Very basic question about scales which is rather bothering me

[aragonaise]

I have long wondered about this:
why does the Western scale as we know it come in this form:
tone-tone-semitone-tone-tone-tone-semitone-tone

Why does it sound so natural, as if there are some innate fundamental qualities to this form? Or have we conditioned ourselves so much to this Western scale that we have come to accept music as such?

I am aware that not all cultures employ the same scales. There are, for instance, the pentatonic scale, which sounds rather odd (to me), confined to the realms of exotic music. To my ear, only the Western scale is the true scale. But why???

[mad_max2024]

A very interesting question
https://www.greenwych.ca/natbasis.htm
I stumbled across this site a while back that offered some explanation, apparently it has to do with overtones
Should make a fascinating investigation though, maybe I'll try to look into it

[preludium]

Actually it's quite simple. You build a tonal system out of 2 definitions: one interval with a frequency ratio of 1:2 and another with 2:3. These are the most consonant intervals you can find, because they are made of the smallest whole numbers available. I don't want to go into the details about how this system comes up, but the result is that the 1:2 interval is called octave and the 2:3 interval fifth. When you stack fifths starting at, say, A you will come to another A after 7 steps. Then you transpose down all to notes you touched along that way by octaves until they all fit into the same octave as the starting note. There you have your major scale, and the positions of half and whole tones appear automatically. It seems somewhat weird that the intervals within one octave have different sizes, so the holes are filled by extending the system to 12 tones. First came the scales that we use today, then the 12 tone system, otherwise the interval that spans 12 notes wouldn't be called octave.

From that you can get pentatonic scales by dropping one note where halftone steps occur, at least in the western music. So you get the C major pentatonic scale as C - D - E - G - A. I don't know if the have a different way to define it in China, but they could. This leads again to how the tonal system is made in the first place, i.e. how you try to solve the problem that a series of octaves and a series of fifths never meet at the same frequency again after the initial fixed starting point. They are close to one another after 5, 8, 12, 53, ... fifths, and in our system we use the 12, whereas you can use anything else, like 5 to get a pentatonic scale. But this will be a bit different from the pentatonic scales that you get from the 12 tone system. If you want to understand this then take a sheet of paper and a pen and calculate.

In cultures where music is plain melody they won't have a scale that can be explained that way, because they don't need to solve the riddle of how you can play as many intervals as possible at the same time or within a short period of time. Western scales are not more or less "true" than others, but they try to solve a problem that arises from the nature of this music.

[danny elfboy]

A very interesting question
https://www.greenwych.ca/natbasis.htm
I stumbled across this site a while back that offered some explanation, apparently it has to do with overtones
Should make a fascinating investigation though, maybe I'll try to look into it

Indeed. Even better than that article by Fink is this one: The Trio Theory
this will probably explain most of the author doubts

Fink is not the only one to say that the scale (part of it at least) come out from the natural series of the overtones. Victor Grauer says that it's a fact that in all musical system and in all societies with primitive musical system there's always a fifth and an octave, they're univeral even in tribal music played with bone flutes. Messiaen even proved that harmonical sounds in nature (like the sound of birds) follow the same scale/scheme of overtones. Oberving the natural overtones (harmonics superior) we can even find the basis for the chords as we know it.
For example the superior harmonics series tell us that the bass part need to have sounds disposed widely while the higher parts need to have sounds disposed more closely (a common principle of voice leading) they also show us which note to double as in the harmonics series the tonic appears 5 times, the fifth 3 times and the third just 1 time. As such we know that it's more natural and common to double the tonic, sometimes the fifth and never the third.

Another hint from where the scale generates come from the theoristi Dionisi
If we observe the first five superior harmonics (including the generator [in this instance C] ) we see that the sounds really different one from the other are: C - G - E
So the superior harmonics themselves provide the basis for the major chord and it's structure
In other words knowing the structure of the C chord it's easy to apply the same construction to other tonics

Let's then consider that the next virtual omnipresent sound and second most udible superior harmonic is the fourth

If we try to construct major chords on the tonic, fourth and fifth we have

C - E - G
G - B - D
F - A - C

As you can see we obtain all the notes of the major scale
According to many ethnomusicologist the difference between our system and other ethnic system are rather small but clearly they all spring from the same natural structure (again consider that the octave and the fifth are universal)

Other than "Origin of Music" by Robert Fink there's also another great book which uses acoustic phenomena facts to explain how tonal harmony has origin in the natural structure of harmonical sounds and it's therefore not solely culture and developed not by chance: Harmonic Experience by William Mathieu

Another good resource to understand the original of the tonal system and the scale is the work of Heinrich Schenker and it's structural analysis

[keyofc]

Aragon,
Some of those answers were over my head.
It is my opinion that we have been preconditioned.
I have an Indian student that brought over music from India.  Their scale is natural to them, just as ours is to us ; but it's very different.
He doesn't know anything about music theory - it's just natural to him.

Even so - the answers that are over my head on this list - I would still say they are supported by the belief that our system is the best.  Any system can be explained theoritically, I suppose.

[danny_sequel]

Aragon,
Some of those answers were over my head.
It is my opinion that we have been preconditioned.
I have an Indian student that brought over music from India.  Their scale is natural to them, just as ours is to us ; but it's very different.
He doesn't know anything about music theory - it's just natural to him.

Even so - the answers that are over my head on this list - I would still say they are supported by the belief that our system is the best.  Any system can be explained theoritically, I suppose.

But you are forgetting that even the indian scale is based on certain harmonical universal principles. According to ethnomusicologists there's no scale system in the world which is devoid of the ratios that form the octave and the fifth
Clearly the western scale is just a "ramification" of what's possible to build from those principles but they're all bond by some universal principle (the same found in other harmonical natural sounds like the singing of birds)
As Mathieu says "either we believe that for a strange coincidence the whole word came up with identical principles of sounds or either there's something inherent to the nature of sound that led all humans to find those universal foundation"
For many this inherent something is the natural superior harmonics that we may not perceive consciously that become subconscious after repetitive listening of whatever harmonic sound (including the voice)

In other words the difference between the western scale as the other scales is just like the difference between yogurt and cheese. They're products of different cultures but they both developed from an universal starting point: i.e. milk is edible and universal

[diminished2nd]

Actually it's quite simple. You build a tonal system out of 2 definitions: one interval with a frequency ratio of 1:2 and another with 2:3. These are the most consonant intervals you can find, because they are made of the smallest whole numbers available. I don't want to go into the details about how this system comes up, but the result is that the 1:2 interval is called octave and the 2:3 interval fifth. When you stack fifths starting at, say, A you will come to another A after 7 steps. Then you transpose down all to notes you touched along that way by octaves until they all fit into the same octave as the starting note. There you have your major scale, and the positions of half and whole tones appear automatically. It seems somewhat weird that the intervals within one octave have different sizes, so the holes are filled by extending the system to 12 tones. First came the scales that we use today, then the 12 tone system, otherwise the interval that spans 12 notes wouldn't be called octave.

From that you can get pentatonic scales by dropping one note where halftone steps occur, at least in the western music. So you get the C major pentatonic scale as C - D - E - G - A. I don't know if the have a different way to define it in China, but they could. This leads again to how the tonal system is made in the first place, i.e. how you try to solve the problem that a series of octaves and a series of fifths never meet at the same frequency again after the initial fixed starting point. They are close to one another after 5, 8, 12, 53, ... fifths, and in our system we use the 12, whereas you can use anything else, like 5 to get a pentatonic scale. But this will be a bit different from the pentatonic scales that you get from the 12 tone system. If you want to understand this then take a sheet of paper and a pen and calculate.

In cultures where music is plain melody they won't have a scale that can be explained that way, because they don't need to solve the riddle of how you can play as many intervals as possible at the same time or within a short period of time. Western scales are not more or less "true" than others, but they try to solve a problem that arises from the nature of this music.

I was reading this thread, and this post specifically. Just when I thought I was starting to understand, I tried working it out on paper, and the 7th one is always a half step sharp. Like if you start on A, wouldn't it go A, E, B, F#, C#, G# (which are all in the A major scale), D#? It should be a D right? I'm confused 

[timothy42b]

I was reading this thread, and this post specifically. Just when I thought I was starting to understand, I tried working it out on paper, and the 7th one is always a half step sharp. Like if you start on A, wouldn't it go A, E, B, F#, C#, G# (which are all in the A major scale), D#? It should be a D right? I'm confused 

You're not confused, you are correct.  You have spotted the difficulty with Pythagorean temperament, and the reason it is never used.

It does not matter which simple ratios you use to build a scale.  Simple ratios (2:1, 3:2, 4:3, etc.) will always give you a harmonious and usually beatless interval.  However there is no way to combine them into a scale.  As you've seen, you always have a little left over.  So you end up spreading that leftover among the scale intervals if you need a scale.  This spreading, or adjusting, or tempering is called temperament.  There are many different types of temperament and all are compromises.  It is mathematically impossible to solve the essential problem. 

[b0mbtrack]

i agree that it is what you are conditioned to is what you will be used to hearing.  You grow up listening to certian songs in pre-school on TV radios etc... What you listened to as a kid is what you will become familiar with.  The question is how did the scale get created in the first place.  How did they know back then without all that science involved what exact notes go into a specific scale? 

[counterpoint]

I don't think, that the C Major scale is more natural than the C Minor scale. Now, if we are looking at step 6 and 7 of the Minor  scale, the position of these notes may vary by 1 halftone, so we get all sorts of intervals. The minor scale is not "natural" (as given by a higher power) and that seems to prove to me, that the major scale isn't either.

[preludium]

There are valid points mentioned in this thread so far, like that the problem cannot be solved completely, but they don't answer the original question of this thread, which was:

why does the Western scale as we know it come in this form:
tone-tone-semitone-tone-tone-tone-semitone-tone

I want to add to my post above and get into some calculations to make things clear. This post will look ugly, but that's the price. ;-)

I was reading this thread, and this post specifically. Just when I thought I was starting to understand, I tried working it out on paper, and the 7th one is always a half step sharp. Like if you start on A, wouldn't it go A, E, B, F#, C#, G# (which are all in the A major scale), D#? It should be a D right? I'm confused 

Yeah, kind of tricky. You're using the finished 12-tone-system (I'll write it as 12TS from now on), but the pattern actually appears first in the 8-tone-system (8TS), whereas the 12TS is just a refinement in that respect, which preserves the basic patterns from the 8TS. And you want to find out where the pattern comes from, so you have to build a tonal system from scratch to find the answer. Otherwise it would be circular reasoning: if you put it in you have to get it out again, but this doesn't prove anything.

Ok, we're interested in ratios only, so we don't have to stick to the definition of a' := 440 Hz or the like. We choose a base note as 1 Hz. This cannot be heard by humans but you can get any frequency you like by multiplying the figures below with a constant factor - the ratios will remain the same. So we start with intervals of the ratios 1:2. At this point it's not clear what the name of this interval should be, but we will find a reasonable name later:

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...

is the series of these intervals. If the base note is 1Hz, the next is 2Hz and so on. Now we create a similar series with the interval 2:3, starting from the same note:

1, 1.5, 2.25, 3.375, 5.0625, 7.59375, 11.390625, 17.0859375, 25.62890625, 38.44335938, 57.66503906, 86.49755859, 129.7463379, ...

These series will never meet at the same frequency again, but we can look for places where they are close to one another. There is a good approximation after 5 intervals with 7.59 being close to 8, then after 8 steps there is 17.09, which is close to 16. This is the one we'll use. We have to squeeze the intervals to get the 17.09 to 16, so we multiply all numbers in the lower series with a factor of 16/17.0859375. This is a brute force method and not suitable to produce sth. that sounds nice. It's just to reveal the pattern of ratios that is already in there:

1, 1.404663923, 2.106995885, 3.160493827, 4.740740741, 7.111111111, 10.66666667, 16

Now we have to transpose the notes down until they fit into the first interval of the 1:2 series, i.e. divide each of these frequencies by 2 until it is not bigger than 2 anymore:

1, 1.404663923, 1.053497942, 1.580246914, 1.185185185, 1.777777778, 1.333333333, 2

The next thing is to sort them in ascending order to get a scale:

1, 1.053497942, 1.185185185, 1.333333333, 1.404663923, 1.580246914, 1.777777778, 2

The only thing that remains to be done now is to find out how these notes relate to one another. So we calculate the ratio of the 2nd to the 1st, the 3rd to the 2nd etc.:

1.053497942, 1.125, 1.125, 1.053497942, 1.125, 1.125, 1.125

So there are two ratios: a big one and a small one. Since we're operating on a logarithmic scale here, doubling a ratio means to square it. 1.0535 ^ 2 = 1.110, i.e. we're not exacly on the spot, but it's surely reasonable to call the small ratio a halftone and the big one a whole tone. Since we have a system of 8 tones we can call the 1:2 interval "octave" and our 2:3 interval "fifth". Well, I'm cheating a bit here, since 1.58 (6th step) is closer to the inverse of 2:3 than 1.40 (5th step), but from a musical point of view this scale is too far off the track anyway. You need to fiddle around with the intervals a bit more to get better fifths at least in some places. At the end of the day you would switch to a 12TS, but only to fill the gaps and smooth the bumps, not in order to introduce sth. completely new.

The positions of the halftones above correspond to the locrian scale (1-2 and 4-5), but not to the ionian (3-4 and 7-8), which would be major these days. Since the difference between modal keys is only which of the notes you consider the root, the halftones actually are in the right place, you just have to start at the 2nd note of the scale we built and you get your major scale with the correct positions of semitones from this 8TS.

[timothy42b]

That was a lot fo confusing math.

I think it is correct, but may not answer the question.

We have known since the time of the early Greeks that simple ratios like 2:1, 3:2, 4:3, etc. make intervals that sound harmonious to most of us.

We also know that way back then they attempted to build a scale using the fifth, the Pythagorean, and that it didn't work.  So thousands of years ago they must have perceived a need for a scale, and it must have related to our modern Western major scale (though major and minor wouldn't be invented for centuries.) 

I don't know the history of the modern scale beyond that, I guess I should go look it up instead of speculating.

But as you point out, not all cultures use a whole-whole-half type of scale.  The scale used in India has many more notes.  Strange thing, though, the Indians who have perfect pitch do not have it for all 17 pitches, but only for those which match the Western major scale.  Clearly the octave interval is hardwired in the brain for whatever reason, and this suggests some of the scale intervals may be as well.