Why are the keys on the piano tuned the way they are? This simple question led to browsing a good number of the vast collection of web sites and library books on music theory, but I still couldn't find an explanation that covered the ground I wanted. In this discussion, which is still evolving, I hope to explore what principles lie behind the construction of the standard Western musical scales and the music that is based upon them.
In fact, I want to try to understand basic music theory, including such deep subjects as tonality and the theory of harmony, from an outsider's perspective, utilizing principles of physics, mathematics and human hearing. (Presumably then, the things that have no such explanation can be attributed to arbitrary cultural conventions). I also want to avoid giving a historical discussion, however worthwhile that may be, and avoid using arcane terminology, like "diesis" or "septimal comma", as much as possible. There are still a number of unanswered questions and gaps in this discussion, and probably some outright errors. If you know how to fix something, please email me .
Contents: (Newer material in Extras section)
The Major Scale from HarmonicsTo get started, consider plucking a taut string on a guitar, for example, or striking a piano string with a hammer. The string will vibrate at a fundamental frequency f, and segments of it vibrate with frequencies 2f, 3f, 4f, ... These are harmonics. (Strictly speaking, a real string is not perfectly flexible, and the vibration rates of the segments deviate a little from pure integer multiples of the fundamental frequency. In acoustics, the term partials is prefered. For solid objects like bells and gongs, the partials differ greatly from integer multiples of the fundamental. However, in the present discussion, we shall avoid this complication and work with harmonics).
Based upon the strength
of the consonances with C (discussed in more detail when we get to the dissonance
curve), we complete the major scale by taking the 3rd harmonic
of C (namely G) and building the major chord there, and then finding the note
below middle C which has middle C as its 3rd harmonic (namely F)
and building another major chord there.
Note that F (frequency 2/3f -- or 4/3f when put in the proper octave)
is not one of the harmonics of C no matter how high you go, but C is a
harmonic of F. To see that F is not a harmonic of C, observe that 2, raised to
any integer power, times 4/3f is not equal to any integer multiple of f.
We derive the G major triad by
[1 5/4 3/2] (3/2f) = [ 3/2 15/8 9/4] = (G-B-D)
and the F major triad
[1 5/4 3/2] (4/3f) = [ 4/3 20/12 12/6] = (F-A-C)
Then, rescaling (this is probably the origin of the word) by 1/2 to bring terms
into the correct octave and reducing
fractions to lowest terms, we have the frequency set for the (Just) major scale:
(C-D-E-F-G-A-B) = [ 1 9/8 5/4 4/3 3/2 5/3 15/8 ]
The above discussion does not reflect what actually happened historically.
The Minor Scale(s)Continuing the above discussion, let's look at their ratios ( extending into the next octave so as to preserve the increasing order of the frequencies i.e. the chords are in root position):
(A-C-E) = [ 5/3 2/1 5/2 ] = 5/3 [ 1 6/5 3/2]
(E-G-B) = [ 5/4 3/2 15/8 ] = 5/4 [ 1 6/5 3/2]
(D-F-A) = [ 9/8 4/3 5/3 ] = 9/8 [ 1 32/27 40/27] (not the same triad)
(B-D-F) = [ 15/8 9/4 8/3 ] = 15/8 [ 1 6/5 64/45] (not the same triad)
1f: 2f 3f 4f 5f 6f 7f ...
(6/5)f: 2(6/5f) 3(6/5f) 4(6/5f) 5(6/5f) 6(6/5f) 7(6/5f) ...
(3/2)f: 2(3/2f) 3(3/2f) 4(3/2f) 5(3/2f) 6(3/2f) 7(3/2f) ...
F: (4/3) [ 1 6/5 3/2 ] = [ 4/3 8/5 2/1 ] = (F - Ab - C) = Fm
G: (3/2) [ 1 6/5 3/2 ] = [ 3/2 9/5 9/4 → 9/8 ] = (G - Bb - D) = Gm
At this point, we should take a look at the Dissonance Curve and Applet. There we can better see why C, F, and G chords are chosen for the major and minor scales.
Why is A minor considered the relative minor to C major?
In equal temperament (discussed later) they share exactly the same set of notes.
Here, in the more fundamental just intonation, they share all but one key:
Apply the minor scale operator to the A from C major (and rescale)
[1 9/8 6/5 4/3 3/2 8/5 9/5] ( 5/3) = [5/3 15/8 2 20/18 5/4 4/3 3/2]
Thus 20/18 = 10/9 is the only one that doesn't match up. The other perfect minor triad in the C major scale was Em = (E-G-B) and it also misses by only 1 note (F#):
[1 9/8 6/5 4/3 3/2 8/5 9/5] ( 5/4) = [5/4 45/32 3/2 5/3 15/8 2 9/4]
Thus we have natural 2 choices for a minor key related to C major; the 10/9 of A minor would have to be corrected by the factor 6/5 = 1.2 to be F=4/3f, whereas 45/32 of E minor must be corrected by the factor 128/135 = 0.948. Thus E minor would seem to be closer, but that is not the way it is: E minor is the relative minor of G major. So we do not have an adequate explanation for this without appealing to equal temperament, or perhaps certain meantone temperaments. But see my remarks about reflections in the Circle of 5ths.
Aside: The Fundamental Bass of RameauIn 1721, Rameau put forward a theory of harmony, which was clarified by d'Alembert in 1762 [Helmholtz 1877 p.232]. One principle finding was that the major chord has a fundamental bass, 2 octaves below.
Changing Keys in Just IntonationGraphical Applet to show changing keys in JI is ugly.
What about Simple Whole Number Ratios?Consider the fractions in the interval (1,2) made by small positive integers. The ancient Greeks (especially Pythagoras ) were very interested in these for musical scales, and the interest continues up to the present day. We have tried to show that the fractions themselves are not fundamental for making a scale; they are a consequence of harmonics and avoiding what the human auditory system percieves as unpleasant dissonance. Still, we have marked in blue the ones that occur in the major scale, and in red the additional ones that occur in the minor scale. The blank spots correspond to fractions that already occurred.
3/2Evidently, 7 is unlucky here!
6/5 7/5 8/5 9/5
8/7 9/7 10/7 11/7 12/7 13/7
9/8 11/8 13/8 15/8
12 Tone Equal TemperamentHow do we have a playable number of physical keys on the piano keyboard, but still be able to modulate into any musical key and have the chords harmonically correct? Well, we can't. We have to weaken the last condition to having the chords almost harmonically correct. To do this, we divide the octave up into 12 equally spaced seqments (called the chromatic scale). It is the fact that the divisions are equal that enables all musical keys to equally accessible. The reason for the choice of the number 12 is that the note locations that we will end up with will pretty closely approximate the pitch ratios of the JI scale (to do significantly better requires many more chromatic scale steps).
Note that the naive way to do this doesn't work: If our base note C has
frequency f, then we might try defining
C# = (1 + 1/12) fAmong other problems, this results in G having a frequency of 1.5833, which is too far off from the 3rd harmonic value of 3/2 = 1.5 to sound good against C (recall the dissonance curve has a steep, narrow valley at 3/2). The generally accepted solution is to build the chromatic scale in intervals determined by iteratively multiplying the base frequency by a factor c, where c is the twelvth root of 2 (approximately 1.059463). This will result in the scale divisions drawn on the horizonal axis of the dissonance curve applet. It should be remarked that the human ear percieves these divisions as being equally spaced in pitch, because the ratio of any 2 adjacent intervals is constant.
D = (1 + 2/12) f
B = (1 + 11/12) f
Then for the equally tempered chromatic scale, we define the notes to be:
C = c0 f
C# = c1 f = Db
B = c11 f
Conclusions and Other scalesWe have derived the major chord from the primary harmonics on a string, the major scale from the notes making up 3 related chords, and the minor scales from other pleasing chords formed in the major scale. This has probably been well known for hundreds of years, but not clearly explained today, even in [Pierce 1983] or [Helmholtz 1877].
From the calculations of the dissonance curve, we could quantify why the C, F and G chords should be chosen, because they include the maximally consonant notes against C. They also include some highly dissonant tones against C, but these have some interesting properties, primarily the ability to form at least one dissonant chord (G-B-D-F) that produces the psychological effect of wanting to "resolve" to C major.
We only sketched some ideas of how to make this leading tone property mathematically precise, but did not do so. Therefore we have established no explanation of why B is a leading tone to tonic C, but C# isn't. We were unable to explain why no other notes should be included in the major or minor scales.
Equal temperament (which is a very clever idea, whatever its faults)
allows moving into any musical key, at the
price of some small loss in harmonic quality. It defines an evenly spaced
chromatic scale, and from here we can form new scales, just
from subsets of the notes on the keyboard. Let's list some other popular
scales [Apel 1961 p.221, 262]:
The pentatonic scale: (C D F G A )
The Blues pentatonic scale: (C D E G A ) the 3rd note has been flatted.
Lydian: (C D E F# G A B)
Whole Tone: (C D E F# G# A#)
The chromatic scale itself.
A more general definition of scale would be: A baseline set of notes that a melody is restricted to, except when it departs from it for purposes of contrast.
Something ExtraJava Applet to study Wave Reflections on a String
References(what was consulted, not all that exist) in alphabetical order:
AcknowledgementsThe author would like to thank the following people for providing detailed and specific comments about portions of this discussion (although the author may not have had the good sense to fully implement them all):
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