The next closest pair is the major third / minor sixth. This has a different flavor. Now the tonic is multiplied and divided by 5.
The overtonal third feels stable and restful, though not quite as much so as the fifth. These notes are a bit farther from the center than the 5 and 4. The reciprocal sixth sounds more dissonant than the 4.
The next closest note to the center is the septimal flatted seventh, or harmonic seventh. The ratio of this note is 7/1, and its mirror twin is 1/7. I have not yet consciously used the mirror-seventh, and it’s not on my drawing of the lattice. The note is the septimal major second, at 231 cents, a dissonant interval indeed. The yellow lens shows where I would put it on the lattice.
Oy! That should put to rest the idea that just intonation is all about consonance! The septimal major second is nastier than anything equal temperament has to offer. I like the word “untempered” for this music because it better captures the wild and wooly nature of JI. “Just Intonation” sounds a bit stuffy to me, and the natural intervals of whole number ratios are anything but academic, they are burned in at a very basic level. Equal temperament is brilliant, but it’s actually the headier and less visceral of the two. IMO.
The next pair is a little further out — each note requires two moves on the lattice.
The ratios are 9/1 and 1/9. I still hear the 2 as stable, though it is less consonant than the previous notes. The b7- is suitably dissonant. It cranks up the tension in dominant-seventh-type chords, the workhorse tension-resolution chords of classical music.
I hear the effect of both tension and resolution diminishing somewhat, as tonal gravity gets weaker farther from the tonic.
These last two videos each contain a minor seventh. One is overtonal, the other reciprocal. The septimal flatted seventh, or harmonic seventh, is a stable, resolved note, the signature of barbershop harmony.
Septimal sevenths abound in this music, and they are sweet and consonant and stable.
The b7-, on the other hand, is dissonant and tense. It makes the ear want to change.
In equal temperament, these two notes are played exactly the same. ET weakens and obscures the difference, but it still can come through because of context.
The common “… and many more” tag, sung at the end of Happy Birthday, is a great example. That last note, “more,” is a harmonic seventh, 7/1, the stable, beautiful barbershop note at 969 cents. If you play “and many more” on a piano, the ear will hear the last note as a septimal seventh, only with less impact, because it is very sharp, at 1000 cents.
Almost all sounds are actually a bundle of waves, of different frequencies. The frequencies in these bundles tend to vibrate at multiples of each other — 2x, 3x, 4x, 5x some base frequency — a harmonic series.
Our ears are highly attuned to such relationships. They help us figure out which frequencies belong together, so we can analyze them and identify the source. If we hear two frequencies in lockstep, three cycles to one, it is very likely they come from the same object.
I think this is why we can hear harmony. It’s a byproduct of our built-in orientation software. The overtone series of a sound is a powerful source of information about the object that made it. The harmonic content tells us whether it’s a barking dog or a friend or rustling leaves. Something in us sorts out and analyzes many harmonic series at once, in real time, identifying sound sources, locating them in space, and even sensing their texture at a distance. It’s a phenomenal processor.
So the processor recognizes the 3-wiggles-to-1 dance of the perfect fourth. But something is wrong — the 4 does not belong in the overtone series of the 1. The overtone series is generated by multiplication, not division.
This is a strange input for the mighty processor.
The 4 has to belong to the 1, because the two are in step with each other at three beats for one. In nature, that’s a dead giveaway.
But the 4 can’t belong to the 1, because the ratio is the wrong way around — natural sounds do not contain 1/3 in their overtone series.
Maybe the 1 actually belongs to the 4. The 1 is three times the 4, so if the 4 were the base, all would make sense.
But every other clue, all the harmonics in the drone (and, importantly, the listener’s memory), are pointing to the 1, hollering “This is the basic frequency!”
What’s a supercomputer to do?
I feel this sensation as unrest or instability, a need for something to change. Either the 4 needs to resolve to an overtonal note such as the 5 or major 3, or the root needs to change. Moving the root to the 4 will resolve the dissonance, introducing a new reciprocal tension — now the root “wants” to go to the 1, because the ear remembers. There are videos of this here.
On the lattice, there are two basic polarities, overtonal and reciprocal. Some notes combine both energies.
Overtonal notes are created by multiplying the base frequency. They appear in the natural harmonic content of sounds. They sound stable, restful, resolved — more so the closer they are to the center of the lattice, where tonal gravity is stronger. Pure overtonal notes are in the Northeast quadrant of the lattice.
Reciprocal notes are created by dividing the base frequency. The mind recognizes them as being related to the base frequency, but they do not appear in its natural overtone series. They sound unstable, restless, tense — and, like the overtonal ones, the effect is stronger closer to the center.
For every note on the lattice (except the 1), there is another note, the same distance away from the center and exactly opposite it. The harmonic moves for the two notes are the same, but the directions are opposite.
Mirror twins are reciprocals of each other. Flipping a note’s ratio upside down will produce its twin.
Listening to these mirror twins helps demonstrate polarity.
The simplest ratios on the lattice, the ones with the smallest numbers in them, are 3:1 and 1:3, the perfect fifth and perfect fourth. Here they are:
Tonal gravity is strong here close to the sun. The fifth sounds remarkably resolved, like it’s part of the drone. In fact, this note shows up so strongly in the natural overtone series that it is part of the drone. If you listen carefully to that tonic drone by itself, you can hear it. The following video shows a 5 by itself, followed by the straight drone on the 1. I hear the fifth appear again, quietly, after the drone has a few seconds to settle in and bloom.
The fifth says, “You’re home, relax.”
The fourth also shows where home is, but in a very different way. Instead of saying, in effect, “Home is here, come on,” It is saying “Home is over there, now go.”
Our built-in audio processor is always looking for mathematical relationships between notes so we can tell which frequencies belong together, identify different sound sources and orient ourselves in our surroundings. I think this is why we hear harmony and why it sounds like a journey — it’s a part of our built-in orientation software.
Three cycles for every one is a “right” ratio. It strongly says, “These frequencies belong together, they are being made by the same thing.” But the ratio is upside down, 1/3, the exact opposite of the “right” sound, 3/1. It’s the shadow version of the 5, yin to its yang.
The 5 feels like this:
The 4 feels like this:
The beautiful stability of the 5, contrasted with the equally beautiful instability of the 4, is what I mean by polarity.
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.
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).
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.
In that experiment, I kept the axis (3) and direction (multiplication, overtonal) the same, and increased the distance.
This time I’ll keep the axis and the distance the same, and switch direction. Each illustration will compare a note with its mirror twin, its reciprocal.
First up is the strongest polarity flip there is, the perfect fourth and fifth. One divides the tonic by 3, the other multiplies it by 3.
The 4 is clearly unstable, it wants to move. The 5 is clearly stable. If a song ends with this interval, I will feel completely satisfied.
The next matchup is the b7- and the 2. The b7- is the crucial note that provides the tension in dominant-type seventh chords and makes their resolution so satisfying. Here it is in undiluted form.
The 2 is fairly stable. Quite a few songs end on this note, and there is a pretty good sense of resolution, maybe with some wistfulness mixed in.
The two notes are about equally harmonious, and of opposite polarity. This is the same pattern as the 4 and 5, only weaker.
Moving outward, we get the b3- and 6+ pair:
The pattern continues — now both notes are rather dissonant, with the b3- weakly unstable and the 6+ weakly stable. It would be rather unsettling to end a song on the 6+, but maybe you could get away with it.
Here are the next two:
These are interesting. They are dissonant, all right, and the b6- is unstable and the 3+ is stable. But I actually hear the polarity a little more strongly than the last pair.
I think my ear is trying to interpret these notes as out-of-tune versions of the b6 (a strongly unstable note) and the 3 (strongly stable).
How is my ear to interpret this 3+ note, the Pythagorean major third? Can I even hear a ratio of 81/64? Maybe not well enough to really recognize it.
Perhaps the ear “decides” that it’s simpler to read this strange note as a badly tuned version of a simpler interval, one I am familiar with. So I hear it as an out-of-tune 5/4 instead of an in-tune 81/64.
This is why equal temperament works, as Mathieu demonstrates so well in Harmonic Experience. A painting doesn’t have to be exactly straight on the wall for the eye to interpret it as straight. Thank goodness! In the same way, a note doesn’t have to be exactly in tune to be heard as that note. The ear is willing to accept “close enough” and hear it as the real thing, though the consonance will not be as good.
Maybe the part of the mind that processes this stuff is like a quantum computer, taking in the sound, trying out all possibilities at once, and spitting out the “most likely” interpretation, which would be the solution with the lowest “potential energy,” the one that is closest to the center, just like real gravity.
We’re probably too far out now to really recognize these intervals as what they are, but for the heck of it:
Suitably nasty, and now the sense of polarity is pretty much gone, I can’t hear it.
In my last post I raised the idea that the consonance of an interval is not just one thing, but has two distinct parts:
The sensation created by the sound of the two notes played together — smooth or rough, pleasant or unpleasant
The sense of stability of the note — does it feel restful, like it has arrived, or restless, unstable, like it “wants” to move?
I propose that #1 is correlated with the harmonic distance between the two notes of the interval, that is, the length of the solid lines that connect them on the lattice.
I think #2 is determined by the polarity of those connections, whether they consist of multiplying (stable) or dividing (unstable), or a combination of the two.
This suggests some experiments.
A mostly good rule in experiments is to change one thing at a time. The prime factors used in an interval’s ratio (3, 5 or 7) strongly affect the sound, so I’ll stick to one axis (x3), and only go in one direction (multiplying, overtonal, East).
This will all happen on the backbone of the lattice, the infinite horizontal line of fifths. The notes on this line are in Pythagorean tuning, which is any tuning that uses only 3 and 2 in its ratios. Pythagorean tuning is used in music, but many of the intervals are sharply dissonant, as we shall see.
If I’m right, according to #1 the notes should get rougher or more unpleasant as I go further from the center, and according to #2 there should be a sense of stability, that gets weaker as the tonal gravity field weakens with distance.
I’ll play all the notes against a drone, several octaves of the tonic. The notes will all be played in a single octave, in between two drone notes. I will octave reduce (divide by 2) as necessary, to put all the notes in this octave.
The first interval is a perfect fifth. To get this, you multiply the frequency of the tonic by 3.
No denying it, this is a consonant interval. Only the drone sounds smoother. It sounds stable too, as it should according to #2. If a song ends like this, I’m perfectly satisfied.
Next note up is the 2. This is two perfect fifths, a factor of 9.
To my ears, this note is still quite consonant. And yes, it still sounds stable. Lots of songs end on this note — I feel wistfulness, gentle yearning, perhaps a sweet sadness, but I don’t feel a powerful need for the note to change.
Next is 3x3x3, the Pythagorean sixth or 6+. I still hear this one as consonant, but just barely. For me the pattern continues — the 6+ is right on the edge of dissonance, and stable, but more weakly. I could hear it resolving down to the 5, but the urge is not strong.
Now 3x3x3x3, the 3+, Pythagorean major third.
I don’t think there’s much disagreement that this is a dissonant note. This major third is 22 cents, an entire comma, sharper than the sweet 5/4 third that appears as “3” on the lattice. It’s has the sharpness of the equal tempered third, on steroids.
For me, the major third is the Achilles Heel of Pythagorean tuning. Any tuning with dissonant major chords is going to be of limited usefulness. I’d go nuts if all songs ended on this note.
Yep, ugly. But still feels somewhat stable. Next?
Ow! And the stability is very weak now; I’d be fine with this note resolving to the 1.
Keep going! Next up is a suitably obnoxious tritone, a version of the good old Devil’s Interval. To my ear, the stability is now lost, and actually the dissonance itself is less profound than it is with the 3+ and the 7+. Tonal gravity is very weak here in the outer solar system.
Just for fun, here are the notes in reverse order. You can hear them coming back to consonance and stability.
To my ear, this experiment supports the idea that more harmonic distance equals more dissonance. The sheer obnoxiousness does seem to fade a bit past the major third or major seventh. I’d say that both consonance and dissonance get weaker as the note gets farther from the center. The ear seems to have less signal to work with as the tonal gravity field weakens.