Why do music key signatures work? Is there science behind why music scales sound good only with the correct notes?

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The scale doesn’t sound right with certain notes, the scale IS the notes.

Scales are a sequence of note distances and can be started at any point. However, in order to maintain the note distances when playing starting from some notes, you will need to add sharps (or flats)

Here is a bit of a crash course on scales, please feel free to ask if anything is unclear. Trust me this is much simpler than it looks, just take your time reading.

Scales are a series of note distances, the smallest distance in most modern music is a half step. So from C to C#.
The distance between C and D is a step.

A scale is a combination of distances, for example, C major is C D E F G A B C.

Now, here is an interesting thing, there is a half step between E and F, and a half step between B and C. So it isn’t just full steps here; if you write down only the note distances of this scale you get this:
(For reference, T is a step, and S is a half step; because they are Tonos and Semi-tonos)

T T S T T T S

This is the distance between notes in a C major scale. Now let’s imagine you want to do G major. You take this note distance, but this time the first note will be G, and we will end on G an octave higher. So let’s see what happens. Remember the order for a major scale.

TTSTTTS

G to A is a step, all good.
A to B is a step, all good. We need a half step next
B to C is a half step, we are still in good track.
C to D is a step, nice.
D to E is a step, we need one more.
E to F is a half step. We have a problem here. In order to get the sound of a major scale, we need a step here. So what can we do? We play F#

E to F# is a step, we need a half to end the scale. Luckily, we already are playing F#, which is just F a half step higher, and F# to G is a half step.

This is why you have to mix and match black keys to get scales that aren’t natural, because a scale is a sequence that can be transposed to any key!

G major is easy enough becausd it has just one black note on a piano, which is F# (F sharp).

However, other scales need different combinations to achieve the same sequence of note distances.
Like D major, which requires both F# as well as C#, or F major, which only uses Bb (B flat)

The rest is sound science and harmony, some frequencies fit in well with others, their peaks and valleys fitting in nicely together. For example, let’s say you play A, in standard tuning that is 440hz. An octave higher, A is 880hz.

If you imagine both sound waves together, you should notice that they will fit in very nicely and that they they sync up. Both waves will be at the same step in their cycle every time the longer one resets. This is perfect harmony. There are different degrees of harmony, the next important one is fifths (think C + G) and then 3rds and 6ths.
This is why C major is C-E-G
1st, 3rd, and 5th. Really nice harmony there.

Within a scale, if established by a chord or melody, any notes that exist within the scale should sound nice (though not necessarily at the same time) as they are part of the same harmony. So if you are playing on the key of C major, your base chord will be C major, and any notes in the scale should sound good in the melody as long as you don’t try to play them all at once and respect some basic guidelines.

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