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Posted by on Aug 7, 2013 in Consonance, Equal Temperament, Just Intonation, Septimal Harmony, The Lattice, The Notes | 0 comments

More Mirror Twins

Mirror twins are pairs of intervals, exactly opposite each other on the lattice.

The two intervals are reciprocals of each other, which means their ratios are flipped — if one is 5/3, the other is 3/5.

Harmonic distance is the same for each interval — the only difference is polarity. Listening to mirror twin pairs gives a good idea of what polarity sounds like.

The clearest example is the fifth/fourth pair, multiplying and dividing the tonic by 3.

Beautiful, consonant notes, one with overtonal energy, and the other with reciprocal energy.

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.

Next: Why Can We Hear Harmony?

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Posted by on Jun 6, 2013 in Just Intonation, Recordings, The Lattice, Tonal Gravity |

Be Love

After two years of working mostly with existing material, I’m happy to be writing songs again. This one took me over completely for a few days, and then I spent another couple of weeks recording it and animating it on the lattice.

Be Love is a simpler song than Flying Dream, and I think it does a better job of illustrating what the lattice is all about. A couple of notable moments:

  • Several times, all the notes suddenly collapse to the 1, the note in the center. Check out the feeling of arrival, or homecoming in the music when it happens. It’s especially powerful going into the first chorus at 1:12. This is a real-time demo of tonal gravity.
  • During the verse, the song hangs out in the left part of the lattice, and then for the chorus it moves to the right. This is an example of a change in mode — the song stays in the same key (1 is still the center) but two of the scale notes change. The language of pure music doesn’t translate literally to English, or to emotion, but it evokes its own sensations that can support, or contradict, the words and feelings in the lyrics.

Next: Home

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Posted by on Oct 28, 2012 in Just Intonation, Recordings, The Lattice | 11 comments

Flying Dream on YouTube

This may have been the most intense art project of my life. Some time early in 2012, I got it into my head to create a stop-motion animation of my song Flying Dream, moving in harmonic space. I’ve spent the past five months working like crazy on it. The song is carefully arranged using the lattice of fifths and thirds, sung and played in just intonation, and animated using colored lenses, rice paper and a flashlight.

If it doesn’t play smoothly, don’t hesitate to lower the resolution. It survives scaling well, and the timing is crucial.

This video encompasses almost everything I’ve learned in the past couple of years. I will be writing much more about it. Meanwhile, here are a couple of external links about the basic subjects:

just intonation

lattice (music)

Next: Back Story


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