In this exercise, we will tune the guitar to an open D major chord in just intonation, or a perfectly pure tuning. This should be good for students of any level to practice tuning their guitar. We will do this with our ears – do NOT use a commercial guitar tuner for this exercise. Once we are tuned up, the guitar will be suitable for single fret bar chords and modal scale passages.

Tune the 6th string, the lowest string, to D.

Tune the 5th string to A. You can tune the perfect fifth by ear, and use the harmonic at the 7th fret on your low D string to produce the A and check your accuracy. Try to listen to the high overtones produced when you pluck, and make sure there are zero waves or beating when both tones are played. When tuning, pluck the strings and let them ring – you really have to tune while the notes are ringing out in order to get them perfect. Just remember – pluck the string and let it ring!

Tune the 4th string to D. Try tuning by ear, and afterward checking it with the harmonic at the 12th fret on the low D string. Tune it absolutely pure, with no waves or beats.

Return to these three low strings and get them absolutely in tune. The tension adjustments you just made may have caused the strings to shift ever so slightly, which will cause a small amount of beating and ruin our pure tuning. Once you are satisfied, move on.

Tune the 3rd string to F#. The F# we want is the just major third from D – we CANNOT use the F# at the 4th fret of the 4th string, it will be too sharp. If experienced, you can tune this easily by ear, but if not, try to produce a nice clear harmonic just behind the 4th fret on the low D string to produce the pure, just intonation F# major third. This is the most difficult note to tune, so if you have trouble don’t worry too much. Tuning is an art and takes a lot of practice to get it right. If you think you’ve got it, the 3rd string tuned to the just major third should be a little flatter than your fretted F# that would be played at the 4th fret of the 3rd string.

Play each of the four strings we have tuned and listen for any unevenness, waves, or beating. Most likely you hear a little thing off, in which case you should go back and work on each string individually again.

Tune the 2nd string down to A. Use the harmonic at the 12th fret of the 5th string to produce the A. Your 5th string and 2nd string should be perfectly in tune, an octave apart.

Tune the 1st string down to D. Use the harmonic at the 12th fret of the 4th string, or the harmonic at the 5th fret of the 6th string to produce the high D harmonic. Your 6th string, 4th string, and 1st string should all be perfectly in tune, each one octave apart.

Check each string again, and try to isolate which string may be causing any beating. Once you have none, or are at least satisfied with what you’ve got, feel free to play around. You may notice that some fretted notes are out of tune, notably with the 3rd string tuned to F#.

I will say that it is impossible to get the guitar perfectly in tune, as there is a limit to what our ears can accomplish. Variations and inconsistencies in your instrument, especially if your strings are worn in, might make this obvious. The important thing is not being perfectly in tune, but being well in tune.

If you thought this was pretty easy, I’ll post another tuning exercise soon, this time tuning the guitar using a flat minor third, a flat minor seventh, and a pure flat tritone.

The foundations of harmony are not based in “tritone substitutions”, or even “tritone resolution”, nor in “proper voice leading”, nor “roman numerals” or even simply in “triads”.

The foundations of harmony, or harmonic theory, lie in the natural overtones which are produced in a musical tone. As a guitarist, you can find these natural overtones by playing harmonics on an open string. The exact position of each harmonic can vary depending on your guitars setup, but try to find the following 12 harmonics on the lowest (E) string:

1: open string
2: above 12th fret
3: above 7th fret
4: above 5th
5: a bit behind 4th fret
6: a bit in front of 3rd fret
7: a bit behind 3rd fret
8: halfway between 2nd and 3rd fret, a bit close to the 2nd fret
9: a bit in front of 2nd fret
10: just a bit behind 2nd fret
11: just a bit behind 10th harmonic
12: just a bit behind 11th harmonic
etc…

The series of overtones can be extended infinitely, but you probably found that they became more difficult to produce the higher you went. We will concern ourselves primarily with the harmonic tones through the 8th harmonic.

In our exercise, we ordered and numbered each harmonic played. The first harmonic was not really a harmonic at all, but simply the open string. The second harmonic was produced at the 12th fret and is “one octave” higher than the first. The important thing here is the relationship between the two different tones. In music, this relationship can be referred to as an “interval”, in mathematics a “ratio”. Western musicians refer to this first interval as an octave, and mathematically the ratio between the two tones is 2-to-1, or 1-to-2, or 2/1, or 1/2, all essentially being the same, being “inversions” of each other.

Each harmonic is related to every other harmonic and can be measured this way. 2/3, the relationship between the second and third harmonic, is called a “perfect fifth” in western theory.

Here is a list of the same harmonics, with the corresponding interval in western theory:

1: fundamental
2: 1st “octave” of the 1st harmonic
3: “perfect fifth”
4: 2nd “octave” of the 1st harmonic
5: “major third”
6: 2nd “octave” of the 3rd harmonic
7: “minor seventh”
8: 3rd “octave” of the 1st harmonic
9: “perfect fifth” of the 3rd harmonic
10: 1st “octave” of the 5th harmonic
11: “tritone”
12: 3rd “octave” of the 3rd harmonic

We established that the relationship of 2/1 is an octave doubling. If we apply this relationship/ratio to any harmonic, we will end up one octave higher. This is an example of what can be called a “derived interval” or the stacking of two or more intervals to produce a “new note” that is not based on a new prime number, or new identity.

Two important things must be understood going forward. When we add two intervals together, say ‘perfect 5th’ and ‘perfect 5th’, we must multiply their ratios, 3/2 * 3/2. Also, when we talk about ratios, we usually want the ratio to fit within one octave – I would rather write 2/1 instead of 4/1, 8/1, 16/1, or 32/1. See Kyle Gann’s great article Just Intonation Explained for further elaboration on working with musical ratios. The best part about Gann’s page is all of the audio demonstrations he has put together. If nothing else, listen to those and let your ear do the work for you.

Let’s look at another example of a compound interval to find a “new note”, the ‘major 9th’. When we multiply the ratio of 3/2 to any harmonic, we will end up a ‘perfect fifth’ higher (in fact, a perfect fifth the next octave higher). Multiply 3/2 by 3/2, and you will have 9/4. Multiply 9/4 by 1/2 to raise it an octave higher and you have 9/8, the 9th harmonic, which is also a ‘perfect fifth’ of the 3rd harmonic, or a ‘perfect fifth’ of a ‘perfect fifth’. Over 1000 years of western music was based entirely on the stacking of perfect fifths – more will be discussed about Pythagorean Theory in another post.

What about intervals based on prime numbers? Go back to the image – see the numbers with the colored circles? Each of these numbers is a prime number. Prime numbers represent new “identities” of tones, sort of like different colors or different moods… 1 is the fundamental. 2 is the octave, both a different note but also the same identity, as the fundamental. 3 is the ‘perfect fifth’, strong and powerful, being closely related to the fundamental. 5 is the ‘major third’, strident, colorful, active, and warm. 7 is the ‘flat minor seventh’ bluesy, soulful, more complex. 11 is a kind of ‘tritone’, and to my ear has a flat and nasal quality that is tough to describe. When we reach prime 13 or above, personally my ear has a tough time understanding and gets lost.

These harmonics do NOT correspond precisely with the notes produced by the frets on your guitar, nor by the notes played by the piano. A explanation and critique of ‘equal temperament’ will follow.

I have begun some volunteer web work for the Orchestra of Our Time. Over the years they have done numerous concerts in New York City, including performances of Erik Satie’s Socrate, staged with a set designed by Alexander Calder, and work with Frank Zappa on his Grammy winning album Zappa’s Universe, release shortly before Zappa passed.

My web design/programming skills are limited, but they are gracious enough to let me join the project. In the planning stages is a call for scores, intended for composers with limited means of having a professional recording of their music made. Not everyone who is writing music is always “on the same team”, but it should be argued that everyone is as equally deserving of having their pieces played and received in some fashion, on a less discriminating basis than the current classical establishment can offer.

Better to play any new piece than to only play Mozart.