Specific question about the basic nature of Sagittal

I'm not sure I really understand Sagittal. My confusion is that I don't know whether Sagittal is based on accidentals which represent specific small JI intervals, and their tempered versions in any regular temperament; or on the other hand if there is always a specific size ordering of Sagittal accidentals that must be preserved regardless of tempering.

As a specific example, consider superpelog temperament: http://x31eq.com/cgi-bin/rt.cgi?ets=9_5c&limit=7

In superpelog, 2 generators is 4/3, so every other note can be represented as the basic chain of fourths in Sagittal. But some new accidentals are needed for the notes that are an odd number of generators. Two specific small intervals you might want to notate as accidentals are +5 generators (99.4 cents in the TOP tuning), and -9 generators (62.9 cents in the TOP tuning).

The larger of these intervals represents 64/63 and the smaller represents 36/35; they're backwards because 81/80 is inverted.

So, my question is, could you have a Sagittal notation system for superpelog in which |) is the 64/63 and /|) is the 36/35, even though their sizes are reversed relative to JI and most other temperaments? Or instead, could you have a system where |) is the 36/35 and /|) is the 64/63, even though they correspond to JI backwards?

Of course, as a practical matter you'd want to avoid this kind of confusion as much as possible, but I'm interested in which one the Sagittal system defines to be technically correct usage.

Let's say we're hundreds of years in the future when xenharmonic music has finally caught on and everyone's into it, and I'm a hotshot trombone player who claims to be able to sightread *anything* in Sagittal perfectly. I show up to a gig and the doofus composer has notated superpelog using these two accidentals. If I see |), which note do I play?

(Perhaps Sagittal is totally agnostic on this point, and says you're allowed to do whatever you want. If so, I regard that as extremely unsatisfying. It should be a consistent system where it's possible to predict with certainty which accidental corresponds to which note.)

Keenan

On Sat, Oct 22, 2011 at 3:03 PM, Keenan Pepper <keenanpepper@...> wrote:
>
> I'm not sure I really understand Sagittal. My confusion is that I don't know whether Sagittal is based on accidentals which represent specific small JI intervals, and their tempered versions in any regular temperament; or on the other hand if there is always a specific size ordering of Sagittal accidentals that must be preserved regardless of tempering.

FWIW, I asked Dave Keenan at one point about how to notate accidentals
for tunings in which the accidental is reversed, like 81/80 for any
decent tuning of mavila. His response was "don't use that accidental."
Only obliquely related to what you asked, but maybe you'll find it
useful.

-Mike

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> I'm not sure I really understand Sagittal. My confusion is that I don't know whether Sagittal is based on accidentals which represent specific small JI intervals, and their tempered versions in any regular temperament;

Yes. Each Sagittal symbol represents a specific (possibly tempered) frequency ratio. That is the primary consideration.

> or on the other hand if there is always a specific size ordering of Sagittal accidentals that must be preserved regardless of tempering.
>

We attempt to preserve the size ordering of the symbols (as per strict JI) whenever possible. So when a temperament gives you a choice of several commatically-valid sets of accidentals, size order is a very important consideration in choosing between them. Choosing a _good_ sagittal notation for a tuning, from among those which are commatically valid, is something of an art. There are several other considerations as well, such as tempered size close to untempered size, consistency of "flag arithmetic" and having the the multi-shaft symbols in the second half-apotome recapitulate the sequence of flag combinations in the first half-apotome, consistency with standard notations for related standard tunings, staying within the smallest standard subset possible e.g Spartan or Athenian, using more popular symbols, etc. etc.

These are not like sagittal _laws_, but are just things that everyone comes to value who thinks about the problems deeply enough with enough different tunings, although folks may disagree on the the fine balance of relative importance of these things.

The abovementioned primary consideration has been broken (or perhaps I should say bent) in a _very_few_ places because of the sheer weight of these secondary considerations.

> As a specific example, consider superpelog temperament: http://x31eq.com/cgi-bin/rt.cgi?ets=9_5c&limit=7
>
> In superpelog, 2 generators is 4/3, so every other note can be represented as the basic chain of fourths in Sagittal. But some new accidentals are needed for the notes that are an odd number of generators. Two specific small intervals you might want to notate as accidentals are +5 generators (99.4 cents in the TOP tuning), and -9 generators (62.9 cents in the TOP tuning).
>
> The larger of these intervals represents 64/63 and the smaller represents 36/35; they're backwards because 81/80 is inverted.
>
> So, my question is, could you have a Sagittal notation system for superpelog in which |) is the 64/63 and /|) is the 36/35, even though their sizes are reversed relative to JI and most other temperaments?

Yes you could. But it is not recommended. I'd go to higher prime-limit extensions of the temperament until I find a pair of valid accidentals that does not involve a size reversal relative to JI. And look at standard notations for related ETs.

What Mike said about not being allowed to use accidentals whose tempered size is zero or negative, should probably be extended to avoiding _pairs_ of symbols that differ by a flag whose standalone comma interpretation has zero or negative size in the temperament, in this case the left barb /|. But such an extension isn't a law, just a good idea.

>Or instead, could you have a system where |) is the 36/35 and /|) is the 64/63, even though they correspond to JI backwards?
>

No. That would be completely outside Sagittal.

> Of course, as a practical matter you'd want to avoid this kind of confusion as much as possible, but I'm interested in which one the Sagittal system defines to be technically correct usage.
>
> Let's say we're hundreds of years in the future when xenharmonic music has finally caught on and everyone's into it, and I'm a hotshot trombone player who claims to be able to sightread *anything* in Sagittal perfectly. I show up to a gig and the doofus composer has notated superpelog using these two accidentals. If I see |), which note do I play?
>
> (Perhaps Sagittal is totally agnostic on this point, and says you're allowed to do whatever you want. If so, I regard that as extremely unsatisfying. It should be a consistent system where it's possible to predict with certainty which accidental corresponds to which note.)
>

Sagittal is definitely not agnostic on this.

|) will represent the 7-comma 63:64

All you have to do is apply the temperament mapping to that comma to find out what to play.

The abovementioned very-occasional bending of this rule involves a few symbols that have a "secondary comma" interpretation. So if the primary comma makes no sense when tempered (i.e. it becomes zero or negative or way out of size order with other symbols) you would try tempering the secondary comma meaning of that symbol.

The secondary comma, for those symbols that have them, is always a more complex comma whose untempered size differs by less than half a cent from the primary comma. The most notable example is /|) whose secondary comma is the 13-medium-diesis 1024:1053.

Automated playing of sagittal is possible given only the temperament mapping and the fixed mapping from symbol to primary comma.

That mapping is called the Sagittal Character Map and is on the sagittal website in several formats, although it needs updating to include the accented symbols.
http://dkeenan.com/sagittal

It doesn't list secondaries. Why not? There is an alternative way of viewing secondary commas. A secondary comma is simply the primary comma for a symbol differing by a mina accent (approx 0.4 c) up or down. The accent has simply been dropped because no other symbols in the notation for that tuning have accents and there is no possibility of confusion. For example 1024:1053 is the primary comma for the /|). symbol. The period or full-stop after the /|) represents a downward right accent in text.

See http://dkeenan.com/sagittal/SagittalJI.gif

Very clear and helpful answer, Dave. I feel like I grok Sagittal now. I'm glad that it's so logically consistent and interacts with regular temperament so well.

Keenan

On 10/22/2011 11:05 PM, dkeenanuqnetau wrote:
> --- In tuning@yahoogroups.com, "Keenan Pepper"<keenanpepper@...>
> wrote:

>> So, my question is, could you have a Sagittal notation system for
>> superpelog in which |) is the 64/63 and /|) is the 36/35, even
>> though their sizes are reversed relative to JI and most other
>> temperaments?
>
> Yes you could. But it is not recommended. I'd go to higher
> prime-limit extensions of the temperament until I find a pair of
> valid accidentals that does not involve a size reversal relative to
> JI. And look at standard notations for related ETs.

In 7-limit you might consider |\\ 28/27 instead of |) 64/63 to notate (-1, +5) in superpelog.

It looks like [<1 2 1 3 3|, <0 -2 6 -1 2|] is a reasonably good 11-limit extension. This adds another alternative, )||~ 22/21.

Another issue with superpelog notation is extending it past a chain of 7 fifths. You can't use the usual /||\ and \!!/ since they go the wrong way. It looks like one option is )||( 25/24, or |\ 55/54 for the 11-limit version. The math works out nicely for |\ + /|) = )||~ in case that matters. Another option is ~|( if that can be interpreted as 245/243, but I don't recall whether that or 126/125 is the preferred 7-limit approximation of ~|( (4131/4096).

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> On 10/22/2011 11:05 PM, dkeenanuqnetau wrote:
> > --- In tuning@yahoogroups.com, "Keenan Pepper"<keenanpepper@>
> > wrote:
>
> >> So, my question is, could you have a Sagittal notation system for
> >> superpelog in which |) is the 64/63 and /|) is the 36/35, even
> >> though their sizes are reversed relative to JI and most other
> >> temperaments?
> >
> > Yes you could. But it is not recommended. I'd go to higher
> > prime-limit extensions of the temperament until I find a pair of
> > valid accidentals that does not involve a size reversal relative to
> > JI. And look at standard notations for related ETs.
>
> In 7-limit you might consider |\\ 28/27 instead of |) 64/63 to notate
> (-1, +5) in superpelog.
>
> It looks like [<1 2 1 3 3|, <0 -2 6 -1 2|] is a reasonably good 11-limit
> extension. This adds another alternative, )||~ 22/21.
>
> Another issue with superpelog notation is extending it past a chain of 7
> fifths. You can't use the usual /||\ and \!!/ since they go the wrong
> way. It looks like one option is )||( 25/24, or |\ 55/54 for the
> 11-limit version. The math works out nicely for |\ + /|) = )||~ in case
> that matters. Another option is ~|( if that can be interpreted as
> 245/243, but I don't recall whether that or 126/125 is the preferred
> 7-limit approximation of ~|( (4131/4096).

125:126 has a weighted complexity of 30.855 and ranks 50 in popularity, whereas 243:245 has a weighted complexity of 31.162 and ranks 38 in popularity. (For comparison, the ~|( symbol definition, 4096:4131, has a weighted complexity of 20.799 and ranks 20 in popularity.) Weighted complexity is calculated using a rather complicated formula involving the prime factors >3 (and also their distribution in the numerator vs. denominator, such that 5:7 is less complex than 5*7), whereas popularity rank was determined by the frequency of occurrence of the various commas required to notate scales in the Scala archive. There's no clear winner, so the "preferred" use of this symbol in a <17-limit notation would be for temperaments in which the symbol is valid for both 125:126 and 243:245 (a specific example would be eneallimmal, which requires a lot of symbols, such that it's necessary to include this one). For the simpler temperaments it's best to use symbols that represent simpler ratios, wherever possible.

--George

On 10/25/2011 12:05 AM, gdsecor wrote:

> 125:126 has a weighted complexity of 30.855 and ranks 50 in
> popularity, whereas 243:245 has a weighted complexity of 31.162 and
> ranks 38 in popularity. (For comparison, the ~|( symbol definition,
> 4096:4131, has a weighted complexity of 20.799 and ranks 20 in
> popularity.) Weighted complexity is calculated using a rather
> complicated formula involving the prime factors>3 (and also their
> distribution in the numerator vs. denominator, such that 5:7 is less
> complex than 5*7), whereas popularity rank was determined by the
> frequency of occurrence of the various commas required to notate
> scales in the Scala archive. There's no clear winner, so the
> "preferred" use of this symbol in a<17-limit notation would be for
> temperaments in which the symbol is valid for both 125:126 and
> 243:245 (a specific example would be eneallimmal, which requires a
> lot of symbols, such that it's necessary to include this one). For
> the simpler temperaments it's best to use symbols that represent
> simpler! ratios, wherever possible.

Thanks. Is there a list of (at least the most "popular") ratios with their weighted complexity and popularity? (Or a spreadsheet with a formula for weighted complexity?)

Herman

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> On 10/25/2011 12:05 AM, gdsecor wrote:
>
> > 125:126 has a weighted complexity of 30.855 and ranks 50 in
> > popularity, whereas 243:245 has a weighted complexity of 31.162 and
> > ranks 38 in popularity. (For comparison, the ~|( symbol definition,
> > 4096:4131, has a weighted complexity of 20.799 and ranks 20 in
> > popularity.) Weighted complexity is calculated using a rather
> > complicated formula involving the prime factors>3 (and also their
> > distribution in the numerator vs. denominator, such that 5:7 is less
> > complex than 5*7), whereas popularity rank was determined by the
> > frequency of occurrence of the various commas required to notate
> > scales in the Scala archive. There's no clear winner, so the
> > "preferred" use of this symbol in a<17-limit notation would be for
> > temperaments in which the symbol is valid for both 125:126 and
> > 243:245 (a specific example would be eneallimmal, which requires a
> > lot of symbols, such that it's necessary to include this one). For
> > the simpler temperaments it's best to use symbols that represent
> > simpler! ratios, wherever possible.
>
> Thanks. Is there a list of (at least the most "popular") ratios with
> their weighted complexity and popularity? (Or a spreadsheet with a
> formula for weighted complexity?)
>
> Herman

Yes, there is a spreadsheet listing over 1800 commas in the Sagittal single-shaft symbol range (<=68 cents); go to:
/tuning-math/files/secor/notation
Filename is CommaExp.xls

The formulas for cols. G and beyond are only in row 3 and the last two rows. (The other rows have the calculated values pasted in. Note: I'm not sure, but you may have to install Excel's math toolpack in order for all of the functions to work.) You can enter a ratio in cells E1872 and F1872, and the apotome complement will appear in the row above (you may have to fiddle with the numbers in cols. E & F in the row below to get the complement ratio into lowest terms, or you can add a formula that checks cells Q1872 and R1872 to come up with the desired values).

As you examine the various cells that contribute to the weighted complexity in col. M, you'll see that it's not a simple calculation; several different things were taken into account (prime limit, number of prime factors and their distribution in numerator vs. denominator, the value of the 3-exponent, and the comma "slope"), and the last three were weighted by the factors in row 1. The factors were determined subjectively (by trial & error) to get the simplest commas to sort in what Dave & I agreed to be the most appropriate order; e.g., Didymus' (syntonic) comma (5C) and Archytas' (septimal) comma (7C) should be less complicated than the schisma (5s) or the pythagorean comma (1C). The color-coding for the comma names & ratios is: magenta for a ratio that defines an athenian-level symbol; orange for a promethean-level symbol (all unaccented symbols not athenian-level); yellow for herculean-level symbols (left-accented only); green for olympian-level symbols (right-accented, for the remaining "minas", defined by 233-EDA boundaries); cyan for ratios that could be assigned unique symbols not previously defined (although we have chosen not to do this, so that the notation does not become more complicated than it already is).

There are two columns (A & B) listing popularity rank, because that was determined twice, using slightly different methods; the second method, Pop2, is the one we're currently using. Popularity was calculated only for intervals less than 1/2-apotome, so the larger of each apotome-complement pair was assigned the same popularity ranking as the smaller (plus .1, so they'll sort with the smaller one first). Note that apotome-pairs will have differing weighted complexity, e.g., 9.162 for 27:28 vs. 15.328 for 57344:59049.

--George