EF see-GEE dee-AY ee-BEE
F C G D A E B
You should include 1/1 in the list, and I think you mean they don't require sharps or flats. They certainly require comma accidentals — for 5/4, 7/4, 11/8 and 13/8.
It's fairly easy to prove it doesn't work for any other 1/1 nominal (with that particular choice of notational commas for the primes 5, 7, 11, 13) by induction on the chain of fifths. It uses every bare nominal, F C G D A E B. So every shift of one position on the chain-of-fifths adds another sharp as you go fifthward, or another flat as you go fourthward.but you wouldn't necessarily get that for other nominals as 1/1... you don't get it for the couple others I checked).
It depends what you mean by "this". In any case, it wouldn't hurt to read Doty.Where can I learn more of this? I'm embarrassed to say I haven't read Doty's Just Intonation Primer yet.
No. If you're willing to choose any of C G D A E to notate your 1/1, there's no reason to go outside of the range 400 Hz to 480 Hz for 1/1. That's ±10%. That's because the largest step between those nominals is a Pythagorean minor third (A:C or E:G), which is 27:32 = 1.185, and 480/400 = 1.2.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?
Whether performers are OK with an A that differs from 440 Hz depends on the nature of their instrument — whether it is made for that reference, or how easily it can be retuned (and possibly whether the performer is cursed with absolute pitch perception ).
Sure. That's what's usually done.Cool idea [using a 3-comma accidental]. Why not just keep adding conventional sharp signs though? Wouldn't it be fairly intuitive to indicate a triple sharp like ?
You've got it wrong. is a shift of 7 fifths, while is a shift of 12 fifths. So F (3×7-3 = 18 fifths from D) would not be F (2×12+7 = 31 fifths from D) but G (12 + 7 - 1 = 18 fifths from D). I chose D as my zero reference here because it is the point of symmetry of the chain of fifths (as conventionally-notated). With 3-comma accidentals the chain could be notated:I feel like I'd rather see that than myself.
E B F C G D A E B F C G E B F C G D A E B F C G E B F C G D A E B F C G
But it's probably not such a great idea, since it uses an accented symbol. And the symbol is based on the 5-comma symbol. The 5-nesses of the core and the accent cancel each other out.
If it aint broke, don't fix it. Leave it.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?
Are you confusing the prime factor notation (which is the topic of this thread) with the Olympian notation (which is what you get from George's spreadsheet)? The Olympian notation has multiple symbols for notating 13, the prime factor notation has only one.Sorry, I should have given an example of what I mean. is the multi-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?
Or are you just asking (slightly off-topic) why George's spreadsheet offers 8:13 as D : B but not as A : F. And conversely why it offers 8:13 as A : F and not D : B?
The reason is, that George believed that one should not combine an apotome symbol with another symbol pointing in the opposite direction if that symbol is larger than a half-apotome. It also stems from the primacy of the pure Sagittal (at least in George's mind). It enforces only one mixed combination for every pure sagittal symbol.
But in the prime-factor notation you have no choice. There is only one symbol for prime 13 so 8:11 has to be D : B and A : F.
Let's use the example of prime 13. Why did we choose instead of to notate 13 in the prime factor notation? With C as 1/1, 13 could be A or A. On the chain of fifths they are Ab Eb Bb F C G D A. We chose A because A is closer to C on the chain of fifths. The thinking behind that was that this would minimise the number of sharps and flats for the "average" choice of 1/1, which we took to be D, the symmetry point.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".
But there is definitely an argument to be had, that the "average" choice of 1/1 is closer to G, in which case there would be fewer sharps and flats on average if we allow the 3 exponents of the prime commas to range from -4 to +7 instead of -5 to +6. This doesn't affect the choice for 13 (or any primes below it), but it does affect 17. It would lead us to prefer the 17-kleisma (2187/2176) to the 17-comma (4131/4096). Dave Ryan prefers the 17-kleisma for other reasons. I actually think it's a good choice too, because it is a visually-simpler symbol. But George preferred to stick to the symmetrical -6 to +6 (which is defacto -5 to +6 for the primes we've looked at so far (I think)).
I think it would be great if the choice between 17C and 17k was a parameter for the prime factor notation calculator, and hence the choice between -5 to +6 and -4 to +7.
That's good to know. I'm sure there are many others like you. And it puts you in a good position to explain the Sagittal system to others, once you feel you understand it sufficiently yourself.[Regarding factors of 3] 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.
That's priceless, for helping me understand where you're coming from.
- 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
You only have to memorise FCGDAEB. There's nearly always a scrap of paper on my desk somewhere with a chain of at least 35 fifths written on it for counting purposes.
Even though Yer tuning doesn't have any pure fifths, it has some near-misses, of 692c, 711c and 718c. Pure fifths are such strong attractors, it could be useful, in effect, to have the notation tell you where they are so you can avoid them.
Oh sure. We've got to get you up to understanding the reasons behind the instructions. Not just following them blindly. I'm sure it will click soon and you'll realise it's simpler than you thought.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!).
One thing you've made me realise is that we really should have the exponent of 3 (and hence the offset along the chain of fifths) shown for each of those prime commas in the first post of this topic.
Dave Ryan told me what to do, but I made the chart. But the interesting thing is that he doesn't agree with -5 to 6 or -6 to 6. In fact he doesn't place any a priori limits like that, but has a complicated algorithm for deciding the best offset for any prime. As mentioned, he prefers +7 for prime 17. That's the first prime where we disagree. (Although I could easily be convinced to go with +7 too).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.
It is absolutely not codified in Sagittal in general. It is mentioned for the first time in the first post of this topic (actually as officially -6 to +6, but defacto -5 to +6).Did I miss somewhere where this concept of -5 to +6 fifths is codified for Sagittal in general?
Yes! You got it.It seems like it's only relevant to figuring out these multi-sagittals.
Yes.Otherwise you're free to use whichever sharped and flatted nominals you want.
No. They should work out any way. They are exactly the exponent of 3 in their monzos. You just seem to have made an arithmetic error for the case of 13 above.My sense is that these multi-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.
The symbol is for the 11-medium diesis up = 33/32 = [ -5 1 0 0 1 >. So to notate 11 you have to cancel out that 3¹ in the comma accidental, by having 3⁻¹ in the combination of nominal and sharp or flat, meaning that this must be one place to the left of the nominal for 1/1, on the chain of fifths. So if 1/1 is C then 11 (or 11/8 if you prefer) is F.
The symbol is for the 13-large-diesis down = 26/27 = [ 1 -3 0 0 0 1 >. So to notate 13 you have to cancel out that 3⁻³ in the comma accidental, by having 3³ in the combination of nominal and sharp or flat, meaning that this must be 3 places to the right of the nominal for 1/1, on the chain of fifths. So if 1/1 is C then 13 (or 13/8 if you prefer) is [thinking "CGDA"] A.
For homework I suggest you work out (or look up) the monzos for all the prime commas listed in the first post, and post them here.