Quoting Gro-Tsen:
"Anyway, for better explanations about this, I refer to JCB's blog post here:
https://johncarlosbaez.wordpress.com/2023/11/06/just-intonation-part-2/
Can you spot how his basic parallelogram appears as an approximate period in my diagram?"
(4/n, n = 4)
Quoting Gro-Tsen:
"And the hexagon to the southeast of any given hexagon is one minor third above (frequency ×6/5) or equivalently, one major sixth below (×3/5); and the one to the northwest is one major sixth above (×5/3) or one minor third below.
The entire grid is known as a “Tonnetz”, as explained in Wikipedia http://en.wikipedia.org/wiki/Tonnetz — except that unfortunately my convention (and JCB's) is up-down-symmetric wrt the one used in the Wikipedia illustration. 🤷
Mathematically, if we talk about the log base 2 of frequencies, modulo 1, we can say that one step to the east adds log₂(3), and one step to the northeast adds log₂(5) (all values being taken modulo 1).
Since log(2), log(3) and log(5) are linearly independent over the rationals (an easy consequence of uniqueness of prime factorization!), NO two notes in the diagram are exactly equal. But they can come very close! And this is what my colors show.
Black hexagons are those which distant from the reference note by more than 1 halftone (where here, “halftone” refers to exactly 1/12 of an octave in log scale), or 100 cents. Intervals between 100 and 50 cents are colored red (bright red for 50 cents), intervals between 50 and 25 cents are colored red-to-yellow (with bright yellow for 25 cents), intervals between 25 and 12.5 cents are colored yellow-to-white (with pure white for 12.5 cents), and below 12.5 cents we move to blue."
(2/n)
Quoting Gro-Tsen:
"(Yes, this is a purely arbitrary color gradient, I didn't give it much thought. It's somewhat reminiscent of star colors.) Anyway, red-to-white are good matches, and white-to-blue are pretty much inaudible differences, with pure blue representing an exact match, … except that the center hexagon has been made green instead so we can easily tell where it is (but in principle it should be pure blue).
The thing about the diagram is that it LOOKS periodic, and it is APPROXIMATELY so, but not exactly!
Because when you have an approximate match (i.e., some combination of fifths and thirds that is nearly an integer number of octaves), by adding it again and again, the errors accumulate, and the quality of the match decreases.
For example, 12 hexagons to the east of the central one, we have a yellow hexagon (quality: 23.5 cents), because 12 perfect fifths gives almost 7 octaves. But 12 hexagons east of that is only reddish (quality: 46.9 cents) because 24 fifths isn't so close to 14 octaves.
For the same reason that log(2), log(3) and log(5) are linearly independent over the rationals, the diagram is never exactly periodic, but there are arbitrarily good approximations, so arbitrarily good “almost periods”.
An important one in music is that 3 just fifths plus 1 minor third, so, 3 steps east and 1 step southeast in my diagram gives (2 octaves plus) a small interval with frequency ratio of 81/80 (that's 21.5 cents) that often gets smeared away when constructing musical scales."
(3/n)
@johncarlosbaez
By looking at the Gro-Tsen coloured Tonnetz one feels to remark that a better "schisma" might be obtained by going one step north and 13 steps north east. Or in ratios (the software doesnt allow to write out highcommas so I write it out) 6/5 to the power of 13 times 3/2 equals 16.0489. Divided by 16 this gives 1.0030613. If this hasnt been given a name yet I'd like to dub it "charismatic schisma".
I posted this also on your blog.
@nad - Hi there!
Thanks, cool! I need to check out what this interval is. I'll talk about it on the blog.
@johncarlosbaez
3/n
Of course one could also try to fill in the rest tones of the 4 twelvetone octaves, in order to get towards a full „charismatic twelvetone tuning“ . This seems however not so straight forward. In order to fill in quite some „omitted tones“ one could tune first the rest of the domain e.g. meander „from above within the domain“ , ie. start at the just fifth above , i.e. at G and go to g‘‘. One obtains: C→7→[G-Bb-db-e<-5<-B-d-f-ab→7→e‘b-g‘b-a‘-c‘‘<-5<-g‘-b‘b-d‘‘b-e‘‘-g‘‘]→-5→C. One sees that e, c‘‘and e‘‘ are double. There one probably has to make a decision, which to take, or take something between the respective two. Some hexagons have a double layering, like being the A in the first tune path and the B in the second. Taking the fifth below one gets still not all notes of the 4 octaves „charismatically tuned“, but one could probably use the other fifths to fill in the gaps. Or one could simply go up and down octaves from the above from the above „charismatic scale“. a.s.o.
@johncarlosbaez
2/n
Now follows a scale, that is a bit more interesting than going up the thirds and then jump by a fifth: If one denotes Ab as A flat etc. and denotes a jump by a minor third (3 halftones, factor 6/5) by „-“ and denoted a jump by a just fifth (7 halftones, factor 3/2) by „– > 7 – >“ and denoted a jump back by a just fourth „< - 5 < - „ (5 halftones back, factor ¾) then the tuning goes e.g. C-Eb-Gb-A - >7 - > e-g-bb-d‘b < - 5 < - ab-b-d‘-f‘ - >7 - > c‘‘-e‘‘b-g‘‘b-a‘‘-c‘‘‘. So between C and c‘‘‘ there is the charismatic schisma, which is connected with the fact that the pure 4 octave jump has to be „distorted“, by multiplying the frequency of C with 16.0489, instead of 16. I would like to call this scale a „charismatic scale“. As one can see this scale meanders through the 3 thirdcycles D-F-Ab-B-D, C-Eb-Gb-A-C, and Db-E-G-Bb-Db that define the usual Tonnetz domain. So one could also think about „flattening“ this scale to one octave, by omitting either C-Eb-Gb-A at the beginning or c‘‘-e‘‘b-g‘‘b-a‘‘ at the end, or by finding a „compromise“. But unfortunately one looses a lot of „charisma“ in that process, that is I could imagine that the step from B to a higher octave C could sound strange.
@johncarlosbaez Yeah I reply there and post some copied content here.
1/n
OK now thinking a bit further one sees that the charismatic schisma allows for an interesting tuning that gives a different tuning per octave. This comment here gets now a bit out of proportion but let me cast that out more. I actually, just this week, wanted to learn a bit more about Gregorian modes, as a free time project, which shall explain my sudden interest in this topic.
The charismatic schisma defines a small 2 rows-14 column fundamental domain.
One can get tuning scales into that domain, which could eventually -with a grain of salt- be seen as „modes“, as, with 4 octaves, they are roughly within the usual singing range.
A tuning scale meanders in the fundamental domain.
Quoting Gro-Tsen:
"Anyway, for better explanations about this, I refer to JCB's blog post here:
https://johncarlosbaez.wordpress.com/2023/11/06/just-intonation-part-2/
Can you spot how his basic parallelogram appears as an approximate period in my diagram?"
(4/n, n = 4)
@johncarlosbaez
4/n
As it turned out the tones in the charismatic scale in comment 2/n obtained by flattening with a power of two are far off their corresponding names. In fact the are about "double as far" from the base tone. As a consequence I noticed that there is yet another schisma. It is not as close to 1 as the above charismatic schisma, but almost. Since it is related to minor thirds and the "continued" (keep meandering) charismatic scale I dub this schisma "charismatic diesis". In short the charismatic diesis is 2 to the power of ten divided by 6/5 to the power of 38, which is 1.003257943. I constructed a scale based on this charismatic diesis that looked more or less straight forward. You can hear it at https://soundcloud.com/cat-caspari/microtonal
@johncarlosbaez Are you familiar with the "harmonic table" / "melodic table" note layout for keyboards? The modern version popularized by C-Thru has fifths going north and major thirds going northeast-ish, but it's been floating around in various forms since Euler since it's just a reorientation of the Tonnetz diagram. I own one with the modern orientation from when I was trying my hand at composition.
@heptapodEnthusiast - I've seen them on YouTube but I've never tried one! In an alternative reality I would be a huge music gearhead. In this reality I have an electronic piano 2500 miles away and I've never figured out how to use a DAW. 😢 I just have too much fun doing math instead.
@johncarlosbaez I could see you getting into algorithmic music construction!
thanks for sharing this heptapod, had never seen it... it is making me wonder about a wind synth instrument with some kind of similar interface. I've been playing an EWI for a while and have been dreaming about alternate control methods and ways to futz with temperament and microtones in improvisation @heptapodEnthusiast
