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.
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.
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”.
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)