A thought about tritones and harmonic series
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- Опубліковано 10 лют 2025
- Here, my Lumatone is set to a layout for 31 tones per octave of quarter-comma meantone, in a Wicki-Hayden arrangement. White is natural, gold is flat, light blue is sharp, orange is double flat, darker blue is double sharp, and purple is the overlap between double flats and double sharps. The circle of fifths is on the / diagonals, while perfect fourths are on the \ diagonals and whole tones go across on the shallow rising diagonals.
I give a little demo of a relationship where we go from the harmonic series over C with an approximately 7/5 tritone between its 5th and 7th harmonic, to a dominant seventh chord on C, with a minor seventh instead. The tritone that minor seventh makes with the major third can either be thought of as (4/3)^2/(5/4) = 64/45 (i.e. the ratio from the major third to the perfect fourth of the perfect fourth), or as 10/7, the octave inversion of the 7/5 tritone. (The comma between 10/7 and 64/45 is 225/224, a quantity known as the septimal kleisma, and it's effectively tempered out by this tuning.)
Then I re-invert that 10/7 tritone, and show how it's part of the harmonic series over Gb.
A question to think about is whether this relationship might be considered part of what the tritone substitution arises from, with 12 equal divisions of the octave erasing the distinction between the two sorts of tritone in play.
I don't have a strong opinion on that yet, and it's not the sort of question likely to have a definite answer, but it's interesting to think about.
#lumatone
This harmonic series would feel incredibly at home on a harp
i love this so much
Amazing!
Ah, beautiful
I so love the (approximate) 7/5 tritone or rather dithird of 31edo (only 1,9 cents i.e. 3,8 mina wrong compared to Just Intonation). 😁
Because it is so beautifully consonant in all its tension! 🥰🎵
So A sharp and B flat aren't the same in 31-tet?
That's right. To name the notes in 31 equal we can use the circle of fifths. Because there are more notes, the circle of fifths takes longer to wrap around, but it happens that 31 equal has a single circle of fifths. So, we can follow the pattern F C G D A E B going up by fifths, and then repeat this with sharps on the right, and flats on the left to get Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B#, and continue this further with double flats on the left and double sharps on the right, and we'll finally end up finding some enharmonics at that point, with Cbb = A## for instance -- the notes 31 steps apart will be the same.
The relationship between major/minor thirds and perfect fifths in 31 equal happens to be the same as it is for 12 equal, if you go 4 steps up on the circle of fifths, you find the major third, and if you go three steps down, you find the minor third. So for instance, G# is a major third above E, and Ab is a minor third above F, and Ab is about 38 cents higher than G#. If you use the wrong one, you get something a bit more dissonant, but there's still some harmony hiding there.
While the major third is an approximation to a 5/4 frequency ratio (to within less than 1 cent), and the minor third is an approximation to the 6/5 frequency ratio (about 6 cents flat), the interval from F to G# is approximately a septimal minor third, a 7/6 ratio (about 4 cents sharp). This interval shows up in a lot of Middle Eastern music, and can be quite beautiful.
The interval from E to Ab on the other hand, is an approximation to a 9/7 septimal major third, though it's about 9 cents out in 31 equal, so I feel it's not quite as convincing, that's not a good enough approximation for 9/7's subtle consonance to shine through. Still, it's possible to make that interval work out nicely in a chord with other notes in context.
I counted a few thoughts here
Ok yeah neato I like the way you're playing with microtones, the effect is nice and it's a very cool thought overall with the switch back and forth from C and F# as the root playing with the harmonic series and their tuning and it's all very complex and you're smart and stuff but you have to tell me what the hell is this device you're using???
Haha, it's a Lumatone! The keys are programmable to produce basically whatever MIDI notes on whatever channel you like, and can change colour. The nicest thing about learning to play it is that you can map notes onto the keyboard so that the same musical interval is always the same physical distance and direction in space. That makes all key signatures play the same way, and means that if you learn to play some chord or scale, you can move it around and play it wherever you want without having to re-learn how your fingers need to go for each root. It also makes it easy to switch to having more notes per octave in various ways, and if your basic intervals are mapped the same way, basically keep the same muscle memory.
@@cgibbard that's rad, thanks!
@@cgibbard did it cost you $5,000 of dollars? Because I want one but not that bad
@@The_SOB_II $4000... and it was probably the best $4000 I ever spent on anything other than my degree in mathematics in terms of the amount of mind-expanding awesomeness I've gotten from it, but yeah, it's quite a harsh barrier to entry. I'm really hoping that in the future they'll be able to build some models that will be more widely accessible.
Is there an easy for dummies way to explain this? I get they are microtones away from each other but is there anything you are trying to accomplish with this?
There isn't exactly a specific goal in mind, but perhaps I can do a better job of explaining the thought I had when I made the video. When we make a tritone substitution, we're replacing a 7 chord for one whose root is a tritone away, and in so doing, keeping the third and minor seventh the same, though they swap roles in the chord. What I'm exploring here is the idea that in some sense, this new tonic was already implied by the tritone in the initial dominant 7th chord.
In a larger tuning system, you can sort of understand a little more about what's making that substitution work (maybe). 12 tone equal temperament obscures the distinction between the 16/9 minor seventh (which it does a decent job of approximating) and 7/4 harmonic seventh (which occurs in the harmonic series, but isn't very well approximated in 12 equal). In a larger tuning system, these become distinct intervals, and when we keep the third and seventh fixed and move the root and fifth, we actually change what type of chord we have, swapping between harmonic seventh chords and dominant seventh chords.
So maybe the essence of what's going on in a tritone sub is that we're taking the tritone which occurs in a dominant 7th chord, inverting it, and finding a new tonic that, to a much better approximation, has the inversion of that interval as part of its harmonic series. For example, in a C7 chord, we have a ((16/9)/(5/4) ~= 10/7) tritone between the E and Bb. If we invert that interval so the Bb is on the bottom (which does nothing to its width in 12 equal, but is actually changing the width of the interval in just intonation or 31 equal), it becomes a 7/5 tritone which occurs naturally in the harmonic series over Gb, as the interval between its 5th harmonic Bb, and 7th harmonic E.
So the E and Bb that occur in a C dominant seventh chord were already in some sense implying Gb to begin with. When you hit the Gb key on your piano, those pitches (with some octave rearrangement) are already happening as overtones.
It works the other way as well: If we start with a C harmonic seventh chord and take the sweeter-than-normal 7/5 tritone between its major third E and harmonic seventh A#, that inverts to a very good approximation to the slightly sharper and more dissonant 10/7 tritone in an F# dominant seventh chord (F# A# C# E).
Not sure if that's any clearer, haha. Hope it helps.
@@cgibbard It does help. This did kind of remind me of the harmonic series so that part makes sense. Experimenting out side of 12 tone equal temperament is a very cool concept and I'd like to hear more songs written in it. Maybe some specially designed instruments such as the natural horn could be used more as well.
Than you for the response! :)
C7
starting sounded creepy as heck