I see that you, like Blaise Pascal, did not have time to make it shorter.
Exactly... I've exposed you to dangerous levels of my writing-to-myself writing...
Those four choices of nominal for 1/1 (C G D A) are indeed special. They are by far the most common choices. C gives a major heptatonic with no sharps or flats. A does the same for the natural minor, and to some it seems logical that the first letter of the alphabet should be 1/1. G gives a complete 13-limit otonality with no sharps or flats, and D gives symmetry, so that any chord and its inversion exchange sharps for flats. There is little incentive to use any other nominals for 1/1 in JI, as you can always just declare your A (or other nominal) to have a frequency other than 440 Hz, or to correspond to some other note in standard 12-equal tuning, similar to what happens with transposing instruments. Minimising accidentals is usually the name of the game.
This chunk is a godsend for me. That's so cool that it works out that way for G (I worked it out and I assume what you mean is that none of 3/2, 5/4, 7/4, 9/8, 11/8, or 13/8 require accidentals when G is 1/1, but you wouldn't necessarily get that for other nominals as 1/1... you don't get it for the couple others I checked). Where can I learn more of this? I'm embarrassed to say I haven't read Doty's Just Intonation Primer yet.
I've heard about setting A to something else than 440 Hz, but usually it's pretty close to 440 Hz. Is the choice of letter really just all about the accidentals? Meaning I could set it to A if that gave the nicest accidentals, but if I actually wanted my piece to be in a different register than that I could just say that A is something crazy like 311.13 Hz, and most performers would be okay with that?
Something that I don't think has been mentioned in this thread so far, is that even the more-distant Pythagoreans in the chain, could be notated using only 12 central Pythagoreans, by using a Sagittal accidental for prime 3, namely
, the accidental for the Pythagorean comma, equivalent to a shift of 12 positions on the chain — an enharmonic shift.
Cool idea. Why not just keep adding conventional sharp signs though? Wouldn't it be fairly intuitive to indicate a triple sharp like
? I feel like I'd rather see that than
myself.
Yes. It would be weird to set any sharped or flatted nominal as 1/1. But a calculator might as well allow it if it's easy to do so. But no need to allow those double-flats or double-sharps, for the reason you gave.
The Sagittal JI calculator spreadsheet does allow setting the double-flats and double-sharps as 1/1, and seems to work alright. Is that something I should fix then, when I do the work to have it help you out with choosing the most conventional options it suggests? Or do you think I may as well leave it that way?
It doesn't matter. We're just minimising the absolute value of the offset from 1/1 along the chain of fifths. I don't understand why you say the symbol for 13 changes depending on the nominal. The point of this notation is to standardise on one symbol for one comma for each prime.
Sorry, I should have given an example of what I mean.
is the prime-factor-sagittal for 13. If I go to the Sagittal JI Calculator and put in 13, leaving C as the default 1/1, indeed I get
as my result (on an A, and technically it's
but let's just ignore the diacritics for now). G and D as 1/1 also agree with
. But if you use A as your 1/1,
is not one of the options; the only option with a bare nominal is F, and the accidental is
. E and B also give this result. (F agrees with C, G, and D). So it's split roughly evenly, 4 against 3. Was
preferred because 4 > 3? Or is there something deeper I'm not getting yet?
I definitely don't understand the statement "We're just minimising the absolute value of the offset from 1/1 along the chain of fifths".
It's not a convention to go from -5 to 6. It's not even a convention to limit it to 12 Pythagorean notes. It just seemed like a good idea to George and I. But I note that this is only the case for notating ratios that do not have any factors of 3. Factors of 3 shift us along the chain in the obvious manner, and so may take us outside of the -5 to +6 range from 1/1.
True. But factors of 3 may result in the other bare nominals being used.
Right. But only when there are no factors of 3.
It's simpler than that. You know right from the start that you're going to put both symbols on, because that's the nature of this notation. It uses one sagittal symbol for every prime factor above 3.
11 =
3⁻¹
13 =
3³
11×13=
3²
with C = 1/1, 3² is a D
So it's
D
So... I do think there may be some fundamental inner workings of Sagittal (and extended H-E systems in general) that I don't quite intuit yet. I'll try to take you through my experience so far. Keep in mind that I don't have a strong musical background -- I've been writing music as long as I can remember, but I've never mastered an instrument or studied it academically.
- Ah, interesting. Commatic alterations. That makes sense. Because each of the primes has a different deviation from standard tuning.
- Okay, each comma can have a bunch of powers of 2 in it. Fine, I understand octave equivalency.
- ...Wait, what...? These commas have a bunch of powers of 3 in them, too! I don't like that. I almost never use fifths. They also aren't pitch class equivalent, so now I'll be limited by that nature. And I'll probably have to memorize the circle of fifths, etc...
- (Ignores JI notation systems for many years, focusing on writing music for computers that don't need to deal with all this nonsense)
- (Decides he'd like to have some of his music performed by humans, starts trying to figure out JI notation systems again)
- I still don't really understand the powers of 3. They seem to magically work out a lot of the time
Part of the reason I included that snippet of me working out 11*13, if only semi-consciously, is that I expected you might say something like this: that I should have just known that right off the bat. But I don't think I get yet why those powers of 3 just work themselves out. I still feel like I have to do it manually. And that's why I'm so uncomfortable with the idea of anything other than -5 to +6 fifths and just following instructions someone who knows what they're doing has told me will work. Either that or I really do just need to learn this by working out enough examples until things click (or by implementing it in code, although I'd rather work out the bugs in my understanding before I try programming them... there'll be room enough for bugs later in just the implementation errors!).
I'm trying to figure out all the thoughts behind the sentences: "It's not a convention to go from -5 to 6. It's not even a convention to limit it to 12 Pythagorean notes. It just seemed like a good idea to George and I." The chart above seems to have been provided by a guy named Dave Ryan. Did I miss somewhere where this concept of -5 to +6 fifths is codified for Sagittal in general? It seems like it's only relevant to figuring out these prime-factor sagittals. Otherwise you're free to use whichever sharped and flatted nominals you want. My sense is that these prime-factor-sagittal might only "work out" with respect to the count of 3's in their monzos as long as you followed this exact -5 to +6 chain.
Where are you getting these counts of three? 3⁻¹ for 11, 3³ for 13? They don't seem to correspond to the monzos for
or
, which have powers of 1 and -5, respectively, not -1 and 3.
Again, I apologize for Just Intonation 101 level questions.
pao 5 / \ 80/81 [ 4 -4 1 ⟩ U+E303
tao 7 f t 63/64 [ -6 2 0 1 ⟩ U+E305
sanai 17 e o 4131/4096 [-12 5 0..0 1 ⟩ U+E342
rai 19 ; r 513/512 [ -9 3 0..0 1 ⟩ U+E390
zai 23 ~ z 736/729 [ 5 -6 0..0 1 ⟩ U+E370
jai 29 ? j 261/256 [ -8 2 0..0 1 ⟩ U+E346
jpao 31 31/32 [ -5 0 0..0 1 ⟩ U+E3A5
phrai 37 37/36 [ -2 -2 0..0 1 ⟩ U+E3A0
satao 47 47/48 [ -4 -1 0..0 1 ⟩ U+E39D
kao 53 53/54 [ -1 -3 0..0 1 ⟩ U+E345
bodai 59 531/512 [ -9 2 0..0 1 ⟩ U+E3F3 U+E30E
binai 61 244/243 [ 2 -5 0..0 1 ⟩ U+E3F2 U+E300 ↑ Short ASCII representation
