Derrick Scott Van Heerden - Mathemagical Music Scales, 2013

Mathemagical Music Scales Harmonic Tuning Guide

By Derrick Scott van Heerden

Content copyright © 2013 Derrick Scott van Heerden. All rights reserved. This publication contains material protected under International and Federal Copyright laws and treaties. Any unauthorized reprint or use of this material is prohibited. No part of this book may be reproduced or transmitted in any form without written permission from the author, except in the manner of brief quotations embodied in critical articles and reviews. Please respect the author and the law and do not participate in or encourage piracy of copyrighted materials.

Contents Introduction: ......................................................................................................................................4 Tools needed for these tutorials: .....................................................................................................5 Universal Harmonics:..........................................................................................................................8 Sounds with matching overtones: ....................................................................................................9 Sounds with strange overtones:..................................................................................................... 10 Analyzing the mono-aural beats in various scales: .............................................................................. 13 Harmonic scale: ............................................................................................................................ 14 Dwarf scale:.................................................................................................................................. 16 Just intonation scale:..................................................................................................................... 18 Pythagorean scale:........................................................................................................................ 19 Equal temperament: ..................................................................................................................... 20 Highly composite numbers:............................................................................................................... 22 Ancient mathematics:....................................................................................................................... 25 Making Tuning files in Scala:.............................................................................................................. 31 Harmonic and Dwarf scales: .......................................................................................................... 31 Just intonation scales: ................................................................................................................... 37 Pythagorean scales: ...................................................................................................................... 40 How to use tuning files: .................................................................................................................... 47 Third party VST synthesizers:......................................................................................................... 47 Albino:...................................................................................................................................... 47 Cronox:..................................................................................................................................... 47 Omnisphere: ............................................................................................................................. 47 Kontakt:.................................................................................................................................... 47 Solutions for problems with .tun files: ............................................................................................ 48 Native synthesizers in Logic: .......................................................................................................... 49 Native synthesizers in Cubase:....................................................................................................... 50 Tuning using cent based sliders or dials: ......................................................................................... 51

Introduction: This book has one purpose and that is to enable computer-based music producers to start making micro-tuned music with as little effort and as possible. The first two chapters delve into the ancient world of harmony and mathematics, giving you a good starting point for making music scales with some amazing properties. After that it moves into a set of screen-shot based tutorials that teach you how to make your own tuning files using the free software “Scala” (links for the PC and MAC versions included). These tuning files retune the individual notes of your synthesizers to exact Hz frequencies , and are compatible with a wide variety of popular music production software and synthesizers , including Cubase (native and third party VSTS), Logic (native EFM1, ESM, ES1, ES2, ESE, ESP, EVB3, EVD6, EVOC 20, EVP88, Garage band Instruments, Sculpture, EXS24 and third party VSTS), or any DAW that can run micro-tunable VST synthesizers such as Kontakt, Omnisphere, Alchemy, CRX4, Cronox 1, 2 and 3, Albino, Octopus and quite a few others. There are also some very good hardware synthesizers that can use these tuning files such as the legendary Yamaha DX 7 and the more current Moog Slim Phatty. Also included is an exclusive tuning file pack, containing most of the scales in this book in the correct formats for all of the software and hardware synthesizers mentioned above. Instructions to open it or download it again are at the end of the next chapter, along with my contact details, so if the pack will not open or if the link breaks in future I can guarantee that you will still get these tuning files. Because most of these scales sound best when played in their root key, I have made one file for each scale and then used each note in that scale as a reference pitch to make a whole set of files covering many keys. Because I did all of this in a 5 different formats, to cover many types of synths, I ended up with 200 tuning files (see list below). 4 Pythagorean scales in 432 Hz / C = 256 Hz. (20 tuning files) 1 just intonation scale in 12 keys. (49 tuning files) 1 dwarf scale in 12 keys. (49 tuning files) 1 Harmonic scale (harmonics 4 to 16) in 8 keys. (33 tuning files) 1 Harmonic scale (harmonics 12 to 24) in 12 keys. (49 tuning files) All scales come in 3 .tun formats (for various VST synths), 1 .scl format (for Logic / Moog) and 1 Kontakt script format (for Kontakt)

To really understand how frequency works and have true freedom in your music tuning however, you really should learn to make scales rather than only using presets or scales made by other people. By working with the free software “Scala” and the tutorials in this book, you will learn how to do just that. At the same time you will also gain a deep and personalized understanding of why certain numbers that have been considered to be superior for the last few thousand years still work well for simplifying calculations.

Tools needed for these tutorials: 1: A pen and paper or any software that can make tables and charts like the one below. I use “Microsoft Word” for this.

2: A calculator. I use the free one in Windows which you can find in “Start”- “All programs” “Accessories”. This one is good because you can copy and paste numbers between it and your frequency tables or tuning software, eliminating typing errors. 3: The free scale making software “Scala” available here: http://www.huygensfokker.org/scala/ for PC and MAC (follow installation instructions on the Scala website). 4: Be sure to download the Scala archive from here: http://www.huygensfokker.org/docs/scales.zip , as this contains some of the scales in this book that you will need to load and edit. Unzip this and save it on your computer remembering where you save it, because you will need to browse for it from Scala when you want to load certain scales. 5: Any music workstation that can run VST synthesizers like Cubase, Logic or Ableton. (Logic is a particularly powerful workstation for micro-tuning, as all of its native synthesizers can be tuned from one control panel in the project setup settings). 6: Micro-tunable VST or other synthesizers. I recommend Linplug’s VST synthesizers like Albino, Cronox, CRX4 and Octopus as a starting point, because they are very stable and easy to tune. Below is a list of other software synthesizers that can be micro-tuned using the .tun and .scl files that you can make in Scala. These lists are a bit old and do not include some newer synths.

7: A guitar tuner plugin to check the Hz frequency of your notes. This is the only way to be sure that your micro-tuning actually works; so it is quite important. There are many of these online under the name “guitar tuners”. If you play a note on a synth with one of these on its effect channel, it should tell you the exact Hz frequency of the note playing.

The Hz read out is often small and not easy to see at first, but should be there if it is a decent tuner like these ones: http://tuneit.free.fr/Site/Download.html http://www.gvst.co.uk/gtune.htm

Native Cubase tuner plugin:

Tune it:

If you want my tuning file pack or are having trouble getting any of these things, feel free to contact me using the information below: Direct link to tuning file pack: http://www.mediafire.com/download/pdp0zzxra1l0te7/Mathemagical+Tuning+Files.zip My Facebook group: https://www.facebook.com/groups/345636055517218/ Or use the request form at the bottom of my website: http://mathemagicalmusic.weebly.com/

Universal Harmonics: Before I start with the tuning tutorials, I think I should first explain some things about the harmonic series. All of the scales that you will learn to make in this book have their roots firmly in this series, so if you know how it works then you will understand how all of the scales connect and are really just different expressions of the same basic laws of vibration. Due to the fractal nature of the universe, the best place to learn about this is actually in the structure of single sounds. Most musical tones are really made from a single tone called a “fundamental frequency” and a series of higher tones called harmonics or overtones . The fundamental frequency (first harmonic) and the harmonics above it are all a type of sound called a “sine wave” which is also the only waveform that does not have any harmonic overtones of its own. What this means is that although it may not sound like it at first, if you listen very carefully you will hear that most single musical tones are in fact chords and not really single tones at all. What is even more amazing is that the intervals between these overtones are almost always the same. There are obviously certain musical instruments where the series gets distorted, stretched or compressed, or may even have a totally unique arrangement of harmonics. But generally the overtone series in most musical sounding instruments does seem to follow these same harmonic intervals as closely as possible. We can visualize these intervals using these musical sounds and a spectrum analyzer. The following images are all made using the free software spectrum analyzer “Voxengo Span” available for free here: http://www.voxengo.com/product/span/. All of the sounds in the following images are playing the same note and so have the same fundamental frequency on the far left. Playing a higher or lower note will not change the intervals between the harmonics, but will simply shift the whole harmonic structure to the left or right on the spectrum read out.

Sounds with matching overtones: Digital sine wave: (no harmonics)

Digital saw tooth wave:

Digital square wave: (every second harmonic is missing)

Acoustic string:

Human voice:

Flute:

Piano:

Sounds with strange overtones: Some sounds, like church bells and gamelans, have always been a problem for me to keep in tune with other sounds like flutes or violins. Now I can see that it is because they have “strange” harmonics that are not the same as most of the sounds that I normally use. (Following images) Church Bell:

Gamelan:

Indonesian xylophone:

In the “sounds with matching overtones” images you can clearly see how they all have very similar intervals between their overtones while the volumes of each overtone varies a lot (some overtones have 0 volume). As I mentioned, these intervals can be slightly distorted, stretched or compressed with certain instruments, but generally they do seem to follow the perfect mathematical overtone series as closely as possible in most tunable musical sounds. The intervals between these overtones are very interesting because they not unique to the harmonic series, but are in fact full of the same musical intervals that we use to make music scales. Although many of these intervals have the same names as the ones we use in music scales, they are not exactly the same. The reason for this is that the scale we use today (the equal temperament scale) was originally based on harmonic intervals but was later adjusted slightly, making the notes equally spaced and not harmonically spaced anymore. Hence the name “equal temperament”. These harmonic intervals are much purer and cleaner sounding intervals than their equal temperament counterparts, because their vibrations are all perfect whole-number multiples of the fundamental frequency. So they all interlock and vibrate together perfectly. When these intervals are played using sounds that have the same intervals in there harmonics , the resulting music will sound very good with all the vibrations mirroring each other like a fractal.

This is not a new idea but is really how choirs or non-fretted stringed instrument players tend to tune themselves when they follow their own harmonics by ear. It is probably where music scales actually came from in the first place. Because this series is so important in music, I will explain each part of it in as much detail as I can, starting with the next chart. This chart contains the first 7 harmonics shown in the “sounds with matching harmonics” images above. In the left column you can see how each overtone is the fundamental multiplied by 1, 2, 3, 4, 5, 6 and 7. If you look at the blue part of the column on the right of this chart, you will see that these harmonics contain a perfect major chord and a harmonic seventh to make a harmonic major seventh chord.

This harmonic major seventh chord is not just in the beginning of the harmonic series , but repeats over octaves though the entire series. Each octave of the fundamental frequency in the harmonic series starts with the first note in this chord and actually has the same intervals as the octave below, only with an extra harmonic in between each one. The chart on the next page expands on the previous one, showing you how all of the intervals used in our modern day music can be found in their harmonic forms when the series is divided into these octaves. These very nice sounding octave portions of the harmonic series are called “harmonic scales” and are easy to generate in Scala (tutorials later in this book).

The 8 tones in the third octave actually sound very much like, but not exactly the same as, the well-known do-re-mi-fa-sol-la-ti major scale, while the fourth octave has all the intervals to make a harmonic version of the 12 tone equal temperament scale (with some extra notes). If you know the names of the 12 standard western musical intervals , you will be able recognize their harmonic versions in these octaves. The next octave above the fourth one contains the same scale but with 32 tones, and the next has 64, then the next has 128 and so on, but I have not included them here.

Because harmonics are all perfect multiples of the fundamental frequency, you will find that they have a very special sound when used in music. Playing glissando’s or chords with these intervals will create perfectly timed, stable and beautiful vibrations in the background when notes decays overlap or when they play chords together. This sound is known as a “beat” in tuning circles, and if you play the harmonic scales above you will find that these beats sound much better and unified than the ones in most other music scales. If you read Mathemagical Music Production (book 1) you will know that the beats between frequencies can affect your brainwaves and are called “mono-aural beats” in that field. So if you are using fractal logic to make your music more entraining, this is another aspect that can add more symmetry to your fractal.

Analyzing the mono-aural beats in various scales: When I recently took a closer look at these beats, I realized that music scales can actually be graded according to vibrational harmony using them as a guide. By looking at the beats between the notes in a scale, we are really looking at the way it vibrates internally which is pretty important because these vibrations tend to get mirrored in our brainwaves and actually do affect our mood. To find the exact beat frequency between any two frequencies, just subtract the lower one from the higher one and your answer will be the beat frequency between them. In the next part of this book I will analyze some of the scales in this book according to beat harmony, and make charts with all of the results. I have used a different reference pitch for each scale so that you can see how the beat frequencies vary when you do this, and how the same scale with different reference pitches, or in different octaves, will create different brainwave states. If you want to know what brainwave state a certain beat frequency will make, just check to see into which state it falls in the brainwave chart on the next page.

Harmonic scale: There is obviously nothing more harmonic than the harmonic series itself, so I will begin harmonic scales as they are octave portions directly from it. The following chart shows the same first four octaves of the harmonic series again, but with Hz frequencies for each harmonic. I did not add the beats between each successive harmonic in this chart as I did with the ones after it, because with the harmonic series these beats are all the same as the fundamental frequency. So, because this chart contains the harmonic series for 12 Hz, there will be a 12 Hz beat between each successive harmonic from the start to the end of this chart. In the “beat between each harmonic and the first note in each octave” column, you can see the beat frequency between each harmonic and the first frequency in that octave. As you can see, they mirror the actual frequencies of the series itself perfectly in each octave. The beat between the first and second note in any octave is always 12 Hz. The beat between the first and third note is always 24 Hz, and the beat between the first and fourth note is always 36 Hz… and so on, making this a real vibrating sound fractal if ever there was one.

The harmonic scales I have pointed out so far all start on octaves of the fundamental (harmonics 1, 2, 4, 8, 16 and so on). Although these octaves do sound the “best” musically there are other octaves that also make interesting scales such as harmonics 12 to 24, which gives you a 12 tone scale and also harmonics 14 to 30 which has 14 tones.

Dwarf scale: This scale is really the 16th to 32nd harmonics in the chart above (16 tone harmonic scale) with some harmonics removed to make a 12 tone scale. Because these harmonics have been removed, making some double-sized intervals, the beat frequencies between some successive notes are an octave higher than the rest. If you use one octave from the harmonic series and repeat it over octaves , as you do with these scales, the beat frequencies will also be octaves higher or lower when the scale is played in different octaves. This means that a lower octave will give more relaxed brain wave frequencies while a higher octave will give more exiting ones. This fits well with the way our brainwave states are naturally divided into octaves, and is an interesting parameter to consider if you want to make entraining music (Remember that to find the exact beat frequency between any two frequencies, just subtract the lower one from the higher one). Because this scale is based on harmonics 16 to 32 (starting on 16), you can always divide the perfect prime of this scale (288 Hz in this case) by 16 and you will get the fundamental of the harmonic series from which this octave portion originally came (288 / 16 = 18 Hz). In the “beats in relation to perfect prime” column, you can see that because this is an octave of the harmonic series of 18 Hz with some harmonics removed, the beats in relation to perfect prime are also all harmonics of 18 Hz with some harmonics missing. The actual frequencies for each note are in the black rows, while the beats between each of them are in the blue rows. As you can see, the beats between each note are almost all 18 Hz with a few an octave higher (36 Hz) because of the “missing” harmonics. In the “octaves of perfect prime” column you can see which beats, in relation to perfect prime, are also octaves of perfect prime, and so have a particularly powerful vibration. If I am not explaining this properly, it works like this: the beat between 288 Hz and 432 Hz is 144 Hz and 144 Hz is an octave of 288 Hz. (Image on next page).

This scale is a very good option if you want the vibrational perfection of the harmonic series , but in a 12 tone scale. Some of the intervals are double the size that they are in the full series . But this raises frequencies by perfect octaves which does not change things too much from a harmony point of view.

Just intonation scale: As an example of this, I will use my favorite just intonation scale; the same one that I use repeatedly in this and my previous book (Basic JI with 7-limit tri-tone, Robert Rich: Geometry), this time it has a reference pitch of 240 Hz. Most but not all of the beats between its successive notes are octaves or octaves of harmonics of various frequencies in the scale. The “beats in relation to perfect prime” also has many frequencies that are octaves or harmonics of perfect prime or other notes in the scale. While its vibrations are not all as unified as the harmonic series itself, this is still a very good scale containing many intervals that are directly from the harmonic series.

Pythagorean scale: In the Pythagorean scale, there is a mixture of harmonious and not so harmonious beats. But it is hard to tell exactly what is going on with all of them, because the numbers are not all whole. If you took the beat between each fifth however (not done in this chart), you would find the extreme octave harmony that you can see in the one perfect fifth that is in the chart. So, this may be an unfair way to look at this scale because, while the other scales have more harmony in relation to perfect prime or when played in there root key, the Pythagorean scale will have better harmony in many keys because it has all of these perfect fifths throughout the scale.

Equal temperament: And last, I have calculated the beats in the equal temperament scale just so that you can see how they are “all over the place” from all angles, so are likely to create scrambled brainwaves. It seems odd that the vibrations between 12 equally spaced notes increase as you go higher up the scale and are not equal as well. But that is just the way vibration works.

So, the general picture here seems to be that the further away from the harmonic series you deviate, the less perfect the vibrations between your notes will be. This perfection in the vibrations of your music is not something to be overlooked. The effects of these vibrations are quite entraining and can take your music to another level if you want it to have powerful effects on its listeners.

If you have ever used sound to vibrate sand, water or other matter (Cymatics), you will already know that perfect harmonic intervals will make the most stable patterns while the intervals found in the equal temperament scale will make more chaotic ones. So, it stands to reason that these intervals will do something similar when they touch your body or ears. There is actually a piece of software called "Hilbert Scope" which is a part of the program "Piano Tuner" for Mac Os X (by Katsura shareware) which makes amazing geometric patterns which are stable for perfect chords and unstable for “bad” chords. This kind of software is nice because it works in any key whereas, with physical cymatics, the size of your vibrating plate or dish has to match the frequency of your sound in order to work properly. These harmonic laws of vibrational stability do not only apply to sound, but actually apply to most things that cycle or vibrate. This is important to know because almost everything that exists tends to cycle or vibrate on some level, including: sound, light, the atoms and other tiny particles that matter is made of, our brainwaves, the orbits of planets, manmade machines and the alternating electric current on which they often run. That fact that the harmonic series lies at the core of so many vibratory things may explain why it sounds so familiar and healthy to the ear and brain (which also functions using vibration). If you want to get a better picture of how deep this goes, you can do your own research on Google by searching for things like brainwaves, chakras, electronics, atoms, planetary orbits etc. and add the word “harmonics” at the end of your search query (examples below). http://en.wikipedia.org/wiki/Harmonics_(electrical_power) http://en.wikipedia.org/wiki/Mechanical_resonance http://en.wikipedia.org/wiki/Atomic_orbital Now that you know some useful things about the intervals between the notes in various music scales, it is time to think about which frequencies might be best to use as a reference pitch or perfect prime for making music scales. Some people like to use “cosmic” numbers for this, while others like mathematically useful numbers. In the next chapter, you will see that “cosmic” and “useful” are really one and the same, and that there is no need to choose one word over the other.

Highly composite numbers: If you work with a computer, you may have noticed that when you enter Hz frequencies or BPMS in any software, you will have a limit on the amount of decimals or numbers you can enter after a comma. Most music software has a limit of between one and four decimal places. So, while a number like 128 or even 128.28 will be fine to enter, a number like 128,47366747 will not. A good example of how this works can be found in my tuning file pack. In it I took various scales and made many tuning files, using each of the notes in each scale as reference pitches to make 12 tuning files for each 12 tone scale. For this to work I needed all 12 notes in each scale to have fairly whole numbers so that I could enter each one into Scala as a new reference pitch for each of the 12 tuning files. A similar thing to this is found in the harmonic BPM + brainwaves system described in my first book “Mathemagical Music Production”. This system requires a 12 tone scale, but with the same low decimal count in all 12 very low octaves (around 1 to 8 Hz), for entering into a binaural beat plugin and also in the harmonic BPMs which you get by multiplying one of these 12 low octaves by 60. Because the scale I used for this (basic JI with 7-limit tri-tone. Robert Rich: Geometry) repeats over octaves, the reference pitch could be any frequency in the top row (3, 6, 12, 24, 48, 96, 192, 384) and the other octaves will still have the same frequencies.

There are only very few numbers that you can use as a base for something this complex without spawning many decimals. Some of the best of these are known as “highly composite numbers”, and are often used for this exact reason by mathematicians (to simplify equations). I did not know about the name “highly composite numbers” when I wrote Mathemagical Music Production (book 1), but if you did read it you may remember that after lots of experimenting with low decimal just intonation, harmonic BPMS and very low Hz brainwave frequencies, I found 12 Hz to be the best reference pitch for this, with 60 Hz and 360 Hz coming close behind. Well, now I see that they worked so well for this because they are all highly composite numbers. There are an infinite amount of highly composite numbers. The first few, starting from the smallest, are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260… Among these numbers there are some “elite” numbers. They are called “superior” highly composite numbers and they give the best results of all. The first few of these are 2, 6, 12, 60, 120, 360, 2520… http://en.wikipedia.org/wiki/Highly_composite_number http://en.wikipedia.org/wiki/Superior_highly_composite_number When using highly composite numbers as reference pitches in the above mentioned situations, the ones below 840 (1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720) seem to work the best as far as low decimals is concerned. If you are using a scale that repeats over octaves, you can also use higher octaves of the very low numbers to get the same scale. Taking into account that many of them are octaves of each other, you are left with 5 unique reference pitches that represent all of the highly composite numbers below 840 (see following image).

These 5 frequencies are also all harmonically connected to each other. So, most of the time you will find that using one of them as a reference pitch for a scale with harmonic intervals, will include most if not all of the other ones in the same scale, whereas using 840 and the numbers above it will not. I am not saying that highly composite numbers above 840, or even other nonhighly composite numbers will not also work for this. I’m really just outlining a very handy way of working, where everything is simple and connects together very well. If you have read my first book you will already know about the amazing things that you can do with your music production if you use the harmonic BPM system. If you haven’t, here is a link to

a video that I made about some of them: https://www.youtube.com/watch?v=SsvXHpY2hM&list=UUqRs4UJUoa-NtpvcH30TPHQ Another very interesting factor, that I have not looked into properly before, is the connection between your reference pitch and the sample rate of your digital workstation. I am not an expert on this and will quote my friend Scott Dakota here: “If one is doing *close* comparisons, in many cases it actually does matter what the relationship is between the target pitches and the sample rate - and how good or bad the aliasing of a soft-synth or sampler is - assuming soft-synths or re-sampling is being used. Nyquist very much applies - and there are a lot of other factors, this isn't something to oversimplify - and it varies case by case. I know this by a lot of applied experimentation and analysis because I'm really touchy about unintended aliasing grit. Anyone is welcome to actually study this with FFT analyzers set to high resolution, and close listening.” “And so it's said, in digital synthesis it is often true that certain selections of JI that are in simple relationship with the sample rate will outperform in clarity an equal tempered set also with one pitch tuned to the sample rate - unless the digital synthesis being used is of a superb quality and already incorporates a pile of anti-aliasing, oversampling, and smoothing in the code. Is the difference going to be make-or-break for most people doing electronic production? No, not at all. But if it matters to someone, it is a factor to be aware of. Again, I'm not blue sky talking I've examined this closely because it matters to me for certain texture things.” “And add quantization noise, when wavetables and look-up tables and related math crunching is stretched around and decimals are rounded off or truncated - the grit of it sounds similar to aliasing noise. Yes, it does in some circumstances interact with the sampling rate, and pitching things to the sample rate can minimize certain ugly artifacts. This will matter to people that are OCD about audio engineering and sound design details - but won't necessarily be a big deal to people otherwise. Depends on how far you want to take things.” What this means is that if (for example) you use a reference pitch of 48 Hz in combination with a sample rate of 48 KHz, you should then be able to explore the things that Scott is talking about (1 KHz = 1000 Hz so 48 Hz still divides perfectly into 48 KHz). Most modern sound cards have 48 KHz, 96 KHz, and 192 KHz which are all octaves of each other, so any one of them should work quite well with music based on any octave of 48 Hz, 96 Hz, and 192 Hz. Although octaves of these will work the best, the other 4 very useful reference pitches that I mentioned before should also work quite well, because they all have pure harmonic / just intonation relationships to 96 Hz: 120 Hz is its pure major third, 128 Hz is its perfect fourth, 144 Hz is its perfect fifth, 180 Hz is its classic major seventh and 192 Hz is obviously its octave.

As you can see, those 5 tones are actually a perfect little just intonation scale in themselves, so everything here works well together in more ways than those that are immediately obvious.

Ancient mathematics: Highly composite numbers have a very interesting and extremely long history stretching back over thousands of years. Their first historically known use is by the ancient Sumerians. We know this because they left many clay tablets behind that show us exactly how they did their mathematics. These tablets show us that they were aware of, and specifically used, highly composite numbers to make many of their calculations simpler. The ancient Sumerians actually invented many of the units of measure that we still partly use today, such as the sexagesimal system that has 60 as its base, the duodecimal system which has 12 as its base and also the division of a circle into 360 degrees (all superior highly composite numbers). Modern uses include 60 sec = 1 minute / 60 minutes = 1 hour, angles and geometry, map coordinates, units of in dozens and gross, 1 Hz = 60 BPM and so on. Sumerians also had an interesting manuscript called the “kings list” which is now in our hands. I am not sure what this list really is, but to me it makes no sense that it could measure the “length of rulership” of actual kings, because all of the numbers recorded are suspiciously convenient for large calculations, with more repetition and musical ratios than you would expect as simple “lengths of rulership”. The following chart shows one of these “kings lists”. The reigns are measured in Sumerian units known as Sars (units of 3600), Ners (units of 600), and Sosses (units of 60). This list starts with the kingship descending from heaven and ends with the great flood.

I am not a big fan of history and only included this list to observe the conveniently composite and harmonic sort of mathematics used to make it. 28800, 36000 and 43200 for example make a perfect harmonic D major chord with 3600 being a lower octave of its root note (28800). This is so similar to a music scale because multiplying 3600 by 5, 8, 10 or 12 is how you get the very musical harmonics 5, 8, 10 or 12 of 3600 Hz. The Mayans had something similar in their “long count calendar” which they used to measure what they called “universal time”. They believed that the universe was destroyed and recreated at the end / start of the 2880000 day cycle, and that everything on earth was directly affected by the smaller cycles in between.

This is also a very composite and harmonic calendar that seems to use a system quite similar to the Sumerian “kings list”. There are many ways to dissect this musically. For example, if you remove the 0’s then 36, 72, 144 and 288 are all octaves of 36 (a highly composite number). So, because each 0 represents frequency x 10 (tenth harmonic) they will all be harmonics of this octave set when you add the 0’s back. Personally I just look at the numbers as a pattern, and can see right away that they are of a similar system to the kings list. The Hindu religion also has a very similar long count calendar called the “Yugas”. The Yugas make up 4 ages within a larger cycle called a Manvantara, which was made up of 71 times the amount of years in all of the Yugas. These are parts of smaller and larger cycles which are confusing to me, but at some point also marked the destruction / creation of the universe. The Yugas are said to be like seasons that create golden and dark ages that effect mankind, making us more or less conscious, and also increasing and decreasing our life span and physical size. The 4 Yugas also have lengths that are very convenient for these types of large measures: 4 Yugas = 4800 years + 3600 years + 2400 years + 1200 years for a total of 12000 years for one arc, or 24,000 years for one full precession of the equinoxes. This is quite close to most modern estimates for this, which range from 26000, 25920 to 25772 years.

This is worth noting because the precession of the equinoxes is thought to cause ice ages and warm ages which do have an effect on all life on earth.

If these numbers were Hz frequencies then 1200, 2400, 3600 and 4800 would be the perfect harmonic series of 1200. As you know, 12, 24, 36, 48 are 4 sequential highly composite numbers and make great reference pitches for low decimal harmonic or just intonation scales. So, there is a whole lot of mathematical “harmony” going on here, too. Another interpretation from the “Srimad Bhagavatam” says that the Yugas are measured in “god years”, and that one god year is equal to 360 human years. If this is true, you need to multiply god years by the convenient highly composite number of 360 to calculate the human years. When you do this, however, the 4 Yugas no longer fit in with one arc of the precession of equinoxes, but instead would be 360 arcs or 180 full precessions.

360 is a very useful number and can also be divided up in many ways without making decimals, which is why it was chosen by the Sumerians to divide the circle and why we still do this today.

If the angles in the pentagram were Hz frequencies, you would have the first three harmonics of the highly composite number 36 (36 – 72 – 108). The number 108 (36 x 3) seems to have been an especially popular measure for a long time in many places: the outer circle of stones at Stonehenge is about 108 feet in diameter, the Vatican occupies 108 acres of land, the Yogis, Buddhists, Taoists, Hindus and even some Catholics have strings of 108 prayer beads , and in Japan the Buddhist temples sound a bell 108 times at the end of each year to remind them of each of the 108 evils that man must overcome to attain enlightenment. According to modern day measures, the distance from the Sun to the Earth is 108 times the diameter of the Sun, and the distance from the Earth to the moon is 108 times the diameter of the moon. Add that to the information below and I can’t help but scratch my head… (All of these are very closely approximated but not exact numbers) Moon's diameter = 2160 miles / Sun's diameter = 864000 miles. Moons radius 1080 miles / Suns radius = 432000 miles. 43200 seconds = 720 minutes = 12 hours = half rotation of earth. 86400 seconds = 1440 minutes = 24 hours = full rotation of earth. Earth takes 72 years = 864 months to move through one degree of arc in the precession of the equinoxes. Earth takes 2160 years = 25920 months to move through one zodiacal age. Earth takes 25920 years to move through all 12 zodiacal ages, or one precession of equinoxes.

I am sure you have noticed the same mathematical patterns being repeated in all of these things. This is quite strange because the Sumerians, Mayans, Hindus, Stonehenge builders, Catholics etc. were not actually meant to have been in contact with each other. While I can understand that human-made measures might have been created for convenience, the ratios between certain aspects of the Sun, Earth and Moon are a bit harder to explain. I only noticed these repeating numbers because they are exactly the same ones that you get when you make scales with harmonic intervals (the scales in this book), using any highly composite number below 840 as a reference pitch. To demonstrate this I have made a chart using my favorite just intonation scale with a reference pitch of 120 Hz. This chart goes into the higher octaves (not often in my charts) where the bigger numbers are. As you can see, this scale is full of cosmic and mathematically useful numbers like 36, 360, 3600, 72, 108, 144, 1440, 144000, 288, 2880, 216, 2160, 432, 4320 and many more, including those 5 super reference pitches which represent all of the highly composite numbers below 840. (Note: the word cosmic by definition means “relating to the universe or cosmos, especially as distinct from the earth”). 5 super reference pitches:

Using almost any note in a scale with harmonic intervals (like the one above) as a new reference pitch for the same scale, will result in the same frequencies with only a few shifts here and there. A good example of this is the “just intonation with harmonic BPMS” chart in the previous chapter which is the same scale as the one above but with 192 Hz as a reference pitch. That chart goes into the very low octaves, so in it you can also see all of the actual highly composite numbers below 840 (1, 2, 4, 6, 12, 24) that are not included in the chart above. Because both charts mentioned above contain all 5 of those very useful reference pitches, you can be sure that they are a very good place to start when making low decimal or cosmic scales. Some people say that music containing these frequencies is more connected to the universe. Personally I don’t worry about this too much because I already use the highly composite numbers below 840 as a base to work from when I make these scales. So the resulting cosmic / useful numbers are in pretty much all of my music by default. As I said before, I am sure there are many other mathematically useful numbers and scales out there that will work for this kind of thing. The ones I have suggested are just nice because they are all in harmony with each other, and so make a very tidy interlocking system to work from. Using this method should work well for everybody because those who are into cosmic type frequencies will automatically get all of the right ones, while those who are strictly logical and don’t like cosmic stuff will also get all the numbers that they would want for simplified / more accurate music scales and digital conversions. So now that you have some nice information on what types of intervals and numbers have worked well musically and mathematically, for the last few thousand years, you should be able to make some nice scales using the tutorials that fill the rest of this book.

Making Tuning files in Scala: In the next part of this book I will teach you how to make 3 types of scales: 1: Harmonic and Dwarf scales that are made from portions of the harmonic series. 2: Just Intonation scales made from harmonic ratios of a root frequency. 3: Pythagorean or “circle of fifths” scales made from a stack of fifths. These scales have very different playing properties which are good to know about if you want to choose the best scale for the music that you are making: 1: Harmonic and Dwarf scales only sound good in their root key and its relative keys. So, with these scales you need to make new tuning file every time you want to make music in new key. If you want to play a synthesizer along with a monochord, didgeridoo or other fixed frequency instrument with very loud natural overtones, these are very good scales. 2: Just intonation scales generally sound best in their root and relative keys, but can also sound good in other keys depending on the specific scale. There are an infinite amount of possible variations with these scales, so they can be custom made to suit all kinds of music. 3: The Pythagorean scale sounds good in most keys and can be adjusted in various ways to sound good in even more keys. With this scale you only need to make one tuning file for music in any key, making it a better scale for classical music or complex music where the root key changes a lot. So, when you choose a scale, you should first think about the music that you plan on making so that you can select the best type of scale.

Harmonic and Dwarf scales: If you read the first half of this book, you will know that the harmonic series contains many octave portions that can be used to make music scales with amazingly unified vibrations. These are easy to generate in Scala and, because it repeats any octaves that you enter, you only need to make one octave to have Scala generate the rest of them automatically. Here is a tutorial to make these scales: 1: Open Scala. 2: Go to “File” then “New” and select “Harmonic Scale”. Here you can select what portion of the harmonic series you want to use (see first and last harmonic in following image).

You can type “show scale” in the command box at the bottom of Scala, or click “show scale” or “show scale by frequencies” under “view” in the main window to see your scale as ratios or Hz frequencies. If you click on the “Open Play dialog” tab, you can try your scale (just click on the notes to hear them). If you hear nothing when you play it, read step 13 in this tutorial. 3: Select the first and last harmonic for your scale. The default setting is harmonics 4 to 16. This uses two successive octaves of the harmonic series and happens to have 12 notes in it, making the two octaves fit perfectly into one octave on a keyboard, and so make a perfect repeating scale. This scale starts with a major 7th chord and then repeats it again with a new frequency in-between each note. This portion of the harmonic series is actually the “scale” that you most often hear when you listen to overtone singing or monochord music. It has that “happy” type of sound you would expect from a scale that contains two major seventh chords over two octaves. Harmonics 4 to 16

If you select harmonics 16 to 32, you will get that nice 16 tone scale with its harmonic versions of all the intervals in our modern day 12 tone equal temperament scale. Harmonics 16 to 32

If you want harmonics 16 to 32, but with only 12 notes, just open a “Dwarf scale” or a “Hobbit scale” instead of a harmonic scale. To do this go to “File” then “New” and select “Dwarf scale” or “Hobbit scale”. If you leave the settings on default, both of these scales will give you the same 12 selected tones from the above “harmonics 16 to 32” scale. Dwarf Scale / hobbit scale

There are many octave portions of the harmonic series that can be used as nice scales. Another one that I quite like is harmonics 12 to 24, which is a more exotic but still musically useful scale that also has 12 tones. Harmonics 12 to 24

4: Once you have selected your first and last harmonic, or opened a default dwarf / hobbit scale, click “Apply” and “OK”. 5: Go to “View” on the top Scala’s main window and select “show scale” or “show scale by frequencies” to make sure your scale is correct. 6: Go to “Edit” (next to file on top of the main window) and click “preferences” right at the bottom of the “Edit” drop down menu. This will open the “User Options” window. On the left of the User Options window there are tabs; “output”, “General”, “edit dialog” etc. to change the part of the user options menu you want to change. 7: Click the top “Output” tab to set your “base frequency” (reference pitch). Using any octave of any highly composite number below 840, or any other note in this scale when it has one of them as reference pitch, will work the best for this if you want “nice” numbers.

8: Click the “MIDI” tab (in the user options window) and change your “reference frequency” so that it is the same as your fundamental frequency. 9: Below “reference frequency” you can change “reference key” and “key for 1/1”. Here you can choose what note will be the starting note or fundamental on your keyboard.

10: Click “Apply” and “Ok” then close the “User Options” window. 11: Go to “Edit” (main window) and select “Edit scale”. This should open the “Edit Current Scale” window. If you want to make the 16 tone harmonic scale or another long scale into a 12 tone one, you can remove some notes by clicking the “remove” tab at the bottom of the window, or by right clicking it and selecting “remove” in the drop down menu.

12 When you have edited your scale to your taste, hit “Apply” and “Ok” to close the “Edit Current Scale” window. 13: If you want to listen to your scale before saving / exporting, you can use Scala’s built in mouse triggered keyboard. To open the keyboard, click the “Play” tab with the small keyboard on it (circled in following image).

To select a synth for testing your scale, click the “sound settings” tab on the keyboard. You should find at least one synth that works in the “MIDI Output Device” drop down menu.

Close the keyboard when you are happy with your scale and are ready to save / export. 14: Open the “User Options” menu again (“Edit” then “Preferences”) and click on the “Synth” tab. Select your tuning files format; 112 will give you a .tun file. There are other formats here for Kontakt, Reaktor and many other synths, just browse to see what is there. 15: Click “Apply” and “Ok” then close the “User Options” window. 16: To save it as .scl (for Logic, Moog slim Phatty and some other synths), go to “File” and “Save Scale As” and select “.scl”. Remember to choose a destination that you can find again. 17: To export your scale as a .tun or other format that you selected in step 10, go to “File” and “Export Synth Tuning”. Name your file at the top where it says “Name”, choose an export destination and click “Ok”. Your file should now be exported and ready for you to use in your synth or workstation.

Just intonation scales: The way these scales are derived from the harmonic series is all in the ratios. Each ratio tells you how that note relates harmonically to “perfect prime” or the first note in the scale. The ratio for a major third (5/4), for example, means that you must multiply the frequency of your perfect prime by 5 and divide it by 4 to get the frequency for that note. For some reason, small numbered ratios like this usually sound better than ones with big numbers , which is why the goal of much just intonation is to have them as small as possible. Multiplying a frequency by 5 will give you it’s fifth overtone, and dividing it by 4 will give you the fourth undertone of that overtone. So, these ratios can also be looked at as an overtone divided by an undertone. Undertones do not occur in sound itself but do occur in buzzers on marimbas, paper held to a tuning fork and other situations where a vibrating object makes another object vibrate through close but loose contact. (You can make an undertone scale by using the “Swap” button in the settings for harmonic scales in Scala).

If you want to try the just intonation scale that I use so often (“Basic JI with 7-limit tri-tone. Robert Rich: Geometry”), you can find it in the Scala archives as “ji_12.scl”. There are many other just intonation scales in the archives which you can find alphabetically under “JI”.

If you want to make your own just intonation scale, or any scale for that matter, using Hz, ratios or cent data from a chart, you can simply start with an empty scale and enter your notes or intervals one at a time. Here is a tutorial for this: 1: Open Scala and go to “File” – “New” and select “Scale” or hold “Ctrl” and click “I” as a keyboard short cut to do the same thing. This will open a new empty scale that only contains one note (perfect prime). 2: Double click the frequency of your perfect prime and change it to whatever frequency you want your reference pitch to be.

Using the highly composite numbers below 840 Hz will also work best as a base frequency for most just intonation scales, if you want low decimals or “cosmic” numbers throughout your scale. Remember that because we are working with a scale that repeats over octaves, you can use the octaves that I have mentioned a few times already.

The following chart is of the “Basic JI with 7-limit tri-tone. Robert Rich: Geometry” with 240 Hz (octave of 120) as perfect prime. This should come in handy if you are looking for more reference pitches.

3: Now you can simply enter the data for each note in the white bar where I have written “enter your data here” (see previous image of Scala’s “edit current scale window”). To enter a frequency as Hz, you have to add a “z” before your number. So, for 288 Hz you would enter “z288” and hit “enter”. To enter a ratio, just type it in. So, for a major third, just type 5/4 and hit “enter”. To enter a note using the amounts of cents that it is above perfect prime, just enter the numbers like this: 386.3137 (cents above perfect prime) and hit enter. When you hit enter you will see your new note appear below “perfect prime”. Continue like this, adding more notes until you have built your whole scale.

4: When your scale is ready, hit “Apply” and “OK” and click “show scale” under “view” in the main window to make sure your scale is loaded. 4: Go to “Edit” (top of main window) and click “preferences” right at the bottom of the “Edit” drop down menu. This will open the “User Options” window. 5: Click the top “Output” tab and make sure that your “base frequency” is correct. 6: Click the “MIDI” tab (in the user options window) and change your “reference frequency” to match your base frequency. 7: Below “reference frequency”, you can change “reference key” and “key for 1/1” to the correct note for your reference pitch. 8: Click “Apply” and “Ok” then close the “User Options” window. 9: To save it as .scl, you must go “File” and “Save Scale As” and select “.scl” 10: To export your scale as .tun or another format, go to “File” and “Export Synth Tuning”. 11: Select export destination and hit “Ok”. Your file should now be saved or exported and be ready to use in your synth.

Pythagorean scales: The Pythagorean scale can be made by repeating the third harmonic (octave + fifth or frequency x 3) 11 times to reach high F, and then raising or lowering each note by the correct amount of octaves to fit them all into one octave.

In the following chart, the middle row starting with 1 Hz shows the repeating stack of frequency x 3 (third harmonic) to the right, with octaves of these frequencies above and below them.

If you want to try this scale, just find it in the scala archives under “pyth_12.scl” (12-tone Pythagorean scale) and use the harmonic scale tutorial to load it. In the above chart, the yellow blocks are the 12 notes that you would find in one octave of the actual 12 tone Pythagorean scale (before they are re-arranged into the final scale). If you play music with the completed scale, you will find that the notes that originally came from the far right side of the stack of 5ths (D#,A#,F etc) don't sound very nice in relation to the notes that were closer to C on the left side (C,G,D etc). This is because of a crazy thing called the "Pythagorean error" which I will now explain: With this scale, you normally go up to F in the stack of 5ths and then you use an octave of the first C to "close the circle". If you look at the far right of the chart below, however, I have added the last pure 5th after F to bring it back to C again. As you can see, it is not the same as the first C at all. You would think that a stack of frequency x 3 and a stack of octaves would line up quite well, but they do not. They actually slowly drift apart as you go higher with the two stacks and never actually meet up at all.

This drift creates disharmony between some of the notes in the final reduced scale. These bad notes are called “wolves” and can be dealt with in various ways. (I had to trim some decimals to fit all numbers in following chart)

“Fixing” the Pythagorean error: To fix these “wolf” notes, a good trick is to divide the original stack of fifths chart above in half, right in the middle between B and F#. Then, bring F# and all the notes after it closer to B. This pulls the whole right side of the chart a little bit to the left, lowering all of those frequencies by a few Hz. Depending on exactly how you do this, you can bring C closer to or exactly to 512 Hz, making the scale sound much better than the standard Pythagorean scale, with notes like F that did not fit in well before, now sounding much better in relation to some of the other notes. To make your own variation of the Pythagorean scale using this method, you will need to start by making a table like the ones below (or use the finished ones at the end of this chapter).

To start your scale, fill in a 1 for the first C (on the far left), then multiply it by 3 and fill in a 3 next to the 1. Now multiply your answer (3) by 3 and keep repeating 12 times. In the windows calculator, you can just hit enter 12 times after the first 1 x 3 (see following image).

To start “fixing” your scale, you will want to lower one of the frequencies near the middle of the stack, deleting and re-calculating all the numbers to the right of this frequency, trying to get that last C on the far right closer to an octave of 512 Hz (exact octave would be 524288 Hz). Although you will use an octave of the first C and not this C in the final scale, getting this C as close as possible to an octave, the first C helps align the other notes so that it sounds a bit better. My favorite way to do this is to lower the 729 Hz F# down to 720 Hz (a highly composite number) which changes the last C to 512.578125 Hz, which is very close to 512. Changing F# to 720 Hz also makes your D major and A major chords into a pure just intonation major chords. So, from a musical perspective it is easy to see why this version can sound better in those keys than the original version (see following chart).

If you change F# to 719.18792866941015089163237311385 Hz (impossible on a computer), that last C will be exactly 512 Hz. Changing 729 Hz to 724 Hz also sounds very nice and is the method used by Maria Renold to make her amazing scale. How you do this is up to you, but when you have your frequency x 3 stack finished, all you need to do is fill in the rest of the chart, lowering and raising each frequency by octaves to bring them into one octave where they will make a perfect 12 tone scale. To save time on filling in the whole chart, you can just compare it to an octave of one of the Pythagorean scales in this book using a calculator to raise or lower each frequency by octaves, until it is in the right one for that note. If you do that, then you can just fill one frequency for each note into your chart, as I have done below.

After that you just re-arrange the notes to make a 12 tone scale, which is also easy to do by comparing it to an octave of the standard Pythagorean scale. You need to make this chart with one octave of the scale because you have to enter the new frequencies manually when you make it in Scala. I will not include a tutorial for this because you can use the just intonation chapters tutorial on entering each note individually. If you don’t feel like doing all of the above, you can just use the “Hertz” sections in the following charts and enter their numbers into Scala.

Pythagorean scale F#= 720 Hz variation:

Pythagorean scale F# = 719.1879 Hz variation:

Pythagorean scale F# = 724 Hz variation:

How to use tuning files: Third party VST synthesizers: First make a folder somewhere on your PC called ".tun files" to keep the .tun files in. Remember where they are so you can browse and easily find them again from your various VST's scale browsers. Remember that .tun files will over-ride your synthesizer’s master tune, so you should always leave this on 440 Hz.

Albino: Just click on the word 'Albino' on the bottom right of the synth to see the back of the synth. At the bottom right, above the fake stereo out plugs, is a box. Click the load button and browse for the .tun file that you want.

Cronox: Cronox is the same as Albino. Just click on the word 'Cronox' to see the back of the synth, or click “settings” on the top right and browse for your .tun files from there.

Omnisphere: For Omnisphere, you have to copy and paste the .tun files into the Omnisphere program files using this file path: program files -- spectrasonics -- steam -- omnisphere -- settings library -presets -- tuning file. Make a new folder in this tuning file folder and call it "my tuning files" (or something) and then paste your new .tun files into this folder. To load them into the synth, just open Omnisphere and look in the middle of the main front window, a bit to the left. You should see a box called "scale". You will now find your new folder and files there. Omnisphere does not like too many folders inside of other folders, so be careful with this.

Kontakt: As with Omnisphere, go to: program files -- Native Instruments – Kontakt – presets – scripts – tuning, and paste your files there. For Mac, go to: Library – application support -- Native Instruments – Kontakt -- presets -- tuning. All synthesizers will work in a similar way. If you cannot load files directly in the synth, then you need to go to its program files and look for a folder with a name like “tuning”.

Solutions for problems with .tun files: Some people have problems with .tun files that make synths do some crazy things. I had problems with Omnisphere and solved them by opening one of the preset .tun files that come with Omnisphere using WordPad, and then comparing it to one of my own also in WordPad. When I looked at the “code”, I found lots of stuff in my one that was not in the preset one. This is not hard to do. All the junk was at the bottom end of the file (the whole “anamark” and “functional tuning” section), so I just deleted the whole second half and saved the new version (all in WordPad), leaving only the top part and the end part that looks like this:

If you are having problems with any synth not accepting tuning files, I would recommend doing the above and comparing its preset tuning files to your own in a text editor. Sometimes Omnisphere also refuses to play the base frequency, playing all the notes perfectly except for the reference note itself. To solve this problem, I set my base frequency and reference pitch to an octave that is just below hearing range. So, for 432 Hz I would use 27 Hz and for 192 Hz I would use 24 Hz etc. Then I set the MIDI note to the C0 octave so that it is off my keyboard and no longer a problem. With some synthesizers, you will also find that bass notes sound better with a very low reference pitch while very high notes might sound better with a high one. For this reason I normally make 2 .tun files for each scale; one with a base frequency and reference key in the same octave as the default middle C, and another with them in a very low octave. Some synths will not work with text edited files and will only accept files directly from Scala. So, don’t just edit all of your files unless your synth demands it. If you do edit them, I recommend editing a copy and keeping the original file for other synths. Another thing to remember is that if you are using a non-octave based scale, like the full harmonic series or some other situations, you may have problems when using two oscillators on a single synth that are detuned or set to different octaves. That covers all of the problems that I have found. If you find any more, please feel free to contact me via my website http://mathemagicalmusic.weebly.com/ or on my Facebook group Life, the Universe and 432 Hz .

Native synthesizers in Logic: (EFM1, ESM, ES1, ES2, ESE, ESP, EVB3, EVD6, EVOC 20, EVP88, Garage band Instruments, Sculpture, EXS24). Logic has very powerful micro-tuning possibilities exclusively for its native synthesizers. It does, however, accept third party VST synthesizers. So, using .tun files with these that match the .scl files that are tuning Logics synthesizers will work perfectly. To access the settings for native synthesizers, just go to your “project settings” window and open the “Tuning” section. Here you can change your “software instrument scale” from “equal tempered” to “fixed”. When you have done that, you can select a scale from the “type” menu (there are already a few scales from the Scala archives there). If you want to use you own .scl tuning files, you will need to copy them to your Logic program files so that they appear in the scale type drop down menu in your tuning section. File path: Pro-app – Contents – resources – Tuning tables. Copy your .scl files into the “tuning tables” folder. They should appear in your project settings tuning options. .scl files don’t over-ride your master tune and root key like .tun files do, so you only need one file to play your scale in any key. You can find the master tune slider and root key settings in the image below.

If you don’t have Scala, you can open any of the .scl files that came with Logic and just edit them in a text editor. They work in ratios, so you can use any ratio based scale charts to make your own .scl files from the ones that come with Logic. Right at the bottom of the tuning options window, you can also select “hermode tuning” which is a self-adapting system that tries to keep your music in just intonation even if you change keys. There are there 3 general variations of this that you can choose from: “baroque” which gives you pure thirds and fifths, “Classic pure” which has slightly tempered thirds and fifths , and “Pop jazz” which has that harmonic seventh that is so important in the overtone series. The depth slider adjusts the intensity of the tuning in relation to equal temperament. There are a few other settings, too, so I am sure that this system can give good results if set up carefully. Another nice function is the “copy to user” function on the right of the scale “type” menu. There you can copy the “fixed” scale to the cent based sliders in the “user” section. This is nice if you have a scale where some notes sound a bit off; you can just use your ears to make a quick adjustment.

Native synthesizers in Cubase: Cubase 7 and later versions also have Hermode tuning which works in the same way as it does in Logic, but with less options. You can find the settings in “project” – “project setup” (only works with native vst 3 synthesizers). In older versions of Cubase, most of the native vst3 synthesizers can also be micro tuned via the Cubase Micro-Tuner plugin. This is a MIDI plugin and must be inserted on a MIDI insert channel, not a regular effect channel. It works in the same way as the “user” scale in Logic, so the following tutorial will work for both programs.

These sliders work with “cents” (percent of 100). You can use the “cents” data from any scale that you have made in Scala, although you do need to make some calculations to get it to work.

You will also need to trim long numbers because cents do not work the same “whole number magic” with harmonic music scales that Hz do. Only the equal temperament scale gives you whole numbers with cents because it is based on equal divisions of 100 cents per semitone. The reason why you need to do some calculations is because the sliders adjust each note in relation to that note’s frequency in the standard 12 tone equal temperament scale, whereas the cents data in Scala tells how many cents each note is above the first or root note of the scale. To calculate the settings for the sliders from any scale in Scala, use the following tutorial.

Tuning using cent based sliders or dials: 1: Open your scale in Scala. Do all your editing and make your scale ready for export. 2: Go to “view” (main Scala window) and click “show scale” to see your scales cents and ratios. 3: Go to “File” – “New”. Open a 12 tone equal temperament scale and click “show scale” again. You will now have the cents values for both scales, and can easily see the difference between each note. This difference is what you need to use in your sliders / cents based Tuner plugin.

In the image above, the top part is the Pythagorean scale while the bottom is the equal temperament scale. The numbers on the right of both show you the cents values for both scales (the equal temperament part is the same as your micro-tuner with all the sliders set to 0). 4: All you need to do now is see if the number for each note in your scale is higher or lower than the same note in the equal temperament scale, and then adjust your slider to the difference between the two numbers. To do this fast, I make a word table with 12 rows and the names of the 12 notes (C to C) so that I can fill in the value for each of the 12 sliders. Then I look to see if each of the 12 numbers is higher or lower than their equal temperament counterparts, adding a “+” or “–“to each of the 12 notes in my chart. Then I take the two numbers (equal temp and Pyth) and subtract the smaller one from the larger one, adding the answer (the difference between each note) into the row for that note. Once your chart is finished, just use it’s data to adjust your sliders. There are many synths and micro tuners that use these cents based settings and this tutorial should work for all of them. Remember, only a guitar tuner plugin can tell you if it really works , so get one and use it to check all of your scales. Even the ones in my tuning file pack. I will let Leonardo da Vinci have the last word and end this book with some of his wise words: “Principles for the Development of a Complete Mind” 1) Study the science of art. 2) Study the art of science. 3) Develop your senses. 4) Realize that everything connects to everything else.