The major scale - why and how?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.)

Answer 6

I believe those that say it begins with using the 3:2 ratio are correct. Schoenberg is also correct, but a 7-note system already existed before the major scale was recognized, so his explanation starts in the "middle", so to speak. We call the 3:2 frequency ratio the frequency ratio for a 5th, but, of course, a scale or mode or system of pitches had to exist before anyone could call it a 5th, since this refers to the relationship of the two pitches within a scale (or mode or system).

By stacking 5th's and telescoping them down so that the pitches are within an octave, or, perhaps better, by alternating up a 5th, down a 4th, up a 5th, down a 4th, it's relatively easy to see why one would stop when they get to the 8th pitch, because it gives a note outside of the octave. When the note is brought within the octave, it results in a note very close to the starting note, and it creates a step-wise interval that doesn't exist within the first seven. Let me explain by an example:

Any frequency can be used to begin with, but it is easiest to think of the beginning note as the F in our modern system, because that will create the 7 natural notes (white keys on the piano). Proceeding up a 5th, down a 4th (or multiplying the frequency by 3/2, and then the next one by 3/4) results in the following: F C G D A E B. Lining these up from lowest to highest gives: F G A B C D E. We could cap it off with another F, which would come from multiplying the starting frequency by 2, which gives a note an octave higher. In fact, it gives us the reason for calling it octave, because it is the 8th note in the list. When the math is applied to the frequencies in this list, the intervals created between adjacent notes are in two sizes. Let's call them L (for large) and S (for small). This is instead of whole-step, half-step, which comes later. The pattern is LLLSLLS. Adding the next note above B in the first list, you get the note that we would call F#, but it is outside of the octave in the up a 5th, down a 4th scheme, and when brought into the octave, the F to F# interval, turns out to be neither the L or the S step interval. So, there is reason to stop at the note we call B.

Adding octave duplications of these 7 notes, extends the basic system. Eventually, the notes get named and additional notes, accidentals, get added. We have been using the names as we know them, but at one time, those names didn't exist yet. The medieval modes exist with in this extended system. The major 3rds FA, CE, and GB within this system were quite dissonant on most instruments because fast "beating" would occur between near coincidental harmonics. The frequency ratio of these major 3rds is 81:64. A much more consonant interval occurs if the ratio is 80:64 which reduces to 5:4. This is a ratio that occurs naturally in the overtone series. The desire to use this interval is what leads to the concept of temperament, which is a whole new topic. But for the current purpose, we can see that once the major 3rd is made more usable through temperament, the major triad gets "discovered" and then Schoenberg's theory for the foundation of the major scale takes over.

I would like to add that I have tried to keep the mathematics to a minimum here, but the math sure helps when trying to dig deeper into this topic, and the math required is mostly just arithmetic with fractions. The math will also help one to understand what is meant by "beating between near coincidental harmonics". The development theorized here is the Western Scale system. It seems reasonable to me that something similar could have occurred in India, but other paths were followed along the way to account for the many differences between the systems.

Answer 7

Sa, Re, Ga, Ma, Pa, Dha, Ni, Sa is the raga equivalent of the western Do re me fa so la te do.

Major scale has the same intervals as Shankarabharanam.
Minor Harmonic scale has the same intervals as Keeravani
Minor Melodic scale has the same intervals as Gourimanohari

But we have 12 versions of each scale! that would be like 12 Shankarabharanam ragas all staring on different notes. Each different Shankarabharanam would have its own Sa Re Ga

All of the western scales originated in India and were transfered either to the Agean, Anatolia and eventually North africa by the Roma people starting during the Greek_indian confluences 10 centrury BC. The west got them from Greece, Anatolia, Egypt and Arabia.

THey are ALL Indian scales. Look to the Indian and Arabic scholars for the ultimate origin of all western Music.

(https://www.amazon.com/Raga-Guide-Survey-Hindustani-Ragas/dp/B00000JT5P)

For a narrower Western view see Helmholtz (https://books.google.co.uk/books?id=2CiqYQXZjIYC&pg=PA15#v=onepage&q&f=false)

Answer 8

There is another (structural/mathematical) reason to favor the major scale, related to evenness of the scale.

To understand this, allow me a small detour, combining historical and structural observations.

The first basic point I want to make is that we need to distinguish between the major scale and the pitch class associated with it. The major scale contains a number of pitches, in certain ratios, together with one pitch which is the tonic. In C major, the tonic is C. Now, we could also look at the pitch class {C, D, E, F, G, A, B} but without emphasizing the idea that any one of these pitches must be the tonic. This generalizes your question to include all the modes of the major scale, which are, after all, used very frequently in various musical traditions.

Now, it is easy to imagine Ancient humans singing a single pitch, and then gradually discovering that other pitches sound well with them. The overtone series does explain fairly well why fifths and fourths above/below a note, and perhaps major thirds above/below a note sound good. But as many have pointed out here, it seems spurious to use the overtone series to motivate the entire major scale. Such explanations feel easy to make afterwards, but they don't really plausibly explain how Ancient humans went on this particular musical voyage. After all, they did not have the concepts that we have today, so it would not have seemed obvious to them at all.

That said, if an Ancient culture embraces the musical intervals of the fifth/fourth and the major/minor thirds/sixts, this has huge mathematical implications, since the use of those intervals implies some sort of division of the octave into 12 tones (although it need not be an equally tempered division). I say 'implied' here because in all likelihood, it would have taken a long time for anyone to actually tease out these implications.

What seems plausible is the following: it is very likely that Ancient humans experimenting with singing would try to add more pitches to their repertoire as they went along. We can imagine them creating songs that use 3, 4, or 5 different pitches. Of course, we cannot know what pitches they used, but it is entirely reasonable to presuppose that they expanded their vocabularies gradually.

In terms of the music of the world that is known today, there is one scale (or pitch class) that certainly stands out: the pentatonic scale. Many many musical cultures of the world use this scale. So before asking 'why the diatonic scale?' we might ask 'why the pentatonic scale?'. The first thing to observe, by the way, is that the pentatonic scale can be used with any of its five scale positions as a key center (for example, if that pitch is a drone that the rest of the scale is played over). What is interesting about the pentatonic scale is this: it is the most even 5-note scale (choosing 5 notes of the 12 in the octave) that you can come up with. The intervals are most evenly spread out. But also, the intervals that exist between the 5 notes of the scale have interesting features. For example, it has a lot of fifth/fourth type intervals, no tritone, and no semitone. All this makes it a very consonant scale, which is rather easy to sing. Any other closely related 5-note scale is much less consonant.

We now come to the diatonic scale. I think it's attractiveness can be explained on similar grounds. I like to imagine Ancient humans simply trying to add extra notes to the pentatonic scale experimentally. (Indeed, you only need to add two to get a major scale.) The resulting major scale is again the most even of all the possible 7-note scales (as in: the intervals are evenly spread out). Recall the pattern w - w - h - w - w - w - h. This is the most even way to spread out seven notes in the octave. Of the more evenly spread scales (compare the melodic minor scale), it is also the one with the fewest tritones. It has a lot of fourths and fifths which are easy to sing. Also, all kinds of pentatonic-style melodies can be sung as a part of the major scale. Furthermore, the major scale gives rise to tertiary harmony, which again may be motivated by appealing to the overtone series. Even scales also have many voice leading possibilities harmonically. (You can go to different chords by only moving one or two of the voices by a small step.)

The concept of evenness is explained in more detailed in Ian Ring's excellent website A Study of Scales: https://ianring.com/musictheory/scales/#evenness

Answer 9

The major triad of C is CEG (3rd, 4th, and 5th harmonics of C). C falls to F (perfect cadence). The major triad of F is FAC (3rd, 4th, and 5th harmonics of F). Now we have 4 notes--C,E,G,F,and A. G falls to C (perfect cadence). G's triad is G,B,and D. Now there are 7 notes: C,E,G,F,A,B,and D.

In summary, there is a family of notes associated with C (sort of like everyone has a home to go to, and everyone is a home to someone else). C falls to F, and G falls to C. This is the origin of the 7 note scale.

Answer 10

From what I can tell the two systems don't seem to have a common origin.

The two systems have a common organizing principle, which is the performance of a melody over a drone. This is no longer a prominent element of European music, but it was dominant in the middle ages and perhaps before. While this may not account precisely for every tone in the major scale, it does explain the prominence of those pitches that are related to the drone pitch by simple numeric ratios. That gives us (at least) the root, the octave, and the pitches we now know as the third and fifth. That these are simple numeric ratios means that they are relatively low on the harmonic series, so they interfere directly and audibly with the overtones of the drone pitch.

I say "at least" because the pitches between the four "major triad" pitches are somewhat more dissonant, but the harmonic series can still explain their precise tuning. The pitch between 1:1 and 5:4 rings more vibrantly if it is tuned to 9:8. The same is true of the pitch between 5:4 and 3:2 if it is tuned to 4:3.

Whatever tuning might be preferred in the absence of a drone, therefore, the pitches will gravitate towards these ratios with the addition of a bass drone. I don't think that the major scale will appear naturally in any musical system, but I suspect that there is at least a very strong bias in favor of the major scale in musical systems that feature the performance of melodies with drones.

Given the linguistic and anthropological connections between the Indian subcontinent and western Europe, I suspect that there is in fact a common origin for these systems, although the details of these commonalities and their transmission through western Eurasia are obscured by the passage of millennia.

Answer 11

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.

Answer 12

The true answer is that it is an organisation of the first seven tones of a series dependent on a given quantity of separate tones that have the closest mathematical relation. For instance, the first two tones in the series are 2:3, whilst the first four are 2:3:4.5:6.75, and so on.

The first seven tones in this series are significant because they contain all the mathematical intervals present in the chromatic system (which is also justified, but I won’t go into it.)

If you were to attempt organising these tones into an ascending scale that has the least composite dissonance, the end result is the major scale. The simpler, consonant sound of this result/organisation causes many people to interpret tonal relations in terms of this scale (and its minor derivative).

PS: trying to justify it with the overtone series has resulted in a wonderous failure of mental gymnastics. Often the cause of this is simply because they share the basis in whole number ratios.