171-ET notation for JI and temperaments

I've been doing some experimentation with a 171-ET-based notation, and it does have its useful points. Only 13 accidental pairs are needed, and 171-ET has a good 7-limit consistency rating. But the end result is quirky and uses the Sagittal symbols in a really unorthodox way. Maybe this could be a good example of how NOT to use Sagittal. :-)

I went through my notes on possible notations for temperaments using the simplest JI intervals, and made a table of how they can be notated with 171-ET notation. There were a few overlaps (e.g. 60/49 comes up in meantone and orwell, but 49/40 in beatles and augene). It's likely that other overlaps will come up once more temperaments are added to the list.

... D 0 E 29 F 42 G 71 A 100 B 129 C 142
-13 16/15 8/5 !!/
-12 40/21 15/14 10/7 45/28 !!)
-11 \!!
-10 48/25 27/25 32/25 36/25 )!!(
-9 27/14 (8/7) 9/7 (12/7) (!/
-8 49/30 \!/
-7 35/18 35/32 35/27 35/24 \!)
-6 \\!
-5 49/25 98/75 72/49 !/
-4 675/343 7/6 21/16 7/4 !)
-3 10/9 5/3 \!
-2 125/63 125/112 ~!(
-1 448/225 28/25 112/75 42/25 !(
0 1/1 9/8 4/3 3/2 27/16 16/9
+1 225/224 25/21 75/56 25/14 |(
+2 126/125 224/125 ~|(
+3 6/5 27/20 9/5 /|
+4 686/675 8/7 32/21 12/7 |)
+5 50/49 49/36 75/49 49/27 |\
+6 //|
+7 36/35 48/35 54/35 64/35 /|)
+8 60/49 /|\
+9 28/27 (7/6) 14/9 (7/4) (|\
+10 25/24 25/18 25/16 )||(
+11 /||
+12 21/20 56/45 7/5 28/15 ||)
+13 5/4 45/32 15/8 ||\

*D-2 486/245
*D+2 245/243
*F+8 49/40

This looks fine for notating these JI intervals, but gets tricky when you notate temperaments with it. An example of how this might be used: 7-limit meantone. In a few cases, the literal meaning of the Sagittal notation doesn't match the simpler intervals used in meantone (and in two cases, the Sagittal symbols stand for a simpler 11-limit interval).

B\!/ 49/30 (+7, -15) ~ 18/11
E\!) 35/32 (+6, -14)
A\!) 35/24 (+6, -13)
D\!) 35/18 (+6, -12)
G!) 21/16 (+5, -11)
C!) 7/4 (+5, -10)
F!) 7/6 (+4, -9)
A(|\ 14/9 (+4, -8) ~ 25515/16384
D||) 21/20 (+3, -7) ~ 137781/131072
G||) 7/5 (+3, -6) ~ 45927/32768
C||\ 15/8 (+3, -5)
F||\ 5/4 (+2, -4)
B\! 5/3 (+2, -3)
E 9/8 (+1, -2)
A 3/2 (+1, -1)
D 1/1 (+0, +0)
G 4/3 (+0, +1)
C/| 9/5 (+0, +2)
F/| 6/5 (-1, +3)
B!!/ 8/5 (-1, +4)
E!!) 15/14 (-2, +5) ~ 16384/15309
A!!) 10/7 (-2, +6) ~ 65536/45927
D(!/ 27/14 (-2, +7) ~ 16384/8505
G(!/ 9/7 (-3, +8) ~ 32768/25515
B|) 12/7 (-3, +9)
E|) 8/7 (-4, +10)
A|) 32/21 (-4, +11)
D/|) 36/35 (-5, +12)
G/|) 48/35 (-5, +13)
C/|) 64/35 (-5, +14)
F/|\ 60/49 (-6, +15) ~ 11/9

While this system has a definite appeal (find what JI interval you want to approximate, find the closest 171-ET note, and notate that), it does have an odd sort of feel to it, and it really stretches the usage of the Sagittal symbols. They still have a consistent meaning, but the meaning refers to 171-ET -- two levels of indirection from the temperament you're trying to represent. And 171-ET isn't ideal for anything beyond 7-limit temperaments.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> I've been doing some experimentation with a 171-ET-based notation,
and
> it does have its useful points. Only 13 accidental pairs are
needed, and
> 171-ET has a good 7-limit consistency rating. But the end result is
> quirky and uses the Sagittal symbols in a really unorthodox way.
Maybe
> this could be a good example of how NOT to use Sagittal. :-)

The Sagittal paper has this symbol set for 171:

171a: |( ~|( /| |) |\ //| /|) /|\ (|\ )||( /|| ||) ||\
(||( /||) /||\

but with a single change of symbol (and its apotome-complement),
everything can be defined and/or interpreted at the 7 limit:

171b: |( ~|( /| |) ~|) //| /|) /|\ (|\ )||( ~~|| ||)
||\ (||( /||) /||\

~|), defined as 48:49, replaces |\, 54:55. This is advisable,
because in your table, below, you're using |\ and !/ to represent
ratios containing 7^2, whereas, in athenian-level JI, 48:49 is
represented by (|(, which amounts to a different number of steps than
|\ (in athenian), which makes this notation inconsistent or ambiguous
with respect to athenian; using ~|) removes the ambiguity. While ~|)
is arguably more complicated than |\, it does introduce a mnemonic
into the symbol sequence in that |( is followed by ~|( and |) is
followed by ~|), in addition to removing the lateral confusability
between /| and |\ . If you're going with a 171-ET-based notation,
then I recommend introducing this one non-athenian symbol (and its
apotome-complement), because there are multiple advantages for doing
so. (Dave & I thought it important enough to assign it an ASCII
shorthand character pair, h = down and p = up.)

~|( represents 125:126 or 243:245, in a secondary role. (It's also
valid as 17C, 3969:4096, its primary definition.)

/|\ can be used for either 3069:4096 or 392:405 (in secondary roles);
essentially, you can think of this as a half-apotome. If you wanted
to specify 392:405, then )/|\ could replace it; however, that would
be a more complicated symbol, and an important objective here is
simplicity.

As with the reasonably simple JI notations in my previous message,
the 5-schisma, 32768:32805, vanishes, so ratios that define left-
accented symbols are notated by dropping the left accents. Also, (|\
is 3 times as many steps as /|, as required for 5^3.

So you indeed have a very nice 7-limit notation if you map the
symbols to 171-ET (alias 16-EDA).

> I went through my notes on possible notations for temperaments
using the
> simplest JI intervals, and made a table of how they can be notated
with
> 171-ET notation. There were a few overlaps (e.g. 60/49 comes up in
> meantone and orwell, but 49/40 in beatles and augene).

Those two are merely apotome-complements of one another, differing by
<1c. If you wanted, you could distinguish them by using /|\ for
392:405 and (|) for 1280:1323; however, introduction of an additional
symbol would probably be an unwelcome complication, because if you
wanted more symbols, you could use the 21-EDA, 7-limit notation I
mentioned yesterday:

21-EDA, 7-limit: |( ~| ~|( /| |) (| ~|) //| /|) (/|
|\) (|\ )||( ~~|| )||~ ||) ||\ (||( )||\\ /||) /||\

One thing I didn't mention yesterday is that (| represents 49:50 in a
secondary role; the symbol '(| is defined as 49:50, so I've simply
dropped the left accent.

I would think long and hard about using this though, because there
are enough non-athenian symbols to give me second thoughts. :-(

You could, of course, replace (/| and |\) with /|\ and (|), to make
it more athenian:
21-EDA, 7-friendly: |( ~| ~|( /| |) (| ~|) //| /|) /|\
(|) (|\ )||( ~~|| )||~ ||) ||\ (||( )||\\ /||) /||\
which also makes it a literal expansion of the symbol set in 171-ET
(with the ~|) modification I recommended). It's nice that the
mnemonic for the 171 symbol sequence involving the ~| flag is not
only retained, but that the ~| symbol itself occurs in the sequence.

> It's likely that
> other overlaps will come up once more temperaments are added to the
list.

Yes, but they're likely to be very small, because the 7-limit error
of 171-ET is so small. If you can tolerate a 5-schisma error, then a
notation based on 171-ET (or 16-EDA) should serve you well.

> ... D 0 E 29 F 42 G 71 A 100 B 129 C 142
> -13 16/15 8/5 !!/
> -12 40/21 15/14 10/7 45/28 !!)
> -11 \!!
> -10 48/25 27/25 32/25 36/25 )!!(
> -9 27/14 (8/7) 9/7 (12/7) (!/
> -8 49/30 \!/
> -7 35/18 35/32 35/27 35/24 \!)
> -6 \\!
> -5 49/25 98/75 72/49 !/
> -4 675/343 7/6 21/16 7/4 !)
> -3 10/9 5/3 \!
> -2 125/63 125/112 ~!(
> -1 448/225 28/25 112/75 42/25 !(
> 0 1/1 9/8 4/3 3/2 27/16 16/9
> +1 225/224 25/21 75/56 25/14 |(
> +2 126/125 224/125 ~|(
> +3 6/5 27/20 9/5 /|
> +4 686/675 8/7 32/21 12/7 |)
> +5 50/49 49/36 75/49 49/27 |\
> +6 //|
> +7 36/35 48/35 54/35 64/35 /|)
> +8 60/49 /|\
> +9 28/27 (7/6) 14/9 (7/4) (|\
> +10 25/24 25/18 25/16 )||(
> +11 /||
> +12 21/20 56/45 7/5 28/15 ||)
> +13 5/4 45/32 15/8 ||\
>
> *D-2 486/245
> *D+2 245/243
> *F+8 49/40

7/5 may use either ||) or !!!(, depending on whether you wish to
spell the interval as a large fourth or small fifth (and everything
else can also have alternate spellings).

> This looks fine for notating these JI intervals, but gets tricky
when
> you notate temperaments with it. An example of how this might be
used:
> 7-limit meantone. In a few cases, the literal meaning of the
Sagittal
> notation doesn't match the simpler intervals used in meantone (and
in
> two cases, the Sagittal symbols stand for a simpler 11-limit
interval).
>
> B\!/ 49/30 (+7, -15) ~ 18/11
> E\!) 35/32 (+6, -14)
> A\!) 35/24 (+6, -13)
> D\!) 35/18 (+6, -12)
> G!) 21/16 (+5, -11)
> C!) 7/4 (+5, -10)
> F!) 7/6 (+4, -9)
> A(|\ 14/9 (+4, -8) ~ 25515/16384
> D||) 21/20 (+3, -7) ~ 137781/131072
> G||) 7/5 (+3, -6) ~ 45927/32768
> C||\ 15/8 (+3, -5)
> F||\ 5/4 (+2, -4)
> B\! 5/3 (+2, -3)
> E 9/8 (+1, -2)
> A 3/2 (+1, -1)
> D 1/1 (+0, +0)
> G 4/3 (+0, +1)
> C/| 9/5 (+0, +2)
> F/| 6/5 (-1, +3)
> B!!/ 8/5 (-1, +4)
> E!!) 15/14 (-2, +5) ~ 16384/15309
> A!!) 10/7 (-2, +6) ~ 65536/45927
> D(!/ 27/14 (-2, +7) ~ 16384/8505
> G(!/ 9/7 (-3, +8) ~ 32768/25515
> B|) 12/7 (-3, +9)
> E|) 8/7 (-4, +10)
> A|) 32/21 (-4, +11)
> D/|) 36/35 (-5, +12)
> G/|) 48/35 (-5, +13)
> C/|) 64/35 (-5, +14)
> F/|\ 60/49 (-6, +15) ~ 11/9
>
> While this system has a definite appeal (find what JI interval you
want
> to approximate, find the closest 171-ET note, and notate that), it
does
> have an odd sort of feel to it, and it really stretches the usage
of the
> Sagittal symbols. They still have a consistent meaning, but the
meaning
> refers to 171-ET -- two levels of indirection from the temperament
> you're trying to represent.

The above notation also suffers from being D-centered. If my tonic
note is C, I might prefer to see B!!!) and E!!!) instead of A(|\ and
D||), and I would also be asking why A(|\ and D||) have different
accidentals, if they're a fifth apart. There are two issues here:

1) Since the whole point of a temperament is to have a single pitch
represent multiple ratios, it becomes more difficult and complicated
if different accidentals are used to notate the same tempered
interval. Did you intend for the accidentals to change if your tonic
key changed? I recommend that only one symbol (accidental) be
selected for each small interval in the temperament (up to the
apotome, since everything larger than that would follow logically).

2) A temperament can benefit from alternate spellings. For example,
you might want to spell a (tempered) 5:7 as either a (large) fourth
G||) or a (small) fifth A!!!), depending on the context. Although
this would require no additional accidentals in the symbol set, these
spellings would need to be taken into account in the process of
selecting the symbol set.

Since the 5-comma vanishes in meantone, I don't see much point in
using any ratio containing a 5-factor in the semantics of the
notation.

> And 171-ET isn't ideal for anything beyond
> 7-limit temperaments.

True. When Dave & I were putting together a general-purpose
(athenian-level) JI symbol set, we wanted something that would work
well at least through the 13 limit, and we decided that there would
need to be enough symbols to notate a 15-limit tonality diamond
without incurring an error of more than a cent (for preferred
spellings) where symbols had to be re-used, and as it turned out,
only ratios of 13 needed to be approximated. Alternate spellings
will generally result in a 2-cent (5-schisma) error, and a 5's
exponent <6 or >-6 generally won't exceed that.

Perhaps the 21-EDA, 7-friendly symbol set, which is also 11- and 13-
friendly, will be worth a look, since it is much better above the 7
limit, yet may be as good for the 7 limit as 171-ET.

--George

George D. Secor wrote:
> 1) Since the whole point of a temperament is to have a single pitch > represent multiple ratios, it becomes more difficult and complicated > if different accidentals are used to notate the same tempered > interval. Did you intend for the accidentals to change if your tonic > key changed? I recommend that only one symbol (accidental) be > selected for each small interval in the temperament (up to the > apotome, since everything larger than that would follow logically).

Often when I'm trying to decide on a notation, I'll start by representing the simplest interval for each note (with the notation centered on D). That helps me to see which accidentals are used more frequently than others for the same interval. But I've come to the same conclusion that in a well-refined notation, it makes things easier to have one spelling for each of the intervals smaller than an apotome.

> 2) A temperament can benefit from alternate spellings. For example, > you might want to spell a (tempered) 5:7 as either a (large) fourth > G||) or a (small) fifth A!!!), depending on the context. Although > this would require no additional accidentals in the symbol set, these > spellings would need to be taken into account in the process of > selecting the symbol set.

With standard nominals, there are two cases of these: apotome complements, and pairs of symbols that add up to 243:256.

> Since the 5-comma vanishes in meantone, I don't see much point in > using any ratio containing a 5-factor in the semantics of the > notation.

Certainly there's no need for /| -- but ||\ seems a useful one to have.

> Perhaps the 21-EDA, 7-friendly symbol set, which is also 11- and 13-
> friendly, will be worth a look, since it is much better above the 7 > limit, yet may be as good for the 7 limit as 171-ET.

I'll give that a try and see how it works out.