Notes on Sagittal notation for regular temperaments

It looks like it's about time for another discussion on applying Sagittal notation to regular temperaments. There are various factors involved in deciding how to notate a temperament, which may conflict with each other.

* Minimizing the number of unique symbols (for a typical number of notes)

* Minimizing the number of accents that need to be used

* Representing simpler ratios when more than one possibility is available

* Representing more accurate ratios when more than one possibility is available

* Compatibility with a standard ET notation for any ET consistent with the temperament

Some examples from a 7-limit meantone:

It may be useful to represent G B\! as a major third. But this introduces a new accidental \! which is not otherwise needed (in fact, it's tempered out in meantone).

The note G# (+3, -6) is a 7/5 above D. But writing this as G.||) requires an accent. Instead, writing it as G||\ (5/3 above E) may be preferable.

The basic 7-note scale is easy to notate since it's a chain of fourths: B E A D G C F. Beyond that, !!/ (or any accidental that's equivalent in meantone) can be used to extend the series.

B||\ E||\ A||\ D||\ G||\ C||\ F||\
B E A D G C F
B!!/ E!!/ A!!/ D!!/ G!!/ C!!/ F!!/

Where to go beyond there? Two possible options for a (-5, +12) accidental are 36/35 /|) and 64/63 |) . 5-limit meantone would need something like 128/125 .//| for this.

Many temperaments, such as porcupine, have gaps between the notes in the chain of fifths or fourths. It may be convenient to label these notes in such a way that the common major and minor thirds are recognizable.

A/| B C||\
D/| E F||\
G/| A B\!
C/| D E\!
F/| G A\!
B!!/ C D\!
E!!/ F G\!

With one extra note ( which could be G||\ or A!!/ ), this gives you a notation for porcupine[22].

Things can get complicated when there is more than one chain of fifths. Here is one possible notation for lemba[26].

C D|) | F.||) A!!/
F!) G A|) | B\! C.||) E!!/
C!) D E|) | F||\ G.||) B!!/
G!) A B|) | C||\ E'!!) F/|
D!) E | G||\ B'!!)

Herman Miller wrote:
> It looks like it's about time for another discussion on applying > Sagittal notation to regular temperaments. There are various factors > involved in deciding how to notate a temperament, which may conflict > with each other.
> > * Minimizing the number of unique symbols (for a typical number of notes)
> > * Minimizing the number of accents that need to be used

What's an accent?

> * Representing simpler ratios when more than one possibility is available

I don't really care about this. The goal is for the minimum number of symbols with a consistent range. Ideally all from the Spartan set, hopefully none outside Athenian.

> * Representing more accurate ratios when more than one possibility is > available

Rather than this, consistent range of symbols. So if you list all the symbols used you can put them in order of interval size with (ideally) no overlap for the most common tunings of the most important temperament classes.

> * Compatibility with a standard ET notation for any ET consistent with > the temperament

Most importantly for the best-tuned consistent ETs. And lets remember that the current ET standards aren't set in stone.

> Some examples from a 7-limit meantone:
> > It may be useful to represent G B\! as a major third. But this > introduces a new accidental \! which is not otherwise needed (in fact, > it's tempered out in meantone).

Maybe some musicians would find this helpful. But we can leave it up to them to make whatever alterations to the simple notations they require.

> The note G# (+3, -6) is a 7/5 above D. But writing this as G.||) > requires an accent. Instead, writing it as G||\ (5/3 above E) may be > preferable.

At least preserve the mapping from # to a two shaft symbol.

> The basic 7-note scale is easy to notate since it's a chain of fourths: > B E A D G C F. Beyond that, !!/ (or any accidental that's equivalent in > meantone) can be used to extend the series.
> > B||\ E||\ A||\ D||\ G||\ C||\ F||\
> B E A D G C F
> B!!/ E!!/ A!!/ D!!/ G!!/ C!!/ F!!/
> > Where to go beyond there? Two possible options for a (-5, +12) > accidental are 36/35 /|) and 64/63 |) . 5-limit meantone would need > something like 128/125 .//| for this.

I'd naturally use |) (the same symbol as 2/72) for 31-equal. For other meantones I suggest a different symbol for tunings either side of 31.

> Many temperaments, such as porcupine, have gaps between the notes in the > chain of fifths or fourths. It may be convenient to label these notes in > such a way that the common major and minor thirds are recognizable.
> > A/| B C||\
> D/| E F||\
> G/| A B\!
> C/| D E\!
> F/| G A\!
> B!!/ C D\!
> E!!/ F G\!
> > With one extra note ( which could be G||\ or A!!/ ), this gives you a > notation for porcupine[22].

Yes. Preserve major and minor thirds where possible. I'm not really familiar with porcupine.

> Things can get complicated when there is more than one chain of fifths. > Here is one possible notation for lemba[26].
<snip>

Or lemba. To what extent would the difficult temperament classes fit a decimal staff?

Graham

Graham Breed wrote:
> Herman Miller wrote:
>> It looks like it's about time for another discussion on applying >> Sagittal notation to regular temperaments. There are various factors >> involved in deciding how to notate a temperament, which may conflict >> with each other.
>>
>> * Minimizing the number of unique symbols (for a typical number of notes)
>>
>> * Minimizing the number of accents that need to be used
> > What's an accent?

A mark added to the left or right side of the Sagittal symbol. On the left, it represents a schisma, and the optional right-hand accent mark represents an interval around the size of 4096/4095.

>> Things can get complicated when there is more than one chain of fifths. >> Here is one possible notation for lemba[26].
> <snip>
> > Or lemba. To what extent would the difficult temperament > classes fit a decimal staff?

If the 10 nominals don't have to be equally spaced, anything with a 10-note MOS could be adapted: blacksmith, pajara, injera, negrisept, lemba, magic, doublewide, beatles, miracle, muggles, and of course decimal. Of the temperaments having two chains of fifths, hedgehog is the main one that doesn't have a 10-note MOS.

I haven't yet examined the temperaments in much detail to see what issues might come up when you try to minimize the number of different accidentals or optimize for recognizable thirds. The last I looked at notating temperaments, I was just trying to find _some_ notation that works, not necessarily the best one. There's still a couple of odd temperaments (vishnu, luna) that continue to resist Sagittal notation, but for the most part notation is at least possible.

Herman Miller wrote:
> Graham Breed wrote:
> >>Herman Miller wrote:
>>
>>>* Minimizing the number of accents that need to be used
>>
>>What's an accent?
> > A mark added to the left or right side of the Sagittal symbol. On the > left, it represents a schisma, and the optional right-hand accent mark > represents an interval around the size of 4096/4095.

Okay. It may be a good idea to use them, but no need to minimize them.

>>>Things can get complicated when there is more than one chain of fifths. >>>Here is one possible notation for lemba[26].
>>
>><snip>
>>
>>Or lemba. To what extent would the difficult temperament >>classes fit a decimal staff?
> > If the 10 nominals don't have to be equally spaced, anything with a > 10-note MOS could be adapted: blacksmith, pajara, injera, negrisept, > lemba, magic, doublewide, beatles, miracle, muggles, and of course > decimal. Of the temperaments having two chains of fifths, hedgehog is > the main one that doesn't have a 10-note MOS.

They don't have to be equally spaced but the natural scale should at least be proper. The decimal scale in miracle is very close to equal. I've adapted decimal notation to magic, because it's what I was most familiar with at the time, but it's no more consistent than fitting it to the usual staff.

> I haven't yet examined the temperaments in much detail to see what > issues might come up when you try to minimize the number of different > accidentals or optimize for recognizable thirds. The last I looked at > notating temperaments, I was just trying to find _some_ notation that > works, not necessarily the best one. There's still a couple of odd > temperaments (vishnu, luna) that continue to resist Sagittal notation, > but for the most part notation is at least possible.

Are those the very accurate temperaments?

Graham

Graham Breed wrote:
> They don't have to be equally spaced but the natural scale > should at least be proper. The decimal scale in miracle is > very close to equal. I've adapted decimal notation to > magic, because it's what I was most familiar with at the > time, but it's no more consistent than fitting it to the > usual staff.

Pajara[10], lemba[10], decimal[10], and supersharp[10] are strictly proper. Injera[10] and doublewide[10] are not proper, but they are Constant Structures.

>> I haven't yet examined the temperaments in much detail to see what >> issues might come up when you try to minimize the number of different >> accidentals or optimize for recognizable thirds. The last I looked at >> notating temperaments, I was just trying to find _some_ notation that >> works, not necessarily the best one. There's still a couple of odd >> temperaments (vishnu, luna) that continue to resist Sagittal notation, >> but for the most part notation is at least possible.
> > Are those the very accurate temperaments?

Yes, those are the two "bonus" 5-limit temperaments in Paul's "Middle Path" paper. Also known as semisuper and hemithirds. Vishnu/semisuper is a temperament with two periods per octave -- but unlike most half-octave temperaments where the half-octave is an approximation of something simple like 7/5, the simplest approximation for the vishnu half-octave is 78125/55296. The period of luna/hemithirds is an octave, but the generator is an approximation of 262144/234375.

Herman Miller wrote:

> Pajara[10], lemba[10], decimal[10], and supersharp[10] are strictly > proper. Injera[10] and doublewide[10] are not proper, but they are > Constant Structures.

What I'm getting at is: how many different staves would a musician have to learn to get the mainstream temperament classes? 7 note diatonics are obviously the mainstream. 10 notes also works because they fit miracle well, and by extension 11-limit harmony. It's also natural for pajara and other diaschismics, and neutral third generated scales.

12 is an obvious number but chromatic musicians these days aren't clamoring for a new system. Divisions of 12 are always written on the conventional staff.

Maybe 9 would be useful. It covers the equal temperament behind ennealimmal. It should do orwell as well. And how about mavila? If it's the standard 9-limit staff, perhaps magic will fit it. It has a strong 3 note MOS so somehow we need 3 of them.

Probably learning a new set of nominals is as hard as learning a new staff anyway. So we have to consider how complicated it gets to use notate the nominals relative to the standard for that number of notes.

> Yes, those are the two "bonus" 5-limit temperaments in Paul's "Middle > Path" paper. Also known as semisuper and hemithirds. Vishnu/semisuper is > a temperament with two periods per octave -- but unlike most half-octave > temperaments where the half-octave is an approximation of something > simple like 7/5, the simplest approximation for the vishnu half-octave > is 78125/55296. The period of luna/hemithirds is an octave, but the > generator is an approximation of 262144/234375.

I expect these would sort themselves out as and when somebody wants to use them. Would you ever expect acoustic musicians to distinguish them from 5-limit JI?

Graham

Graham Breed wrote:
> Herman Miller wrote:
> >> Pajara[10], lemba[10], decimal[10], and supersharp[10] are strictly >> proper. Injera[10] and doublewide[10] are not proper, but they are >> Constant Structures.
> > What I'm getting at is: how many different staves would a > musician have to learn to get the mainstream temperament > classes? 7 note diatonics are obviously the mainstream. 10 > notes also works because they fit miracle well, and by > extension 11-limit harmony. It's also natural for pajara > and other diaschismics, and neutral third generated scales.

Depends on what you consider "mainstream temperament classes". As a first approximation, I assume that would be some subset of the 5-limit and 7-limit temperaments in Paul's "Middle Path" paper? There could of course be 11- or 13-limit temperaments that aren't listed there (hemiw�rschmidt comes to mind), but that's a good place to start. Hemiw�rschmidt by the way has a 7-note MOS, with one very small 36.7 cent step, but it still manages to be strictly proper. I don't like MOS with large/small step ratios much greater than 2 or 3, but the seventh note could just be omitted. A set of 7 nominals with accidentals for -6, -12, +19, and +25 would get you a pretty good range of notation for hemiw�rschmidt.

Not all temperaments have a 7- or 10-note MOS, though. Examples (all except the last two 7-limit ones are listed in the Middle Path paper):

5-limit:
father (5, 8)
bug (5, 9)
augmented (6, 9, 12)
dimipent (8, 12)
sensipent (5, 8, 11)
compton (12)
orson (5, 9)

7-limit:
dimisept (8, 12)
august (6, 9, 12)
semaphore (5, 9)
augene (6, 9, 12)
catler (12)
hedgehog (6, 8)
sensisept (5, 8, 11)
cynder (5, 6, 11)
orwell (5, 9)
ennealimmal (9)
superpelog (5, 9)
gorgo (5, 6, 11)

> 12 is an obvious number but chromatic musicians these days > aren't clamoring for a new system. Divisions of 12 are > always written on the conventional staff.

Compton, catler, and the few scales based on 1/3 or 1/4 octave could use 12-ET notation.

> Maybe 9 would be useful. It covers the equal temperament > behind ennealimmal. It should do orwell as well. And how > about mavila? If it's the standard 9-limit staff, perhaps > magic will fit it. It has a strong 3 note MOS so somehow we > need 3 of them.

9 does come up quite a bit in the list above. Mavila can be notated with 7 nominals, as long as you don't use the conventional sharps and flats (which are reversed in mavila). I don't see any need to go beyond 23 notes for mavila notation, and that's only because it's convenient for 23-ET (otherwise no more than 16 notes would likely be needed).

That leaves father (which only has 8 usable notes anyway, and probably no one beyond me is likely to write for it), sensi, hedgehog, cynder, and gorgo as temperaments without a 7-, 9-, 10-, or 12-note MOS. Of these, sensi and cynder (aka mothra) are probably the most important.

> Probably learning a new set of nominals is as hard as > learning a new staff anyway. So we have to consider how > complicated it gets to use notate the nominals relative to > the standard for that number of notes.

Learning how the nominals work for each different temperament does seem to be one of the big issues. I used a lemba notation with the 10 nominals D L T U N O P I J R. It's usable, but you have to relearn all the interval relationships. A D major tetrad is D U Pb J. You can think of U as a sort of F# and I as a sort of Bb -- so, a fifth below U is I#, and that happens to work. But if you put a series of lemba fifths on a 10-nominal staff, and then compare with a series of miracle fifths, the pattern won't be anything alike.

In a sagittal notation for lemba based on a chain of fifths, on the other hand, many of the intervals are easily recognizable. Some, like G!) B!!/ aren't as obvious (this is a major third), but at least it looks like some kind of third. If you learn that !) and )!!( are equivalent in lemba, it becomes clearer. I suspect that relearning enharmonic equivalents (as in 19-ET where Cx = Db) may be easier than relearning the whole system of nominals, accidentals, and intervals.

>> Yes, those are the two "bonus" 5-limit temperaments in Paul's "Middle >> Path" paper. Also known as semisuper and hemithirds. Vishnu/semisuper is >> a temperament with two periods per octave -- but unlike most half-octave >> temperaments where the half-octave is an approximation of something >> simple like 7/5, the simplest approximation for the vishnu half-octave >> is 78125/55296. The period of luna/hemithirds is an octave, but the >> generator is an approximation of 262144/234375.
> > I expect these would sort themselves out as and when > somebody wants to use them. Would you ever expect acoustic > musicians to distinguish them from 5-limit JI?

I expect only a few electronic musicians with an interest in exploring vast areas of the 5-limit lattice (like Gene with his 99-ET piece that uses all 99 notes) will even consider using these.

Herman Miller wrote:
> Graham Breed wrote:
> >>What I'm getting at is: how many different staves would a >>musician have to learn to get the mainstream temperament >>classes? 7 note diatonics are obviously the mainstream. 10 >>notes also works because they fit miracle well, and by >>extension 11-limit harmony. It's also natural for pajara >>and other diaschismics, and neutral third generated scales.
> > Depends on what you consider "mainstream temperament classes". As a > first approximation, I assume that would be some subset of the 5-limit > and 7-limit temperaments in Paul's "Middle Path" paper? There could of > course be 11- or 13-limit temperaments that aren't listed there > (hemiw�rschmidt comes to mind), but that's a good place to start. > Hemiw�rschmidt by the way has a 7-note MOS, with one very small 36.7 > cent step, but it still manages to be strictly proper. I don't like MOS > with large/small step ratios much greater than 2 or 3, but the seventh > note could just be omitted. A set of 7 nominals with accidentals for -6, > -12, +19, and +25 would get you a pretty good range of notation for > hemiw�rschmidt.

Mainstream temperament classes are whatever musicians who worked with different temperament classes would be expected to know. Which temperament classes they turn out to be is something we have to second guess as much as which notations they'll use.

Yes, it's the large/small step ratio that's important rather than being proper.

> Not all temperaments have a 7- or 10-note MOS, though. Examples (all > except the last two 7-limit ones are listed in the Middle Path paper):

Some of them cover 5 or 12 notes, though. Here's what's left:

> 7-limit:
> hedgehog (6, 8)
> ennealimmal (9)

>>Maybe 9 would be useful. It covers the equal temperament >>behind ennealimmal. It should do orwell as well. And how >>about mavila? If it's the standard 9-limit staff, perhaps >>magic will fit it. It has a strong 3 note MOS so somehow we >>need 3 of them.
> > 9 does come up quite a bit in the list above. Mavila can be notated with > 7 nominals, as long as you don't use the conventional sharps and flats > (which are reversed in mavila). I don't see any need to go beyond 23 > notes for mavila notation, and that's only because it's convenient for > 23-ET (otherwise no more than 16 notes would likely be needed).

I worked out a 9-based notation for magic, and I like it very much. I'll post details here maybe tomorrow. It's not far from the natural Orwell notation. Three nominals different for 22-equal.

> That leaves father (which only has 8 usable notes anyway, and probably > no one beyond me is likely to write for it), sensi, hedgehog, cynder, > and gorgo as temperaments without a 7-, 9-, 10-, or 12-note MOS. Of > these, sensi and cynder (aka mothra) are probably the most important.

Father works fine as alternate notes in decimal notation (and I've used something from that general family). Hedgehog looks like the tricky one.

>>Probably learning a new set of nominals is as hard as >>learning a new staff anyway. So we have to consider how >>complicated it gets to use notate the nominals relative to >>the standard for that number of notes.
> > Learning how the nominals work for each different temperament does seem > to be one of the big issues. I used a lemba notation with the 10 > nominals D L T U N O P I J R. It's usable, but you have to relearn all > the interval relationships. A D major tetrad is D U Pb J. You can think > of U as a sort of F# and I as a sort of Bb -- so, a fifth below U is I#, > and that happens to work. But if you put a series of lemba fifths on a > 10-nominal staff, and then compare with a series of miracle fifths, the > pattern won't be anything alike.
> > In a sagittal notation for lemba based on a chain of fifths, on the > other hand, many of the intervals are easily recognizable. Some, like > G!) B!!/ aren't as obvious (this is a major third), but at least it > looks like some kind of third. If you learn that !) and )!!( are > equivalent in lemba, it becomes clearer. I suspect that relearning > enharmonic equivalents (as in 19-ET where Cx = Db) may be easier than > relearning the whole system of nominals, accidentals, and intervals.

I don't think performers would have to learn enharmonic equivalents. I want the accidentals to be consistent between systems. So that only leaves the nominals to relearn each time.

Graham

Graham Breed wrote:
> Which temperament classes they turn out to be is something we
> have to second guess as much as which notations they'll use.

Why do we have to guess either?

> Yes, it's the large/small step ratio that's important rather
> than being proper.

Important to what?

Herman wrote:
> Learning how the nominals work for each different temperament does
> seem to be one of the big issues. I used a lemba notation with the
> 10 nominals D L T U N O P I J R. It's usable, but you have to relearn
> all the interval relationships.

That's true, but in exchange for that I'd say you get novelty.
Listening back over things -- at least in my case -- I wrote more
compelling music when I didn't know what I was doing.

> In a sagittal notation for lemba based on a chain of fifths, on the
> other hand, many of the intervals are easily recognizable. Some, like
> G!) B!!/ aren't as obvious (this is a major third), but at least it
> looks like some kind of third. If you learn that !) and )!!( are
> equivalent in lemba, it becomes clearer. I suspect that relearning
> enharmonic equivalents (as in 19-ET where Cx = Db) may be easier than
> relearning the whole system of nominals, accidentals, and intervals.

If G-B is a third and all, I think I'd tend to write normal
music in it. This is maybe a lame argument, but I feel it's
true for me.

Graham again (I think):
> I don't think performers would have to learn enharmonic
> equivalents. I want the accidentals to be consistent
> between systems. So that only leaves the nominals to
> relearn each time.

That sounds like Dave and my's idea of using 'natural' nominals
with sagittal accidentals. It's all fine and well except the
sagittal accidentals are hard to read, harder to read on screen,
too wide, and hard to distinguish from one another. Basically,
they burned up too much of shape space by making everything
arrows.

The other exception is that there's little or no value at all
in having a universal set of accidentals. Musicians are not
going to think... ok, here I am playing the lemba scale, oh,
and this notes flat by a ... it's a 36/35, and that's about, er,
this much flat. Instead, they are going to hear the accidentals
in each system they can play and perform them that way. Even
if knowing which comma is which is terribly important to them,
they can memorize "# is 36/35 in lemba and 25/24 in meantone" as
easily as they can "^!!/ is 36/35 and ^\\!!/ is 25/24".

-Carl

Carl Lumma wrote:
> Herman wrote:
>> Learning how the nominals work for each different temperament does
>> seem to be one of the big issues. I used a lemba notation with the
>> 10 nominals D L T U N O P I J R. It's usable, but you have to relearn
>> all the interval relationships.
> > That's true, but in exchange for that I'd say you get novelty.
> Listening back over things -- at least in my case -- I wrote more
> compelling music when I didn't know what I was doing.

I see that I have an illustration of my lemba notation, which I rarely used, so this is probably the only fully notated example. I never got used to the lemba staff, so more typically I'd just notate the rhythms and write the note name for each note.

http://www.io.com/~hmiller/png/lemba-notation.png

This chart of interval sizes uses a compound nominal notation with semiflats and semisharps, but it represents the same basic 10-note scale.

http://www.io.com/~hmiller/png/lemba-intervals.png

>> In a sagittal notation for lemba based on a chain of fifths, on the >> other hand, many of the intervals are easily recognizable. Some, like >> G!) B!!/ aren't as obvious (this is a major third), but at least it >> looks like some kind of third. If you learn that !) and )!!( are >> equivalent in lemba, it becomes clearer. I suspect that relearning >> enharmonic equivalents (as in 19-ET where Cx = Db) may be easier than >> relearning the whole system of nominals, accidentals, and intervals.
> > If G-B is a third and all, I think I'd tend to write normal
> music in it. This is maybe a lame argument, but I feel it's
> true for me.

I like to mix "normal" and "strange" in the same music.

> Graham again (I think):
>> I don't think performers would have to learn enharmonic >> equivalents. I want the accidentals to be consistent >> between systems. So that only leaves the nominals to >> relearn each time.
> > That sounds like Dave and my's idea of using 'natural' nominals
> with sagittal accidentals. It's all fine and well except the
> sagittal accidentals are hard to read, harder to read on screen,
> too wide, and hard to distinguish from one another. Basically,
> they burned up too much of shape space by making everything
> arrows.
> > The other exception is that there's little or no value at all
> in having a universal set of accidentals. Musicians are not
> going to think... ok, here I am playing the lemba scale, oh,
> and this notes flat by a ... it's a 36/35, and that's about, er,
> this much flat. Instead, they are going to hear the accidentals
> in each system they can play and perform them that way. Even
> if knowing which comma is which is terribly important to them,
> they can memorize "# is 36/35 in lemba and 25/24 in meantone" as
> easily as they can "^!!/ is 36/35 and ^\\!!/ is 25/24".

Actually it's worse than that, since it's a _tempered_ 25/24 in meantone. The actual size depends on the tuning of the fifths (and potentially the octaves).

One option would simply be to adapt traditional notation for various ET's, and use an appropriate ET to notate regular temperaments (26-ET for lemba, 31-ET for meantone, 72-ET for miracle).

Carl Lumma wrote:
> Graham Breed wrote:
> >>Which temperament classes they turn out to be is something we
>>have to second guess as much as which notations they'll use.
> > Why do we have to guess either?

Because we're designing a general system of notation for regular temperaments.

>>Yes, it's the large/small step ratio that's important rather >>than being proper.
> > Important to what?

Choosing the nominals. The closer the nominal scale is to an equal temperament, the less chance there is of accidentals overlapping. And if you use different sets of nominals that are all close to being equally spaced there'll be some consistency between them. Particularly for decimal notation, where the standard nominals are very close to equal steps, a notably unequal scale will behave a lot differently.

> Graham again (I think):
> >>I don't think performers would have to learn enharmonic >>equivalents. I want the accidentals to be consistent >>between systems. So that only leaves the nominals to >>relearn each time.
> > That sounds like Dave and my's idea of using 'natural' nominals
> with sagittal accidentals. It's all fine and well except the
> sagittal accidentals are hard to read, harder to read on screen,
> too wide, and hard to distinguish from one another. Basically,
> they burned up too much of shape space by making everything
> arrows.

If the accidentals are hard to distinguish, from all being arrows, that's a problem with the sagittal system.

> The other exception is that there's little or no value at all
> in having a universal set of accidentals. Musicians are not
> going to think... ok, here I am playing the lemba scale, oh,
> and this notes flat by a ... it's a 36/35, and that's about, er,
> this much flat. Instead, they are going to hear the accidentals
> in each system they can play and perform them that way. Even
> if knowing which comma is which is terribly important to them,
> they can memorize "# is 36/35 in lemba and 25/24 in meantone" as
> easily as they can "^!!/ is 36/35 and ^\\!!/ is 25/24".

If they're going to learn each system anyway is there any value in making the staves consistent? How about a new, distinctively different staff for each temperament class.

I don't agree with your "as easily". For the former fact to be useful they'd need to remember what temperament they're playing in -- and keep it permanently in their head while playing. I don't think this should be the case. 9-limit music will have similar pitches whether it's intended for miracle, magic, meantone, orwell, schismatic, or whatever. They can think "I'm playing 9-limit music" and have the notation guide them when it comes to equivalences.

Graham

>>>Which temperament classes they turn out to be is something we
>>>have to second guess as much as which notations they'll use.
>>
>> Why do we have to guess either?
>
>Because we're designing a general system of notation for
>regular temperaments.

What does what they'll use have to do with that design?
Accident and fate, and if we're lucky a bit of merit, will
decide what they use. So let's design on merit.

>>>Yes, it's the large/small step ratio that's important rather
>>>than being proper.
>>
>> Important to what?
>
>Choosing the nominals. The closer the nominal scale is to
>an equal temperament, the less chance there is of
>accidentals overlapping.

You mean like A-up being higher than B-down? I'm not sure
how bad that is, but in the type of notations I'm advocating,
propriety maps 1:1 with whether it happens.

>>>I don't think performers would have to learn enharmonic
>>>equivalents. I want the accidentals to be consistent
>>>between systems. So that only leaves the nominals to
>>>relearn each time.
>>
>> That sounds like Dave and my's idea of using 'natural' nominals
>> with sagittal accidentals. It's all fine and well except the
>> sagittal accidentals are hard to read, harder to read on screen,
>> too wide, and hard to distinguish from one another. Basically,
>> they burned up too much of shape space by making everything
>> arrows.
>
>If the accidentals are hard to distinguish, from all being
>arrows, that's a problem with the sagittal system.

Yes.

>> The other exception is that there's little or no value at all
>> in having a universal set of accidentals. Musicians are not
>> going to think... ok, here I am playing the lemba scale, oh,
>> and this notes flat by a ... it's a 36/35, and that's about, er,
>> this much flat. Instead, they are going to hear the accidentals
>> in each system they can play and perform them that way. Even
>> if knowing which comma is which is terribly important to them,
>> they can memorize "# is 36/35 in lemba and 25/24 in meantone" as
>> easily as they can "^!!/ is 36/35 and ^\\!!/ is 25/24".
>
>If they're going to learn each system anyway is there any
>value in making the staves consistent?

For 5-10 nominals, there are 6 staves to learn. There are
14 currently in use
http://en.wikipedia.org/wiki/Clef#The_positions_of_the_clefs
so they shouldn't be that hard. Granted, the diatonic
intervals stay the same here since they're just clefs.

-Carl

Graham Breed wrote:
> Carl Lumma wrote:
>> Graham Breed wrote:
>>
>>> Which temperament classes they turn out to be is something we
>>> have to second guess as much as which notations they'll use.

("they" being "the mainstream temperament classes" - HM)

>> Why do we have to guess either?
> > Because we're designing a general system of notation for > regular temperaments.

I've noticed something about the "good" temperaments. They tend to have many different consistent ET combinations with them. Meantone (7-limit) has 12, 19, 31, 43, 50, 81; almost any combination of these (from 12&19 to 50&81) is a meantone temperament. Similarly porcupine with 15, 22, 37, 59; orwell with 9, 22, 31, 53; miracle with 10, 31, 41, 72; ennealimmal with 27, 45, 72, 99. Compare:

hexadecimal 9&16
supersharp 10&18
gorgo 5&16
august 9&12 (as opposed to augene: 12, 15, 27, 42)
dominant 5&12
semaphore 5&19
hedgehog (a variety of 8&14, not even consistent)

In between are temperaments like blackwood (5, 10, 15) and lemba (10, 16, 26), which at least have more than one consistent ET pair, but not as many as the "better" temperaments. But there are temperaments with 4 or more consistent ETs and their combinations that haven't had much attention.

27&29
<<5, 18, 17, 17, 13, -11||
[<1, 4, 11, 11|, <0, -5, -18, -17|]
P = 1197.945680, G = 577.314340
ET's: 27, 29, 56, 83

This might be explained by the fact that meantone is both simpler and more accurate.

>>> Yes, it's the large/small step ratio that's important rather >>> than being proper.
>> Important to what?
> > Choosing the nominals. The closer the nominal scale is to > an equal temperament, the less chance there is of > accidentals overlapping. And if you use different sets of > nominals that are all close to being equally spaced there'll > be some consistency between them. Particularly for decimal > notation, where the standard nominals are very close to > equal steps, a notably unequal scale will behave a lot > differently.

I used the step ratio size as a guide back when I was working with the 24-letter nominal system. One of the problematic temperaments is myna: it does have a 7-note MOS, but the step size ratio is around 6.6. You could either use a 4-nominal system with LOTS of accidentals, or go all the way to 27 or 31 notes with about a 1.6 step size ratio. One possible sagittal notation gives this set of 31 notes, which (although complex) can easily be extended in both directions (and it seems the complexity is pretty much unavoidable with this temperament).

E)\!/ G\!) A.(|\ C.||) E\! G B!!/ D'!!) E|) G/|)
B)\!/ D\!) F!) G.||) B\! D F/| A'!!) B|) D/|)
F)/|\ A\!) C!) D.||) F||\ A C/| E'!!) G'(!/ A/|)
C)/|\

>> Graham again (I think):
>>
>>> I don't think performers would have to learn enharmonic >>> equivalents. I want the accidentals to be consistent >>> between systems. So that only leaves the nominals to >>> relearn each time.
>> That sounds like Dave and my's idea of using 'natural' nominals
>> with sagittal accidentals. It's all fine and well except the
>> sagittal accidentals are hard to read, harder to read on screen,
>> too wide, and hard to distinguish from one another. Basically,
>> they burned up too much of shape space by making everything
>> arrows.
> > If the accidentals are hard to distinguish, from all being > arrows, that's a problem with the sagittal system.

Considering how many different accidentals there are (all built from a small set of flags), most of them (especially the more common ones) seem to me to be fairly easy to tell apart. Some of the ones built from the less common flags are a little confusing at first, but most 5- or 7-limit applications can ignore those.

Maybe a better set of symbols can be found. I personally would like a single unaccented symbol for 21/20. But on the whole, I haven't seen anything more suitable for my purposes than Sagittal.

>> The other exception is that there's little or no value at all
>> in having a universal set of accidentals. Musicians are not
>> going to think... ok, here I am playing the lemba scale, oh,
>> and this notes flat by a ... it's a 36/35, and that's about, er,
>> this much flat. Instead, they are going to hear the accidentals
>> in each system they can play and perform them that way. Even
>> if knowing which comma is which is terribly important to them,
>> they can memorize "# is 36/35 in lemba and 25/24 in meantone" as
>> easily as they can "^!!/ is 36/35 and ^\\!!/ is 25/24".
> > If they're going to learn each system anyway is there any > value in making the staves consistent? How about a new, > distinctively different staff for each temperament class.

There are dozens of them. Potentially hundreds, but probably only a handful are likely to have much use.

> I don't agree with your "as easily". For the former fact to > be useful they'd need to remember what temperament they're > playing in -- and keep it permanently in their head while > playing. I don't think this should be the case. 9-limit > music will have similar pitches whether it's intended for > miracle, magic, meantone, orwell, schismatic, or whatever. > They can think "I'm playing 9-limit music" and have the > notation guide them when it comes to equivalences.

That's one of the reasons I'm leaning more in the direction of a consistent sagittal notation for all regular temperaments. Theoretically you could write 7-limit music and play it in any 7-limit temperament. Practically, though, some accidentals are more suitable for some temperaments than others. Unsuitable accidentals are ones that are tempered out or the reverse of their usual direction.

Herman Miller wrote:

> Actually it's worse than that, since it's a _tempered_ 25/24 in > meantone. The actual size depends on the tuning of the fifths (and > potentially the octaves).

A sharp notates the difference between a 5-limit major and minor third in meantone. In that sense using the symbol for a 25:24 gives the correct harmonic cue. But a sharp is also an offset used to extend the spiral of fifths. Using a 25:24 for that would give the wrong cue, but perhaps you could guess a meantone context and so interpret it correctly. Anyway, I suggested a compromise symbol, between a pythagorean sharp and a 5-limit 25:24 for meantone.

> One option would simply be to adapt traditional notation for various > ET's, and use an appropriate ET to notate regular temperaments (26-ET > for lemba, 31-ET for meantone, 72-ET for miracle).

I think the ET notation should be informed by the regular temperaments.

Graham

Carl Lumma wrote:

> What does what they'll use have to do with that design?
> Accident and fate, and if we're lucky a bit of merit, will
> decide what they use. So let's design on merit.

How do we judge merit without an intended use? It's interesting up to a point to guess what might be generally useful. But we can also design for our own needs, so let's do that as well.

I've thought of a list of core temperament classes that I'd expect to have a dedicated notation for. These are mainly for composition and theoretical purposes. As far as I need to remember what I wrote and get it into the computer they're for performance. Here are my suggestions:

Meantone -- conventional staff (or guitar tablature) with additional diesis symbols. Maybe pure sagittal.

Miracle -- decimal

Magic -- the hot new 9 nominal system

Schismatic -- Wilson's docecatonic staff with sagittal accidentals. This fits my 29 note keyboard mapping well (12 black notes to the octave). I've never really been happy with schismatic on a conventional staff.

Without trying very hard, that gives four distinctively different systems. Along with them I'm likely to be interested in some secondary systems that can borrow notation from the primary ones:

Neutral thirds -- meantone with quartertones, or decimal

Negri -- decimal variant or a 9 note staff

Wonder -- every other note of miracle

Bug -- every other note of negri

Mystery -- dunno, but it has to start with 29-equal

With these systems, I'd expect to remember the order of typical sizes of the sagittal accidentals. Using symbols from sagittal saves having to do my own design in which I'm likely to make mistakes. It also means my scores or examples might be readable by other people. Probably I'll develop simplified symbols for handwriting the accidentals that don't differentiate different 1 or 2 shaft accidentals. (That is I'll find a quick way of writing a 1 shaft symbol, and another way for 2 shafts, and only use the full symbol set for formal printing.)

> You mean like A-up being higher than B-down? I'm not sure
> how bad that is, but in the type of notations I'm advocating,
> propriety maps 1:1 with whether it happens.

I always want pitch contour on the page to match the sound. That's more important for theoreticians than performers but generally nice to have.

>>If they're going to learn each system anyway is there any >>value in making the staves consistent?
> > For 5-10 nominals, there are 6 staves to learn. There are
> 14 currently in use
> http://en.wikipedia.org/wiki/Clef#The_positions_of_the_clefs
> so they shouldn't be that hard. Granted, the diatonic
> intervals stay the same here since they're just clefs.

Where do you get 14? I see 4 in common use from a total set of 9 (or a redundant set of 15). Microtonal staves are different in all kinds of ways that break the analogy.

Graham

Herman Miller wrote:

> I've noticed something about the "good" temperaments. They tend to have > many different consistent ET combinations with them. Meantone (7-limit) > has 12, 19, 31, 43, 50, 81; almost any combination of these (from 12&19 > to 50&81) is a meantone temperament. Similarly porcupine with 15, 22, > 37, 59; orwell with 9, 22, 31, 53; miracle with 10, 31, 41, 72; > ennealimmal with 27, 45, 72, 99. Compare:

That sort of ties in with why I combine ETs to look for "good" temperaments. I even have some algebra to back it up. For a reasonable (STD) choice of error, you can write the error of a linear temperament as a quadratic function of the generator size. The greater the complexity (also STD) the steeper the curve. From diophantine approximation theory, all numbers have rational approximations, and so there will always be simple ETs close to the optimal tuning. The lower the error for that tuning, and the lower the complexity, the more ETs you'll find within a given badness.

Consistency is a special case of a badness cutoff.

Bike chain temperaments are a bit of an exception. For example, mystery involves 29-equal so ETs that fit it must be a multiple of 29 notes. So there can only be one smaller than 58.

Graham

Graham wrote:
>> What does what they'll use have to do with that design?
>> Accident and fate, and if we're lucky a bit of merit, will
>> decide what they use. So let's design on merit.
>
>How do we judge merit without an intended use?

Like I said on tuning, it's an information visualization
problem, which is mainly governed by principles of
cognitive psychology. It will matter whether we expect
to support traditional orchestral instruments, or only
computers, etc.

>But we can also design for our own needs, so let's
>do that as well.

That's the only kind of use that:
1. We can really understand.
2. We can be sure will ever happen.

Really so far this message sounds like I'm on my usual rant
about the destructiveness of use cases and user testing in
UI design...

>I've thought of a list of core temperament classes that I'd
>expect to have a dedicated notation for. These are mainly
>for composition and theoretical purposes. As far as I need
>to remember what I wrote and get it into the computer
>they're for performance. Here are my suggestions:
>
>Meantone -- conventional staff (or guitar tablature) with
>additional diesis symbols. Maybe pure sagittal.
>
>Miracle -- decimal
>
>Magic -- the hot new 9 nominal system
>
>Schismatic -- Wilson's docecatonic staff with sagittal
>accidentals. This fits my 29 note keyboard mapping well (12
>black notes to the octave). I've never really been happy
>with schismatic on a conventional staff.
>
>Without trying very hard, that gives four distinctively
>different systems.

I would want to have pajara and ennealimmal -- I suppose you
would use the magic system for the latter. What are you
suggesting for pajara?

>With these systems, I'd expect to remember the order of
>typical sizes of the sagittal accidentals. Using symbols
>from sagittal saves having to do my own design in which I'm
>likely to make mistakes.

Or you could use the sagittal accidentals and be guaranteed
to make mistakes.

I don't know if they were designed to be hand-writable, but
have a look at
http://dkeenan.com/sagittal/Sagittal2_character_map.pdf
zoom in, and tell me you want to transcribe with these.

>Probably I'll
>develop simplified symbols for handwriting the accidentals
>that don't differentiate different 1 or 2 shaft accidentals.

Yeah. But you still have the arrow part to cope with. I
suppose the answer is, you wouldn't need more than one or two
arrow heads per score, and therefore it's manageable. I can
only say I hardly see the point of differentiating them anywhere
if they aren't different enough to articulate in a single score.

>> You mean like A-up being higher than B-down? I'm not sure
>> how bad that is, but in the type of notations I'm advocating,
>> propriety maps 1:1 with whether it happens.
>
>I always want pitch contour on the page to match the sound.
>That's more important for theoreticians than performers but
>generally nice to have.

OK.

>>>If they're going to learn each system anyway is there any
>>>value in making the staves consistent?
>>
>> For 5-10 nominals, there are 6 staves to learn. There are
>> 14 currently in use
>> http://en.wikipedia.org/wiki/Clef#The_positions_of_the_clefs
>> so they shouldn't be that hard. Granted, the diatonic
>> intervals stay the same here since they're just clefs.
>
>Where do you get 14? I see 4 in common use from a total set
>of 9 (or a redundant set of 15).

Sorry I did count the common four twice. But it's a total set
of 10.

>Microtonal staves are
>different in all kinds of ways that break the analogy.

Maybe. They're enough different from one another that performers
have to learn each one independently, so I think there's some
validity there. In fact they're doing the very thing you claim
is worst -- using the same thing (a note at a position on the staff)
to mean something completely different in a different context. And
they often change throughout a piece.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
> I don't know if they were designed to be hand-writable, but
> have a look at
>
http://dkeenan.com/sagittal/Sagittal2_character_map.pdf
> zoom in, and tell me you want to transcribe with these.

Hi Carl,

I have to admit, if my first introduction to sagittal was the first
page of that spreadsheet, I'd probably run screaming and never come
back. But if I did, I'd be making the "equal importance" and "equally
likely to be used together" fallacies.

The point is to _not_ zoom in. But instead to ignore the symbols that
are set in small type. Also, I wish I could arrange so it opened on
page 5, centered on the natural. And even then you should really only
be looking at the symbols with a green background (the Athenian set).

To decide the relative importance or likelihood of use of a symbol I
used statistics from the Scala archive plus a bunch of ad-hoc-ery. If
I'd actually typeset their size in proportion to their importance then
the 5-comma symbols would have a whole page to themselves, and the
X-shaft symbols and many others would be mistaken for dirt spots on
the page. There's more than 1000:1 difference in importance there.

So I mapped the log of the importance to the point-size, and even then
had to have a minimum size cutoff.

This spreadsheet needs to exist, for people like Hudson Lacerda who
need to implement the whole font in their notation software. But we
sure need some better introductory material. :-)

-- Dave Keenan

>This spreadsheet needs to exist, for people like Hudson Lacerda who
>need to implement the whole font in their notation software. But we
>sure need some better introductory material. :-)

Do you have a table (png image, preferably) of the most important
accidentals?

-Carl

Graham Breed wrote:
> Herman Miller wrote:
>> One option would simply be to adapt traditional notation for various >> ET's, and use an appropriate ET to notate regular temperaments (26-ET >> for lemba, 31-ET for meantone, 72-ET for miracle).
> > I think the ET notation should be informed by the regular > temperaments.

Hmm... Well, many ET's are themselves regular temperaments.

E.g. 31-ET, TOP step size = 38.757 cents
<31, 49, 72] (5-limit)
<31, 49, 72, 87] (7-limit)
<31, 49, 72, 87, 107] (11-limit)

If you look at 5-limit 31-ET without considering how it relates to meantone, you can examine which accidentals might be useful for notating it. E.g.

1 step: .//| 128/125 /|) 36/35 /|\ 33/32 (|( 45/44
2 steps: )||( 25/24 .||) 21/20 )||~ 22/21

In meantone, .//| /|) and /|\ represent (-5, +12), while (|( represents (+8, -19); )||( and .||) represent (+3, -7), while )||~ represents (-10, +24). Of course, there are other possibilities (especially for meantone, since many of the Sagittal symbols differ by a syntonic comma). The recommended "standard set" of accidentals for 31-ET is /|\ for one step and /||\ for 2 steps.

Carl Lumma wrote:

> Like I said on tuning, it's an information visualization
> problem, which is mainly governed by principles of
> cognitive psychology. It will matter whether we expect
> to support traditional orchestral instruments, or only
> computers, etc.

It depends more importantly on whether or not we expect anybody else to read it at all and whether we expect them to play music written in it.

>>But we can also design for our own needs, so let's >>do that as well.
> > That's the only kind of use that:
> 1. We can really understand.
> 2. We can be sure will ever happen.

I can't even predict with any certainty what temperament classes I'll use in the future or what instruments I'll want to play them on.

> Really so far this message sounds like I'm on my usual rant
> about the destructiveness of use cases and user testing in
> UI design...

Oh well, if you've got heretical views in this we'd better stay away from it.

> I would want to have pajara and ennealimmal -- I suppose you
> would use the magic system for the latter. What are you
> suggesting for pajara?

I have reservations about using the magic system for ennealimmal. The point for magic is that the nominals are unequally spaced and the staff reflects this. The point of 9 notes for ennealimmal is that they're equally tempered. Also magic is part of the large family that tempers out 225:224 and ennealimmal isn't.

The natural system for ennealimmal would reinforce the equality of 9 nominals. A 4 line octave staff should do it. I'd have to be very familiar with the magic staff to want to adapt that instead.

For pajara I'd use adapted decimal notation. I'd probably also use a 24 note keyboard mapping with the black notes as nominals. There are obviously other ways to do it.

>>With these systems, I'd expect to remember the order of >>typical sizes of the sagittal accidentals. Using symbols >>from sagittal saves having to do my own design in which I'm >>likely to make mistakes.
> > Or you could use the sagittal accidentals and be guaranteed
> to make mistakes.

If you can see mistakes in sagittal, publish your own font with the revisions. Then we'll all benefit.

> I don't know if they were designed to be hand-writable, but
> have a look at
> http://dkeenan.com/sagittal/Sagittal2_character_map.pdf
> zoom in, and tell me you want to transcribe with these.

I can see the Spartan set in the Sagittal paper. Everything I need can come from there.

There generally isn't a close relationship between formal and cursive scripts. Cursives often lose some redundancy.

>>Probably I'll >>develop simplified symbols for handwriting the accidentals >>that don't differentiate different 1 or 2 shaft accidentals.
> > Yeah. But you still have the arrow part to cope with. I
> suppose the answer is, you wouldn't need more than one or two
> arrow heads per score, and therefore it's manageable. I can
> only say I hardly see the point of differentiating them anywhere
> if they aren't different enough to articulate in a single score.

If you've got a computer that's capable of distinguishing hundreds of thousands of different characters, why not use 27 for your accidentals? And you would need more on the same score for JI or temperaments that approach JI in precision.

>>Microtonal staves are >>different in all kinds of ways that break the analogy.
> > Maybe. They're enough different from one another that performers
> have to learn each one independently, so I think there's some
> validity there. In fact they're doing the very thing you claim
> is worst -- using the same thing (a note at a position on the staff)
> to mean something completely different in a different context. And
> they often change throughout a piece.

The evidence is that different clefs are rare, transposing instruments are common, and key signatures are ubiquitous. You could say that microtonal accidentals on a conventional staff are like key signatures but different fairly-good 9-limit temperaments won't have auidibly different contexts the way key signatures do.

I wonder if numerical notation has it right by making all the notes depend on the key so you can never forget the context.

Graham

Herman Miller wrote:

> Hmm... Well, many ET's are themselves regular temperaments.

Touch�!

> E.g. 31-ET, TOP step size = 38.757 cents
> <31, 49, 72, 87, 107] (11-limit)
> > If you look at 5-limit 31-ET without considering how it relates to > meantone, you can examine which accidentals might be useful for notating > it. E.g.
> > 1 step: .//| 128/125 /|) 36/35 /|\ 33/32 (|( 45/44
> 2 steps: )||( 25/24 .||) 21/20 )||~ 22/21

That's reasonable.

> In meantone, .//| /|) and /|\ represent (-5, +12), while (|( represents > (+8, -19); )||( and .||) represent (+3, -7), while )||~ represents (-10, > +24). Of course, there are other possibilities (especially for meantone, > since many of the Sagittal symbols differ by a syntonic comma). The > recommended "standard set" of accidentals for 31-ET is /|\ for one step > and /||\ for 2 steps.

This really highlights my disagreement with the standard set. Other than that I'm quite happy with sagittal.

Graham

Carl Lumma wrote:
> Graham wrote:
>> But we can also design for our own needs, so let's >> do that as well.
> > That's the only kind of use that:
> 1. We can really understand.
> 2. We can be sure will ever happen.

That's also my main focus -- although I'd like as a goal to have a system for notating anything, I've been giving more attention to those aspects of the notation useful for the tunings that I'm interested in (e.g. lemba, porcupine, keemun, orwell) and not given much consideration to how a performer might make sense of it.

>> I've thought of a list of core temperament classes that I'd >> expect to have a dedicated notation for. These are mainly >> for composition and theoretical purposes. As far as I need >> to remember what I wrote and get it into the computer >> they're for performance. Here are my suggestions:
>>
>> Meantone -- conventional staff (or guitar tablature) with >> additional diesis symbols. Maybe pure sagittal.
>>
>> Miracle -- decimal
>>
>> Magic -- the hot new 9 nominal system
>>
>> Schismatic -- Wilson's docecatonic staff with sagittal >> accidentals. This fits my 29 note keyboard mapping well (12 >> black notes to the octave). I've never really been happy >> with schismatic on a conventional staff.
>>
>> Without trying very hard, that gives four distinctively >> different systems.
> > I would want to have pajara and ennealimmal -- I suppose you
> would use the magic system for the latter. What are you
> suggesting for pajara?

Besides the ones mentioned, I'd like to notate at least father, mavila, porcupine, keemun, lemba, and orwell. I do have conventional notations for porcupine (http://www.io.com/~hmiller/music/temp-porcupine.html) and keemun, but by the time I started using lemba, I figured there had to be a better way. I thought of generalizing the diatonic scale so that a typical MOS of around 5-12 notes could have letter names in order (a 9-note orwell MOS could be notated A B C D E F G H I). I'm still interested in alternative notation systems, and your 9-nominal magic system sounds interesting. Still, a system based on Sagittal notation seems to meet my needs.

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> >This spreadsheet needs to exist, for people like Hudson Lacerda who
> >need to implement the whole font in their notation software. But we
> >sure need some better introductory material. :-)
>
> Do you have a table (png image, preferably) of the most important
> accidentals?
>
> -Carl
>

Only figures 2 and 4 of this.
http://dkeenan.com/sagittal/Sagittal.pdf

At 08:21 AM 11/30/2007, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>>
>> >This spreadsheet needs to exist, for people like Hudson Lacerda who
>> >need to implement the whole font in their notation software. But we
>> >sure need some better introductory material. :-)
>>
>> Do you have a table (png image, preferably) of the most important
>> accidentals?
>>
>> -Carl
>>
>
>Only figures 2

Well, the "early sagittal" is a very good set of accidentals
for 72. The curved flags are where you went wrong from my
point of view.

>and 4 of this.
>http://dkeenan.com/sagittal/Sagittal.pdf

Ouch.

-Carl

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Herman Miller wrote:
>
> > Hmm... Well, many ET's are themselves regular temperaments.
>
> Touché!
>
> > E.g. 31-ET, TOP step size = 38.757 cents
> > <31, 49, 72, 87, 107] (11-limit)
> >
> > If you look at 5-limit 31-ET without considering how it relates
to
> > meantone, you can examine which accidentals might be useful for
notating
> > it. E.g.
> >
> > 1 step: .//| 128/125 /|) 36/35 /|\ 33/32 (|( 45/44
> > 2 steps: )||( 25/24 .||) 21/20 )||~ 22/21
>
> That's reasonable.
>
> > In meantone, .//| /|) and /|\ represent (-5, +12), while (|(
represents
> > (+8, -19); )||( and .||) represent (+3, -7), while )||~
represents (-10,
> > +24). Of course, there are other possibilities (especially for
meantone,
> > since many of the Sagittal symbols differ by a syntonic comma).
The
> > recommended "standard set" of accidentals for 31-ET is /|\ for
one step
> > and /||\ for 2 steps.
>
> This really highlights my disagreement with the standard
> set. Other than that I'm quite happy with sagittal.
>
> Graham

Hi Graham,

I've been following this discussion but have been quite busy lately,
so have resisted the temptation to reply -- until now.

First off, I'm naturally very delighted to hear that you're quite
happy with Sagittal. In the past several years Dave & I have gotten
all sorts of criticism about Sagittal. At first it wasn't versatile
enough, so we needed more symbols, then (left) accents, then right
accents. Then some folks said it was too complicated. Can't please
everybody, can you? :-)

Now we have problems such as: Since there are now many different
ways to notate 1 and 2 steps of 31-ET (or 19-ET, or whatever), which
symbols should be in the standard symbol set? You're looking at 31-
ET as a 5-limit tuning, but what about the 7 and 11 limits? Would
you then use different accidentals?

The "standard" symbol sets were intended to make ET notation as
*generally useful and simple* as possible. To accomplish this, we
sought to make use of excellent features found in existing notations,
specifically:
1) the multiple vertical lines in the Tartini fractional sharps,
2) Bosanquet's slanting comma-lines, and
3) the clear directionality (and invertibility) of arrows.

Points 1) and 3) are clearly evident in Figure 1 of
http://dkeenan.com/sagittal/Sagittal.pdf
where the resemblance of pure Sagittal to the Tartini fractional
sharps is unmistakable. The Tartini set notates 31-ET so naturally
that it was a foregone conclusion that the corresponding pure
Sagittal symbols (interpreted as a semi-sharp, sharp, sesqui-sharp,
and double-sharp) would be both obvious and simple, requiring one to
read only the number of shafts in the symbol (since the flags are all
the same).

In your recent discussion about choice of accidentals (both here and
on the main list -- see, e.g.,
/tuning/topicId_74318.html#74536
where ||\ is the preferred "sharp" accidental for 19-ET, even though
it's not in your 31-ET list, above), you point out that there are
other considerations, such as:
1) suitability to the temperament classes to which the tuning
belongs -- useful for understanding tonal relationships within the
tuning -- and
2) the desire to use an accidental whose (JI) definition approximates
the actual size of the interval in the tuning -- useful for helping
the player of a conventional flexible-pitch instrument get to the
appropriate pitch.

I was very interested in consideration #2 some years ago, before I
had as many symbols to play with as now, and before I thought of
defining the symbols in terms of JI intervals. Thus I was very
interested to take note of your observation that using the ||\ symbol
for a "sharp" in meantone tunings (e.g., for 2deg31) would give the
player a better clue to its proper size. It also points up an
advantage of pure over mixed Sagittal: while #\! is definitely
perceived as more complicated than #, ||\ would not necessarily be
(on the basis of its appearance) more complicated than /||\ --
indeed, you could easily think of them as different kinds
of "sharps". (In working out pajara notation, I observed that F||\
and G!!/ are equivalents for the tone 1/2 octave from C, so the
accidental actually represents the 12-ET sharp.)

Anyway, I want to encourage you to seek and advocate whatever
alternatives to the "standard" symbols may happen to appeal to you.
One of the strengths of Sagittal is that it offers options to meet
the needs of a variety of applications, so go ahead and use whatever
works best for you.

--George

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> At 08:21 AM 11/30/2007, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <carl@> wrote:
> >>
> >> >This spreadsheet needs to exist, for people like Hudson Lacerda
who
> >> >need to implement the whole font in their notation software.
But we
> >> >sure need some better introductory material. :-)
> >>
> >> Do you have a table (png image, preferably) of the most important
> >> accidentals?
> >>
> >> -Carl
> >
> >Only figures 2
>
> Well, the "early sagittal" is a very good set of accidentals
> for 72. The curved flags are where you went wrong from my
> point of view.

Hi Carl,

There are two reasons for the curved flags for notating 72-ET.

1) The curved flags eliminate the problem of lateral confusability.
Is it easier to distinguish
/| from |\
or
/| from |) ?

2) Accidentals altering by 2deg72 notate 7-limit consonances, so it's
appropriate to use the 7-comma (63:64) symbol |) for this purpose.
The |\ symbol represents 54:55, an 11-diesis (32:33) /|\ less a 5-
comma (80:81) /|; although it's valid, it would notate more complex
pitches.

> >and 4 of this.
> >http://dkeenan.com/sagittal/Sagittal.pdf
>
> Ouch.

Hey, cut me some slack here. There are enough symbols in
this "spartan" set to notate literally dozens of ET's (the most
complicated being 130 and 142)!

In the 142-ET symbol sequence (which uses all of them) the flag
arithmetic is completely consistent:
|( = 1deg, the symbol is defined as |) minus /|
/| = 2deg
|) = 3deg
|\ or (| = 4deg
(The 130-ET set omits (|) and counts (| as 3deg, and it still has
consistent flag arithmetic.)

So what problem(s) do you have with this, and what alternatives might
you suggest?

Do the symbols look too much like one another? Look at the symbols
in this sequence:
0 1 2 3 4 5 6 7 8 9
Do you easily distinguish 3 from 5, 4 from 9, 5 from 6, or 8 from
9? Do they look more or less distinct from one another than the
symbols in the spartan sequence (as they appear in the pdf file)?

If you're not notating anything as complicated as 130 or 142, then
you won't need all of those symbols. For example, 7-limit JI that
allows a 5-exponent up to +-5 and 7-exponent up to +-1 can be mapped
to 118-ET, using the following (pure) Sagittal accidentals (a subset
of athenian-level JI) up to the apotome:

)| /| |) //| /|) (|\ )||( ||) ||\ /||) /||\

If the foregoing exponent limitations are observed, the resulting
notation will be exactly the same as with the athenian-level symbol
set,
|( )|( ~|( /| |) (| (|( //| /|) /|\ (|) (|\ )||( ~||( )
||~ ||) ||\ (||( //|| /||) /||\
and it's an illustratration that you don't need to learn the symbols
you're not using.

--George

On 30/11/2007, George D. Secor <gdsecor@yahoo.com> wrote:

> First off, I'm naturally very delighted to hear that you're quite
> happy with Sagittal. In the past several years Dave & I have gotten
> all sorts of criticism about Sagittal. At first it wasn't versatile
> enough, so we needed more symbols, then (left) accents, then right
> accents. Then some folks said it was too complicated. Can't please
> everybody, can you? :-)

Of course you can't, and that's always going to be a problem with a
comprehensive system.

> Now we have problems such as: Since there are now many different
> ways to notate 1 and 2 steps of 31-ET (or 19-ET, or whatever), which
> symbols should be in the standard symbol set? You're looking at 31-
> ET as a 5-limit tuning, but what about the 7 and 11 limits? Would
> you then use different accidentals?

The 7-limit should naturally follow the 5-limit for temperaments with
225:224 as a unison vector. The difference for the 11-limit is that
I'd consider the half-sharp to literally be half a sharp so that I
could use 31-equal equivalences.

> In your recent discussion about choice of accidentals (both here and
> on the main list -- see, e.g.,
> /tuning/topicId_74318.html#74536
> where ||\ is the preferred "sharp" accidental for 19-ET, even though
> it's not in your 31-ET list, above), you point out that there are
> other considerations, such as:
> 1) suitability to the temperament classes to which the tuning
> belongs -- useful for understanding tonal relationships within the
> tuning -- and
> 2) the desire to use an accidental whose (JI) definition approximates
> the actual size of the interval in the tuning -- useful for helping
> the player of a conventional flexible-pitch instrument get to the
> appropriate pitch.

I haven't decided what "sharp" should be used for 19-ET. And I don't
have strong opinions anyway. I happened to respond to Dave on the
issue of small inconsistencies being more confusing than large ones.
The main point is, whichever symbol is best, the same as a Pythagorean
sharp is way too big.

> Anyway, I want to encourage you to seek and advocate whatever
> alternatives to the "standard" symbols may happen to appeal to you.
> One of the strengths of Sagittal is that it offers options to meet
> the needs of a variety of applications, so go ahead and use whatever
> works best for you.

Musicians are going to make their own choice anyway, and I assume
they'll make the best one because they'll know what they want. As far
as standards, the main thing is to outline how a consistent system
based on interval sizes could work and see what the problems are.

Graham

>> Well, the "early sagittal" is a very good set of accidentals
>> for 72. The curved flags are where you went wrong from my
>> point of view.
>
>Hi Carl,
>
>There are two reasons for the curved flags for notating 72-ET.
>
>1) The curved flags eliminate the problem of lateral confusability.
>Is it easier to distinguish
> /| from |\
>or
> /| from |) ?

It's easier to distinguish anything not involving a curved flag.

>2) Accidentals altering by 2deg72 notate 7-limit consonances, so it's
>appropriate to use the 7-comma (63:64) symbol |) for this purpose.
>The |\ symbol represents 54:55, an 11-diesis (32:33) /|\ less a 5-
>comma (80:81) /|; although it's valid, it would notate more complex
>pitches.
>
>> >and 4 of this.
>> >http://dkeenan.com/sagittal/Sagittal.pdf
>>
>> Ouch.
>
>Hey, cut me some slack here. There are enough symbols in
>this "spartan" set to notate literally dozens of ET's (the most
>complicated being 130 and 142)!

It's not the number of symbols, but rather the sight of those
curved flags from which eyes are still recovering.

Personally I can see no need to notate more than 72 tones at
once. Chain-of-generator notation is great for tunings like
171, etc. (using multiple instances of a single accidental
up or down, with the understanding you won't use more than
one most of the time, and never more than three).

>In the 142-ET symbol sequence (which uses all of them) the flag
>arithmetic is completely consistent:
>|( = 1deg, the symbol is defined as |) minus /|
>/| = 2deg
>|) = 3deg
>|\ or (| = 4deg
>(The 130-ET set omits (|) and counts (| as 3deg, and it still has
>consistent flag arithmetic.)
>
>So what problem(s) do you have with this, and what alternatives might
>you suggest?

The ASCII version has always been unreadable to me.
Is (| up or down?

I suggest sticking with the original Sagittal accidentals from
figure 2.

>Do the symbols look too much like one another?

Yes.

>Look at the symbols in this sequence:
>0 1 2 3 4 5 6 7 8 9
>Do you easily distinguish 3 from 5, 4 from 9, 5 from 6, or 8 from
>9? Do they look more or less distinct from one another than the
>symbols in the spartan sequence (as they appear in the pdf file)?

Much more distinct. Because they're not all arrows, but rather
can be any shape. As far as a list of 200 arrows, I'm sure yours
are as distinct as possible.

>If you're not notating anything as complicated as 130 or 142, then
>you won't need all of those symbols.

In light of the recent thread about notations for a 'core' number
of rank 2 temperaments, I would suggest you and Dave do a poster
showing accidental sets and an ascending MOS <= 10 tones for each.

-Carl

Graham Breed wrote:

> The 7-limit should naturally follow the 5-limit for temperaments with
> 225:224 as a unison vector. The difference for the 11-limit is that
> I'd consider the half-sharp to literally be half a sharp so that I
> could use 31-equal equivalences.

That's one advantage of /|\ for 1 step and /||\ for 2 steps of 31-ET. On the other hand, if you use ||\ for 2 steps of 31-ET, there's an 11-limit symbol |\ (55/54) that maps to 1 step of 31-ET.

> I haven't decided what "sharp" should be used for 19-ET. And I don't
> have strong opinions anyway. I happened to respond to Dave on the
> issue of small inconsistencies being more confusing than large ones.
> The main point is, whichever symbol is best, the same as a Pythagorean
> sharp is way too big.

I agree. A step of 19-ET is only 63.36 cents; even the 92.18 cent ||\ is a bit on the large side. My choice would be )||( for its relative familiarity in comparison with the other good option )/|\ (405/392).

On the other hand, the fifths in ET's like 19-ET are far enough from just that the cumulative discrepancy in an interval like D-F adds up to the point where F||\ might be a better notation than F)||( for the major third above D. Consider that the notated "F" is pretty close to F/| already.

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> >> Well, the "early sagittal" is a very good set of accidentals
> >> for 72. The curved flags are where you went wrong from my
> >> point of view.
> >
> >Hi Carl,
> >
> >There are two reasons for the curved flags for notating 72-ET.
> >
> >1) The curved flags eliminate the problem of lateral
confusability.
> >Is it easier to distinguish
> > /| from |\
> >or
> > /| from |) ?
>
> It's easier to distinguish anything not involving a curved flag.

From what? Are you saying that changing the flag in the right-hand
symbol from straight to curved would make it more difficult to
distinguish that symbol from the left-hand one (with a straight left
flag)? I don't understand how you could think that making two
symbols more unlike one another would make them more difficult to
tell apart.

Or were you playing the politician and avoiding my question? :-)

> >2) Accidentals altering by 2deg72 notate 7-limit consonances, so
it's
> >appropriate to use the 7-comma (63:64) symbol |) for this
purpose.
> >The |\ symbol represents 54:55, an 11-diesis (32:33) /|\ less a 5-
> >comma (80:81) /|; although it's valid, it would notate more
complex
> >pitches.
> >
> >> >and 4 of this.
> >> >http://dkeenan.com/sagittal/Sagittal.pdf
> >>
> >> Ouch.
> >
> >Hey, cut me some slack here. There are enough symbols in
> >this "spartan" set to notate literally dozens of ET's (the most
> >complicated being 130 and 142)!
>
> It's not the number of symbols, but rather the sight of those
> curved flags from which eyes are still recovering.

Why is that? Don't most ASCII characters contain curved lines? Does
reading music in flat keys hurt your eyes? ;-)

> Personally I can see no need to notate more than 72 tones at
> once.

Then you're basically taking issue with which symbols to use for 72-
ET, for which only 1/3 of the symbols contain curved flags.

> Chain-of-generator notation is great for tunings like
> 171, etc. (using multiple instances of a single accidental
> up or down, with the understanding you won't use more than
> one most of the time, and never more than three).
>
> >In the 142-ET symbol sequence (which uses all of them) the flag
> >arithmetic is completely consistent:
> >|( = 1deg, the symbol is defined as |) minus /|
> >/| = 2deg
> >|) = 3deg
> >|\ or (| = 4deg
> >(The 130-ET set omits (|) and counts (| as 3deg, and it still has
> >consistent flag arithmetic.)
> >
> >So what problem(s) do you have with this, and what alternatives
might
> >you suggest?
>
> The ASCII version has always been unreadable to me.
> Is (| up or down?

With | it's up, and with ! it's down. The actual Sagittal font
should be use for a manuscript, so that's not an issue.

> I suggest sticking with the original Sagittal accidentals from
> figure 2.
>
> >Do the symbols look too much like one another?
>
> Yes.
>
> >Look at the symbols in this sequence:
> >0 1 2 3 4 5 6 7 8 9
> >Do you easily distinguish 3 from 5, 4 from 9, 5 from 6, or 8 from
> >9? Do they look more or less distinct from one another than the
> >symbols in the spartan sequence (as they appear in the pdf file)?
>
> Much more distinct. Because they're not all arrows, but rather
> can be any shape.

That's not the answer I had in mind, because I've frequently found
that some numerals are difficult to distinguish from one another,
particularly on an analog television screen; e.g., in the graphic
display (at the top of the screen) for a baseball game, I've
sometimes been unable to tell whether a game was in the 5th vs. the
6th inning, or the 8th vs. the 9th. However, I don't think I've ever
had trouble distinguishing between 9 and 4, because one is curved and
the other isn't.

> As far as a list of 200 arrows, I'm sure yours
> are as distinct as possible.

Okay, then, because our objective was to have only arrows, and to
make them as distinct as possible from one another. The purpose of
having arrows was to make directionality obvious (a problem that
Joseph Pehrson often encountered with the Sims 1/2-sharp/flat, BTW).

> >If you're not notating anything as complicated as 130 or 142, then
> >you won't need all of those symbols.
>
> In light of the recent thread about notations for a 'core' number
> of rank 2 temperaments, I would suggest you and Dave do a poster
> showing accidental sets and an ascending MOS <= 10 tones for each.

Easier said than done, in light of the recent discussions concerning
the intricacies in selecting appropriate accidentals for
temperaments. If Herman and Graham would look at some of these and
agree on sets of accidentals, I'd be happy to prepare a diagram.

--George

>> It's easier to distinguish anything not involving a curved flag.
>
>From what? Are you saying that changing the flag in the right-hand
>symbol from straight to curved would make it more difficult to
>distinguish that symbol from the left-hand one (with a straight left
>flag)? I don't understand how you could think that making two
>symbols more unlike one another would make them more difficult to
>tell apart.

The curved flags are very hard to read to my eye, even alone.
Put them on a staff and it's worse.

>Or were you playing the politician and avoiding my question? :-)

I'm trying to be as direct and honest as possible, without
being offensive or nonconstructive. I think you should use only
straight flags, or redraw your curved flags in a big way.

>> >> >and 4 of this.
>> >> >http://dkeenan.com/sagittal/Sagittal.pdf
>> >>
>> >> Ouch.
>> >
>> >Hey, cut me some slack here. There are enough symbols in
>> >this "spartan" set to notate literally dozens of ET's (the most
>> >complicated being 130 and 142)!
>>
>> It's not the number of symbols, but rather the sight of those
>> curved flags from which eyes are still recovering.
>
>Why is that? Don't most ASCII characters contain curved lines? Does
>reading music in flat keys hurt your eyes? ;-)

There's something about the curves that manages to be lumpy and
ungraceful.

>> Personally I can see no need to notate more than 72 tones at
>> once.
>
>Then you're basically taking issue with which symbols to use for 72-
>ET, for which only 1/3 of the symbols contain curved flags.

I especially don't like the look of having some curved and some
straight. I would say that I like the 'original sagittal'
approach to 72 pretty well. One issue with arrows is that they
can be hard to place on the staff in a chord. Blackwood drew
his arrows through circles for this reason, but I liked that
approach even less. Sagittal arrows seem pretty good in that
they are compact, not long like Blackwood's were.

>> >Look at the symbols in this sequence:
>> >0 1 2 3 4 5 6 7 8 9
>> >Do you easily distinguish 3 from 5, 4 from 9, 5 from 6, or 8 from
>> >9? Do they look more or less distinct from one another than the
>> >symbols in the spartan sequence (as they appear in the pdf file)?
>>
>> Much more distinct. Because they're not all arrows, but rather
>> can be any shape.
>
>That's not the answer I had in mind, because I've frequently found
>that some numerals are difficult to distinguish from one another,
>particularly on an analog television screen; e.g., in the graphic
>display (at the top of the screen) for a baseball game, I've
>sometimes been unable to tell whether a game was in the 5th vs. the
>6th inning, or the 8th vs. the 9th. However, I don't think I've ever
>had trouble distinguishing between 9 and 4, because one is curved and
>the other isn't.

I've never had trouble distinguishing numbers. But then again
I wasn't reading many of them back in the age of awful display
technology.

>> As far as a list of 200 arrows, I'm sure yours
>> are as distinct as possible.
>
>Okay, then, because our objective was to have only arrows, and to
>make them as distinct as possible from one another. The purpose of
>having arrows was to make directionality obvious (a problem that
>Joseph Pehrson often encountered with the Sims 1/2-sharp/flat, BTW).

I guess I would say I've never met a musician who confused the
directionality of b and #. I agree that arrows may be easier
out-of-the-box. But I expect bb b // # x to cover most of my
microtonal notation needs anyway.

>> >If you're not notating anything as complicated as 130 or 142, then
>> >you won't need all of those symbols.
>>
>> In light of the recent thread about notations for a 'core' number
>> of rank 2 temperaments, I would suggest you and Dave do a poster
>> showing accidental sets and an ascending MOS <= 10 tones for each.
>
>Easier said than done, in light of the recent discussions concerning
>the intricacies in selecting appropriate accidentals for
>temperaments. If Herman and Graham would look at some of these and
>agree on sets of accidentals, I'd be happy to prepare a diagram.

Hopefully they'll agree then!

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> >> It's easier to distinguish anything not involving a curved flag.
> >
> >From what? Are you saying that changing the flag in the right-
hand
> >symbol from straight to curved would make it more difficult to
> >distinguish that symbol from the left-hand one (with a straight
left
> >flag)? I don't understand how you could think that making two
> >symbols more unlike one another would make them more difficult to
> >tell apart.
>
> The curved flags are very hard to read to my eye, even alone.
> Put them on a staff and it's worse.
>
> >Or were you playing the politician and avoiding my question? :-)
>
> I'm trying to be as direct and honest as possible, without
> being offensive or nonconstructive. I think you should use only
> straight flags,

I don't know how we might do that. At one point, when we were
looking for a 4th type of flag, I suggested a right-angle flag, but
both Dave & I agreed that the horizontal component tended to get lost
in the staff lines. I also considered filled-in triangle flags, but
it was too time-consuming to draw them by hand, and they tended to
look too much like notes.

> or redraw your curved flags in a big way.

We already did that, in response to Joseph Pehrson's criticism that
the convex flags (arcs) were not curved sufficiently to distinguish
them from straight-line flags (barbs).

> >> >> >and 4 of this.
> >> >> >http://dkeenan.com/sagittal/Sagittal.pdf
> >> >>
> >> >> Ouch.
> >> >
> >> >Hey, cut me some slack here. There are enough symbols in
> >> >this "spartan" set to notate literally dozens of ET's (the most
> >> >complicated being 130 and 142)!
> >>
> >> It's not the number of symbols, but rather the sight of those
> >> curved flags from which eyes are still recovering.
> >
> >Why is that? Don't most ASCII characters contain curved lines?
Does
> >reading music in flat keys hurt your eyes? ;-)
>
> There's something about the curves that manages to be lumpy and
> ungraceful.

Well, I don't know what else to say, except that AFAIK no one else
has objected to them.

> >> Personally I can see no need to notate more than 72 tones at
> >> once.
> >
> >Then you're basically taking issue with which symbols to use for
72-
> >ET, for which only 1/3 of the symbols contain curved flags.
>
> I especially don't like the look of having some curved and some
> straight. I would say that I like the 'original sagittal'
> approach to 72 pretty well. One issue with arrows is that they
> can be hard to place on the staff in a chord. Blackwood drew
> his arrows through circles for this reason, but I liked that
> approach even less. Sagittal arrows seem pretty good in that
> they are compact, not long like Blackwood's were.
>
> >> >Look at the symbols in this sequence:
> >> >0 1 2 3 4 5 6 7 8 9
> >> >Do you easily distinguish 3 from 5, 4 from 9, 5 from 6, or 8
from
> >> >9? Do they look more or less distinct from one another than
the
> >> >symbols in the spartan sequence (as they appear in the pdf
file)?
> >>
> >> Much more distinct. Because they're not all arrows, but rather
> >> can be any shape.
> >
> >That's not the answer I had in mind, because I've frequently found
> >that some numerals are difficult to distinguish from one another,
> >particularly on an analog television screen; e.g., in the graphic
> >display (at the top of the screen) for a baseball game, I've
> >sometimes been unable to tell whether a game was in the 5th vs.
the
> >6th inning, or the 8th vs. the 9th. However, I don't think I've
ever
> >had trouble distinguishing between 9 and 4, because one is curved
and
> >the other isn't.
>
> I've never had trouble distinguishing numbers. But then again
> I wasn't reading many of them back in the age of awful display
> technology.
>
> >> As far as a list of 200 arrows, I'm sure yours
> >> are as distinct as possible.
> >
> >Okay, then, because our objective was to have only arrows, and to
> >make them as distinct as possible from one another. The purpose
of
> >having arrows was to make directionality obvious (a problem that
> >Joseph Pehrson often encountered with the Sims 1/2-sharp/flat,
BTW).
>
> I guess I would say I've never met a musician who confused the
> directionality of b and #. I agree that arrows may be easier
> out-of-the-box.

Their best feature is that you need only invert (vertically mirror)
them in order to get the opposite direction, so not only is the
direction immediately obvious, but memorization is also easier, since
you need only remember the up-symbols (since their conversion to down
is trivial).

> But I expect bb b // # x to cover most of my
> microtonal notation needs anyway.

You can notate 17, 22, 31, and 41-ET in Sagittal using only straight
flags, so perhaps curved flags will be a non-issue for you.

> >> >If you're not notating anything as complicated as 130 or 142,
then
> >> >you won't need all of those symbols.
> >>
> >> In light of the recent thread about notations for a 'core' number
> >> of rank 2 temperaments, I would suggest you and Dave do a poster
> >> showing accidental sets and an ascending MOS <= 10 tones for
each.
> >
> >Easier said than done, in light of the recent discussions
concerning
> >the intricacies in selecting appropriate accidentals for
> >temperaments. If Herman and Graham would look at some of these
and
> >agree on sets of accidentals, I'd be happy to prepare a diagram.
>
> Hopefully they'll agree then!

The uncertainties seem to revolve around the question of how to
notate temperaments as *tunings* so as to take into account that they
are members of more than a single temperament class.

Anyway, I'll have to take a closer look at Herman's notation for
orwell.

--George

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> >> It's easier to distinguish anything not involving a curved flag.
> >
> >From what? Are you saying that changing the flag in the right-hand
> >symbol from straight to curved would make it more difficult to
> >distinguish that symbol from the left-hand one (with a straight left
> >flag)? I don't understand how you could think that making two
> >symbols more unlike one another would make them more difficult to
> >tell apart.
>
> The curved flags are very hard to read to my eye, even alone.
> Put them on a staff and it's worse.

Hi Carl,

Thanks for this. It would be good if you could explain this in more
detail. Can you get more specific about why it is hard or how it might
be improved? Do you just find them ugly? Do certain strokes or parts
of strokes get lost against the staff? Do they look too much like some
other staff symbols? Assume you're talking to people who are
completely blind to whatever effect you're seeing, but who understand
that if you see it that way, there will be many others who do too.

> I'm trying to be as direct and honest as possible, without
> being offensive or nonconstructive. I think you should use only
> straight flags, or redraw your curved flags in a big way.

I'm afraid using only straight flags isn't possible in the overall
system (but certainly possible in many of the most common tunings as
George pointed out).

So please suggest ways we might redraw them.

> >> It's not the number of symbols, but rather the sight of those
> >> curved flags from which eyes are still recovering.
> >
> >Why is that? Don't most ASCII characters contain curved lines? Does
> >reading music in flat keys hurt your eyes? ;-)
>
> There's something about the curves that manages to be lumpy and
> ungraceful.

OK. This is good. Can you nail it down any more than that?

I tried to use the "style" of the conventional flat symbols's curves,
while being distinct from it. As with the flat, the stroke is fatter
where it is horizontal and thinnner where vertical. This is necessary
anyway to avoid getting lost against the staff lines.

> I especially don't like the look of having some curved and some
> straight. I would say that I like the 'original sagittal'
> approach to 72 pretty well.

I note that it remains a valid sagittal notation for 72. However, as
far as suggesting a _standard_ notation for 72 goes, unless we can
obtain evidence that curve haters outnumber lateral confusers I think
we'd best avoid it.

It can be useful to look at existing alphabets and numerals (what
symbols do occur in the same alphabet and what don't) to get an idea
of what is generally considered distinct, as opposed to "distinct to
me". Consider how rare lateral reversed pairs are, and consider the
expression "mind your 'p's and 'q's". And consider, as George did that
the distinction between '5' and '6' is almost entirely in a curve
replacing a right-angle. Similarly 'D' and 'O'. Similarly '4' and some
versions of '9'.

But you seem to be agreeing that the curves are maximally distinct
from the straight flags. You seem to be saying that the curves are
just painful to look at or hard to perceive in and of themselves. Is
this perhaps because you've never seen anything like them before.

What about the symbol with two arcs and those with one arc and one
straight?

> One issue with arrows is that they
> can be hard to place on the staff in a chord. Blackwood drew
> his arrows through circles for this reason, but I liked that
> approach even less. Sagittal arrows seem pretty good in that
> they are compact, not long like Blackwood's were.

Thanks for that observation.

-- Dave Keenan

At 01:54 PM 12/5/2007, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>>
>> >> It's easier to distinguish anything not involving a curved flag.
>> >
>> >From what? Are you saying that changing the flag in the right-hand
>> >symbol from straight to curved would make it more difficult to
>> >distinguish that symbol from the left-hand one (with a straight left
>> >flag)? I don't understand how you could think that making two
>> >symbols more unlike one another would make them more difficult to
>> >tell apart.
>>
>> The curved flags are very hard to read to my eye, even alone.
>> Put them on a staff and it's worse.
>
>Hi Carl,
>
>Thanks for this. It would be good if you could explain this in more
>detail. Can you get more specific about why it is hard or how it might
>be improved? Do you just find them ugly?

Yes, it's largely an aesthetic thing. But I also think it would
be hard to read. Like, I can't even immediately count the number
of flag types on pg. 8. And I like to think I have a decent
appreciation for and understanding of typography.

>Do certain strokes or parts
>of strokes get lost against the staff?

Yes, that's my fear. I guess I'd have to look at some scores to
be sure... letssee, I've got Walking To Class, and hey, this is
MUCH better, and oh, here's the problem: Sagittal.pdf is using
raster mode. Ouch ouch.

There are still problems, though. The last beat of measure 8
has a curved flag crossing a staff line, that looks like it would
easily be confused for a straight flag on a printout. The main
difference isn't the curve in the flag, but rather that the entire
symbol is lighter than the straight version. And this score
combines standard flats and sharps with sagittal accidentals, which
seems like a type of hell to me.

Black Laundry uses triple flags, and I completely agree with you
about these. Anyone reading this on a music stand is headed
straight for an optometrist afterwards.

O Europae is pretty simple, with few accidentals.

All of these are one-voice-per-staff scores. I haven't seen
any sagittal piano music...

>So please suggest ways we might redraw them.

Let's get a vector version of them first. I suggest you DELETE that
pdf from the website in the meantime.

>I note that it remains a valid sagittal notation for 72. However, as
>far as suggesting a _standard_ notation for 72 goes, unless we can
>obtain evidence that curve haters outnumber lateral confusers I think
>we'd best avoid it.

Is there any literature on "lateral confirmability"? I've never
heard of it and I don't fancy it existing.

>It can be useful to look at existing alphabets and numerals (what
>symbols do occur in the same alphabet and what don't) to get an idea
>of what is generally considered distinct, as opposed to "distinct to
>me". Consider how rare lateral reversed pairs are, and consider the
>expression "mind your 'p's and 'q's".

I never understood that expression. Is that its origin?

>What about the symbol with two arcs and those with one arc and one
>straight?

I'm not making any more comments until I can see a real version
of these. Wasn't there a TT font around somewhere?

-Carl

Dave Keenan wrote:

> Thanks for this. It would be good if you could explain this in more
> detail. Can you get more specific about why it is hard or how it might
> be improved? Do you just find them ugly? Do certain strokes or parts
> of strokes get lost against the staff? Do they look too much like some
> other staff symbols? Assume you're talking to people who are
> completely blind to whatever effect you're seeing, but who understand
> that if you see it that way, there will be many others who do too.

I agree that some of the accidentals look ugly. For example, the 7C and 35M (as they're labeled in Sagittal.pdf) from Spartan. They're unbalanced. The solution is to get a talented calligrapher to redraw them, or allow competition between fonts. I don't consider it a fundamental flaw in the system. Curves are not unknown in calligraphy, after all.

> It can be useful to look at existing alphabets and numerals (what
> symbols do occur in the same alphabet and what don't) to get an idea
> of what is generally considered distinct, as opposed to "distinct to
> me". Consider how rare lateral reversed pairs are, and consider the
> expression "mind your 'p's and 'q's". And consider, as George did that
> the distinction between '5' and '6' is almost entirely in a curve
> replacing a right-angle. Similarly 'D' and 'O'. Similarly '4' and some
> versions of '9'.

You know, I'm finding it difficult to think of examples in existing writing systems (no need to restrict it to alphabets and numerals) where curves or straight lines are the *only* distinguishing feature. Yes, there's D and O but a D becomes a triangle with straight lines and an O a rectangle. There's also U and V which originally were the same character but can look different with straight lines. 5 and 6 are topologically distinct and can be distinguished on digital watches with straight lines. Similarly 4 has a tick that 9 doesn't. In a world where 土 and 士 or 人 and 入 are distinct characters there's plenty of difference between 4 and 9. 2 and Z is a better example, but then some of us do write a Z with a cross because of bitter experience with algebra. (And numerals and letters needn't be distinct: remember O/0 and 1/I/l.)

I don't know how much this says straight lines and curves aren't easily distinguishable. It may be more that certain styles of writing depend on straight lines and so characters have to be distinguishable with straight lines.

Roman numerals do result in laterally reversed pairs: IV and VI for example. Otherwise it is rare, I admit. The p and q business is a sidetrack. The phrase comes from printing where you look at mirror images of the characters, so it's easy to make the mistake. I notice that my terminal font (which is designed to make characters distinct) shows p and q as mirror images. Now, maybe you can point to research showing that some people do find this confusing.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Dave Keenan wrote:
>
> > Thanks for this. It would be good if you could explain this in
more
> > detail. Can you get more specific about why it is hard or how it
might
> > be improved? Do you just find them ugly? Do certain strokes or
parts
> > of strokes get lost against the staff? Do they look too much like
some
> > other staff symbols? Assume you're talking to people who are
> > completely blind to whatever effect you're seeing, but who
understand
> > that if you see it that way, there will be many others who do too.
>
> I agree that some of the accidentals look ugly. For example, the
7C and
> 35M (as they're labeled in Sagittal.pdf) from Spartan. They're
> unbalanced. The solution is to get a talented calligrapher to
redraw
> them, or allow competition between fonts. I don't consider it a
> fundamental flaw in the system. Curves are not unknown in
calligraphy,
> after all.

Could you clarify what you mean by unbalanced? I think the problem
is that, because the figures are bitmapped, you need to view them
at "actual" (100%) size on your computer screen. (This is not a
problem if you print hard copy.)

Even if we had used the scalable Sagittal font this problem wouldn't
be completely resolved, because the (moderately-priced) font creation
software Dave used didn't provide for hinting (and subsequent efforts
to fix that were not successful).

--George

George D. Secor wrote:

> Could you clarify what you mean by unbalanced? I think the problem > is that, because the figures are bitmapped, you need to view them > at "actual" (100%) size on your computer screen. (This is not a > problem if you print hard copy.)

I have PDFs by default at 125%, which is smaller than actual size. Oh the joys of screen resolutions! Also, the fonts in Sagittal.pdf get pixellated if I enlarge it.

Anyway, I have the character map PDF instead. The one I mean seems to be decimal code 166, /|) n. It's unbalanced in that there's more weight to the right of the shaft than the the left of it, but it naturally centers on the shaft.

> Even if we had used the scalable Sagittal font this problem wouldn't > be completely resolved, because the (moderately-priced) font creation > software Dave used didn't provide for hinting (and subsequent efforts > to fix that were not successful).

AIUI, creating a professional quality font is very difficult. So I don't expect perfection on that front. But the problem here is more a matter of calligraphy. Not a simple issue either, especially for new characters. Not a fundamental flaw in the system either way.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> George D. Secor wrote:
>
> > Could you clarify what you mean by unbalanced? I think the
problem
> > is that, because the figures are bitmapped, you need to view them
> > at "actual" (100%) size on your computer screen. (This is not a
> > problem if you print hard copy.)
>
> I have PDFs by default at 125%, which is smaller than actual size.
Oh
> the joys of screen resolutions! Also, the fonts in Sagittal.pdf
get
> pixellated if I enlarge it.
>
> Anyway, I have the character map PDF instead. The one I mean seems
to
> be decimal code 166, /|) n. It's unbalanced in that there's more
weight
> to the right of the shaft than the the left of it, but it naturally
> centers on the shaft.

The curved flags |) and (| in both /|) and (|\, respectively, singly
represent larger ratios than their accompanying straight flags, /|
and |\, so it's not inappropriate for them to span a larger area,
giving the appearance of imbalance.

> > Even if we had used the scalable Sagittal font this problem
wouldn't
> > be completely resolved, because the (moderately-priced) font
creation
> > software Dave used didn't provide for hinting (and subsequent
efforts
> > to fix that were not successful).
>
> AIUI, creating a professional quality font is very difficult. So I
> don't expect perfection on that front. But the problem here is
more a
> matter of calligraphy. Not a simple issue either, especially for
new
> characters. Not a fundamental flaw in the system either way.

With limited resources we do the best we can. :-)

--George