Pythagorean Scale Part 1

Before I get into discussion of the Pythagorean Scale, there are some basics that I want to review.
First off, basic modern music notation. We’ll start with the basic C Major Scale: C,D,E,F,G,A,B,C. If we look at this scale on the piano, we notice that there are black notes in between C and D, D and E, F and G, G and A, A and B, but not E and F or B and C. These black notes are known as Sharps or Flats, depending on context. The smallest space between two notes on the piano is called a Half Step (E to F, B to C, F to F#, C# to D). If we put two Half Steps together, we get a Whole Step (C to D, D to E, F to G, B to C#, Bb to C, E to F#, Eb to F). The C Major scale is made up of a pattern of Whole Steps and Half Steps, specifically: Whole, Whole, Half, Whole, Whole, Whole, Half. We can also create a Major scale by starting on any note and applying this formula. To review: C (Whole) D (Whole) E (Half) F (Whole) G (Whole) A (Whole) B (Half) C. Now I am going to discuss intervals. An interval is measuring the distance between two notes. There are two elements to an interval, quality and number. The number refers to the distance between letter names of the notes, the quality refers to the alteration of the notes from its original appearance in the Major scale. For example, C to D is a 2nd, because if we start on C and count C as 1, D would be 2. C to E is a 3rd, C to F is a 4th, C to G is a 5th, C to A is a 6th, C to B is a 7th, C to C is called an Octave (8). Also, if we started on C and played the same C again, it would be called a Unison. As far as quality, there are five basics: Perfect, Major, minor, diminished and Augmented. In a Major scale, Unisons, 4ths, 5ths and Octaves start out as Perfect. 2nds, 3rds, 6ths and 7ths start out as Major. If you make a Major interval smaller, it becomes minor. If you make a Perfect interval smaller, it becomes diminished. If you make a Perfect interval bigger, it becomes Augmented. So in the C Major scale: C to C is a Perfect Unison (PU), C to D is a Major 2nd (M2), C to E is a Major 3rd (M3), C to F is a Perfect 4th (P4), C to G is a Perfect 5th (P5), C to A is a Major 6th (M6), C to B is a Major 7th (M7), C to C is a Perfect Octave (P8). Also, C to Db is a minor 2nd (m2), C to Eb is a minor 3rd (m3), C to F# is an Augmented 4th (A4), C to Gb is a diminished 5th (d5), C to Ab is a minor 6th (m6), C to Bb is a minor 7th (m7). I will refer to these notes and intervals often.
Next is the Overtone series. The Overtone Series is a series of notes created naturally whenever a note is sounded. Usually these notes, referred to as “Harmonics”, are hard to hear, but there are ways to bring them out. They are based on specific mathematic ratios in relation to the original, or ‘fundamental’ note. Each overtone gets higher and higher, and as they go up, the space in between them gets smaller and smaller. If we were to start on C, the overtone series would be (1)C, (2)C, (3)G, (4)C, (5)E, (6)G, (7)B flat (slightly flat from the one on the piano, or should I say, the one on the piano is slightly sharp), (8)C, (9)D, (10)E, (11)F sharp (slightly flat from the one on the piano), (12)G, (13)a note in between A flat and A, (14)B flat (slightly flat from the one on the piano), (15)B, (16)C. So the intervals in between the overtones are 1C (PO) 2C (P5) 3G (P4) 4C (M3) 5E (m3) 6G (smaller m3) 7flat Bb (bigger M2) 8C (M2) 9D (M2) 10E (smaller M2) 11flat F# (bigger m2) 12G (bigger m2) 13between Ab and A (bigger m2) 14flat Bb (bigger m2) 15B (m2) 16C. (Notice how numbers of the binary sequence (1,2,4,8,16) are all C’s. Binaries starting on 3 (3,6,12) are G’s, 5’s (5,10) are E’s, etc. This is where ratios of intervals come from. A Perfect Fifth, or C up to G is 3/2, because it’s the 3rd partial over the 2nd partial, or the frequency of G is three times, half the frequency of C.