rr.com/jmjensen/musicTheory.html#major


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