There are potentially endless ways to make “number music,” but here, I’ll explain how to make scales and chords by making frequency relationships with rational numbers. This might be the most initially boring thing about the God Chord, but I think after it is understood and applied, it is one of the most fascinating and infinite musical ideas I’ve explored.
Also, feel free to let me know here if there is anything I am unclear on, or leave out any explanations. Most of this information is stuff I’ve scoured from books and websites, so there are certainly patchy spots in my knowledge.
More Overtones
We can begin with a familiar starting point for musicians: the major scale. As I explained in my previous blog, the major scale that most musicians use is different than a “pure” major scale derived from the overtone series. It is a scale built from a 12-tone to the octave chromatic scale based on an irrational number (two to the x/12 power). We’ll try to make a major scale that is as closely related to the overtone series as possible. This is a loose way of saying that we are trying to make a scale with “just intonation.”
Since we are using the overtone series as our basic material–we really are just working with a bunch of integers. Seeing the overtone series from a mathematical standpoint, that means that we can use any integer number, because, theoretically, the overtone series extends to infinity. The overtone series is just a fundamental tone multiplied by the ratio x/1, where X increases by one for each successive overtone.
The practical limit to the overtone series is human hearing. Since the overtone series is a linear progression, and humans perceive pitch on an exponential scale, the higher you proceed up the overtone series, the “closer” the sounds are in pitch, to the point where they are difficult to distinguish. I made a simple reaktor patch that moves up the overtone series, tone by tone:

I made an audio example with this patch:
click here to hear overtone series audio example:
It shows how the overtone series sounds. the fundamental frequency here is 100 hertz. I move up and down the series stepwise in a few different ways to give a feel for how the series sounds. All I was doing was moving the value output by the “overtone” knob up and down by a value of one for each different pitch.
You can hear how the notes move, in the beginning, by big jumps, but as they move higher, become “closer” together. If you’re just looking at the frequencies, the notes are all the exact same distance apart, as shown by this image, which is the previous sound graphed on a pitch versus frequency graph, where pitch is shown on a linear scale:

Click here to see the full size image:
but in our hearing, they become closer and closer together. It is a basic concept. Think about a piano. Start with “C,” and play each higher “C.” Each “C” note has a frequency which is double the previous frequency. That is how we hear pitch–exponentially. The overtone series is just frequencies stacked: for instance, an overtone series based on the frequency 100 would be: 100, 200, 300, 400, 500, and so on.
Getting into the scale
Because of these properties of the overtone series, the lower overtones are normally used to create scales. By extension–that means that the ratios we are using to define scales and pitches will be based only on smaller numbers. The major scale here is a “5-limit” scale, which means I didn’t use any numbers higher than 5 in these ratios. This is a basic Just intonation major scale, spelled out with ratios. I’m going to use a C-major scale here. I use the N/D (numerator/denominator) notation here. Plenty of other people use a colon, like “N:D” to denote ratios, and it means the same thing.
C: 1/1
D: 9/8
E: 5/4
F: 4/3
G: 3/2
A: 5/3
B: 15/8
It might look like I contradicted myself here–there are plenty of numbers higher than 5 use in those ratios. But there are a few ways you can manipulate these “5-limit” numbers. If we multiply two of these numbers by each other–for instance 3*3 = 9 — we arrive at the 9 that is used in the 9/8 ratio for the note D.
A note: when adding musical ratios, you *multiply* the ratios, not add them. When subtracting ratios from each other, you divide them.
Another way to manipulate these numbers is to multiply or divide them by two. If you multiply or divide the numerator or denominator of a fraction by two, you don’t change its harmonic identity–you only change which octave the note is in. For instance, if I take the ratio 9/8 and divide the denominator by two, I get the ratio 9/4. This simply places the ratio an octave higher. Multiplying the numerator by two would produce the exact same pitch, but it would be written as 19/8.
When working with ratios, we generally place them within the same octave as the fundamental tone, so they are most useful for scale and chord construction. For instance, 6/1 is the exact same note as 3/2–but 2 octaves higher. When making a single octave scale we would bring 6/1 down two octaves and write is as 3/2 to make things consistent.
Using these basic operations, we can get all those notes in the major scale fairly simply. I’ll do them one by one. After going through a few explanations, hopefully it will be pretty apparent how these operations work.
C: 1/1
1/1 is one. If the fundamental frequency is 200, and we multiply it with this ratio, we get the same frequency, of course. We can get the C an octave higher with 2/1, and another octave higher with 4/1, or an octave lower with 1/2, and so on.
G: 3/2
This note is based on the third overtone of the overtone series, which would be written as 3/1. All we have to do is place that note an octave lower by dividing the ratio by two, arriving at 3/2. The third overtone, or the “fifth” is the lowest note that is not just an octave displacement of the fundamental frequency (like the second overtone: 2/1). Some other notes in the major scale can be arrived at by starting with this note, for instance, D, or a “major second:”
D: 9/8
Stack two “Gs,” or fifths, on top of each other and you arrive at this note. 3/2 * 3/2 = 9/4. place it an octave lower: 9/8.
E: 5/4
This is the second lowest overtone to create a “new” note that is not an octave displacement of the fundamental, and so it is the second ‘strongest’ harmony after the fifth. It is based on the fifth overtone: 5/1. Bring 5/1 down two octaves: 5/2, then 5/4–and you arrive at E, or a “major third.” This is the middle note of a major chord. In equal temperament these notes are sharp–about 19 cents sharp if I remember!
F: 4/3
This note is essentially a fifth subtracted from an octave. If you divide 1/1 with 3/2 (G, or a ‘fifth’), you get the ratio 2/3. Multiply that by two to move it up an octave, and you get the ratio 4/3.
A: 5/3
This is made by adding an “E” and an “F,” or a fourth and a major third. The ratios for the two, respectively: 4/3 and 5/4. Multiply them and you arrive at 20/12. Reduce that fraction, and you get 5/3.
B: 15/8
This “major seventh” is an “E” above “G,” or a third above the fifth. To add intervals we multiply, so: 3/2 * 5/4 = 15/8.
Numbers and Notes
We just slapped together the most common scale in the western music vocabulary with numbers. But we don’t have to stick that basic list of ratios. In fact, just about any two numbers under seven, when combined in a ratio, will make a useful musical interval. Somehow, this just works. When beginning to work with higher numbers, then things get strange, and sometimes very interesting. But, just for example, here are a bunch of number combinations, and the intervals they make. I explain them in terms of a scale whose root is C.
6/5: minor third, or E flat
3/5: minor third
4/5: minor sixth, or A flat
7/5: tritone, or G flat
3/4: fifth, or “G”
5/4: third
6/4: fifth
7/4: minor seventh, or “B flat.” This is a ‘harmonic minor seventh’ based on the overtone series, as opposed to a normal minor seventh, which is more like a minor third stacked on a fifth (an E flat stacked on a G).
2/3: fourth
4/3: fourth
5/3: sixth
7/3: an interesting one–a note between D and E flat.
3/2: fifth, or G
4/2: octave