Intonation and Supplemented Equal Temperament

Estimated reading time 12 minutes

Introducing Supplemented Equal Temperament (SET)

Tired of having your guitar or synth sound out of tune? Do you want your chords to sound amazing? Intonation and Supplemented Equal Temperament describes the creation of an exciting new tuning system.

This post describes my journey as I developed SET. SET is the tuning system that underpins, to one degree or another, all of the posts in the Intonation and Singing in Tune blog categories.

For practical examples with play-along songs, see:

This article uses technical musical terms. For definitions, see the Glossary at the end of the post.

What is Intonation?

The New Harvard Dictionary of Music gives this first definition for intonation. “(1) The degree to which pitch is accurately produced in performance…” It goes on to give a second definition. “(2) a system of tuning, such as just intonation…”

At its heart, intonation is about how musicians tune. The first definition shows that many instruments have the flexibility to change pitch on the fly in performance. Thus, intonation studies look at how flexible pitch instruments tune while playing.

The second definition shows that some instruments tune before the concert and remain in the same tuning throughout the show. Tuning does not have a role in how fixed pitch instruments play music.

The first group of instruments includes violins, woodwinds, brass, fretless bass, and voices. The second group includes pianos, accordions, organs, and mallet percussion. Guitars hold a place in the twilight between both groups.

Mostly, guitarists tune the guitar before a show without changing the tuning during the concert. However, the ability to bend strings gives the guitar some pitch flexibility while playing. Conversely, if a guitarist uses a slide, the guitar changes into a variable pitch instrument.

Microtonality versus Tuning Theory

Microtonality and tuning theory both fall under the umbrella of intonation.

On this website, microtonality and tuning theory are treated as distinct, but related subjects. Microtonality mostly entails slicing equal tempered steps into smaller chunks. The Boston Microtonal Society creates microtonal music. The smaller intervals are audibly different from what most people would consider “normal” music.

On the other hand, in tuning theory, different intonation systems tune instruments off from equal temperament (ET). (Pianos are tuned to ET.) Yet, the sound may not be obviously different to the casual listener. For instance, just intonation, meantone, and the well temperaments all tune off from ET. However, when carefully written, the music will sound normal enough to most people.

Tuning theory also studies how flexible pitch instruments tune as they play.

I have composed some microtonal music, which I may share at a later point. Nevertheless, this site will focus mostly on tuning theory with the goal of getting instruments (particularly keyboard and guitar using Supplemented Equal Tempered intonation) better in tune.

Intonation and Supplemented Equal Temperament

Throughout the Intonation and Singing in Tune categories, we will talk about intonation and Supplemented Equal Temperament. SET is a temperament that I created that blends equal temperament and just intonation. It avoids most of the pitfalls that you find in many microtonal systems.

It can be played, to one degree or another, on any western instrument.
SET does not need special notation.
It does not need special ear training.
In fact, SET can provide a good sound model for harmonic ear training.
Moreover, the sharps and flats label the notes in the normal way, though there are no enharmonic notes.
Also, the music can modulate around the circle of fifths.

If carefully written, music in SET yields triads that sound fantastic.

Intonation as a Hobby

Studying intonation started as a hobby. Also, it did not start as SET. It started as just intonation.

Back in the 1990s, I had just finished creating my own quartertone composition theory. (Do not let the baby-faced photo fool you, I am in my 50s.) Below is a sound track of one of the quartertone songs I made (it fades in, so give it a few seconds):

Oh, My Love:

The key changes on “Oh, My Love” were smooth enough that it made the quartertones acceptable to most people. (I have included an Alois Haba piece as an example of what quartertone music usually sounds like.)

However, “Oh, My Love” was a lot of work for a very little microtonal effect, so I moved on from quartertones. (We may end up talking about my quartertone theories waaaaaay down the road.)

Around this time, I got the idea to compare Fibonacci numbers to the harmonic series. Low and behold, I found a link. It seemed that Fibonacci numbers and geometrical progressions wove their way through the harmonic series.

At this point, I was hooked on the idea that the harmonic series proved that the just intonation system with small-numbered ratios created a “natural” scale. Harry Partch was a big influence during this time, and his work still remains at the heart of SET.

Just Intonation and the Logical Fallacy

One of the problems with just intonation, since it is based on ratios, is it technically has an infinite number of notes. Ratios like 2/1, 3/2, 4/3, and 5/4 all have prime numbers in them.

Prime numbers make it impossible to have the notes cycle around, because they can only be divided by the prime number and the number 1. (Trust me on this one.)

However, this is a little like the old Greek logical fallacy. It goes like this: There are an infinite number of points on a line. This means you cannot cross your kitchen because of the infinite distance created by the infinite number of points.

The logic breaks down because the infinite number of points becomes infinitely small. Likewise, the infinite number of intervals between the infinite number of notes also becomes infinitely small.

While discussing just intonation with my dad (the engineer), I explained some of the numbers in the just intonation system. He replied, “You are thinking like a mathematician. In engineering, we set a limit.”

There is a limit to the smallest size of interval that the human ear can hear. It only makes sense to limit the cent resolution of the tuning. For fully in-tune triads, a 2 cent resolution worked great. This yielded 600 notes per octave.

The Very Slightly Tempered Just Intonation System

Mobile users: for best results reading the examples, tilt your screen 90^o to the right.

This, admittedly, was a lot of notes. However, basic set theory says that when you have a closed data set, the patterns will cycle around. Therefore, the 600-note system was fully circular in all directions.

It also contained all of the equal tempered notes. However, the equal tempered notes were not yet arranged usefully. In other words, it had not gotten as far as Supplemented Equal Tempered intonation yet.

Also, issues remained: What instruments could play it, how to notate it, and how to deal with enharmonics for 600 notes? Above is a partial table of the 600-note just intonation tonality/modality chart.

The numbers are the cent values of the notes, which sit to the right of their own specific note letters. The blue cell is the note C . On the bottom row, I broke the pattern of the higher cells to show the minor thirds from the parallel minor scale.

A note, such a G, has more than one cent value on the full chart. With flexible pitch instruments, a note contains more than one pitch. Just intonation (and also SET) has this feature built into it. We will discuss this more when we talk about pitch clusters in flexible-pitch-instrument performance.

The Challenge of Ethnomusicology

I had my just intonation scheme. However, it was not Supplemented Equal Tempered intonation yet. Also, doubts crept in about the whole just-intonation-is-the-natural-scale thing. It started to seem like music theorists tried to defend the naturalness their own culture’s music.

This can be seen in Rameau’s work. The father of classical harmonic theory, Rameau jumps through hoops trying to justify the “naturalness” of the subdominant note.

Just intonation works just fine in the context of Western culture. However, the little I read of ethnomusicology at this point in the story made me doubt the natural scale. I had questions like:

If the ratios in the harmonic overtones train the ear to hear this so-called natural scale. What about cultures that have monophonic music like the traditional music from the plains native Americans? Does just intonation apply here?
What about music that tunes its scales around instruments that have an inharmonic spectrum, such as the gamelan? Just intonation cannot apply to this music.
How do you explain the singers in eastern Europe who sing harmonic intervals a second apart? How could just intonation work for music that is dissonant?

I began to realize that much of musical acoustics, music psychology, and tuning theory was based on western listeners. However, I wanted to know how people tuned, not just westerners.

I started a five-year topical study of ethnomusicology in the Garland Encyclopedia of World Music. I also read many articles in the New Grove Dictionary of Music and Musicians (in 29 volumes!). Without meaning to, I started to create a unified theory of how people tune.

(We may discuss the unified-theory-of-how-people-tune later down the road. Nevertheless, for right now we will focus on SET and ear training for music in western cultures. After all, we have to start somewhere.)

The Breakthrough to Supplemented Equal Tempered Intonation

The guitar and the Kurzweil K2000 synth both helped me make the breakthrough to Supplemented Equal Tempered intonation.

By the early 2000s, I had resigned myself that the instruments that could play my 600-note just intonation system were the synth and flexible pitch instruments.

I play guitar, and I wanted to be able to play the guitar within the just intonation system. I did not think it was going to happen, but the desire was there.

The guitar

Harmonics

The breakthrough, for both the guitar and for SET, came after reading information on antique tunings by Owen Jorgensen. I wanted to try to tune a pure major third by ear. However, I did not want to retune my piano to do it. I decided to tune the third string G and the second string B on the guitar using the fourth and fifth fret harmonics.

The following video discusses the string fundamental and how harmonics force the string to vibrate in multiples of the fundamental.

Tuning the Guitar

The harmonic at the 4th fret of the 3rd string makes the G string vibrate in five equal parts. This produces the note B. The harmonic at the 5th fret of the 2nd string makes the B string vibrate in four equal parts. This also produces the note B.

Basically, I tuned the 2nd string B harmonic to the third string B harmonic. This forced the open B string into an approximation of a 5/4 frequency ratio. It was not exact, because guitar strings are slightly inharmonic.

Regardless, the interval sounded good. However, all of the other strings were tuned using frets. This means that they were tuned to ET. I was curious, so I played an open A chord, and it sounded awesome.

Then, I played A-form barre chords with the root on the 5^th string. They also sounded great. Right away, I started to make cent charts of the guitar neck so that I could figure out how it worked.

(Below I have provided a sound track that contains guitars tuned similarly to what I have described.)

Guitars Tuned Using Harmonics:

The K2000

One of the issues with the 600-note just intonation tuning system, is it did not fit on the synth very well. The K2000 has an intona note function. It lets the player take a globally tuned intonation table and center it on a new key with just a couple of button presses.

This way a keyboardist could change keys around the circle of fifths. However, all of the intona notes were tuned to ET.

In just intonation, the just fifths are just a little bit wider than the fifths in ET. To go around the circle of fifths, I would have needed to retune the whole synth in addition to using the intonation tables and intona notes.

The phrase pain-in-the-neck springs to mind.

When I tuned the guitar to a blend of just and tempered intervals, I instantly saw the benefit for the K2000. With Supplemented Equal Tempered Intonation, I could skip the step where I retuned the whole keyboard.

Finally, I reordered the 600-note chart so that the key tonics could cycle around the twelve notes of the circle of fifths. Here is a partial version of the chart:

Intonation and Supplemented Equal Temperament and the Tonality/Modality Chart

This is a partial SET chart. The full chart has 600 notes. However, that is more notes than most players will need.

The chart below has 60 notes. It sets the practical limit for writing music. (I have rotated the chart so mobile users can read it on their phone.)

The numbers are the cent values of the notes, which sit to the right of their own specific note letters. The yellow cells are the equal tempered notes. The blue cells are the various values of the note C. I have broken the pattern for the bottom row to show the minor thirds from the parallel minor.

Notice that SET is a series of mutually exclusive, but interlocking circles of fifths. The tonal notes (the fourths, fifths, and octaves) are equal tempered. This makes the octaves pure and the fourths and fifths near pure. The modal notes (the thirds, sixths, and sevenths) are just intervals.

SET Keyboard Example:

Intonation and Supplemented Equal Temperament: The Tonality/Modality Chart (Rotated for Mobile Users)

Intonation and Supplemented Equal Temperament in the Real World (sort of)

As I worked on intonation and Supplemented Equal Temperament, I kept on studying ethnomusicology. Bit by bit, I began to grasp that guitarists often tune their guitars in similar ways as the SET tuned guitar. In other words, guitarists often tune without pressing down any frets, yielding cent values similar to SET.

However, guitarists will usually tune by comparing unison notes between open strings and fretted notes. As an example, in standard tuning the open D string is compared to the D on the 5^th string, 5^th fret. The fret forces the open D string into ET.

While most guitarist tune using unison comparison, many compare adjacent open strings, mostly when tuning open tunings. For example, Keith Richards tunes open G tuning by comparing open strings.

However, guitarists sometimes hold a chord, such as E major, when in standard tuning, and tune while strumming the whole chord. Many rock, blues, folk, Celtic, and Hawaiian guitarists tune this way.

Guitarists have been tuning using the adjacent open strings for over 150 years. The 19^th century anonymous guitar method book, Guitar without a Master, suggests tuning standard tuning using only the open strings. For a further discussion of Universal Tuning Elements:

Tune Like an Authentic Bluegrass Banjo Player

Four Options for Tuning the Guitar

We have seen that guitarists tune open strings to frets and open strings to open strings. Many guitarists tune using another option.

They tune using harmonics.

We will explore more about how tuning to the adjacent open strings impacts each style of music mentioned above. Nevertheless, for right now, we will focus on tuning using harmonics as a stepping-stone to tuning in the manner of Keith Richards.

Therefore, the guitarist has three options for tuning by ear: unison comparison, comparing open strings and harmonics. (Holding a chord while strumming the strings yields basically the same results as comparing adjacent open strings.)

Nevertheless, the last option for tuning is to use an electronic tuner. Most electronic tuners put the guitar in ET. However, strobe tuners have extreme accuracy. Some of the Supplemented Equal Tempered intonation guitar tunings that we will look at will need a strobe tuner.

The SET Ensemble

With Supplemented Equal Tempered intonation I reached the goals stated in section 4. SET takes no special notation.

ET instruments can always play at least some notes that work with SET tuned guitars and keyboards.
Any fretted, stringed instrument can be tuned to SET.
Flexible pitch instruments, of course, just tune up by ear as they always do.
Indefinite pitch percussion need no special tuning at all.

Taken together, this instrument selection creates the SET ensemble. For more information:

SET and Styles of Music

This form of Supplemented Equal Tempered intonation is optimized for triads. Baroque, classical, folk, country, bluegrass, world music, Hawaiian, blues, rock, and pop all work well in SET. I also have a seventh chord version that is designed for blues and funk. However, it is not optimized for jazz or 21^st century classical. For more read:

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geoff@successmusicstudio.com

Intonation and Supplemented Equal Temperament

Estimated reading time 12 minutes

Table of Contents

Introducing Supplemented Equal Temperament (SET)

What is Intonation?

Microtonality versus Tuning Theory

Intonation and Supplemented Equal Temperament

Intonation as a Hobby

Just Intonation and the Logical Fallacy

The Very Slightly Tempered Just Intonation System

The Challenge of Ethnomusicology

The Breakthrough to Supplemented Equal Tempered Intonation

The guitar

Harmonics

Tuning the Guitar

The K2000

Intonation and Supplemented Equal Temperament and the Tonality/Modality Chart

Intonation and Supplemented Equal Temperament: The Tonality/Modality Chart (Rotated for Mobile Users)

Intonation and Supplemented Equal Temperament in the Real World (sort of)

Four Options for Tuning the Guitar

The SET Ensemble

SET and Styles of Music

Glossary

Exploring the Music Themes and Leitmotifs of Star Wars

Song Structure, Musical Phrases, Musical Structures and Forms

Learn the Secret Relationship of Timbre and Harmony

What Should I Look for in a “Piano Teacher Near Me”?

Guitar • Bass • Banjo • Ukulele • Mandolin • Autoharp • Keyboard • Piano

Voice#: 978-440-9966

Text#: 508-561-9724

geoff@successmusicstudio.com

Sudbury, MA