## Music is Math

Many years ago, I developed an urge to make music that is in harmony with earth, other planets, or even the whole universe. I started my journey on the web, where I discovered 432 Hz, the solfeggio 528 Hz thing, various websites with frequencies for various planets and loads of other conflicting information. It soon became obvious there was no way to discover the actual "frequency of the universe" by reading all of these web pages, so I started to read about binaural beats and isochronic tones instead. This was more scientific, had been proven in real labs, and actually made sense. While reading about this I learned that your brain mirrors all sounds that it hears, and that it prefers sounds that follow the harmonic series as closely as possible. To put it simply, sounds that are in good vibrational harmony with each other are more pleasant, while sounds that do not connect to each other well are not so great. So I decided that if I could not find the true frequency of the universe, I would simply try to tune all aspects of my music to each other. Then everything would work together creating more unified brainwaves, and therefore more pleasant music.

This seemed simple enough until I tried it in my DAW (Cubase)... When I did I realized that all software has a decimal limit, for example you can enter 120.123 BPM into cubase, but not 120.12345. Other software like the Hz based plugin that I use to make binaural beats can only have 2 decimals, so a brainwave frequency of 1.12 Hz would work while 1.123 was too long.

So, what I needed was a scale with some very specific properties.

1: It needed to have 12 keys that are in good harmony with each other so that I could make a nice sounding album with it.

2: These 12 keys needed to have less than 4 decimals as Hz frequencies so that I could enter them into my scale making software.

3: They also needed low decimals in their very low octaves so that I could enter them into my binaural plugins.

4: And they needed 12 harmonic BPMS (Hz x 60 = bpm) that have less than 3 decimals for entering into Cubase.

It took some time, but I eventually designed such a system and even wrote a book about it. I would like to say that I invented all of this, but after further study, it seems like the mathematics behind it is really as old as time itself...

Here is a break down of how it all works:

Start with the first 15 harmonics in the harmonic series: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15. If you remove octaves to get only unique musical notes, you will have: 1, 3, 5, 7, 9, 11, 13 and 15.

Higher octaves of 7, 11 and 13 do not share harmonics with higher octaves of 1, 3, 5, 9 and 15, higher octaves of 1, 3, 5, 9 and 15 however do share many harmonics with each other (see image at end of this text).

This is why higher octaves of 7, 11 and 13 will not sound good with higher octaves of 1, 3, 5, 9 and 15, while higher octaves of 1, 3, 5, 9 and 15 will sound good together. It is also why intervals with ratios that contain 7, 11 and 13 don't sound very good, while intervals that contain 1, 2, 3, 4, 5, 6 etc in theirs do.

So it seems wise to throw 7, 11 and 13 away and to keep 1, 3, 5, 9 and 15. Higher octaves of these would actually make a perfect just intonation G major scale, only A and F# are missing... If you add 27 as A and 45 as F# however, then you will have a perfect G major scale. This is actually an ancient scale called "Ptolemy's intense diatonic scale", and not some random invention of mine. Although I don't know of anybody who has looked at its very low Hz frequencies from this angle before.

The following image shows this scale with higher octaves of 1, 3, 5, 7, 9, 11, 13, 15, 27 and 45 as reference pitches. Higher octaves of 1, 3, 5, 9, 15, 27 and 45 have been color coded to show shared notes / harmonics, while octaves of 7, 11 and 13 have not. This is because 7, 11 and 13 share nothing with the other numbers or each other, and so there is not really anything to color code.

As you can see the scale in G has all of its blocks colored. This means that it contains only octaves of 1, 3, 5, 9, 15, 27 and 45 which include 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 27 and 45. If you look at the ratios on the left, you will see that they contain only 1, 2, 3, 4, 5, 8, 9 and 15. So we have a very nice mirroring of the same simple numbers between the ratios and Hz frequencies.

This seemed simple enough until I tried it in my DAW (Cubase)... When I did I realized that all software has a decimal limit, for example you can enter 120.123 BPM into cubase, but not 120.12345. Other software like the Hz based plugin that I use to make binaural beats can only have 2 decimals, so a brainwave frequency of 1.12 Hz would work while 1.123 was too long.

So, what I needed was a scale with some very specific properties.

1: It needed to have 12 keys that are in good harmony with each other so that I could make a nice sounding album with it.

2: These 12 keys needed to have less than 4 decimals as Hz frequencies so that I could enter them into my scale making software.

3: They also needed low decimals in their very low octaves so that I could enter them into my binaural plugins.

4: And they needed 12 harmonic BPMS (Hz x 60 = bpm) that have less than 3 decimals for entering into Cubase.

It took some time, but I eventually designed such a system and even wrote a book about it. I would like to say that I invented all of this, but after further study, it seems like the mathematics behind it is really as old as time itself...

Here is a break down of how it all works:

Start with the first 15 harmonics in the harmonic series: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15. If you remove octaves to get only unique musical notes, you will have: 1, 3, 5, 7, 9, 11, 13 and 15.

Higher octaves of 7, 11 and 13 do not share harmonics with higher octaves of 1, 3, 5, 9 and 15, higher octaves of 1, 3, 5, 9 and 15 however do share many harmonics with each other (see image at end of this text).

This is why higher octaves of 7, 11 and 13 will not sound good with higher octaves of 1, 3, 5, 9 and 15, while higher octaves of 1, 3, 5, 9 and 15 will sound good together. It is also why intervals with ratios that contain 7, 11 and 13 don't sound very good, while intervals that contain 1, 2, 3, 4, 5, 6 etc in theirs do.

So it seems wise to throw 7, 11 and 13 away and to keep 1, 3, 5, 9 and 15. Higher octaves of these would actually make a perfect just intonation G major scale, only A and F# are missing... If you add 27 as A and 45 as F# however, then you will have a perfect G major scale. This is actually an ancient scale called "Ptolemy's intense diatonic scale", and not some random invention of mine. Although I don't know of anybody who has looked at its very low Hz frequencies from this angle before.

The following image shows this scale with higher octaves of 1, 3, 5, 7, 9, 11, 13, 15, 27 and 45 as reference pitches. Higher octaves of 1, 3, 5, 9, 15, 27 and 45 have been color coded to show shared notes / harmonics, while octaves of 7, 11 and 13 have not. This is because 7, 11 and 13 share nothing with the other numbers or each other, and so there is not really anything to color code.

As you can see the scale in G has all of its blocks colored. This means that it contains only octaves of 1, 3, 5, 9, 15, 27 and 45 which include 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 27 and 45. If you look at the ratios on the left, you will see that they contain only 1, 2, 3, 4, 5, 8, 9 and 15. So we have a very nice mirroring of the same simple numbers between the ratios and Hz frequencies.

If you are wondering if this scale actually sounds good in real life, here is some chill-out music made using it. As you will hear, it does not sound very different to the standard equal temperament scale.

If we want to extend this to make a 12 tone scale, all we need to do is to add a decimal to these 7 numbers. Adding this decimal lowers each frequency by a three octaves and a 4/5 -major third, so these new numbers will be in good harmony with the other 7 magic numbers. (A 4/5 -major third is exactly the same as a 5/4 major third, only the top note is the root instead of the bottom one). For numbers like 320 with a 0 at the end, you just remove the 0.

When you do this 1, 3, 5, 9, 15, 27 and 45 become 0.1, 0.3, 0.5, 0.9, 1.5, 2.7 and 4.5. While their higher octaves, 256, 384, 320, 288, 240, 216 and 360 become 25.6, 38.4, 32, 28.8, 24, 21.6 and 36. If you raise 25.6, 38.4, 32, 28.8, 24, 21.6 and 36 by a few octaves, you get 102.4, 307.2, 256, 230.4, 384, 345.6 and 288. Since 256, 384 and 288 are already in the original set of numbers, we now have 11 unique numbers: 256, 384, 320, 288, 240, 216, 360, 102.4, 307.2, 230.4 and 345.6. With an octave added, these make a perfect 12 tone just intonation scale with G = 192 Hz as reference pitch.

When you do this 1, 3, 5, 9, 15, 27 and 45 become 0.1, 0.3, 0.5, 0.9, 1.5, 2.7 and 4.5. While their higher octaves, 256, 384, 320, 288, 240, 216 and 360 become 25.6, 38.4, 32, 28.8, 24, 21.6 and 36. If you raise 25.6, 38.4, 32, 28.8, 24, 21.6 and 36 by a few octaves, you get 102.4, 307.2, 256, 230.4, 384, 345.6 and 288. Since 256, 384 and 288 are already in the original set of numbers, we now have 11 unique numbers: 256, 384, 320, 288, 240, 216, 360, 102.4, 307.2, 230.4 and 345.6. With an octave added, these make a perfect 12 tone just intonation scale with G = 192 Hz as reference pitch.

As you can see 7, 11 and 13 are nowhere to be found, except for the 7/5 ratio between G and C#. G to C# however has been called the "devils interval" since ancient times, so it makes sense to have a dissonant 7/5 ratio in this position.

In its low octaves the scale contains: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 27 and 45, and also the same numbers with a decimal added: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 0.9, 1.0, 1.2, 1.5, 1.6, 2.7 and 4.5. If you are looking for a simple system with low decimals that also sounds good, you really can't get much better than this. The way the same numbers are mirrored in the ratios, Hz frequencies and bpms is also very nice, as it makes visualizing and remembering things much easier.

Because the standard 12 tone equal temperament scale used in modern music is based on this type of just intonation, the above mathematics is really the true source of modern music.

Because the standard 12 tone equal temperament scale used in modern music is based on this type of just intonation, the above mathematics is really the true source of modern music.