Why are there only N scales and 12 notes values that can be root note

Question

I just started reading music theory book with no previous musical background. I understand that a scale is defined as a set of musical frequencies. for example major scale with root note C

# Ratios between root and scale tones in C Major
C:C = 1:1
C:D = 9:8
C:E = 5:4
C:F = 4:3
C:G = 3:2
C:A = 5:3
C:B = 15:8

you can shift root note from C to any other frequency and construct the rest of frequencies in major scale using the formula above.

My question is why are there only handful of scales ? Why didn't people come up with many many more.

Second question is why do I have to choose between multiples of 12 standard frequency keys defined on the piano. Why can't I start at whatever frequency I feel like as the root note and construct the scale from that.

Answer 1

What is a scale, exactly? Let's steal a definition from Wikipedia:

a scale is any set of musical notes ordered by fundamental frequency or pitch

So, under that definition, there are practically infinite scales. But we only tend to give names to scales that are used often (note: often is a relative term). I can invent a new scale, and give it a name, but it's probably not going to be that useful.

So, to answer your first question, people do come up with loads of scales, but we only tend to name the ones that are useful.

As to your second question, of course you can start on any frequency, and construct a scale out of as many intervals as you feel like. In fact, it's been done many times (e.g. 19-TET). However, it's probably not going to sound that good to your average Western ears (music is deeply cultural). Many people have loads of fun experimenting with these things, so if you're interested, go for it. There are plenty of questions around this site on alternate musical systems.

Answer 2

The modern major/minor scale is a compromise. Maybe it is better understood if put as the following. For brevity, denote K as \sqrt[12]{2}=1.05946, the twelfth root of 2 (sorry but no LaTeX support here). Verbally, to divide an octave as 12 equal-sized intervals. And use scientific notation, that is the middlemost octave is C4 to B4, with C5 to B5 the higher, and C3 to B3 the lower, and so on.

Than, starting with C2, denote its frequency to be f_0. The harmonic series, those human most easily perceive, with a particular note (here C2) given. We see:

C3 =2f_0
G3 =3f_0
C4 =4f_0
E4 =5f_0
G4 =6f_0
B♭4 =7f_0
C5 =8f_0
D5 =9f_0

And so on. The introduction of constant K seems to be a good approximation of

K^7 ~ 3/2
K^4 ~ 5/4
K^3 ~ 7/6
K^2 ~ 8/7

And so on. The rest story is that the major/minor scale is gradually accepted within European music community. You may refer the relevant history by looking up in any serviceable music history book, the Renaissance section.

Why aren't there other scales around the world? Because there are! When you listen to a traditional Japanese folksong, or Chinese folksong, they each use a scale different from the major/minor scale. The exoticness gives us the flavor of Chinese-ness or Japanese-ness, but while the difference is true, the relation is artificial. Indeed, a scale is not intrinsic Western European, or Japanese, or Chinese, but because people there use it, and we are informed so. In the end, the classical conditioning results, I suppose.

The view that a modern piano always use K as the freq. for adjacent keys --- the so-called equal temperament --- seems not to be true. A tuning is often a compromise between equal temperament and just temperament (a strict adherence of the proportions you said and I repeated above). Different tuners apply different methods too. You has urged me to check relevant source or ask a question myself. Correction welcomed.

Answer 3

Musicians have come up with many more scales than those listed on the page but many of them are not in common use. You can calculate all possible sets of scales of length N (where N is between 1 and 12) using the power set. The final number is 2048 scales. Obviously this will include 2 and 3 note scales which would be interpreted as intervals and triads respectively. Around the world, 4, 5, 6, 7, and 8 tone scales are the most common. Of these, you will see 5 and 7 note scales most often. The others exist but they are rarer.

Western music theory for functional harmony has become dominant in many parts of the world and typically uses the major and minor scales only (with a few accessory scales for specific chord functions). Modern 20th century music that does not use functional harmony based music theory is where you are most likely to find other kinds of scales in common use (for example, modal jazz). However, around the world you are more likely to find cultures that used many more scales historically. The reason for this is that Western theory basis its principles of harmony on the construction of chords which restricts the number of scales that can accompany them and harmonize in a pleasing way. Many cultures around the world expanded in the melodic direction and have a simpler sense of chord structure (for example, drone notes or intervals rather than triads). Thus, there are many more scale combinations that can harmonize without creating clashing notes.

As for your question about frequencies, prior to the invention of the tuning fork, cities throughout Europe played their music tuning to a variety of different root frequencies. This caused difficulties if musicians from different areas were to play together, especially if an instrument was not easy to retune. Eventually standard were developed to come up with a standard frequency to tune to.

Outside of this, you can also divide the octave into many other intervals besides 12 equal divisions of the octave (Equal Temperament). The reason this is standard in Western harmony (though not around the world) is a pretty complicated topic. Here is a link to a great website on the topic. Wikipedia also has a lot of useful information and links for further reading. The basic, simplified idea is that the construction of scales that can be shifted to different keys but still follow the frequencies of the overtone series turns out to be rather difficult task. Many alternative divisions of the octave were used over the last 2500 years to find a solution to this issue and as Western theory moved toward more complex harmony when thirds were considered consonant, this became even more complicated. Our modern music system is the result of many years of trying different tuning systems that could accommodate both complex harmony and also easily switch between keys without having to retune your instrument. The sacrifice is that the intervals we have chosen are "out of tune" to a degree relative to the overtone series. Since we have adopted this music system (12 tone equal temperament) the other alternative systems fell out of common use.

Answer 4

I understand that a scale is defined as a set of musical frequencies.

That's a slightly restrictive definition. Some 'musical scales' might work in terms of ranges of frequencies through which notes can bend and inflect. The blues scale is one of the more obvious examples.

My question is why are there only handful of scales ? why didn't people come up with many many more.

On the page you linked to, there are more than 60 - I'm not sure how easy a scale is to pick up, but that seems like more than a 'handful' to me!

Scales tend to be constructed so that at least some notes are concordant with the root - sounding 'clean' when played together. With most instruments, these intervals are the ones with simple frequency ratios. If you look up the ratios, you'll see why many scales have the notes of the ratios 4:3 and 3:2 of the root note. These correspond to the fourth and fifth in the diatonic scale. It's then a question of filling in the gaps according to the musical 'feel' you want, and the harmonic possibilities you want to create.

One thing that's great about the diatonic scale and its variants is that most notes have a strong relationship with some of the other notes in the scale. This characteristic enables different chords to be played. The 12-tone equal temperament scale expands on this strength.

Second question is why do I have to choose between multiples of 12 standard frequency keys defined on the piano.

If you're playing the piano, it's 'cos it's hard to re-tune the piano. If you're playing an instrument that's easily re-tuned, you can start wherever you like.