## 13-limit JI

### 13-limit JI

Hi, I'm trying to get to grips with notating JI in Mixed Sagittal. Is there no dedicated symbol for the 13-comma (27:26)? As I understand it now has to be notated as 36:35 up from a flat which is confusing. Sure, that's almost the same pitch but that notation doesn't tell you how to find it or where it is in the lattice - which is the whole point of the JI notation systems of Johnston, Sabat etc.

Juhani

Juhani

### Re: 13-limit JI

Let me add that I do know that the recommended notation for 13 is the apotome complement of , i.e. from 27 but I find that even more confusing. The logic of the accidental is not obvious for me. Why is it the horizontal mirror of 36:35? I don't see another example of an accidental whose apotome complement is its horizontal (-cum-vertical) mirror. I thought this was a pure-sagittal symbol only, but it's recommended for mixed Sagittal as well. (By the way, the chart on page 9 of the Sagittal article would be much clearer if the ratios were actually written in the correct order, i.e. the accidentals were not 'up' when the ratios are 'down' - is 33:32, not 32:33 and so on).

- Dave Keenan
- Site Admin
**Posts:**1940**Joined:**Tue Sep 01, 2015 2:59 pm**Location:**Brisbane, Queensland, Australia-
**Contact:**

### Re: 13-limit JI

Hi Juhani,

Sorry to take so long to respond. I apparently still haven't figured out how to get email notification of new posts or private messages in this forum. Good questions.

I should first explain that I consider "35:36" to denote an interval or dyad, not a pitch. And while the pitch 36/35 is quite different from the pitch 35/36, I consider 35:36 and 36:35 to be exactly the same undirected interval or dyad. In the Sagittal documentation, we usually consider an accidental as representing a comma, which is usually considered a small interval rather than a pitch. And for intervals we standardised on the form with the smallest number on the left, because this is the universal convention for triads and larger extended ratios, e.g. 4:5:6 for a JI major triad. We see no reason to make an exception for dyads. For example, why should a JI major third be notated 4:5 when it forms part of a triad, but 5:4 as a bare dyad or interval?

http://sagittal.org/SagittalJI.gif

But you can certainly use Sagittal in the same way as those notations if you wish, i.e. one symbol per prime number above 3. We designed it so it could be used that way too. The symbols to use for that purpose have their descriptions given in

Correction: Table 1 gives two symbols for some primes. If you insist on only one symbol for each prime, you would use the one with the simplest ratio and so would not use at all. You would only use , as 26:27 up.

Sorry to take so long to respond. I apparently still haven't figured out how to get email notification of new posts or private messages in this forum. Good questions.

I should first explain that I consider "35:36" to denote an interval or dyad, not a pitch. And while the pitch 36/35 is quite different from the pitch 35/36, I consider 35:36 and 36:35 to be exactly the same undirected interval or dyad. In the Sagittal documentation, we usually consider an accidental as representing a comma, which is usually considered a small interval rather than a pitch. And for intervals we standardised on the form with the smallest number on the left, because this is the universal convention for triads and larger extended ratios, e.g. 4:5:6 for a JI major triad. We see no reason to make an exception for dyads. For example, why should a JI major third be notated 4:5 when it forms part of a triad, but 5:4 as a bare dyad or interval?

Yes, there is a dedicated symbol for that. It is as shown on the lower right of this chart.Juhani wrote:Hi, I'm trying to get to grips with notating JI in Mixed Sagittal. Is there no dedicated symbol for the 13-comma (27:26)?

http://sagittal.org/SagittalJI.gif

I assume you mean 27:26As I understand it now has to be notated as 36:35 up from a flat which is confusing.

*down*. Although would be a valid spelling, you can see the preferred spelling used to notate the pitch 18/13 in Figure 7 on page 14 of http://sagittal.org/sagittal.pdfOf course Johnston and Sabat et al notations don't have a single symbol for 35:36 or any of the other combinations of primes for which Sagittal has single symbols. They require stacking up more than two accidentals horizontally, often pointing in opposite directions, which can be confusing for much the same reason that is confusing.Sure, that's almost the same pitch but that notation doesn't tell you how to find it or where it is in the lattice - which is the whole point of the JI notation systems of Johnston, Sabat etc.

But you can certainly use Sagittal in the same way as those notations if you wish, i.e. one symbol per prime number above 3. We designed it so it could be used that way too. The symbols to use for that purpose have their descriptions given in

**bold**in Table 1 on page 9. If you prefer to use Sagittal in that manner, then you would notate sharp-26:27-down as and 35:36 as or .Correction: Table 1 gives two symbols for some primes. If you insist on only one symbol for each prime, you would use the one with the simplest ratio and so would not use at all. You would only use , as 26:27 up.

- Dave Keenan
- Site Admin
**Posts:**1940**Joined:**Tue Sep 01, 2015 2:59 pm**Location:**Brisbane, Queensland, Australia-
**Contact:**

### Re: 13-limit JI

In that case, I don't understand why you said it "has to be notated as 36:35 up from a flat ".Juhani wrote:Let me add that I do know that the recommended notation for 13 is the apotome complement of , i.e. from 27

That might best be considered as an accident due to our assignment of commas to flags. Horizontal mirroring has no meaning in Sagittal. We'd prefer that nothing was ever the horizontal mirror of anything else, but this is difficult to maintain. We thought we did well to delay it until prime 13. We have tried to make them distinct by making the one that represents the smaller pitch-alteration noticeably narrrower. versus . I note that horizontally-mirrored flats are commonly used to representbut I find that even more confusing. The logic of the accidental is not obvious for me. Why is it the horizontal mirror of 36:35? I don't see another example of an accidental whose apotome complement is its horizontal (-cum-vertical) mirror.

*half*-flats in other notations, with no "logic" that I can see.

The fact that is the apotome complement of is an unavoidable consequence of having made the complement of .

= +

= ( + ) + ( + )

= ( + ) + ( + ) second and fourth symbols swapped

= +

=

### Re: 13-limit JI

Hi,

thanks! I'll read your explanations carefully and may come back with questions or comments. About the interval notation - "why should a JI major third be notated 4:5 when it forms part of a triad, but 5:4 as a bare dyad or interval?" - I know, but that's a very common convention in any case. But what I mean is simply this: means 33:32 up = multiply by 33:32. means multiply by 32:33. Although in Sagittal symbols the direction is usually obvious (as opposed to Johnston's, for example, where an inverted seven L means 36:35 up), it would still be clearer if the ratio of the comma and its direction (less or more than 1) would match.

Juhani

thanks! I'll read your explanations carefully and may come back with questions or comments. About the interval notation - "why should a JI major third be notated 4:5 when it forms part of a triad, but 5:4 as a bare dyad or interval?" - I know, but that's a very common convention in any case. But what I mean is simply this: means 33:32 up = multiply by 33:32. means multiply by 32:33. Although in Sagittal symbols the direction is usually obvious (as opposed to Johnston's, for example, where an inverted seven L means 36:35 up), it would still be clearer if the ratio of the comma and its direction (less or more than 1) would match.

Juhani

### Re: 13-limit JI

"In that case, I don't understand why you said it "has to be notated as 36:35 up from a flat "."

OK, that was because I assumed understanding the logic of the symbol requires familiarity with pure Sagittal - which I don't want to refer to in my scores - but of course one can simply memorize that symbol.

OK, that was because I assumed understanding the logic of the symbol requires familiarity with pure Sagittal - which I don't want to refer to in my scores - but of course one can simply memorize that symbol.

Last edited by Juhani on Tue Nov 10, 2015 12:31 am, edited 1 time in total.

- Dave Keenan
- Site Admin
**Posts:**1940**Joined:**Tue Sep 01, 2015 2:59 pm**Location:**Brisbane, Queensland, Australia-
**Contact:**

### Re: 13-limit JI

It appears that you and I think of ratios expressed with a colon ":" in very different ways. You seem to treat them as synonymous with fractions expressed with a slash "/", where I do not.

Yes it's very common for people to put the larger number to the left, in the case of a dyad, e.g. 5:4, even though we would never think of writing it E:C, but would always write it C:E. E:C would be 5:8. Because it's so common, I'm happy to accept it written either way, 4:5 or 5:4. But when I write it myself, I choose to use 4:5, as that is more consistent with 4:5:6 and C:E.

So I say that means 32:33 up, just the same as it means 33:32 up, because 32:33 and 33:32 are exactly the same thing in musical terms. I'm sorry that you find that confusing. But in terms of multiplication, it is neither "multiply by 32:33" nor "multiply by 33:32" because multiplication is not defined on colon ratios. means multiply by 33/32. means multiply by 32/33.

It's safe to assume that, in Sagittal, an upward-pointing arrow will always correspond to multiplying by a fraction greater than one, irrespective of the "crazy" way we write our colon ratios.

"Otic"? Do you mean "acoustics"?

Yes, all single-shaft symbols are intended for use in mixed Sagittal, including those greater than a half-apotome, like .

Yes it's very common for people to put the larger number to the left, in the case of a dyad, e.g. 5:4, even though we would never think of writing it E:C, but would always write it C:E. E:C would be 5:8. Because it's so common, I'm happy to accept it written either way, 4:5 or 5:4. But when I write it myself, I choose to use 4:5, as that is more consistent with 4:5:6 and C:E.

So I say that means 32:33 up, just the same as it means 33:32 up, because 32:33 and 33:32 are exactly the same thing in musical terms. I'm sorry that you find that confusing. But in terms of multiplication, it is neither "multiply by 32:33" nor "multiply by 33:32" because multiplication is not defined on colon ratios. means multiply by 33/32. means multiply by 32/33.

It's safe to assume that, in Sagittal, an upward-pointing arrow will always correspond to multiplying by a fraction greater than one, irrespective of the "crazy" way we write our colon ratios.

"Otic"? Do you mean "acoustics"?

Yes, all single-shaft symbols are intended for use in mixed Sagittal, including those greater than a half-apotome, like .

### Re: 13-limit JI

Hi,

"Sabat et al notations don't have a single symbol for 35:36 or any of the other combinations of primes for which Sagittal has single symbols."

Johnston has 5-limit nominals, so actually the single symbol for prime 7 is indeed for 35:36 but then 63:64 is notated with a combination of 35:36 and 80:81.

Even if Sagittal has more symbols for various commas, and they're single symbols, I still read them as following the logic of a combination of other comma symbols (two 's make a and so on). See below.

"it is neither "multiply by 32:33" nor "multiply by 33:32" because multiplication is not defined on colon ratios. means multiply by 33/32. means multiply by 32/33."

OK, I read now that that's how it is in mathematics but I learned the colon ratios differently in school (ie. as ordered, not unordered), and I also have ratio calculator software where a:b is used for a/b.

"Otic"? Do you mean "acoustics"? Sorry, odd typo. Logic.

I assume means 8192:8505 corrected by a 4095:4096 comma? But then the accidental is still derived in a complicated path from a 7-limit interval. That's what I mean by the logic of the accidentals. Assuming I never use the 7-limit comma 8192:8505 I can tell the musicians that this is the accidental used for 13-limit intervals. All the other symbols for low-prime-limit intervals have their own symbol (5-comma, 7-comma, 11-comma), and how signs are combined to a single symbol is often clear ( for two syntonic commas etc.). I would have liked a separate symbol for 13 that is distinct from a combination of lower-limit intervals. I predict musicians won't always be satisfied with learning a symbol by heart but will try to understand it, like me, as a combination of symbols (intervals). And that's not how a 13:8 limit interval is played or sung. A musician playing in just intonation, from a Johnston score, for example, sees a 13-accidental and tunes the 13/8 interval by ear - NOT by tuning two 3/2's, one 5/4 and one 7/4 down, as the Sagittal symbol seems to be saying. But as I said, I can simply tell the musicians that this is used for 13, don't worry why it looks like that. Maybe I'd add the number 13 above the note to remind that you're supposed to find the 13th harmonic here. I don't find that ideal, though.

"The symbols to use for that purpose have their descriptions given in bold in Table 1 on page 9. If you prefer to use Sagittal in that manner, then you would notate sharp-26:27-down as and 35:36 as or ." Table 1 doesn't show examples of those combinations of separate glyphs. It seems that's indeed what I'm after but then I'd need loads of new combined symbols in the font.

"Sabat et al notations don't have a single symbol for 35:36 or any of the other combinations of primes for which Sagittal has single symbols."

Johnston has 5-limit nominals, so actually the single symbol for prime 7 is indeed for 35:36 but then 63:64 is notated with a combination of 35:36 and 80:81.

Even if Sagittal has more symbols for various commas, and they're single symbols, I still read them as following the logic of a combination of other comma symbols (two 's make a and so on). See below.

"it is neither "multiply by 32:33" nor "multiply by 33:32" because multiplication is not defined on colon ratios. means multiply by 33/32. means multiply by 32/33."

OK, I read now that that's how it is in mathematics but I learned the colon ratios differently in school (ie. as ordered, not unordered), and I also have ratio calculator software where a:b is used for a/b.

"Otic"? Do you mean "acoustics"? Sorry, odd typo. Logic.

I assume means 8192:8505 corrected by a 4095:4096 comma? But then the accidental is still derived in a complicated path from a 7-limit interval. That's what I mean by the logic of the accidentals. Assuming I never use the 7-limit comma 8192:8505 I can tell the musicians that this is the accidental used for 13-limit intervals. All the other symbols for low-prime-limit intervals have their own symbol (5-comma, 7-comma, 11-comma), and how signs are combined to a single symbol is often clear ( for two syntonic commas etc.). I would have liked a separate symbol for 13 that is distinct from a combination of lower-limit intervals. I predict musicians won't always be satisfied with learning a symbol by heart but will try to understand it, like me, as a combination of symbols (intervals). And that's not how a 13:8 limit interval is played or sung. A musician playing in just intonation, from a Johnston score, for example, sees a 13-accidental and tunes the 13/8 interval by ear - NOT by tuning two 3/2's, one 5/4 and one 7/4 down, as the Sagittal symbol seems to be saying. But as I said, I can simply tell the musicians that this is used for 13, don't worry why it looks like that. Maybe I'd add the number 13 above the note to remind that you're supposed to find the 13th harmonic here. I don't find that ideal, though.

"The symbols to use for that purpose have their descriptions given in bold in Table 1 on page 9. If you prefer to use Sagittal in that manner, then you would notate sharp-26:27-down as and 35:36 as or ." Table 1 doesn't show examples of those combinations of separate glyphs. It seems that's indeed what I'm after but then I'd need loads of new combined symbols in the font.

- Dave Keenan
- Site Admin
**Posts:**1940**Joined:**Tue Sep 01, 2015 2:59 pm**Location:**Brisbane, Queensland, Australia-
**Contact:**

### Re: 13-limit JI

Ah, yes. Thank you for reminding me of that.Juhani wrote:Johnston has 5-limit nominals, so actually the single symbol for prime 7 is indeed for 35:36 but then 63:64 is notated with a combination of 35:36 and 80:81.

Yes. That is one of the most powerful features of Sagittal.Even if Sagittal has more symbols for various commas, and they're single symbols, I still read them as following the logic of a combination of other comma symbols (two 's make a and so on).

Yes. That is one literal reading. But I tend to read it like this: That's a 35 or 13 symbol, but they have added a right accent, so for some reason theyI assume means 8192:8505 corrected by a 4095:4096 comma? But then the accidental is still derived in a complicated path from a 7-limit interval. That's what I mean by the logic of the accidentals.

*really*care about that 0.4 cent difference (which usually no one can hear) and they want an

*extremely*precise 13.

I would not recommend using the accented symbol for 13, as in any given situation only one of the 35 or 13 meanings will be tunable by ear, and only one of them will make a simple ratio with the local tonic or the root of the chord, and if you got the wrong one, no one would know anyway since it's only 0.4 of a cent. And I understand that it's not really the kind of symbol you had in mind when you asked if there was a dedicated symbol for 13. So in that sense, no, there is no dedicated symbol for 13.

Yes, certainly. That would be a good thing to do. You could even define as a the 35-comma (35:36) and as the 13-comma (26:27).Assuming I never use the 7-limit comma 8192:8505 I can tell the musicians that this is the accidental used for 13-limit intervals.

I agree with all that you say here. But you can see that we had a difficult job, to satisfy everyone. Some people say there are too many symbols. By ignoring the tiny interval of 4095:4096, and other even smaller intervals, we were able to achieve a significant reduction in the number of symbols needed, and hence the number of different flag types that needed to be distinguished. Then we added the right accents for those whoAll the other symbols for low-prime-limit intervals have their own symbol (5-comma, 7-comma, 11-comma), and how signs are combined to a single symbol is often clear ( for two syntonic commas etc.). I would have liked a separate symbol for 13 that is distinct from a combination of lower-limit intervals. ... But as I said, I can simply tell the musicians that this is used for 13, don't worry why it looks like that. Maybe I'd add the number 13 above the note to remind that you're supposed to find the 13th harmonic here. I don't find that ideal, though.

*thought*they really needed to distinguish between things that were 0.4 cents apart.

I note that is not a combination of a 7-comma flag and a 5-comma flag. is a 7:11-comma and is a 55-comma. But of course the result is still logically a 35-comma and you just have to learn that it's also the 13-comma (26:27).

Although we wanted to show that Johnston-like one-symbol-per-prime usage was possible for those whoTable 1 doesn't show examples of those combinations of separate glyphs. It seems that's indeed what I'm after but then I'd need loads of new combined symbols in the font.

*thought*they needed it, we didn't really want to encourage it, because we don't believe it is the best way. For example, while something like is fine, what about the poor performer who sees . It is far from obvious what direction the overall alteration is. So we defined = . There's even a kind of visual logic to that, and it's then obvious that it represents a small upward alteration.

### Re: 13-limit JI

"But you can see that we had a difficult job, to satisfy everyone. Some people say there are too many symbol." I understand this. But instead of the cumbersome clustering of symbols that you find in JI notation systems such as Johnston's there are now all these unique symbols to memorize. By reading a page of instructions from a Johnston score, or that of Sabat, one is instantly capable of notating and reading an almost arbitrarily complex just interval (at least up to 31-prime-limit beyond which Johnston never went), but I've come back to the Sagittal manual countless times over the years, and while this is the first time I haven't immediately given up, all the dedicated symbols for different commas of the same p-limit do frustrate me and I still have no idea how to notate even such a simple interval as the seventh harmonic of the seventh harmonic (49/32) which in Johnston would be two 7's horizontally and in Sabat two 7's vertically placed - nothing to memorize there.

Some years ago I wrote a piece for chamber choir in 11-limit JI, using Johnston's notation. I made rehearsal tapes for the conductor and met with him, but to teach the notation to the choir (they had no previous experience of extended just intonation, only modernist works with quarter-tones) took only an initial meeting of 45 minutes. When I later went to hear rehearsals, intonation was discussed, of course, but not the notation - even with the admittedly confusing 81/80 symbol logic of Johnston's 5-limit nominals. I wonder if this would be the case with Sagittal. Perhaps, judging from the success of Jacob Barton's praxis camps.

I do agree with you, I wouldn't want nor need dedicated symbols for very many prime limits, either, but even if Partch stopped at 11, many just intonation composers do use 13 and expect it to be tuned by ear - hence the need for the notation to show it clearly.

"You could even define as a the 35-comma (35:36) and as the 13-comma (26:27)." This is fine - but need I tell you how much more easily Johnston's symbol for the 13th harmonic, which is, well, 13, is memorized than a symbol that looks very much like another one that designates a completely different (7-limit) interval?

"It's safe to assume that, in Sagittal, an upward-pointing arrow will always correspond to multiplying by a fraction greater than one" - & "what about the poor performer who sees . It is far from obvious what direction the overall alteration is" Yes, this is indeed an improvement on Johnston's accidentals and their combinations.

It worries me a little bit that no-one has taken to take the trouble of transcribing even a single movement of the existing JI masterpieces - Johnston's string quartets - to Sagittal. A JI cello piece of mine, originally written in Johnston's notation is going to be published and I was thinking of including additional versions in both Sabat's system (popular in Germany) and Mixed Sagittal but for the latter, I have to see if I can notate even the fairly simple ratios that are used in it, and how it looks like.

Some years ago I wrote a piece for chamber choir in 11-limit JI, using Johnston's notation. I made rehearsal tapes for the conductor and met with him, but to teach the notation to the choir (they had no previous experience of extended just intonation, only modernist works with quarter-tones) took only an initial meeting of 45 minutes. When I later went to hear rehearsals, intonation was discussed, of course, but not the notation - even with the admittedly confusing 81/80 symbol logic of Johnston's 5-limit nominals. I wonder if this would be the case with Sagittal. Perhaps, judging from the success of Jacob Barton's praxis camps.

I do agree with you, I wouldn't want nor need dedicated symbols for very many prime limits, either, but even if Partch stopped at 11, many just intonation composers do use 13 and expect it to be tuned by ear - hence the need for the notation to show it clearly.

"You could even define as a the 35-comma (35:36) and as the 13-comma (26:27)." This is fine - but need I tell you how much more easily Johnston's symbol for the 13th harmonic, which is, well, 13, is memorized than a symbol that looks very much like another one that designates a completely different (7-limit) interval?

"It's safe to assume that, in Sagittal, an upward-pointing arrow will always correspond to multiplying by a fraction greater than one" - & "what about the poor performer who sees . It is far from obvious what direction the overall alteration is" Yes, this is indeed an improvement on Johnston's accidentals and their combinations.

It worries me a little bit that no-one has taken to take the trouble of transcribing even a single movement of the existing JI masterpieces - Johnston's string quartets - to Sagittal. A JI cello piece of mine, originally written in Johnston's notation is going to be published and I was thinking of including additional versions in both Sabat's system (popular in Germany) and Mixed Sagittal but for the latter, I have to see if I can notate even the fairly simple ratios that are used in it, and how it looks like.

Last edited by Juhani on Tue Jan 12, 2016 2:25 am, edited 2 times in total.