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background: I am an amateur guitarist. I was trained in Indian classical music as a kid. Most of what I know in music theory is from books or Google. So please feel free to correct me or tell me if I'm wrong.

One thing that always bothered me is how the major scale came to be. As an Indian classical music student I started out with Sa, Re, Ga, Ma, Pa, Dha, Ni, Sa' which are analogous to the Western C, D, E, F, G, A, B, C' (Note that the default key for both is C). From what I can tell the two systems don't seem to have a common origin. So I have always wondered if the major scale will naturally appear in any musical system.

I always believed there has to be some mathematical reason as to how the major scale came to be. I have tried to derive it myself using the perfect interval as the first stepping stone, but I could never arrive at anything. Books and the web tell you of properties derived from the major scale, but not the other way around.

So I was wondering if anyone knows how the major scale came to be? Is it some natural mathematical conclusion to the way we perceive music or is it just an age old establishment? Is it really a coincidence that two culturally different musical systems have the same technical basis?

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Did you read this and this? Their answers maybe does not answers your culturally related question exactly, but might be of help. –  Ulf Åkerstedt Dec 27 '12 at 14:13
    
I'm surprised you didn't include G (Pa) –  Dave Dec 27 '12 at 17:40
    
@Dave I meant the whole series. I edited the original question to avoid further confusion. –  tinkerbeast Dec 28 '12 at 7:28
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The book: The Harmonic Experience by W. A. Matthieu may be useful for answering your questions. –  Dave Dec 28 '12 at 17:13
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6 Answers

The cultural sources of the diatonic scales are prehistoric, and attempts to find their origins have included studies of ancient bone flutes. Diatonic scales, including the major mode, do occur in multiple cultures, but do not occur in all cultures. For example, Indonesian gamelan music uses scales called slendro and pelog, which are nothing like the major scale or any other diatonic scale. Triads and tonal harmony are much newer than diatonic scales, dating back to Renaissance Europe. In the common practice period (ca. 1600-1900) based on the European musical tradition, we have the major-minor system.

These facts suggest that we should be very skeptical about attempts to derive the major scale based on mathematical principles. It's one of a variety of musical tools that have been used at various times as parts of various techniques and musical cultures. In any attempt to explain the major scale based on the overtone series or the circle of fifths, we have the problem that the various forms of the minor scale don't really fit.

There are certainly reasons why musical techniques such as Gregorian chant, polyphony, and tonal harmony work better with a major scale than with an Indonesian pelog scale. These reasons do have mathematical origins. However, they can't explain the origin of the diatonic scale or what we now call the major mode, since those are much older.

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If you played 2 notes that sound the most harmonious you'd choose notes that are octaves apart. These have a frequency ratio of 2:1 (or 4:1 or 8:1 etc.) but this doesn't get you very far in deriving a scale so the next "most pleasant" musical interval is when the notes are in the frequency ratio 3:2 - this will yield a G from a C. Most answers above have alluded to this and folk sometimes refer to this as a circle of fifths.

Starting from F you get the C maj scale and, if you carry on it progressively forms all the major scales in all the keys on a standard fixed-pitch instrument like a piano or guitar. I find this makes the major scale very special.

If you took the next most pleasing interval you'd choose 4:3 and this is a perfect 4th. You can get the same result as a perfect 5th but going down in pitch. This should be obvious to a few folk!

Forgive me. I'm a newbie and if i've repeated parts of other answers I'm sorry but I don't know of any other scale that is based on such simple ratios of frequencies.

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Arnold Shoenberg, in his Theory of Harmony, shows that the major diatonic scale is constituted by the major triads of the tonic, dominant and sub-dominant tones. For example, this means that the C major diatonic scale can be constructed with just the tones in the C major, G major, and F major chords.

He points out that the notes of a major triad are exactly the notes of the first three unique harmonics, indicating a possible source of the power of major triads. It has also been recognized as early as Pythagoras that tones whose frequencies lie approximately at small integer ratios sound "consonant" and pleasing when played simultaneously or soon after one another.

As to why such a revered scale should be based around the tonic, dominant, and sub-dominant, we can again take note of the harmonic series and Pythagoras. The dominant sits at a frequency ratio of 3/2 with respect to the tonic, taking after the second unique harmonic in the harmonic series based on the tonic. Relatedly, the tonic lies at a frequency ratio of 3/2 to the sub-dominant.

If you have access to a standard piano keyboard, try playing just the C (tonic), F (sub-dominant), and G (dominant) tones in various sequences and rhythms. Try comparing the affect of the tonic alongside various other pairs of tones (C, A and C#, for example). I've found that no other combination of just three tones better serves to instil the feeling of "home" in the note C.

Also, play the C major scale, one note at a time, up and down in a variety of rhythms. Since the major scale consists entirely of notes from the tonic, dominant, and sub-dominant major triads, try modifying the activity as follows: alongside each note of the scale, play a chord from either the C, F, or G major chords, choosing only a chord that possesses the note being played at the time. So you can play C with CEG, D with GBD, E with CEG, F with FAC, G with CEG, A with FAC, B with GBD, and finally C with CEG, for example. I find that playing alongside the chords feels like a very filled-in version of playing the lone scale. Try substituting the tonic, sub-dominant, and dominant major chords with other chords in the C major scale (or even chords from the chromatic scale), still only playing a chord when it possesses the tone currently being played in the scale. It seems like playing with the tonic, sub-dominant and dominant major triads form the most agreeable and representative expansion of the lone major scale than any other three chords (even though other combinations are nice and interesting).

In light of this, playing the diatonic scale in step-wise motion seems to have an effect of alternating between the "three areas" (tonic, dominant, and sub-dominant) of a key, providing harmonic contour of fundamental importance somewhat automatically.

As a brief indication of another important factor, note the inherent constitution of minor chords and their potential activity in the major scale. The second and third tones of the major scale can be viewed as the sub-dominant and dominant, respectively, of the sixth tone. Try the exercises again, but using the sixth tone as your tonic and minor chords in place of the major tonic, dominant, and sub-dominant chords (so playing the minor scale with respect to the sixth tone in our original major scale).

Those are some points that stand out in my mind as to how the major diatonic scale might be so important in music.

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This is the only correct answer. Any answer referring to the overtone series only is bs as that would yield something more like the mixolydian mode. –  tjb1982 May 11 '13 at 22:09
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This answer is incorrect. The diatonic scale dates back to prehistory, and attempts to find its origins have included studies of ancient bone flutes. The organization of harmony around tonic, dominant, and subdominant, began in the common practice period, i.e., starting around 1600. Therefore there's no way that the origin of the major scale can be explained by these triads and harmonic relationships. –  Ben Crowell Mar 21 at 22:33
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tl;dr The simple answer is "The major scale comes from the overtone series."

I don't know the history, which I suspect goes something like "the major scale is that way because people liked the sound of it." But I do know the mathematics, which may help to explain why people like the sound of it.

Let's start from first principles, many of which you probably already know. To begin with, sound is vibration. We perceive sound through the vibration of the eardrum and the tiny bones (the hammer, anvil, and stirrup) in our ears. Most commonly, we hear vibrations in the air, although when we speak, our own heads also vibrate, contributing to the sound we perceive as our own voices---this is why our voices sound different to ourselves than to other people and why most people don't like the recorded sound of their own voice.

So a note in its purest form is some vibration at some frequency, measured in Hertz (Hz). For example, by convention in the United States, orchestras tune to a pitch at 440 Hz (an A).

Now, when we say that a pitch has a certain frequency, like A is 440 Hz, that's mostly a theoretical simplification. Most sounds aren't composed of a single frequency. If you sing an A or play an A on a trumpet or a violin, the resulting sound is actually composed of multiple frequencies simultaneously---the fundamental frequency (the 440 Hz) and a series of related frequencies known as the overtone series. Which members of the overtone series are present, and in what proportions, determines the timbre (or tone quality) of the sound. It's what makes the voice sound different from the trumpet and the violin. So again, when we hear a sound, we are most often hearing a collage of related frequencies.

What is the overtone series? The frequencies that make up the overtone series are integer multiples of the fundamental frequency. In the case of A440, the overtone series includes 880 Hz, 1320 Hz, 1760 Hz, 2200 Hz, 2640 Hz, .... Taking octave equivalence into consideration, we can generate closely-related frequencies within an octave of 440 Hz:

  • A: 440 Hz
  • B: 440 * (9/8) = 495 Hz
  • C#: 440 * (5/4) = 550 Hz
  • D: 440 * (4/3) = 586.67 Hz
  • E: 440 * (3/2) = 660 Hz
  • F#: 440 * (5/3) = 733.33 Hz
  • G#: 440 * (15/8) = 825 Hz
  • A: 440 * 2 = 880 Hz

All of these frequencies are present in the overtone series, and so they occur naturally, to varying degrees, when you play an A. Contrast these frequencies with the frequencies for the major scale pitches in Equal Temperament:

  • A: 440 Hz
  • B: 494 Hz
  • C#: 554 Hz
  • D: 587 Hz
  • E: 659 Hz
  • F#: 740 Hz
  • G#: 831 Hz
  • A: 880 Hz

Pretty close, right? And in fact, Equal Temperament is a relatively recent compromise developed in response to issues like this. The point is: the notes of the major scale come from the naturally-occurring overtones present in sound.

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"All of these frequencies are present in the overtone series"... Yes, but theoretically all pitches are present in the series, and in the A series a pitch closer to G-natural comes way before (and more strongly than) G-sharp. Your explanation doesn't account for this at all. @oliTUTilo 's explanation comes closest, but the reality is that it's as much a product of cultural evolution as the physics of sound. –  tjb1982 May 11 '13 at 22:17
    
This answer is incorrect, for the reasons explained in my answer. –  Ben Crowell Mar 21 at 22:51
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The major triad is closely linked to the overtone series series of a given fundamental note

  • Taking c as the root,
  • c' (the c one octave up) has a frequency 2x that of the root,
  • g' has a frequency 3x that of the root,
  • c'' has a frequency 4x that of the root,
  • e'' has a frequency 5x that of the root, and that our ears respond in a particular way to these kinds of frequencies that are related by simple integer multiples.

Another physiological feature is that we identify all notes whose frequency ratios are powers of 2 (2, 4, 8, 16...) as the same note -- it's widely reported that the music of all cultures identify the different octaves of a given tone as the same note. This also means that we can octave-reduce the notes indicated above to put them all into the same octave.

The next most basic relation is the fifth, which corresponds to a frequency ratio of 3x.

Usually, the 4th is identified as the note whose 5th is the tonic, note that the fifth above F is C. Thus the 4th is the inverse of the 5th.

The fifth of the fifth is D (the fifth of C is G, the fifth of G is D), so this note is obtained by simple compounding of a simple interval.

In principle, all of the notes can be generated by compounding fifths, yielding Pythagorean tuning.

However, most modern theories of tones take the major third, corresponding to a frequency ratio of 5x, as fundamental as well. Compounding fifths, thirds and their inverses (fourths and sixths) yields Just intonation.

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It's quite easy to "construct" the major scale by adding perfect fifths (and adjusting octaves to keep the resulting frequencies close together). Start from C, and you get (in this order) G, D, A, E, H and F sharp, which is the entire G-major scale. (To get C major, you have to start from F instead.) Note that if you stop after five steps, you get a standard pentatonic scale.

Now, this scale has some nice practical properties - it fills the octave with a manageable number of frequencies (7), it contains just two different intervals, and those two are not too different from each other. (In practice, however, this is probably not how it was initially created - western tonal music is derived from earlier systems with fewer tones (four, five or six), and those were not constructible by piling up perfect fifths, but 'adjacent' notes (major or minor seconds). For instance, it was much more common to use C D E F G rather than C D E G A, so the "perfect interval" construction is probably a post-facto rationalization of a scale that practitioners had already found without mathematical considerations. I think it's nice that there is more than one explanation for something so fundamental as the tone repertoire that people whistle, hum and sing in.)

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