Why does the scale have seven (or five) notes? Why not six?

Question

I'm a programmer by trade, and I have always felt that music was arbitrarily difficult. Please forgive my inexperience with musical notation. I had a little thought experiment with my wife today, and I wanted to ask why we don't do it the way I thought up.

My wife explained to me that a scale(octave?) is made up of seven notes, which we typically call ABCDEFG or Do-Re-Mi-Fa-So-La-Ti(-Do). From this answer: https://music.stackexchange.com/a/3004 we know that those 7(8) notes are this progression:

Every major scale has seven notes. They all start on a root note and proceed to go up in the following pattern: Whole Step, Whole Step, Half Step, Whole Step, Whole Step, Whole Step, and then a final Half Step returns to the root note (an octave above where we started).

Why go up by a half step twice? Why not go up a whole step every time? It seems like having B# be C and Cb be B (and same with E/F) is arbitrarily complicated. Was this done just to make pianos easier to play by feel? Is there a mathematical root?

If you will suspend your disbelief with me for a minute, what if we had a scale made up of 7 lines? The spaces in between each line represent the notes (I'll call them 1-6, to avoid confusion with A-G). The lines themselves represent sharps and flats. So a 1# is a 2b, etc.

The piano would have to change to having black keys in between every white key. To offset this, the 1 keys would be wider on the left, and the 6 keys would be wider on the right so that one could still determine octaves (septaves?) by feel.

What problems does this present? Is there a good reason not to go to an easier to remember system? If not, why has no one done it?

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Questions I've already looked at to make sure this isn't a duplicate:

• What is the theory behind scales?
• Why Seven Principal Tones?
• The major scale - why and how?

Answer 1

I think your question is largely about the chosen notation for the Western system, which most answers haven't really addressed.

The notation we have is actually pretty natural and logical, for a simple reason: there are twelve different notes in the Western system, but only a subset of these -- seven, in fact -- are used in a given scale such as the major scale.

Let's use individual semitones as the basis for a notation as you suggest; so, let's say the note A is still denoted by A, but now A# (or Bb) is denoted by B, and then the remaining notes are C, D, E, F, G, H, I, J, K, and L (twelve in total).

I understand why you'd want to do this; it removes synonyms. But at what cost? What does an actual key look like now? Take C major as an example. In the new notation, the notes are D, F, H, I, K, A, C. This is confusing and hard to remember. Compare with C major in normal notation: C, D, E, F, G, A, B. It just cycles through the seven letters.

What about other keys? Let's take F major as another example. I won't write it all out in the new notation again because you just get another confusing list of letters, but in normal notation, it's F, G, A, Bb, C, D, E.

Hopefully now you see the benefit of this notation: it's easy to think about every key, because, ignoring accidentals (i.e. the flat on the B) they just cycle through our seven letters.

You lose uniqueness of note names -- though in fact, not really in practice, for example you'd never call Bb "A#" when talking about the F major key -- and the usefulness of this feature of the notation far outweighs this minor problem.

Answer 2

You can divide up the octave however you want, but it turns out that doing what you suggest doesn't really make good sounding music, at least to our western ears.

It all has to do with overtones and pleasant ratios of pitches. An interval sounds consonant to us when the ratio of the frequencies is mathematically simple. It causes the waveforms line up and produce constructive interference.

If I take C as a base from which to construct the overtone series, I quickly find G and E to have simple ratios (3:1 and 5:1, and by shifting octaves to get them closer together, 3:2 and 5:4). Stack two fifths and drop the octave to create D = 9:8, and go a fifth down and an octave up to create F = 4:3. Now we have the beginning of a scale: C D E F G, and the notes aren't evenly spaced (E-F is roughly half the distance of the others). This is the beginning of Pythagorean tuning, and various ways to construct the remaining notes of the major scale and fill in the gaps result in a huge number of ratio-based tunings.

In short: it's the way it is because it sounds good. Sure, it's a bit screwy in some ways, but we don't want to force an art form to conform to some notion of mathematical simplicity.

Answer 3

The reason is that dividing an octave into 12 notes sounds the best for a very mathematical reason! The frequency of each semi-tone is 2^1/12 away from its neighbours.

Note    C × ?   Fraction    Note    C × ?   Fraction
C       1       1/1         C       2       2/1
C♯/D♭   1.059   18/17       B       1.888   17/9
D       1.122   9/8         A♯/B♭   1.782   16/9
D♯/E♭   1.189   6/5         A       1.682   5/3
E       1.260   5/4         G♯/A♭   1.587   8/5
F       1.335   4/3         G       1.498   3/2
F♯/G♭   1.414   7/5         F♯/G♭   1.414   10/7
G       1.498   3/2         F       1.335   4/3
G♯/A♭   1.587   8/5         E       1.260   5/4
A       1.682   5/3         D♯/E♭   1.189   6/5
A♯/B♭   1.782   16/9        D       1.122   9/8
B       1.888   17/9        C♯/D♭   1.059   18/17
C       2       2/1         C       1       1/1

Notice how each fraction on the right hand side (descending) is almost the inverse of the left hand side (ascending)? The difference is one of the numbers is doubled or halved each time. The smaller the two numbers are and the smaller the difference between them the better they sound to us. This is because the parts of the waveforms they produce agree very often.

When the peaks often coincide they produce a chord, or an agreement. When the peaks rarely coincide they are discordant and the sound is disagreeable! So we can see from the table that C and G will sound the best together as C has 2 peaks for every 3 peaks that G has. The next best note for C is F, which is actually the inverse ratio of C:G. Then comes E, giving us the C-E-G chord, which we already know sounds very nice! The ratios for C-E-G are (4:5:6)/4. In the minor scale we have C-E♭-G which is 6/(6:5:4).

Either the numerator or the denominator must be able to be multiplied to a common, small value for the two notes to sound good together. You might think that E♭-E would sound good because they both have a 5 but it doesn't work that way. You would either get (24:25)/20 or 30/(25:24), neither of which would sound good because of the high numbers needed to find a common frequency.

Answer 4

Most of the answers here appear to be focusing on why we ended up with a seven note scale in western music.

This is a great area of inquiry; however, it is worth noting that whatever the answer to this question, the seven note scale is a fundamentally arbitrary product of Western culture.

Dissonance and harmony are culturally relative. The idea of the octave appears in almost every society; however, the way in which the octave is split and which combinations of frequencies are pleasing vary entirely by culture.

"Strictly speaking, there are no structural characteristics that have been identified in all known musical systems." - http://www.academia.edu/10684651/Cross-Cultural_Perspectives_on_Music_and_Musicality

So I would argue that although the other answers are mostly correct in identifying reasons why we use a seven note scale, it should be kept in mind that these are fundamentally cultural and historical reasons, not biological or mathematical reasons.

Edit: Just wanted to disambiguate based on the comments. I am referring to the dictionary definition of "harmony," which is "the combination of different musical notes played or sung at the same time to produce a pleasing sound" - http://merriam-webster.com/dictionary/harmony. This definition is not related to any particular mathematical relationship or consonance between the notes: "Harmony" simply means that the resulting sound is pleasing to listener.

Answer 5

The answer to the question "was the diatonic scale designed to make pianos easier to play" is clearly "no", because the diatonic scale precedes the invention of the piano by some thousands of years.

Remember, for the vast majority of the history of music, it was not played on keyboard instruments. It was played on wind or string instruments. If you want to see instruments on which the chromatic scale is clearly laid out, see the neck of any guitar, ukulele, or other fretted stringed instrument.

The answer to the question "why is C sharp enharmonic with D flat" is because it is highly convenient to do so. As other answers have noted, the fundamental relationships in music are ratios of vibrations that are 2:1 or 3:2. But it is impossible to make any combination of 3:2 ratios that works out to a 2:1 ratio! What we do then is we choose twelve notes that are each in a ratio to each other of the twelfth root of two; that number can be raised to an integer power that gives a result very close to 3:2. I wrote a series of articles about this ten years ago (start from the bottom).

The answer to your question "could we have a black key between every white key on the piano?" is yes, and this arrangement would have several nice properties including making it trivial to transpose on a piano (by any number of full tones; transposing half tones is tricky in this layout). The traditional piano keyboard arrangement makes it difficult for even experienced pianists to play a piece known in one key in a different key, say, to accommodate the range of a particular singer. The Wikipedia article on isomorphic keyboards may be of interest to you.

You may also be interested in studying the key layout of the button accordion.

It would be entertaining to build a small piano or organ which had the keyboard layout you propose, and learn how to play scales and chords on it. If I ever build a keyboard I'll give it a try and report back.

The answer to your question "why not just go up whole tones every time and have a six-note scale?" is: You go right ahead and play music like that if you want. If you're watching a movie made in the middle part of the 20th century and a character suddenly goes into a dream sequence, odds are pretty decent that the incidental music uses the scale you are describing. Music written in this scale can have an unsettling and dreamlike quality to it, at least to people accustomed to listening to Western music.

Answer 6

There is no deep reason. Western "folk music" often only used 5-note scales (approximately C D E G A in modern notation). The song "Amazing Grace" is a well known example.

There have been experiments with more notes per octave - 19, 31, and 43 all work quite nicely. People have built playable keyboards for those, and other systems. There are some pictures at http://en.wikipedia.org/wiki/Enharmonic_keyboard.

Non-western music follows different rules. Arabic scales use 24 equal divisions per octave. Turkish scales divide each whole tone into 9 equal parts, but they don't use all of the 54 notes in one scale. Javanese gamelan uses two groups of instruments tuned to different scales with 5 and 7 notes, both different from any notes in the western scale.

Rationalizing western scales with hindsight using "just intonation" intervals like 3:2 and 4:3 is interesting (and was first done at least 2,500 years ago) but given what the rest of the world does, I find it had to accept that there is anything "fundamental" about it. Some very old European monophonic instruments don't even play "octaves" tuned in a 2:1 ratio - for example Scottish bagpipes, though some modern ones are tuned in equal temperament.

In fact, even pianos are not tuned in mathematically equal temperament - Google for "stretched tuning".

Answer 7

There is a scale using tones all the way - it's called a whole tone scale. Just as there's a scale using semitones - a chromatic scale.

Going with your idea of extra black keys - there's no need to change the width of the white ones, a couple of extra blacks would fit in the same way as they do between the existing whites. Trouble is, the pattern is then lost, so there would have to be other landmarks, like on a harp.

Answer 8

Three musical intervals are special: the octave, the perfect fifth, and the perfect fourth. If one plays a note and its first three harmonics, the intervals among those pitches will be an octave, a fifth, and a fourth. Scales tend to sound good if some of their notes have intervals of perfect or near-perfect fifths or fourths between them. A perfect fifth is very close to being 7/12 of an octave and perfect fourth is very close to being 5/12 of an octave. Because these are odd subdivisions, there is no way to divide an octave into fewer than twelve roughly-equal pieces and have it contain a pair of pieces separated by a perfect fourth or fifth.

Because an octave is a perfect fifth plus a perfect fourth, and a perfect fifth is larger than a perfect fourth, it makes sense that there should be more notes between two pitches that are separated by a perfect fifth than the remaining notes in the octave that are separated by a perfect fourth. Unless the subdivisions are about half the size of the difference between a perfect fourth and fifth, however, it doesn't make sense for there to be two more notes in the fifth than in the fourth. If the number of notes within the fifth is one greater than the number within the fourth, that implies the total number of notes will be odd.

Answer 9

The strongest motivation for the ABCDEFGA scale is the SYSTEM of CHORDS which make a major key. For the key of C-Major, the basic chord of C gives us the notes C-E-G-C. Its related chords are F-major, consisting of F-A-C, and G-major, consisting of G-B-D. Putting it all together gives the notes C-D-E-F-G-A-B-C, which are all the white notes on the piano. The same kind of thing can be done for any other Key, and progressively using each of the white notes to form a system of major chords for that key motivates all of the BLACK notes on the piano. As has been said, this is fundamentally a matter of identifying a very specific frequency ratio (4-5-6-8) as being maximally pleasing to our WESTERN and EUROPEAN ears. Given that, it's all in the chord systems for a key.

Answer 10

The piano would have to change to having black keys in between every white key.

That's called a Jankó keyboard. They did not gain the traction needed to become popular in significant numbers. A variant for accordion is the "Beyreuther system". Again, they did not gain significant traction as compared to the now common "chromatic button accordion" which uses 3 rather than 2 non-redundant rows for arranging semitones in a uniform manner (for ease of fingering and transposition, there are additional 0-3 redundant rows, with 2 redundant rows for a total of 5 being the most common variant nowadays).

There is nothing new under the sun...

Answer 11

To reformulate the mathematical reason differently: Two sounds sound harmonic if they share many overtones. For one-dimensional oscillators (such as strings or flutes, but not drums for instance) overtones occur at integer multiples of a base frequency, hence harmony occurs when the quotient of the base frequencies is a fraction with very low numerator and denominator. Among the "best" such fractions are 1/2 and 1/3 (or 2/3). Therefore it should be easy to play notes with this relation, i.e., going a certain number of keys to the right should get us one octave (or one quinte) up. One cannot fulfill both demands at the same time (at least not with finitely only many keys), so one has to rely on approximations.

Mathematically, we need rational approximations to log 3/log 2, and the best such approximations are found by investigating the continued fraction for this number, which is

log 3/log 2 =1+1/(1+1/(1+1/(2+1/(2+1/(3+1/(1+1/(5+...)))))))

The best approximations are found by cutting this infinitely long continued fraction, and that gives us the approximations

1, 2/1, 8/5, 19/12, 65/41, 84/53, 485/306, ...

The most interesting approximant is 19/12 because it leads to our 12 half-tones. Let's try it: We start at a random frequency, 200 Hz say, and repeatedly multiply this by 3, always dividing by 2 when we exceed 400 Hz. Doing this twelve times, we obtain (approximately)

200, 300, 225, 337.5, 253.1, 379.7, 284.8, 213.6, 320.4, 240.3, 360.4, 270.3, (202.7)

and if we for simplicity agree that 202.7 is close enough to the 200 we started with, this is our scale (unsorted).

The previous approximant 8/5 would lead to a smaller scale, but would require us to agree that 379.7 is approximately 400. The next approximant 65/41 on the other hand simply require too many keys on our piano.

Answer 12

I try to explain in my poor english.

You need to satisfy two conditions to obtain what we call a "major scale".

1) FIRST CONDITION: HARMONIC CONNECTION

The strongest consonance of two different notes is made by a "fifth", for example the distance between C and G (C D E F G are five notes apart).

You can create a "cicle of fifths", a chain of notes where every note is distant a fifth. But let me start with Gb, just for this example:

Gb Db Ab Eb Bb F C G D A E B

As you can see, the notes of the C major scale, are all together on the right. So they are connected in a strong way.

2) SECOND CONDITION: DISTANCE

We can represents the octave as a dodecaghon where each side is a semitone, a different note.

Now try to put seven points on the vertex of a dodecaghon at the maximum distance possible. You will get the same configuration of a major scale: W W H W W W H (as your wife told you).

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So, the reason why the major scale (and all their derivates) has seven notes is because it is:

"THE SCALE MADE OF A CERTAIN NUMBER OF NOTES WHICH ARE ALL CONNECTED BY INTERVALS OF FIFTHS AND ARE EQUALLY DISTRIBUTED OVER AN OCTAVE"

In the same way you will also get the pentatonic scale, more diffused than the major scale.

Answer 13

I think 'arbitrary' is the right answer. I suspect that pleasing tones and intervals existed long before scales, keys, and other theories existed. And there's something fundamental in the human organism that allows us to enjoy music. Look at how many great (not just good) musicians don't read music. Then some ridiculously complex theory was created to fit the reality. Here's something to consider: suppose that the treble clef staff and base clef staff in piano music were connected by 2 notes - middle C and 'middle A". Then the notes in both staffs would have the same names - the bass clef staff would be read as e,f,g,a,b,c,d,f, same as treble clef. This would cut the complexity in half. Good luck getting that changed.

Answer 14

Piano keys have to be the same width otherwise piano is not playable. It is to do with the way our muscles learn to go over the keys. Having some keys wider than others to accommodate for black keys everywhere would make it impossible to play piano. We hit piano keys with different fingers at different times, it is nothing like typing on a computer keyboard. The muscle memory would dictate to hit keys in a specific way, but when a key is wider, all that would not work any more, as one would have to adjust to different width at different times... sort of like having your steering wheel on your car steer at different rate randomly depending on what lane of what highway you are in.

Current system of 2 and 3 black keys works wonderfully well - it helps us see everything at once.

And current system is actually very simple - if you think about it, there are only 12 notes to learn: 5 black keys and 7 white ones. Then it is all repeated all over again. Now, as to the way this is written in the staff, that's a bit more complex, but that's a whole different discussion, and to be frank, I too have some issues with it... (don't let my piano performer wife see this :) )