22 EDO

[William Lynch]

Haven't been here in a while. This is my preferred notation for 22 EDO based on Cam's original proposal, it may be identical. I dunno

ACCIDENTALS
Accidentals come in the apotome and 1/2 apotome.
and have a value of 3 Sharp/Flat
and have a value of 1 Pai/Pao

Symbols are combined to form 2 and 4 values. Double sharp and flat can be combined with and to form values of 5.
and or "Vai/Vao" can be used for 11/8 and 16/11 if desired but I see no reason to complicate things beyond a 5 mapping for a small tuning like 22. I do use this in the intervals though.

INTERVALS

1/22 Pai Unison C C , Minor Second C D
2/22 Pai(minor) Second C D
3/22 Pao (Major) Second C D AKA "The Quill"
4/22 Major Second C D
5/22 Minor third, C E Second is it pai-major second? C D
6/22 Pai-(Minor) Third C E
7/22 Pao (maj) Third C E
8/22 Major Third C E
9/22 Natural Fourth C F
10/22 Vai Fourth C F, Vaoaug fourth? C F (Not sure what the difference besides notation) - Alternatively if using just 5 mapping, these could merely be Pai-fourth and Paoaug-fourth.
11/22 Augmented Fourth?? C F
12/22 Vaidim Fifth C G, Vao Fifth C G Again, not sure why we have both of these.
13/22 Natural Fifth C G
14/22 Vaififth C G , Minor Sixth C A
15/22 Pai-Minor Sixth C A
16/22 Pao-Major Sixth C A
17/22 Major Sixth C A
18/22 Minor Seventh C B
19/22 Pai Minor Seventh C B
20/22 Pao-Major Seventh C B
21/22 Major Seventh C B
22/22 Octave C C

Now I realize that perhaps and could be used for 7:5 but this seems a little overkill. May'be I'm wrong, If we use it, would it be like this? C F How do you pronounce this one btw? Natural may be needless but it's better than perfect.

CHORDS

(Tao) Major = M3 + 5
(Tai) Minor = m3 + 5
Pai Major = m3 + 5
Pao Minor = M3 + 5

Most chords could be spelled out interval by interval, I dunno how to handle the various dim chords.

[Dave Keenan]

Hi William. I'm pleased that you prefer the pitch notation for 22-edo that has always been its standard Sagittal notation. It is shown in figure 8, on page 16 of http://sagittal.org/sagittal.pdf. Although to keep the size of the article down, since it was originally published in the paper journal Xenharmonikôn, we only gave the pure-Sagittal form. Here's a sample of the mixed-Sagittal form of the 22-edo notation, which you would be well aware of, but I include it for the benefit of other readers. It shows successive pitches horizontally and enharmonic spellings vertically. I've shown the Sagittal symbols to the right of the sharps and flats here, which are to the right of the letter name, as is conventional in text.

22-edo:
D	D	D	D	D	D	D
E	E	E	E	E	E	E
F	F	F	F	F	F	F

But when it's used on the staff, the Sagittal would be to the left of the sharp or flat which would be to the left of the notehead. e.g.
O
And you may prefer to put the Sagittal to the left of the sharp or flat in text as well, e.g.

22-edo:
D	D	D	D	D	D	D
E	E	E	E	E	E	E
F	F	F	F	F	F	F

I don't understand why you refer to and as 1/2-apotome symbols. In general they do not have a meaning as any particular fraction of the apotome, and in 22-edo they happen to correspond to 1/3-apotome. Their general meaning is the 5-comma (syntonic comma).

I agree that you should not use or in notating 22-edo pitches, particularly if you intend it to be read by anyone else. I don't recommend using them in naming 22-edo intervals either.

In JI, an 8:11 from C could be notated C:F , and as a composer thinking in terms of 11-limit JI, you might notate its closest 22-edo approximation that way privately. But when communicating with others who may not think that way, and particularly when communicating with a performer who shouldn't have to learn multiple symbols for a single step of 22, it should be notated using the symbol for the lowest prime comma that will do the job, namely C:F .

Using would be even worse, because it would point in the opposite direction to the required alteration in 22-edo. In JI, a 5:7 from C could be notated C:G , but in 22-edo it is C:G .

There is already a separate thread on 22-edo interval names, in the "Interval and Chord names" sub-forum. You can see Cam Taylor's scheme in it here: viewtopic.php?p=47#p47

[Xen-Gedankenwelt]

I agree that you should not use or in notating 22-edo pitches, particularly if you intend it to be read by anyone else. I don't recommend using them in naming 22-edo intervals either.

In JI, an 8:11 from C could be notated C:F , and as a composer thinking in terms of 11-limit JI, you might notate its closest 22-edo approximation that way privately. But when communicating with others who may not think that way, and particularly when communicating with a performer who shouldn't have to learn multiple symbols for a single step of 22, it should be notated using the symbol for the lowest prime comma that will do the job, namely C:F .

Sure, you can reduce the number of accidentals if you want a pure 22-edo notation; just like you can ignore enharmonic spellings or completely omit flats and notate everything with sharps in 12-edo.

But a lot of people think in meantone temperament when composing in 12-edo, so they use enharmonics in a way that allows the piece to be played in another meantone temperament.

Similarly, someone who composes in 22-edo may prefer to think in magic or orwell temperament. However, notating the 11/8 above C as F implies that 55:54 is tempered out, which is inconsistent with those temperaments. This also means that even if the piece is intended to be in magic temperament, but notated as you suggest, then someone else wouldn't be able to perform it on a 19- or 41-edo instrument without re-interpreting it first.
(analogously with orwell temperament and 31- / 53-edo)

Edit: What I'm trying to say is that there are cases where your suggestion makes sense, but there are also other cases where a different choice of accidentals would be better.
However, I agree that there shouldn't be too many different accidentals, if possible.

[Dave Keenan]

Hi Xen,

Although I find your analogy somewhat strained, I think you make a good point -- not that we need multiple notations for 22-edo -- we really want to get away from one-notation-per-composer-per-edo -- but that it would be good to standardise Sagittal notations for the most popular linear temperaments.

You have the honour of being the first, and so far only, person to initiate a thread in the Linear Temperaments sub-forum.

There are two main approaches to linear-temperament notation: MOS nominals, which require non-standard staves, and chain-of-fifth nominals, which do not.

[Dave Keenan]

Sorry William. I think I just found the source of the idea that corresponded the 1/2-apotome in 22-edo. Me!

I mistakenly wrote it, about half way through this recent post in another thread. I've corrected it now.
viewtopic.php?p=415#p415

[William Lynch]

Sorry William. I think I just found the source of the idea that corresponded the 1/2-apotome in 22-edo. Me!

I mistakenly wrote it, about half way through this recent post in another thread. I've corrected it now.
viewtopic.php?p=415#p415

Ah ok lol. So should we use the 11 comma or not? I don't care either way but it does make it more JI intuitive to use vai and vao.

[Dave Keenan]

Ah ok lol. So should we use the 11 comma or not? I don't care either way but it does make it more JI intuitive to use vai and vao.

No, don't use vai or vao in 22-edo scores. Vai and vao often have a secondary role as 1/2-apotome, so the fact that 1 step of 22-edo is not 1/2-apotome but 1/3-apotome is a reason not to use them.

By all means, use them privately to write ratios of 11 while composing, if that helps, but when it comes to presenting your work to others, it is best to use the standard 22-edo notation.

[Xen-Gedankenwelt]

Hi Dave,

Hi Xen,

Although I find your analogy somewhat strained, I think you make a good point -- not that we need multiple notations for 22-edo -- we really want to get away from one-notation-per-composer-per-edo -- but that it would be good to standardise Sagittal notations for the most popular linear temperaments.

You have the honour of being the first, and so far only, person to initiate a thread in the Linear Temperaments sub-forum.

There are two main approaches to linear-temperament notation: MOS nominals, which require non-standard staves, and chain-of-fifth nominals, which do not.

Yes, I agree my analogy was far from perfect.

An alternative notation for 11:8 that I like a lot would be as a tempered 48:35. This requires 385:384 to be tempered out, and works with a lot of linear temperaments like orwell, magic, porcupine, shrutar, astrology, hendecatonic, doublewide, sensa, or other important edos like 17, 19, 24, 27, 31, 41, 46, 53, 72, 87, 94, 99 (some of which admittedly have a very ambiguous mapping in the 11-limit, like 17, 19, 99). 48:35 is a perfect fourth sharpened by 36:35, which can be notated using , .

A counter example is hedgehog, which tempers out 55:54, but not 385:384, so 11:8 could still be notated using , . And superpyth and pajara neither temper out 385:384 nor 55:54, so both mentioned notations for 11:8 would be wrong, but their alternative 11-limit extensions suprapyth and pajarous temper out both.

It seems to me that allowing , to notate a single step could be useful in a lot of cases. But maybe I'm about to open a can of worms here, since one could probably find more and more accidentals that would be useful to add, and it might be hard to draw a line which accidentals to add, and which not.

[Dave Keenan]

Similarly, someone who composes in 22-edo may prefer to think in magic or orwell temperament. However, notating the 11/8 above C as F implies that 55:54 is tempered out, which is inconsistent with those temperaments. This also means that even if the piece is intended to be in magic temperament, but notated as you suggest, then someone else wouldn't be able to perform it on a 19- or 41-edo instrument without re-interpreting it first.

To be performed on a 19 or 41-edo instrument it would be best if it was translated into the standard notation for 19 or 41-edo, which have a one-to-one correspondence between numbers of steps plus or minus and Sagittal symbols.

If, when composing, you are thinking in terms of a linear temperament, then I would encourage you not to use random symbols from the full JI set, but to devise a notation specific to that linear temperament (which we should eventually publish standard versions of). It should have a one-to-one correspondence between multiples of a suitable chroma (defined as a certain number of generators) and Sagittal symbols.

I give an example for Porcupine here:
viewtopic.php?f=6&t=162&p=441#p441

[Xen-Gedankenwelt]

To be performed on a 19 or 41-edo instrument it would be best if it was translated into the standard notation for 19 or 41-edo, which have a one-to-one correspondence between numbers of steps plus or minus and Sagittal symbols.

I wonder about that. On one hand, a minimum number of accidentals can improve readability, and they're easier to memorize. On the other hand, it can be hard to imagine how an interval / chord sounds if you think in a specific linear temperament, but the accidentals "just" represent steps in an equal temperament.
Another example: If an interval is intended to be an 11/9 interval in a specific temperament, and the performer plays it in 22-edo from a sheet that uses a 22-edo notation, they won't know whether the interval is considered to be a (consonant) minor third, or a (dissonant) neutral third, which makes it difficult to emphasize it adequately if a faithful performance is intended.

However, those are just the thoughts of someone (i.e. me) who doesn't have any experience with the performance of microtonal works by other composers, so you should probably take that with a grain of salt.

I think that ideally, the composer would create a digital score where the nominals / accidentals are internally managed as monzos in the intended temperament, and anyone could let the notation software convert those into lower-ranked temperament notations if they want, and choose how the notes / accidentals are represented on the staff.

If, when composing, you are thinking in terms of a linear temperament, then I would encourage you not to use random symbols from the full JI set, but to devise a notation specific to that linear temperament (which we should eventually publish standard versions of). It should have a one-to-one correspondence between multiples of a suitable chroma (defined as a certain number of generators) and Sagittal symbols.

I give an example for Porcupine here:
viewtopic.php?f=6&t=162&p=441#p441

Yes, I wasn't expecting that adding 36:35 accidentals that can represent 33:32 will automagically turn a 22-edo notation with nominals that assume 64:63 is tempered out into a magic / orwell temperament notation. Sorry about my misleading wording! And don't worry, I'm familiar with regular temperaments and their notation; I'm aware that we would need specific linear temperament notations for those; or maybe a planar notation as a work-around (e.g. a 7-note marvel or zeus notation for orwell temperament), if we don't find a good linear notation for a regular staff.

I think I went a bit off-topic here, sorry about that! I think I'll get a bit active in the linear temperament subforum.