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Posted by on Jul 30, 2013 in Consonance, The Lattice, The Notes, Tonal Gravity | 0 comments

Harmonic Distance

Harmonic distance is the total length of the connection between two notes on the lattice, as measured on the solid lines. The more tinkertoy sticks you traverse to get from one note to the other, the greater the harmonic distance.

It’s not the same thing as melodic distance, which is a difference in pitch. Two notes can be far apart in harmonic space, but close together in melodic space, or vice versa. This post has a demonstration.

Each solid line on the lattice is a prime factor — 3, 5 or 7. A simple way to put a number on harmonic distance is to multiply together all the prime factors used in the ratio of the interval. Doesn’t matter if you’re multiplying or dividing by the factor, the distance is the same. Twos don’t count; these are octaves and they don’t add distance on the lattice.

The closest intervals on the lattice are the perfect fifth and perfect fourth. To get these intervals, you multiply or divide the original note by 3. The ratio of the fifth is 3/1, and the ratio of the fourth is 1/3. The harmonic distance is 3, in both cases.

The major seventh, or 7, is a more distant interval. Its formula is x3, x5, or 15/1, so its harmonic distance is 15.

The b2- is the reciprocal of the 7. Its formula is ÷3, ÷5, or 1/15, and it is equally distant. The polarity is opposite, but it’s the same distance away from the center.

There are two other notes at this same distance of 15 — the 6 and the b3. Their ratios are 5/3 and 3/5 respectively. They are reciprocals of each other, and have opposite polarities.

Here is the inner lattice, showing the ratios (without any factors of 2), and harmonic distances instead of the note names. The ratio of an interval defines it completely; it would make perfect sense to name the notes by their ratios alone (it’s been done).Harmonic Distance central lattice


In the consonance experiment from a few posts ago, I played intervals in order of harmonic distance, and sure enough, as they got further out, they got more dissonant. I used the Pythagorean axis (multiples of 3) to keep it simple. Pythagorean tuning is somewhat limited musically; harmonic distance increases so fast that there are very few consonant notes.

On the lattice of thirds and fifths, there are more consonant notes to play with. How would that same experiment sound, when you add in these new intervals?

I’ll stick with the overtonal, Northeast quadrant of the lattice. Every ratio involves multiplication only, so there is no reciprocal energy, and I’m not comparing apples to oranges. My intention is to test only one ingredient of consonance, the harmonic distance. The intervals travel away from the center, and back again. Listen and watch a couple of times, and hear what happens.

I think the pattern holds very nicely. At the very end, the #4+ with its distance of 45, I think the dissonance has lost some of its obnoxiousness. It does appear that as the distance gets big enough, both consonance and dissonance start to weaken. The ear has less to go on, the signal is weaker.

Also note how the other component of consonance, stability/instability, changes as we roam farther out and come home again. All these intervals are stable, since they are all overtonal. This sense of stability gets stronger the closer we are to home, as though the ear is receiving a stronger signal and is more and more sure of itself. I start to clearly hear the stability at the major seventh (15/1), and it quickly gets stronger from there on in.

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Posted by on Feb 1, 2013 in Just Intonation, The Lattice, The Notes | 0 comments

The Untempered Major Scale

In harmonic space, the clearest name for each note is its ratio — 5/3, 3/2, 4/3, etc. Precise and unambiguous. But the ratios don’t give a very good idea of how to go about playing or singing those notes.

There are a few instruments, such as the left hand keyboard on the accordion, that are organized for harmonic thinking.


The rows are arranged in fifths, and the first two rows are a major third apart, just like the lattice.

For most music making, though, we need to know the pitch of the note. Instruments and voices tend to live in the world of melodic space — scales and pitches.

Here’s what the simplest, most harmonically consonant major scale looks like on the lattice:

One can convert the ratios to pitches using the formula for cents. The ET values are in parentheses.

1 = 1/1 = 0 cents (0)

2 = 9/8 = 204 (200)

3 = 5/4 = 386 (400)

4 = 4/3 = 498 (500)

5 = 3/2 = 702 (700)

6 = 5/3 = 884 (900)

7 = 15/8 = 1088 (1100)

Note that the 1, 2, 4 and 5 are very close to their ET equivalents. Most ears would be unable to tell the difference.

The third, sixth and seventh, however, are all noticeably flat. Or perhaps I should say their ET namesakes are noticeably sharp.

I think this explains a lot about rock music, which depends heavily on power chords (roots and fifths with no thirds) and 1-4-5 chord progressions. The notes of a 1-4-5 power chord progression are 1, 4, 5 and 2!

As Eddie Van Halen said in his terrific Guitar Magazine interview, “Really, the best songs are still based on I-IV-V, which is so pleasing to the ear. Billy Gibbons [of ZZ Top] calls me now and then, and he always asks, ‘Eddie, have you found that fourth chord yet?’ [Laughs].”

Of course the I-IV-V is inherently satisfying, it’s that great rocking chair between reciprocal and overtonal territory, in the most consonant part of the lattice, as close to the tonic as you can get (harmonically).

But I think the special appeal for rock music is that, in ET (and guitar is essentially an equal-tempered instrument), the 1, 4 and 5, and also the fifth of the 5 (the 2) are all in tune. Whatever one might think of ZZ Top’s simplicity (and some do scoff), it’s undeniable that they are fiercely in tune, and harmonically their music strikes a deep chord in the psyche, pun intended.

If the bass and rhythm guitar stick to those roots and fifths, the voices and lead guitar can play all the other notes, because they can be bent and wiggled until they sound right.

This works just as well in traditional country music. Let the bass player nail down those roots and fifths, and the voices (and the fiddle) can sing in-tune harmonies as sweet as you please.



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Posted by on Nov 12, 2012 in Just Intonation, The Notes | 0 comments

Octave Reduction

Doubling the frequency of a note certainly changes it. The ear hears a higher-pitched note. But there is something in the essence of the note that does not change, a character that stays consistent through the octaves.

This allows a process called octave reduction. When you’re working with notes as ratios, it’s convenient to multiply or divide the raw ratio by 2, as many times as is necessary to bring it into the same octave as the tonic.

3/1 generates a perfect fifth. 3-1

This note is actually an octave plus a fifth above the tonic. Now divide by 2 and you have 3/2, one and a half times the original frequency, and just a fifth above. 3-2

The reference frequency is 1, the octave is 2, so what you want to achieve with octave reduction is a ratio, or fraction, between 1 and 2.

These are the beginnings of a scale, a collection of notes within a single octave. Such a scale can be repeated up and down the octaves to cover the whole range of hearing.

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Posted by on Nov 11, 2012 in Just Intonation, The Lattice, The Notes | 0 comments

Notes As Ratios

Notes are pitched sounds. A given note means little by itself. It could be the tonic of a key, or some member of a scale based on a different tonic. By itself, it generates no tension, resolution or sense of place on the harmonic map.

So when I name a note in this blog, I’ll usually be referring to a ratio, the relationship between the note and a reference note — the tonic, or the root of a chord, or another note in the harmony or melody.

Ratios are fractions. The first number is divided by the second number to give the value of the ratio.

If the tonic is, say, 100 Hz, then another 100 Hz note is related to the tonic by the ratio 1/1. This is the interval of a unison, two identical notes.

Each note name on the lattice represents a unique ratio, relative to the tonic. The 1, at the center, stands for 1/1.

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