What are the first 12 harmonic series of a fundamental (Lets say A (440hz)?
You probably mean (as other answers indicate) the first 12 harmonics of A. I list the frequencies and the pitch name according to the western music scale in scientific notation.
- 1st: A4 (440Hz)
- 2nd: A5 (880Hz)
- 3rd: E6 (1320Hz)
- 4th: A6 (1760Hz)
- 5th: C#7 (2200Hz)
- 6th: E7 (2640Hz)
- 7th: (none) (3080Hz), this is between F#7 and G7
- 8th: A7 (3520Hz)
- 9th: B7 (3960Hz)
- 10th: C#8 (4400Hz)
- 11th: (none) (4840Hz), this is between D8 and D#8
- 12th: E8 (5280Hz)
Do those harmonic series of the fundamental pertain to every note in an equal temperament scale?
You can calculate the harmonic series of every fundamental frequency. But you do not obtain all 12 pitches of the equal temperament scale from the series. In fact, you do not obtain any pitch from the equal temperament except for the different octaves of A exactly, because the ratio between different pitches is irrational (as it is powers of the twelth root of 2), but the harmonic series produces rational ratios. The harmonics listed above coincide with just tuning based on A. The frequencies of the named pitches are different in equal temperament tuning, but they are quite close.
Such as if we use A as the fundamental, do the notes B,C,D,E,F,G exist in the harmonic series of A?
As you see, the harmonic series only has A, B, C# and E (and some frequencies that sound out-of-tune for the western ear). You can obtain further frequencies that can sound in-tune with A by dividing instead of multiplying by integers. In that case you are not looking for frequencies that are in the harmonic series of A, but frequencies that contain A in their own harmonic series:
- The 3rd harmonic of D3 is A4
- The 5th harmonic of F2 is A4
- The 9th harmonic of G1 is A4
Still, this does not provide all pitches from the A major scale (F# and G# missing) or from the A minor scale (C natural missing).
and last question, could you explain to me in a simplified manner, how we take a fundamentals harmonic series and turn them into intervals? How does that process work exactly?
Intervals are frequency ratios. In all western scales, we treat pitches with a ratio of 2 (or a power of two) as "nearly identical", and call a ratio of two an "octave". The exact value of the other intervals depends on the tuning you use. If you base the tuning on harmonics, the intervals will be simple fractions, whereas if you use equal temperament, the intervals are powers of the twelth root of two. The names from the intervals are counting how many steps in a typical western scale they cover, so you first need to settle down on a scale before you can understand the names of the intervals. Yet you can identify frequency ratios you expect to appear in the scale. As western scales contain 7 pitches (within each octave), and interval names are counting the number of steps they take in the scale, with one indicating the first note, i.e. no step, the first interval, with a ratio of 2, is called the octave. If you continue to look into the series, you discover the following ratios that are of use:
- Ratio 3: A4 to E6. If you step down 1 octave, it's 3/2 for A4 to E5. The (perfect) fifth.
- Ratio 5: A4 to C#7. If you step down 2 octaves, it's 5/4 for A4 to C#5. The major third.
- Ratio 9: A4 to B7. If you step down 3 octaves, it's 9/8 for A4 to B4. The major second.
You can step down another octave (i.e. divide by 2), so the intervals are no longer going up, but going down:
- The perfect fifth upwards, A4 to E5 gets the perfect fourth downwards, A4 to E4 with a ratio of 3/4, so the pure fourth upwards from E4 to A4 has a ratio of 4/3.
- The major third upwards, A4 to C#5 gets the minor sixth downwards, A4 to C#4 with a ratio of 5/8, so the minor sixth upwards has a ratio of 8/5.
- The major second upwards, A4 to B4 gets the minor seventh downwards, A4 to B3, with a ratio of 9/16, so the major seventh upwards has a ratio of 16/9.
Note that I did not include any interval based on the 7th harmonic. This ommision is not inherent to how the harmonic series works, but this interval just isn't used in western music and does not appear (even approximately) in the equal temperament scale. This does not mean there are no other scales that do include that interval.
In the end, you need to be aware the just intonation you obtain from the harmonic series and the equal-temperament scale are different. In just intonation, not every major second is equal. We defined that A4-to-B4 is 9/8, and A4-to-C#5 is 5/4. This makes B4-to-C#5 to be 5/4 divided by 9/8, which is 10/9. We call both A4-to-B4 and B4-to-C#5 a "major second", but they turn out a different ratios! Equal temperament is designed as an approximation of the just intonation (which actually sounds more pure in the key it is base on) where the ratio of neighbouring pitches is always exactly the same, so in equal temperament, A4-to-B4 and B4-to-C#5 in fact has the same ratio, which is neither 9/8 nor 10/9, but something inbetween.