Estimated reading time 5 minutes
Table of Contents
Introduction
Can Supplemented Equal Temperament help you coordinate your guitar and synth tunings so they sound good? A wide range of Supplemented Equal Temperament (SET) charts and preexisting EDOs (equal-division-of-the-octave tunings) can fit under the umbrella of SET. This gives you a lot of control over how you tune your guitars, ukuleles, banjos, and synths. This in turn will allow your whole ensemble to sound better in tune. Read more to learn about the flexibility in Supplemented Equal Temperament’s theory.
Flexibility in Supplemented Equal Temperament’s Theory: Links to Practical Posts
Supplemented Equal Temperament’s conceptual flexibility underpins all of the posts in the Intonation and Singing in Tune blog categories.
Today’s post talks about SET theory. For practical examples for learning to play SET guitar, banjo, and ukulele using videos and soundtracks, see:
- Experimental Online Guitar Tuner
- How to Tune Like a Rock Star
- How to Tune Country Blues Guitar
- Tune Your Ukulele Like a Hawaiian
- Play Like a Rock Star with Open G Tuning
- How to Play and Tune Parlor Guitar Style
- Tune Like an Authentic Bluegrass Banjo Player
For learning how to sing in tune using sing-along songs, see:
- Sing in Tune Like a Rock Star
- Ear Training Exercises for Harmonizing in Tune
- Singing in Tune: The Intonation Fight Simulator
For micro tuning synths, see:
- Micro Tune the Keyboard to make the Chords Sound Awesome
- How Chords in Supplemented Equal Temperament Work
- Changing Keys in Supplemented Equal Temperament
- How Supplemented Equal Temperament Minor Keys Work
If you’re interested in how the retuned guitars and synths work together:
Preexisting EDOs Show the Flexibility in Supplemented Equal Temperament’s Theory
Equal Division of the Octave Tunings
Recently, I talked with someone about SET. He asked why I didn’t use 72edo instead of SET. I told him:
“Actually, 72edo would fit under the umbrella of Supplemented Equal Temperament. Really, I think of SET as a class of tunings. The tonal notes (the tonic, subdominant, octave, and to a lesser extent the supertonic) tune to equal temperament (ET). However, the modal notes (the mediant, submediant, and leading tone/subtonic) tune closer to pure than ET does.
“72edo has near pure major 3rds (383.33 cents). Also, it contains all of the equal tempered notes, so it works as an SET type tuning. In addition, this means that 48, 60, and 96edo could also be considered SET tuning systems.” [Click here for a definition of cents.]
I created none of these EDOs – they predate SET. This demonstrates the flexibility in the Supplemented Equal Temperament model to accommodate other tunings.
The charts below show the 48, 60, 72, and 96edo. Also, I’ve included rotated versions of the charts so mobile users can read them.
For best results, rotate your phone 90 degrees.
How to Figure Out the Note Names on the Charts
If you look at the charts below, the blue cells indicate the cells labeled as the note C. The yellow cells contain the ET notes arranged in the circle of 5ths. Also, all the other rows have been arranged in the circle of 5ths. From these clues you can figure out the note names of all of the cells on each chart.
This keeps all the notes uniform from chart to chart. It also avoids strange enharmonic equivalents, which allows the theory to function much like music theory in ET. The only downside to this system is what I call note labeling distortion.
Note labeling distortion has to do with how the notes get labeled, which can be an issue at the extreme edges of the chart. For example, on the 2 cent resolution chart (not shown), all of the blue cells are Cs. This means I have a C as high as 364 cents and as low as 878 cents, which does not meet with Scala’s ideal for note labeling.
However, I did this because it allowed me to have note labeling consistency across the wide range of charts I worked with. Also, it relates to the philosophical underpinning of the system: making SET accessible to the widest range audience possible.
This means that I wanted SET to be playable on ET instruments (all ET instruments can play at least some notes that work with SET). I also wanted the simplest theory possible while being able to use tab and standard notation.
Thus, this shows a different type of flexibility in the Supplemented Equal Temperament model. If you stick to the notes nearest the center of the charts (C 0), all these things remain possible.
EDO Charts
Rotated EDO Charts for Mobile Phones
Multiple SET Charts Show the Flexibility in Supplemented Equal Temperament’s Theory
The SET charts below also demonstrate the flexibility in the Supplemented Equal Temperament concept. Besides the preexisting EDOs discussed above, I have a 10 cent resolution chart that has a 390 cent major 3rd. Also, I have a 5 cent resolution chart with a 385 cent major 3rd. And two 4 cent resolution charts with 384 and 388 cent major 3rds respectively. This doesn’t even take into account the charts that have septimal elements (not shown).
In earlier posts, I’ve used the 2 cent resolution ear training chart as a place to start talking about SET. I like the 2 cent resolution chart, because the major 3rd lies close to the just intonation cent value. It works really well with the guitar tunings I’ve been posting about. In fact, the guitar tunings, especially open tunings, constitute one of the main driving forces behind SET.
Note that the 2 cent resolution chart shown below has been reduced in size. At 600 notes, the full-sized chart would be too large to fit on mobile phones, even with a rotated chart. Therefore, the chart below has the 60 notes that composers will most likely need.
The gray cells (in various shades) on the charts in this post represent the tuning for quartertone and eighth-tone scales. Thus, it’s possible that composers will be able to combine quartertone music with just intonation elements if they so desire.
SET Charts
Rotated SET Charts for Mobile Phones
Concluding Thoughts on the Flexibility in Supplemented Equal Temperament’s Theory
As you can see, the Supplemented Equal Temperament concept has a lot of flexibility for tuning instruments. You just need:
- All of the equal tempered notes to be available.
- The major 3rds need to be somewhere in the ballpark of the just major 3rd. (Though I like some of the well-tempered/meantone sized major 3rds – around 392 to 394 cents.)
- The root of each chord has to be an equal tempered note, and the 5th often has an ET cent value. However, in some tunings the 5th can be tuned pure (not shown).
The closer your major 3rds get to just the better your tuning will sound. Nevertheless, this will also impact the flexibility of your chord voicings on guitar, banjo, ukulele, and really any fretted string instrument. Plus, it can impact how well your tuning fits on your keyboard controller.
Also, some types of music will be better fitted to one tuning over another. I suggest you try out the different tunings shown in the above charts and see what you like best. Have fun writing!
© 2022 Geoffrey Keith
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Glossary
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