Tritone

What's the deal with the tritone? Mi against fa. Very much frowned upon in counterpoint practice. Dissonant, not harmonious. However if you look at compositions you'll find it used, often for dramatic or emotional effect—"War Requiem" by Benjamin Britten comes to mind—and if everything is harmonious, the work risks being a bore, just like a TV series that omits all cliffhangers and other such devices. Certainly some music can get by without drama; the music should after all suit the circumstance, so one might use some highly harmonious (that is, in the not dissonance sense) for a bit of fun in, say, heavy metal where dissonance might well be expected. However! Britten is modern decadence and all that, so how about something older?

"De profundis" by Johann Joseph Fux dates to the Baroque, a random pick from a church music section, and in church music one might expect the devil interval least used, and because Fux is the author of a notable work on counterpoint in the Palestrina style (a work used by Haydn, maybe less diligently Beethoven, etc). Also because the Orlando di Lasso work I already had only had a MusicXML file, and LilyPond crashed and did not manage to convert said MusicXML to MIDI. Oh, well. Anyways, it was faster to look at the score than to devise some tritone interval detector software thing (vertical is easy; complications arise if you want to look for a tritone arising from "close enough" but not simultaneous voices, parallel major thirds, or, tritones outlined by a phrase or worse different voices, none of which programming would much exercise a brain for music).

=> mifa.png | mifa.midi | mifa.mp3

Nine measures in and we have a dissonance, gotten to by a leap. All sorts of bad in this phrase according to software that used a Palestrina ruleset. So mi against fa does get used, here at the end of a measure, where one will often find the most dissonant of chords, ideally resolved on the subsequent strong beat. Tension and release is what some call this, again more or less like the cliffhanger in other media. "Le Comte de Monte-Cristo" readers had to wait for the next publication to come out.

Fux however is Baroque, modern decadence and all that, so how about something older? A random Giovanni Pierluigi da Palestrina Renaissance work had fewer tritones, but not zero, maybe more in passing. Particular intervals may be in or out of fashion, so what is more common in one era might be rare in another. A rules system might capture this, poorly, so again one is likely better off by looking at what composers actually did. Mozart for example spent some time transcribing the works of Bach. Do you really know the material if you cannot rewrite it yourself?

Now, the Renaissance is modern decadence and all that, but you may need to learn how to read weird scores from the Medieval period, or find translations or modern reviews thereof. Probably tritones would be even more rare back then? At some point a mass computer analysis may help categorize tritone use by composer and over time—interesting, but maybe not much help to a modern composer, unless they need to compose in a very particular style, but for that they could also just drown in period works for a while. (David Cope was analyzing and generating music before the present AI craze, so it's been done before.)

How did mi against fa get such a bum rap? Probably inertia and snowballing; enough people in some group did not like it, convinced others not to like it—the ritual action of the elders—and then you have something akin to a Home Owner's Association that tries to enforce the not-use of the interval. It's the devil in music! The grass length is out of tolerance! And, worse, there were attempts to eradicate the tritone, or entire musical traditions, for being too unlike a particular Western tradition being spread around the world ("Who asked the first question". Joseph Jordania. 2006). There can be overlap between music and political order; William Byrd ran into trouble for being too Catholic in the wrong country at the wrong time, just as Aaron "Fanfare for the Common Man" Copland got some flack for being too communistic. Shostakovich had it even worse.

Meanwhile, there are vocal traditions where singing dissonances is normal; if you've heard the "Ghost in the Shell" (1995) soundtrack, there is a Japanese adaptation of Bulgarian singing which uses all sorts of highly dissonant intervals. The scales may even be totally unfamiliar to the Western ear. Gamelan comes to mind. If you grew up with it, it's normal, just as food may be normal or weird depending on what you grew up with. Rotted baby cow food? Yuck! …with live insect larva? Yum!

=> https://www.cpdl.org/wiki/index.php/De_profundis_(Johann_Joseph_Fux) | https://www.cpdl.org/wiki/index.php/De_profundis_(Giovanni_Pierluigi_da_Palestrina) | These INSANE Vocal Harmonies Just Blew Me Away - published by Charles Cornell

So what is a tritone? Some might argue for the definition up front (some math and other such far too orderly types), but here we are, down here. Humanities more explores questions to which there may not actually be answers, for better or worse.

    $ atonal-util pitch2freq b 
    59      246.94
    $ atonal-util pitch2freq f\'
    65      349.23
    $ echo $((65-59))
    6

The tritone is an interval six semitones apart (assuming the Western 12 equal temperament scale, etc). The frequency ratio is not so "perfect" as one will see with an octave (double or half the frequency) or a perfect fifth, though is far enough apart that there is not beat interference like happens with a minor second. There are various formal definitions of dissonance, Plomp-Levelt is one of them.

    $ perl -MMusic::Tension::PlompLevelt \
      -E '$t=Music::Tension::PlompLevelt->new;' \
      -E 'say "$_\t", $t->pitches(59, 59 + $_) for 0..12'
    0       0.0981837518044547
    1       5.32547525736193
    2       3.2506526669889
    3       2.45410655538872
    4       2.67126926621076
    5       1.70066426607742
    6       3.62133526924154
    7       1.10283838392247
    8       2.87290496201737
    9       1.75847652276244
    10      1.78541904699288
    11      2.10578584176398
    12      0.0827272921844735

By this scale the tritone (6) is worse than a major second (2) but not so bad as a minor second (1). Additionally, lower frequencies need larger intervals to avoid sounding "muddy". That is, the voices can be put closer together in higher octaves (soprano, alto) than they can be for the bass or maybe tenor. Another interesting point is that by this measurement the perfect fourth (5) is more consonant than a sixth (9 and 10) though is considered dissonant in some traditions of music theory, while sixths are consonant.

R. Plomp and W. J. M. Levelt, Tonal consonance and critical bandwidth, J. Acoust. Soc. Amer. 38 (4) (1965), 548-560.

Note that I've mostly been talking about vertical (simultaneous) intervals; minor seconds happen a lot horizontally (sequentially) in melodies, provided that any problematic verticals that result are correctly handled, if the style demands such special handling. A final weird point (for now) is that "mi against fa" makes little sense under modern note counting ("do, a deer, a female deer, re", mi, fa, so what's bad about C, D, uh E against F? Hexachords might be something to read up on.