Theory of Musical Scales
The variation of air pressure against the inner ear is what gives rise to the experience we call "sound". Most sound that we recognize as "musical" is dominated by periodic variations rather than random ones, and we call the transmission mechanism as a "sound wave". In a very simple case, listening to the sound of a "sine wave", the air pressure increases and decreases in a regular fashion, and we hear it as a very "pure" tone. The rate at which the air pressure varies governs the "pitch" of the tone, and can be measured in oscillations per second, or Hertz.
Whenever two different pitches are played at the same time, their sound waves interfere with each other - the highs and lows in the air pressure get mixed together to produce a different sound wave. As a result, any given sound wave can contain many different oscillation frequencies; the human ear and brain are capable of isolating these frequencies and hearing them distinctly. When two notes are played, there is a single variation of air pressure at your ear that "contains" the pitches of both voices, and your ear and brain isolates them into two distinct notes.
When the original sound sources are periodic, as most musical instruments tend to be, the interference between any two pitches may cause the listener to hear additional pitches which don't necessarily have a "musical" relationship to the originals. However, whenever one pitch is a simple multiple of the other (1,2,3,4 times the oscillation frequency) the interfence does not generate these any new pitches. Thus, any two pitches related like this sound perfectly "in tune" in that you hear those pitches, and nothing else.
The simplest ratio is clearly 1:1, but this is a trivial case of the same note being played twice. More interesting is the 2:1 ratio. Any two pitches with a 2:1 ratio between them define a difference in frequency (or "interval") that we call an "octave". This is the smallest interval at which two different pitches will be perceived by the listener as being "the same note", precisely because when played together, they sound perfectly "in tune". The human ear can perceive from about 20Hz at the low end to around 20,000Hz at the high end. By starting at 20 and doubling up to 20,000, you can see that the human ear has a range of a little under ten octaves.
There are clearly many other integer ratios, and even though they do not all avoid the generation of additional pitches, when the ear hears any two notes with such a ratio (or close to it), they are perceived to be "in tune".
So, we can now move to the definition of a scale: a way of defining the intervals between each of a set of notes within an given distance. The distance and number of notes is variable, but in the majority of the western classical and popular tradition, twelve notes are used to span a single octave. The intervals between them are:
1:1 unison 21:20 half step / minor second 9:8 whole step / major second 6:5 minor third 5:4 major third 4:3 perfect fourth 7:5 tritone / augmented fourth / diminished fifth 3:2 perfect fifth 8:5 minor sixth 5:3 major sixth 9:5 dominant seventh / flat seventh 17:9 major seventh / natural seventh 2:1 octave
For purposes of tuning we need a reference pitch, something all the instruments can agree on. Usually a 440hz sine wave is used as the reference pitch, as an A natural. Now, according to our table above, we can calculate the pitch of any other note by setting up a simple ratio relationship. For example, if I wanted to calculate the pitch of a perfect fifth from an A440, I would write:
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(X / 440) = (3 / 2)
and solve for X. Simple algebra, right? In the above, X comes to 660. Let's calculate two more:
(X / 440) = (9 / 8); major second = 495
(X / 440) = (5 / 4); major third = 550
The note that a scale centers around is called the tonic. We often use the term "key" for a scale, so the key of A is just a scale with A as the tonic.
Now, what actual pitches do we end up with? If we pick A natural (440Hz) as the tonic, we have a scale containing the following frequencies/pitches:
440.000 A 462.000 A# 495.000 B 528.000 C 550.000 C# 586.667 D 616.000 D# 660.000 E 704.000 F 733.333 F# 792.000 G 831.111 G# 880.000 A
Any scale in which the ratio of any note to the tonic is an integer ratio is called a scale of just intonation. These scales have a very natural-sounding quality to them.
The problem is, they're very difficult to achieve in any stopped or fretted instrument. The difficulty is subtle, but it means big headaches. It's hard to explain, though, so I'll give an example.
The interval of a major second is the "whole step" so common in the western tradition. It defines the distance between A and B, or C and D, among others. The interval of a major third defines the distance between two notes with two "whole steps" between them; for example, the distance between C and E or F and A.
If this is true, then the major second of a major second (that is, two whole steps from a given note) should be the same as the major third (two whole steps from a given note), or:
(X / 495) = (9 / 8)
X should equal 550. But instead, X is 556.875. What has happened here? Well, A(440) was the initial tonic of the scale, and all the intervals we defined above meet the integer ratio condition. But then we took a different note (495) as the tonic, and computed the major second of that. So in effect, we used two different scales and found that after a whole step from the tonic in each case, we end up with a pitch that isn't in the other scale.
This is the problem. Any given scale of just intonation must be tuned to a tonic, which is fine if you only want to play in one scale or "key". However, you have to retune the instrument every time you modulate keys. As many classical composers (and pop ballad writers) will tell you, this has a way of limiting your expressive power.
So what's a keyboard manufacturer to do? The answer is simple: make one note in tune, and space all the other notes equally (logarithmically equally, anyway). This is what happens on most fretted instruments and keyboard instruments. Now, instead of calculating pitch with integer ratios, we just plug an interval into the following equation:
P = 440 * 2 ^ (n / 12)
where n is the number of half steps sharp you want to go (and hey, guess what, negative numbers work as expected; (n == -3) finds the pitch of the major sixth below A440). This is called a scale of even (or equal) temperament, since the distance to any other note is independent of (and consistent across) key centers.
Now, waitasec, doesn't this put everything out of tune? Well, yes it does, but not by very much. Observe:
Note Just Pitch E.T. Pitch (approx.) Error (%) A 440 440 0.0 Bb 462 466.16 +0.9 B 495 493.88 -0.2 C 528 523.25 -0.8 C# 550 554.37 +0.7 D 586.6- 587.33 +0.1 D# 616 622.25 +1.0 E 660 659.26 -0.1 F 704 698.46 -0.7 F# 733.3- 739.99 +0.9 G 792 783.99 -1.0 Ab 831.1- 830.61 -0.1
As you can see, we're never more than 1% out of pitch, which most people can't hear. However, you *are* out of tune inherently, so when your instrument then goes further out of tune you sound *really* bad. The advantage, though, is that you get stopped instruments, which makes composition and playing much easier.
In this system the fifth tone ratio is about 1.4983 instead of 1.5, and the half-tone ratio is 1.059463 instead of 1.05. Only the octave is still 2:1. It was not easy when they first learned to tune the "well tempered clavier" to interpolate between tuning suggested by different keys. Insofar as tuning is still done by ear we are not likely to achieve a half-tone ratio that matches the twelfth root of two in six or seven digits!
( There are also Indian raga scales and other non-standard scales that are "just intonation" scales, but I wanted to present something understandable.)
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[ one thing i neglected to mention in the original article was that many classical composers wrote compositions for just-intonated instruments (wind instruments in particular). however, since these instruments couldn't re-tune to a new tonic, modulating the key of the piece created a tension; it sounded like you were still playing in the original key and wanted to return to it. at least, that's what some people say, and they're usually quite insistent that playing the piece on a JI instrument is the only way to truly hear what the composer intended. i personally don't believe that, but judge for yourself. { ajax } ]
see also [Joseph Schillinger]?