Re: A common notation for JI and ET's

Hello, everyone, and for my first post on tuning-math, I'd like to
propose a new sagittal symbol that might illustrate an approach to the
notation of certain tempered systems not based on an equal division of
an interval such as the octave and involving more than one chain of
fifths.

Before presenting my proposal, I should caution that I am still in my
early beginner's stage of learning sagittal notation, and only within
the last day or so realized that I hadn't been quite clear on the
effect of comma signs in the region between an 11-diesis and a sharp
or flat (apotome-less-comma rather than diesis-plus-comma). Happily,
reading again the relevant portions of the draft by George and Dave
for _Xenharmonikon_ 18 brought this nuance to my attention, so that I
can at least construct my suggested new symbol correctly.

Also, I would caution that especially given my inexperience, this
article is not unlikely to include its share of obvious beginner's
errors, and that I warmly welcome any corrections.

As I discuss in some longer articles which I might want to proofread
and possibly revise accordingly, one approach to the kind of tempered
system I here discuss is a "quasi-JI" transcription which shows
precise or approximate deviations from Pythagorean intonation,
including significant commas which are dispersed in the system.
Consider, for example, this diatonic scale and a possible "quasi-JI"
representation, with octave numbers (C4=middle C) appearing before the
note names and sagittal signs:

4C 4D|( 4E)|( 4F!( 4G 4A|( 4B)|( 5C
1/1 ~44/39 ~14/11 ~4/3 ~3/2 ~22/13 ~21/11 2/1
0 208.19 416.38 495.90 704.10 912.29 1120.48 1200

Mapping a regular temperament to a JI notation, like mapping a
three-dimensional globe on a 2-D surface, inevitably involves some
distortion. Thus the notation for this scale based on a regular
temperament with fifths around 704.096 cents (the Wilson/Pepper "Noble
Fifth") accurately represents the sizes of many intervals, including
the whole-tones near 39:44 -- 8:9 + 351:352 or |( -- and the diatonic
semitones near 21:22 -- 243:256 less 891:896 or )|(.

From a vertical perspective, the notation here also accurately
suggests that the major thirds are very close to 11:14, or 64:81 plus
891:896 -- e.g. 4C-4E)|( or 4F!(-4A|(.

However, certain small vertical and melodic anomalies occur because
the notation does not actually show the fractional commas by which
each fifth or fourth is tempered -- not quite half of a 351:352, or
1/4 of an 891:896. Thus 4C-4G and 4E)|(-4B)|( are arbitrarily shown as
pure, while 4C-4F!( and 4E)|(-4A are shown as tempered by about a full
351:352 (~4.925 cents), although in reality all of these intervals are
tempered by an identical of amount of ~2.141 cents.

Apart from the issue of such unavoidable distortions, the main
drawback of this "quasi-JI" style of notation is that it is in tension
with the usual sagittal principle of notating routine intervals along
a single chain of fifths with routine symbols. Here 4C-4E)|( indeed
suggests that the size of this regular major third is very close to a
pure 11:14 (~417.51 cents), but the extra symbols might distract from
the basic fact that it is the _usual_ major third to be found simply
by pressing the keys 4C-4E.

Of course, for many linear temperaments based on a single chain of
fifths, one might simply pick a nearby equal temperament whose
standard symbol set fits. However, I here consider the system called
Peppermint 24, with two 12-note chains of fifths (~704.096 cents) at a
distance of ~58.680 cents, the "quasi-diesis" which when added to the
regular major second yields a pure 6:7 minor third (~266.871 cents).

My solution for communicating some intonational information about the
system while following the usual conventions of sagittal notation
within a chain of fifths is to use a symbol rather accurately defining
the size of the quasi-diesis as around 88:91 (11-diesis at 32:33 plus
11:13 comma at 351:352, ~58.036 cents) or 117:121 (32:33 plus 363:364
at about 4.763 cents, ~58.198 cents). The ratios of 88:91 and 117:121,
like 351:352 and 363:364, differ by the harmonisma at 10647:10648
(~0.163 cents).

In JI, 88:91 defines the difference for example between 39:44 and 6:7,
7:11 and 8:13, or 13:22 and 4:7; the very slightly larger 117:121
defines the difference between 11:13 and 9:11, or 22:39 and 6:11.

In a tempered system such as Peppermint 24, these 88:91 or 117:121
relationships are closely approximated, with the first interval of
each of the above pairs as a regular interval along the chain of
fifths, and the second as an interval realized by the addition of a
58.68-cent quasi-diesis.

To show the approximate size of the quasi-diesis, and thus to imply
this type of intonational structure, I propose the following sign
showing a modification of 32:33 or /|\ plus 351:352 or |(.

/|\(

I am tempted to call this an "intonational signature," since the
sagittal symbol looks a bit like the usual 11-diesis arrow plus the
sign "(" or "C" associated metrically with "common time" (4/4). Here
the sign could be said to stand for "common temperament," a term
applied to 24-note systems with two chains of fifths at around 704
cents or so spaced so as to optimize certain ratios, and especially
those of 2-3-7-9 (e.g. 6:7, 7:9, 4:7).

In such a scheme, the approximate 88:91 diesis or /|\( serves at once
as the 11-diesis (e.g. 11:12 vs. 8:9) or /|\ in JI, and the septimal
or Archytas semitone at 27:28 (~62.96 cents), or )/|\( as it might be
written in JI (32:33 plus 891:896). The latter interval marks the
difference, for example, between 8:9 and 6:7, or 7:9 and 3:4.

Following this approach, we might notate some characteristic
progressions as follows, with JI approximations given below the
examples:

5E\!!/ 5F 5E\!!/ 5F
4B\!!/ 5C 4B\!!/ 5C
4G 4F 4E 4F 4G 4F/|\( 4F or 4G!!!) 4F
4C 4D\!/( 4E 4D\!/( 4C
16/9 2/1
3/2 4/3 14/11 4/3 3/2 4/3 3/2
1/1 12/11 14/11 12/11 1/1 28/27 1/1

4A/|\( 4G/|\( 4B!!!) 4A!!!) 4A/|\( 4A 4B!!!) 4A
4G 4G/|\( 4G 4A!!!) 4G 4A 4G 4A
4D/|\( 4C/|\( 4E!!!) 4D!!!) 4E/|\( 4D 4F!) 4D
4C 4C/|\( or 4C 4D!!!) 4C 4D or 4C 4D

7/4 14/9 7/4 27/16
3/2 14/9 3/2 27/16
7/6 28/27 21/16 9/8
1/1 28/27 1/1 9/8

From the alternative notations for some of these examples, it can be
seen that in this kind of tempered system, the "limma complement" of
the 88:91 or /|\( is the representation of the 63:64 Archytas comma or
7-comma, !) -- thus D/|\( is equivalent to E!!!) at a 6:7 above C.

One "refinement" -- if that is the right word -- to go along with this
innovation is an "apotome complement" of sorts for /|\(. On this
point, I will propose a poetic liberty with the sagittal system.

Just as /|\( closely approximates the _absolute_ size of the
quasi-diesis in a tempered system like Peppermint 24, so its proposed
apotome complement also represents an absolute size close to that
obtaining in such a system, where the apotome /||\ or \!!/ has a size
of around 13:14 or ~128.30 cents (in Peppermint 24, actually ~128.67
cents).

Thus this "system-specific" apotome complement is equal to about
169:176 (~70.262 cents), the difference between 13:14 and 88:91, and
also the sum of 26:27 (the 13' diesis, ~65.337 cents) and 351:352.
I suggest this symbol for the 169:176, or also 121:126 (~70.100 cents)
as the difference of 13:14 and 117:121, or sum of 26:27 and 363:364;

(|\(

This is the usual 13' diesis sign plus a 351:352. While its use as an
apotome complement, at least in relation to 88:91 or 117:121, is
highly "system specific," the 169:176-like size could also represent
the Pythagorean "tricomma" of ~70.380 cents or (531441:524288)^3, a
ratio of 150094635296999121:144115188075855872. The schisma between
this tricomma and 169:176 is 25365993365192851449:25364273101350633472
or ~0.117 cents, well within usual sagittal tolerances.

A nice touch here, at least for those of us with a taste for this kind
of thing, is that both the 88:91 sign /|\( and its system-specific
apotome complement at around 169:176 of (|\( share a 351:352 comma as
a distinguishing mark, also the intonational signature for a "common
temperament" of this sort. In Peppermint 24, the quasi-diesis and its
apotome complement have sizes of around 58.680 cents and 69.990 cents,
quite close to the advertised rational representations.

Using (|\(, we can conveniently write a progression like the following
using single-symbol notation, with a double-symbol version also shown:

5E\!!/ 5E(!\( 5Eb 5Eb/|\(
5C/|\( 4B(!\( 5C/|\( 5Bb/|\(
4B\!!/ 4B(!\( 4Bb 4Bb/|\(
4F/|\( 4E(!\( or 4F/|\( 4Eb/|\(

27/14 2/1
27/16 3/2
81/56 3/2
9/8 1/1

I must admit that the second style of notation looks much more
intuitive to me: it quickly reveals the location of regular whole-tone
and near-27:28 steps -- e.g. 4F/|\(-4Eb/|\( and 4Bb-4Bb/|\(. This
consideration might motivate the use of some double-symbol notation in
pieces where the single-symbol style is generally preferred.

If we wish to stick with a single-symbol style, another solution is
also possible:

5E\!!/ 5F!!!)
5D!!!) 5C!!!)
4B\!!/ 5C!!!)
4G!!!) 4F!!!)

Here 4F!!!) -- or 4Fb!) in double-symbol style -- is the system's
Archytas comma lower than the diminished fourth 4C-4F\!!/ or 4C-4Fb at
~367.235 cents, yielding an interval of about 346.393 cents, also
written conveniently in double-symbol notation as 4C-4Eb/|\(. Thus
while a regular diminished fourth approximates 14:17 (~365.825 cents),
the diminished fourth less Archytas comma (or minor third plus
quasi-diesis) approximates 9:11 (~347.408 cents). This notational
solution involves more "remote" accidentals like F!!!), but shares
with the double-symbol example the advantage of readily communicating
usual whole-tone steps and approximate 27:28 steps.

Getting back to my main proposal for the symbol /|\( to show the
"quasi-diesis" of around 88:91 in a tempered system with two chains of
fifths at about 704 cents, this approach seeks to analyze some
important intonational and harmonic relationships in the system and
then give them a "shorthand" signature.

For example, /|\( signals that the near-88:91 is serving as a
representation of both 32:33 and 27:28, and that regular major and
minor thirds are rather close to 11:14 and 11:13 or 28:33, regular
whole-tones to 39:44, and diatonic semitones to 21:22. Also implied
are "39:44 + 88:91 = 6:7" and "7:11 + 88:91 = 8:13" -- and so forth.

As compared to a "quasi-JI" style of transcription showing approximate
deviations from Pythagorean intonation, this solution follows the
usual sagittal approach of regarding the reference intervals as those
following the chain of fifths -- thus 4C-4E for a regular major third,
rather than 4C-4E)|( to show that this ~11:14 interval is ~891:896
wider than a Pythagorean 64:81 (~407.820 cents).

As compared to the approach of attempting to map a non-equal system to
some very large equal temperament, the "intonational signature"
approach reflects the structure of two regular chains of fifths placed
an arbitrary distance to produce certain ratios of 2-3-7-9-11-13
complementing the ratios represented within each chain (e.g. 22:26:33
and 14:17:21).

The resulting sagittal notation looks rather like a style of keyboard
notation showing simply which note is played on which keyboard, with a
symbol like the asterisk (*) used to show a note on the upper keyboad
raised by a quasi-diesis, e.g. F*4-Bb4 for a near-7:9 third, written
as 4F/|\(-4B\!!/ or 4F/|\(-4Bb in this sagittal approach. One could
also write 4G!!!)-4B\!!/ to show the represented 7:9 relationship.

In view of the accustomed subject of this thread, "A Common Notation
for JI and ET's," I might seem in a curious position devoting my first
post to a "not-so-common" sagittal approach to a nonequal tempered
system based on two chains of fifths at an arbitrary distance. Such is
what can happen when a good system falls into certain not so
experienced hands.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning-math@yahoogroups.com, "M. Schulter" <mschulter@m...>
wrote:
> Hello, everyone, and for my first post on tuning-math, I'd like to
> propose a new sagittal symbol that might illustrate an approach to
the
> notation of certain tempered systems not based on an equal division
of
> an interval such as the octave and involving more than one chain of
> fifths.
>
> Before presenting my proposal, I should caution that I am still in
my
> early beginner's stage of learning sagittal notation, and only
within
> the last day or so realized that I hadn't been quite clear on the
> effect of comma signs in the region between an 11-diesis and a sharp
> or flat (apotome-less-comma rather than diesis-plus-comma). Happily,
> reading again the relevant portions of the draft by George and Dave
> for _Xenharmonikon_ 18 brought this nuance to my attention, so that
I
> can at least construct my suggested new symbol correctly.

I'll have to express my apologies to just about anyone else who might
be reading this (and who hasn't seen the paper), that this probably
isn't going to mean a whole lot to you, and hopefully I'll have a
presentation on this soon. Margo asked me a question in a private
communication about whether the new notation would provide a way to
do such-and-such, and my explanation wouldn't have been very
meaningful without sending along the latest draft of the paper to
her. An apology is also extended to you, Margo, since I haven't yet
responded to a message from last Thursday -- after looking at what
you had to say, I started getting bogged down in the mathematics and
couldn't find a quick answer.

> Also, I would caution that especially given my inexperience, this
> article is not unlikely to include its share of obvious beginner's
> errors, and that I warmly welcome any corrections.

We're all beginners with this notation, and we're all going to be
learning from one another how best to use it for various tunings.

> As I discuss in some longer articles which I might want to proofread
> and possibly revise accordingly, one approach to the kind of
tempered
> system I here discuss is a "quasi-JI" transcription which shows
> precise or approximate deviations from Pythagorean intonation,
> including significant commas which are dispersed in the system.
> Consider, for example, this diatonic scale and a possible "quasi-JI"
> representation, with octave numbers (C4=middle C) appearing before
the
> note names and sagittal signs:
>
> 4C 4D|( 4E)|( 4F!( 4G 4A|( 4B)|( 5C
> 1/1 ~44/39 ~14/11 ~4/3 ~3/2 ~22/13 ~21/11 2/1
> 0 208.19 416.38 495.90 704.10 912.29 1120.48 1200

This is very close to 46-ET, which is notated very simply using the
naturals (as you also noted):

deg46: 8 16 19 27 35 43 46
C D E F G A B C
0 208.70 417.39 495.65 704.35 913.04 1121.74 1200

Just as meantone can be notated with conventional notation, any
temperament with a regular chain of fifths can be notated similarly.
If you need fractional alteration symbols, the 46-ET notation should
do the job. For example, if you needed to notate the tone 11 fifths
along the chain ~545.10c or ~11/8), instead of E# or E/||\ you could
use the symbol for 21deg46, or F/|\. (But after figuring out
something else below and reading this again, I might now do this
differently -- just a precaution -- as I said, I'm still learning.)

Since the symbols for a 2-digit ET are inevitably going to be much
simpler than the ones for rational intervals (or JI), it would be
much easier to read the ET notation, *provided* that you have
specified how the symbols are being used and that anyone who is to
perform your music reads and understands your explanation.

> Mapping a regular temperament to a JI notation, like mapping a
> three-dimensional globe on a 2-D surface, inevitably involves some
> distortion.

That's why I don't recommend doing that -- it can get very
complicated. But I do recommend mapping JI or a regular temperament
to an ET notation, as long as you define your symbols consistently.
The rational symbols that we have are division-independent, i.e.,
they can be used without mapping to any ET, but if one modulates too
far there won't be enough rational symbols to handle the
requirements, and even if there were, they might not be very
meaningful. So an ET mapping is the only practical way. One of the
purposes of this notation is to keep the symbols as simple as
possible, with much of the complexity of a tuning being put in an
explanation of symbols (or note-symbol combinations) that should
accompany a composition.

If a regular temperament uses a generator other than an approximation
of a fifth, then an ET mapping is almost certainly a foregone
conclusion. For example, the Miracle temperament would be done by
mapping it to 72-ET and using that division's standard symbol set.

> ... Apart from the issue of such unavoidable distortions, the main
> drawback of this "quasi-JI" style of notation is that it is in
tension
> with the usual sagittal principle of notating routine intervals
along
> a single chain of fifths with routine symbols. Here 4C-4E)|( indeed
> suggests that the size of this regular major third is very close to
a
> pure 11:14 (~417.51 cents), but the extra symbols might distract
from
> the basic fact that it is the _usual_ major third to be found simply
> by pressing the keys 4C-4E.

Keyboard, guitar, or other polyphonically scored music is one place
where it's essential to keep the symbols simple; otherwise it would
be very difficult to read. For the one-note-at-a-time flexible-pitch
instruments precise pitch information is necessary. As to whether it
is more useful to have it appearing with each note (per Johnny
Reinhard) or placed in an accompanying explanation (as is required
for sagittal symbols) is dependent on the variables of a particular
situation -- the tuning, the players, their level and type of
experience, the instruments (off-the-shelf vs. specially built or
retuned), so we have decided to supplement the sagittal symbols with
Reinhard 1200-ET (cents) notation to accommodate either methodology
(or a combination of the two).

> Of course, for many linear temperaments based on a single chain of
> fifths, one might simply pick a nearby equal temperament whose
> standard symbol set fits. However, I here consider the system called
> Peppermint 24, with two 12-note chains of fifths (~704.096 cents)
at a
> distance of ~58.680 cents, the "quasi-diesis" which when added to
the
> regular major second yields a pure 6:7 minor third (~266.871 cents).
>
> My solution for communicating some intonational information about
the
> system while following the usual conventions of sagittal notation
> within a chain of fifths is to use a symbol rather accurately
defining
> the size of the quasi-diesis as around 88:91 (11-diesis at 32:33
plus
> 11:13 comma at 351:352, ~58.036 cents) or 117:121 (32:33 plus
363:364
> at about 4.763 cents, ~58.198 cents). The ratios of 88:91 and
117:121,
> like 351:352 and 363:364, differ by the harmonisma at 10647:10648
> (~0.163 cents).
>
> In JI, 88:91 defines the difference for example between 39:44 and
6:7,
> 7:11 and 8:13, or 13:22 and 4:7; the very slightly larger 117:121
> defines the difference between 11:13 and 9:11, or 22:39 and 6:11.
>
> In a tempered system such as Peppermint 24, these 88:91 or 117:121
> relationships are closely approximated, with the first interval of
> each of the above pairs as a regular interval along the chain of
> fifths, and the second as an interval realized by the addition of a
> 58.68-cent quasi-diesis.
>
> To show the approximate size of the quasi-diesis, and thus to imply
> this type of intonational structure, I propose the following sign
> showing a modification of 32:33 or /|\ plus 351:352 or |(.
>
> /|\(
> ...
> One "refinement" -- if that is the right word -- to go along with
this
> innovation is an "apotome complement" of sorts for /|\(. On this
> point, I will propose a poetic liberty with the sagittal system.
>
> Just as /|\( closely approximates the _absolute_ size of the
> quasi-diesis in a tempered system like Peppermint 24, so its
proposed
> apotome complement also represents an absolute size close to that
> obtaining in such a system, where the apotome /||\ or \!!/ has a
size
> of around 13:14 or ~128.30 cents (in Peppermint 24, actually ~128.67
> cents).
>
> Thus this "system-specific" apotome complement is equal to about
> 169:176 (~70.262 cents), the difference between 13:14 and 88:91, and
> also the sum of 26:27 (the 13' diesis, ~65.337 cents) and 351:352.
> I suggest this symbol for the 169:176, or also 121:126 (~70.100
cents)
> as the difference of 13:14 and 117:121, or sum of 26:27 and 363:364;
>
> (|\(

If I understand you correctly, you have:

/|\( at around 58 cents, and
(|\( at around 70 cents

which are supposed to be apotome complements. But in order for this
to be so, they must add up to an apotome, ~113.7 cents. So your
usage of symbols does not seem to be internally consistent. You
mentioned something like a "system-specific apotome complement". We
tried a concept called "alternate complements" for some ET notations,
but that was eventually dropped, because it was much simpler to have
a one-to-one correspondence between a symbol and its complement
without exception.

We are very reluctant to define symbols with three flags unless there
is a very good reason for doing so (as we just did for the 7:17
comma, but that was the first one).

It's not necessary to define new symbols if you consider that it
would still be necessary for you to describe your temperament apart
from the symbols themselves. The 46-ET notation would actually do
quite nicely in this instance, the standard symbol sequence being:

46: /| /|\ (|) ||\ /||\

Since you're not using ratios of 5 and since ratios of 7 are so
prominent in Peppermint, you would probably want to replace the 5-
comma symbols with the 7-comma ones:

46: |) /|\ (|) ||) /||\

These are simple symbols that everyone who used the sagittal notation
would be familiar with.

One problem that you might encounter with this is if you extended one
chain of fifths to D# or D/||\ and wished to respell that as E(!),
whereas you already have an E(!) in the other chain of fifths.
Avoiding the respelling would sidestep the problem, but I don't know
if that would be suitable for your purposes.

But I see another way to avoid this problem, one that involves a
quasi-217 mapping, and I believe that the end result would be
preferable to the above. In the 46-mapped notation, equivalent
pitches in your upper chain of fifths are D/|\ and E!!!). In 217
these equivalents are 47 and 48 degrees, respectively. The
equivalent tones for ~14/11 in the 46 mapping are E and F(!), but in
217 these are 74 and 79 degrees, respectively, far enough off to
indicate that the F(!) is not related to E in the same way as with
the other equivalents. If we were to use the 217 symbol for 75deg217
as the equivalent of E (~14/11), then we would have F\!! or Fb|\ for
this purpose; better yet, using the rational symbol for 14/11 -- F)!!
~ or Fb(| -- which would involve the same number of degrees in 217
(which confirms that its use here would be appropriate).

So then the symbols for your two chains of fifths (single-symbol
version, with "enharmonic" equivalents -- maybe not quite the right
term) would be:

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

D)||~ A)||~ E)||~ B)||~ F)X~ C)X~ G)X~ D)X~ A)X~ E)X~ B)X~
E\!!/ B\!!/ F C G D A E B F/||\ C/||\
G/||\
F)x~ C)x~ G)x~ D)x~ A)x~ E)x~ B)x~ F)!!~ C)!!~ G)!!~ D)!!~
A)!!~

This uses 11-limit rational symbols that are harmonically meaningful
and, taken together with the information that your major thirds are
~11:14 and minor thirds ~11:13, these would be easily understood by
anyone already familiar with the notation.

Otherwise, we already have a couple of symbols between /|\ and (|),
which you might want to look at:

(/| - the 31' comma: 31:32 (~54.964c), and
|\) - the (7+11-5) comma: 1701:1760 (~59.031c)

These are for all practical purposes apotome complements of one
another, and the larger of these is close to /|\(. These two symbols
are in the 494-ET standard set.

But I think the foregoing solution is the one to go with.

Let me know how this works out.

--George

Please let me thank both George and Dave for your most helpful and
engrossing comments, and also take note of my possible wisdom in
writing a first response after George's initial reply, but waiting a
bit to follow the further dialogue before posting.

This process has helped me both to evaluate which parts of my first
draft might be of most interest and relevance here, and to confirm
some of my initial impressions: for example, that 121-ET notation
seemed an attractive choice for Peppermint, albeit with one possibly
not so relevant "glitch" of 121-ET itself.

At the outset, I should respond to your most gracious welcome, George:

> I'll have to express my apologies to just about anyone else who
> might be reading this (and who hasn't seen the paper), that this
> probably isn't going to mean a whole lot to you, and hopefully I'll
> have a presentation on this soon. Margo asked me a question in a
> private communication about whether the new notation would provide a
> way to do such-and-such, and my explanation wouldn't have been very
> meaningful without sending along the latest draft of the paper to
> her. An apology is also extended to you, Margo, since I haven't yet
> responded to a message from last Thursday -- after looking at what
> you had to say, I started getting bogged down in the mathematics and
> couldn't find a quick answer.

Please let me apologize and take full responsibility for possibly
showing more enthusiasm than prudence in excitedly leaping into this
discussion without fully considering the distinction between the
public discussion here so far and our private communications. While I
hope that no harm has been done, I apologize for any confusion or
puzzlement on the part of other participants -- my responsibility, not
yours.

Also, I would like to emphasize your generosity and grace in
responding to my e-mail questions while you are at an important point
in this process -- which hopefully I won't complicate too much with my
suggested "refinements" <grin>.

The reply in which you welcomed me is one example of your generosity,
and of your gifts as a teacher.

Thank you for helping me get oriented with excellent points such as
the drawbacks of attempting to notate a tempered system in "quasi-JI"
style, and the frequent advantages of notating JI using ET symbols.
Some of the commas you used for certain equivalent spellings also give
me an opportunity to become acquainted with more symbols, and I might
pursue some questions in further posts or via e-mail.

However, recalling the length of my first post here, I'll try to be a
bit more concise this time by focusing mainly on the question of
choosing an ET notation for Peppermint 24 -- or for a larger tuning
set based on the same parameters of two chains of fifths in the
Wilson/Pepper temperament (~704.096 cents) at ~58.680 cents apart,
yielding some pure ratios of 6:7.

[As you'll see, I was not so successful in keeping this brief -- but
possibly somewhat briefer than if I had tried to address more topics.]

Since the general focus here is "tuning-math," I have an excuse to
describe one method for looking for such an ET approximation through a
Fibonacci series -- hopefully without wandering too far from our main
concern in this thread with sagittal notation. Where this digressions
leads is my own practical conclusion that 121-ET notation could be
fine for Peppermint 24 or even Peppermint 34 (two 17-note chains),
despite a minor inconsistency or nonuniqueness of 121-ET itself. Here
I invite the opinions of more experienced people.

In the Wilson/Pepper tuning, the ratio of logarithmic sizes (e.g. in
cents) between the whole-tone and chromatic semitone is equal to the
Golden Section, or Phi (~1.618034), and the ratio between whole-tone
and the smaller diatonic semitone equal to Phi + 1 (~2.618034). We can
use a Fibonacci series of equal temperaments -- an actual Fibonacci
series, as opposed to the "Fibonacci-like" series of Yasser's theory
of harmonic evolution (5,7,12,19,31...) -- to produce successively
closer approximations of the Wilson/Pepper temperament.

Here I use the abbreviations of (L) for limma or diatonic semitone;
(A) for apotome or chromatic semitone; and (T) for whole-tone.
Easley Blackwood's R is defined as the ratio between the logarithmic
sizes of whole-tone and limma. The generator is the fifth in the
tuning closest to the Wilson/Pepper fifth of ~704.096 cents, while the
quasi-diesis (Q-D) is equal to the difference between two of these
generators less an octave and the best approximation of 6:7.

ET L A T R Generator Steps Q-D Steps
-----------------------------------------------------------------------
12 1 1 2 2.000 700.000 7 100.000 1
17 1 2 3 3.000 ~705.882 10 ~70.588 1
29 2 3 5 2.500 ~703.448 17 ~41.379 1
46 3 5 8 ~2.667 ~704.348 27 ~52.174 2
75 5 8 13 2.600 704.000 44 64.000 4
121 8 13 21 2.625 ~704.132 71 ~59.504 6
196 13 21 34 ~2.615 ~704.082 115 ~61.224 10
317 21 34 55 ~2.619 ~704.101 186 ~60.568 16
513 34 55 89 ~2.6176 ~704.094 301 ~58.480 25
--------------------------------------------------------------------
Wilson/Pepper ~2.6180 ~704.096 ~58.680
--------------------------------------------------------------------

A complication here is that while successive equal temperaments in
this Fibonacci series yield closer and closer approximations of the
Wilson/Pepper generator, they do not necessarily yield closer
approximations of the quasi-diesis, or of the just 6:7 around which
this interval in Peppermint is defined. Indeed 121 does considerably
better than 196 or 317, with 513 as the first ET in the series coming
really close (quasi-diesis ~0.200 cents narrow of Peppermint, and best
approximation of 6:7 at ~266.667 cents, ~0.204 cents narrow).

Returning to our main theme of sagittal notation, the question before
us is whether 121 is quite accurate enough for our purposes -- or
whether the precise properties of 121-ET might not be so important, as
long as the symbol set can accurately represent Peppermint.

First, let me quote the most attractive symbol set for Peppermint
proposed in this thread by George, to which I am really drawn, with
the number of 121-notation steps shown:

19 35 69 90 106
C/|||\ E(!) F/|||\ G/|||\ B(!)
C/|\ D/|\ E/|\ F/|\ G/|\ A/|\ B/|\ C/|\
6 27 48 56 77 98 119 127
--------------------------------------------------------------------
13 29 63 84 100
C/||\ E\!!/ F/||\ G/||\ B\!!/
C D E F G A B C
0 21 42 50 71 92 113 121

Now for the arguable glitch in 121-ET, which I raise in order to seek
advice from more experienced hands as to whether it is actually a
problem as long as the 121-symbols show what is happening in
Peppermint itself.

One feature of Peppermint is that a fourth less the Archytas comma or
7 comma (2deg121) -- e.g. 4F-4A/|\ -- yields an approximation of 16:21
at ~475.062 cents, or ~4.282 cents narrow, the same absolute deviation
as with 8:9 in the wide direction. This could alternately be spelled
as 4F-4B!!!) using your notational equivalents, George.

In a 34-note version of Peppermint with two chains of 17 notes, the
next MOS size for Wilson/Pepper after 12, we would also have an
approximation for 13:17 -- G\!!/-A/||\ or G(!)-A/|||\ -- formed from a
chain of 16 regular fifths up at ~465.530 cents, or ~1.102 cents wide.

In 121-ET, the ~16:21 of Peppermint would map to 48 steps at ~476.033
cents, and the ~13:17 to 47 steps at ~466.116 cents. We can derive the
first interval as regular major third plus quasi-diesis (42 + 6) or
fourth less 7 comma (50 - 2); and the second as a major third plus the
regular diesis or "Pythagorean-like comma" of 5 steps (42 + 5).

Unfortunately, in 121-ET as opposed to Peppermint, the Archytan comma
or 7 comma at 2 steps is only ~19.835 cents rather than ~20.842 cents,
so that the 48deg121 interval of ~476.033 cents is ~5.252 cents wide,
actually further from 16:21 than 47deg121 (~4.665 cents narrow).

One way of putting the problem is to say that in 121-ET, in contrast
to Peppermint, the best approximation of 4:7 less the best
approximation of 3:4 does not yield the best approximation of 16:21.
Another approach is to say that in 121-ET, the 272:273 comma (~6.353
cents) defining the distinction between 16:21 and 13:17 is tempered
out, while in Peppermint this distinction is observed.

Now for the practical issue: given that the 121-notation seems neatly
to fit what actually happens in Peppermint, where F-A/|\ or F-B!!!) is
indeed the best approximation of 16:21, and G\!!/-A/||\ of 13:17, is
the "inconsistency" or "nonuniqueness" of 121-ET really relevant here?

For example, if we are simply seeking a convenient symbol set for
Peppermint 34 itself, the following sagittal spellings give the best
representations for the isoharmonic chords 9:13:17:21 and 11:16:21:26,
the latter also available in Peppermint 24:

5D/|\ 5F/|\
4A/|||\ 5C/|||\
4G(!) 4A
4C 4D/|\

~9:13:17:21 ~11:16:21:26

Incidentally, speaking of equivalents, if we choose some note on the
upper keyboard as the 1/1, the second sonority could also be written:

5F
5C/||\
4A\!/
4D

Having digested the first lesson that sagittal symbols should be kept
as simple as possible, I would be inclined to go with the 121-notation
for Peppermint 24. Even with Peppermint 34, I might guess that this
notation could present a problem only if someone draws the inference
that 16:21 and 13:17 as notated map accurately in 121-ET -- unless
there are other "glitches" to be found. If we really want an ET yet
more closely modelling Peppermint, then I might look at 513-ET.

Now to some points in your responses, Dave and George. First, Dave,
you gave this alternative ASCII version of a 121-notation:

> The lower chain of 704.1 cent fifths is notated
> F C G D A E B F# C# G# D# A#
> The upper chain, a 58.7 cent quasi-diesis above it, is notated
> F^ C^ G^ D^ A^ FL CL GL DL AL EL BL
> with FL and CL optionally being notated as
> E^ B^

Here I see that ^ saves some typing in comparison with /|\, although
I like the sagittal style of the latter -- a bias possibly related to
my custom of using ^ in nonsagittal notation to show a note raised by
about a Pythagorean comma or Archytan (63:64) comma, as George has
aptly named the second ratio.

> Upon examining 121, I found that ratios of 13 aren't compatible with
> ratios of 7 for the notation (as often occurs with ETs). For
> Margo's benefit I will explain: /| is 3deg and |) is 2deg, so /|)
> would be 5deg by adding the flags, but the 13 comma should be
> 6deg121.

Let's be sure that I'm following this. In Peppermint, or generally in
121-ET, the Archytan comma |) maps to 2deg121, and the Didymic comma
/| to 3deg121, with the 13 comma usually the sum of these or /|) --
with a schisma in JI of 4095:4096. Here, however, it's actually equal
to the quasi-diesis of 6deg121, the difference between a regular minor
sixth and a ~8:13, e.g. 3B-4G and 3B-4G/|\ (~7:11 and ~8:13).

> So one must make a choice between the 7 and 13-comma symbols for the
> symbol set. Even though 13 has less error in 121-ET than either 7
> or 11, 7 is preferred over 13 because 1) it's a lower prime, and 2)
> the 11 and 11' commas are the same number of degrees as the 13 and
> 13' commas, respectively, so the symbols for a lower prime, 11, can
> be used for both 11 and 13 (which is appropriate, since 351:352
> vanishes in both 121-ET and Peppermint).

Why don't I first confirm my own intution that the 7-comma symbols are
indeed best, and offer the additional argument that the one just
interval in Peppermint other than the 1:2 octave is the 6:7 or 7:12.
An interesting point is that while "11" or "13" often suggests to me a
family of intervals sharing a factor (e.g. 11:14, 11:12, 9:11; or
11:13, 8:13, 7:13, 9:13), here we are referring to a single ratio or
comma, thus:

3rd harmonic 2:3 +~2.141 cents
7th harmonic 4:7 +~2.141 cents
9th harmonic 8:9 +~4.282 cents
11th harmonic 8:11 +~3.266 cents
13th harmonic 8:13 +~1.770 cents

As discussed in the paper, this approach permits comparing these
deviations to calculate the accuracy of other ratios -- for example
9:11 at (~3.266c - ~4.282c) or ~1.015 cents narrow in Peppermint 24.

Also, I get the point that the 11-diesis (regular fourth vs. ~8:11) is
identical to the 13-diesis for ~8:13, thus 4D-4G/|\ and 4E-5C/|\
respectively. Thus 351:352 is tempered out -- as is also 891:896, with
the 11-diesis or ~32:33 also representing ~27:28, the difference
between a regular major second and a 6:7 third, e.g. 3B\!!/-4C/|\ or
3B\!!/-4D!!!).

> The one thing that differs from what I suggested is this: Should
> Margo would want to respell E (~14/11 relative to C) as F-something,
> then the symbol I suggested, using the 7:11 comma -- Fb(| or F)!!~
> -- is not in the 121 standard set that Dave has proposed (below).
> His set has the 5:11 comma, which gives Fb(|( or F~!!(. Either one
> of these is valid for 5deg121, but since Peppermint lacks ratios of
> 5, Margo might prefer the 7:11 comma symbol. Something to consider
> when we're mapping something to an ET is that the standard symbol
> set might not be the "best," i.e., the most meaningful or useful
> one.

Here I can quickly understand that the modification upward of F\!!/ in
F)!!~ is here equivalent to the usual diesis or "Pythagorean-like"
comma of 5deg121. To understand this, I consider the 7:11 comma as the
difference between (|) and |). If I define (!) as the difference
between a regular major third C-E and a ~9:11 at C-E(!), this gives
7deg121, which compared to the Archytan comma of 2deg121 places the
7:11 comma at 5deg121. I see the advantage that these relationships
involve intervals actually present in Peppermint 24 -- unlike the
5-comma.

[Dave:]

> I couldn't find anywhere that we agreed on sagittal notation for
> 121-ET. So here's my proposal for its single shaft symbols.

> 121: )| |) /| (|( (|~ /|\ (|)

[George:]

> As you have probably concluded by now, I agree with this for the
> standard set. With rational complements this becomes:

> 121: )| |) /| (|( (|~ /|\ (|) )|| ~||( ||\ ||) (||~ /||\

> I will add this ET to Table 4 in the paper.
> For Peppermint my suggested modification is:

> 121: )| |) /| (| (|~ /|\ (|) )|| )||~ ||\ ||) (||~ /||\

Why don't I take up some of the symbols less familiar to me in another
post, since this one is already longer than I intended -- but
concluding for now by saying that I'm really excited about this symbol
set, and will try some examples to test my usage at this point and get
your feedback.

Most appreciatively,

Margo
mschulter@value.net

Hello, there, Dave and George, and now that I've read the latest
digest, I can share my own puzzlement about the 7:11 comma in 121-ET
or Peppermint, and check my understanding of a few other less familiar
(to me) symbols also.

> At 10:04 AM 20/12/2002 -0800, George Secor wrote:

>> The one thing that differs from what I suggested is this: Should Margo
>> would want to respell E (~14/11 relative to C) as F-something,

> Why would she want to do that? Do you want to do that, Margo?

Really, I'm not sure -- it's not something that would obviously occur
to me. The way I would read it is defining E (~14/11) as F (~4/3) down
by a ~21:22, the same interval as the regular limma of 8deg121. The JI
notation in the paper presents this as 2187:2048 or the usual apotome
down and a 7:11 comma up at 45056:45927 (~33.148 cents), the latter
equal to the difference between 64:81 and 9:11, or 27:32 and 22:27.

As a beginner, maybe I should ask an open question: are there reasons
I might want to use this kind of equivalence, either in certain
musical contexts or else to validate the consistency of a given
mapping, for example?

>> then the symbol I suggested, using the 7:11 comma -- Fb(| or F)!!~
>> -- is not in >the 121 standard set that Dave has proposed (below).
>> His set has the 5:11 comma, which gives Fb(|( or F~!!(. Either
>> one of these is valid for 5deg121, but since Peppermint lacks
>> ratios of 5, Margo might prefer >the 7:11 comma symbol. Something
>> to consider when we're mapping >something to an ET is that the
>> standard symbol set might not be the "best," i.e., the most
>> meaningful or useful one.

An aside to George: this point about the best symbol set for a given
musical application is something I much like, with an example I might
offer stemming from your earlier mention of a possible 46-notation for
Peppermint. Why don't I take this up in due course?

[Dave:]

>>> here's my proposal for its single shaft symbols.
>>>
>>> 121: )| |) /| (|( (|~ /|\ (|)

[George:]

>> As you have probably concluded by now, I agree with this for the
>> standard set. With rational complements this becomes:

>121: )| |) /| (|( (|~ /|\ (|) )|| ~||( ||\ ||) (||~ /||\

[Dave:]

> Agreed.

[George:]

>> I will add this ET to Table 4 in the paper.
>> For Peppermint my suggested modification is:
>>
>> 121: )| |) /| (| (|~ /|\ (|) )|| )||~ ||\ ||) (||~ /||\

[Dave:]

> By my calculations, (| as the 7:11' comma (45056:45927) is 5deg121,
> not 4deg121 as you have it above.

This is what I also wanted to query, with some beginner's caution
since I could simply be puzzled by unfamiliar symbols.

As I understand, the 7:11' comma is the difference between 704:729
(regular major third vs. ~9:11) and the 7 or Archytas comma at 63:64.
In 121-ET, the 704:729 is 42deg121 (~11:14) less 35deg121 (~9:11), or
7d121, and the 63:64 is 2deg121, with 5deg121 as the difference -- the
same as the usual limma.

By the way, I take the 5:11 comma (|( at 44:45 as the difference
between 704:729 and the 5 or Didymus comma at 80:81, so that the 5:11
would define the difference between ~4:5 and ~9:11. In 121-ET, I have
this (|) at 7deg121 less /| at 3deg121, or 4deg121. I can see why it's
a natural choice for the standard 121-ET set, but not of obvious
relevance to Peppermint 24. The ~5:6 and ~4:5 in Pepper/Wilson are
represented by 20 fifths down (~318.088 cents) and 21 fifths up
(~386.008 cents), so the 5 comma comes up in larger sets.

Let me also seek confirmation that the 11:19 diesis (|~ is equal to
the 11 diesis at 32:33 less the 19 comma at 512:513, or the difference
between ~16:19 and ~9:11. In 121-ET, the 512:513 is 1 step, and the
32:33 is 6 steps, so that indeed yields 5 steps for this diesis at
171:176. That explains the 5-step symbol in both versions of the
symbol set for 121.

> The 7:11 comma that is relevant to Peppermint is 891:896 which I
> don't believe we have considered before in regard to the sagittal
> notation. The appropriate symbol for it would be )|(, however it
> vanishes in Peppermint (and 121-ET).

That's how I might describe a temperament like Peppermint -- an
"eventone" at close to 1/4-891:896 to produce a regular ~11:14 major
third, analogous to meantone (which tempers out the 80:81) or to 22-ET
(which tempers out the 63:64).

The symbol )|( appeals to me, since 891:896 is almost exactly twice
351:352 -- actually 891:896 is realized in one 17-note JI system I use
as 351:352 (~4.925c) plus 363:364 (~4.763c), with a small schisma or
harmonisma of 10647:10648 (~0.163c), another example of a schisma which
could be disregarded in sagittal notation.

George, recalling an earlier post of yours, I also realize that
adopting an 891:896 symbol doesn't necessarily mean that I should use
it every time I write 11:14 in a JI context -- you mentioned that it
can be a good idea at times to map JI to an ET symbol set as long as
the symbols are consistently defined and adequately explained.

Getting back to Peppermint for a moment, I've noticed that we might
also say that 891:896 is tempered out in that the same quasi-diesis
(mapped to 6deg121) represents both 32:33 and 27:28.

On the theme of comparing prime factors, I also notice that if we take
the deviations for 4:7 and 8:11 at ~2.14c and ~3.26c wide, the
difference is the amount by which 11:14 is narrow or 7:11 wide, ~1.13c.

> We had better check all existing uses of )|( for agreement with
> this. I don't think )|( appears in the article. But I guess we
> should check for other useful 15-limit dual-prime commas we may have
> missed.

There are some ratios about which I have questions, but I'm not sure
which ones would fall in this category. Why don't I mention some of
these in another post?

Most appreciatively,

Margo
mschulter@value.net

Hello, there, George and Dave and everyone, and Happy Holidays!

Now that we have a symbol set for Peppermint, or at least for the
steps and intervals that I'm likely to use in Peppermint 24, I don't
want to miss the point that the idea of this notation is to put it
into practice.

Of course, the discussion about the fine points of developing a symbol
set have been an opportunity for me to learn some sagittal philosophy,
and also to get more understanding by joining in on the process of
problem-solving with more experienced users. However, this festive day
seems an apt occasion to write down a few progressions and see how
musically understandable they are to others.

In part, this is a fun exercise for me as a beginner, and like a
counterpoint student in one of the classic dialogues I wouldn't be
surprised if I commit a few errors and oversights, warmly inviting
your corrections. At the same time, it's a kind of trying out of the
notation, also, assuming that I can write it correctly -- how does
this progression or passage look to someone else?

To provide a bit of a reference on interval sizes, I've included a
Scala file for Peppermint 24 at the end of this post.

First, here's a scale I find quite beautiful in Peppermint, which I
give in two spellings, and placing octave numbers before note names
(C4 = middle C):

3D 3E 3E/|\ 3G 3A 3B 3B/|\ 4D

3D 3E 3F!) 3G 3A 3B 4C!) 4D

Next is a progression which presents a kind of problem in creative
tempering of intervals for which Peppermint provides an interesting
solution:

4G/||\ 4G/|\ 4B\!!/ 4B(!)
4F/|\ 4G/|\ ------ 4F/|\
4E\!!/ 4F ------ 4F/|\
3B(!) 4C/|\ ------ 3B(!)

Here's a different progression using some of the same intervals, and
one I much like -- can anyone give a 13-prime rational approximation
for the opening sonority?

4B(!) 4B
4F/|||\ 4F/||\
4E 4F/||\
3B/|\ 3B

Now we come to a fingering exercise for my right hand in an historical
scale, originally given in rational form, with the tempered intervals
quite close to the theoretical values. First I'll notate, as above, in
a "keyboard tablature" style taking the lower chain of fifths (on the
lower keyboard) as the reference:

4G/|\ 4F/|\ 4E/|\ 4F/|\ 4G/|\
4C/|\ 4D 4E/|\ 4D 4C/|\

Here's a version with the "1/1" or lowest note of the pentachord as
the reference:

4G 4F 4E 4F 4G
4C 4D\|/ 4E 4D\|/ 4C

By the way, the JI intervals in one rendition:

4G 4F 4E)|( 4F 4G
4C 4D\|/ 4E)|( 4D\|/ 4C

Given some recent "popularity polling" of rational intervals used in
Scala tunings, I can't help contributing this example, featuring a
tempered version of an isoharmonic chord with all intervals within one
cent of just (the resolving sonority is less accurate):

5G/|||\ 5F/|||\
5E\!!/ 5C/|||\
4F/|\ 4F/|||\

Finally, here's a kind of procedure I often use when improvising in
Peppermint, with the following example only a sketch inviting lots of
elaboration (mainly the meandering of "florid" but rather stately
melody in a note-against-note texture, given the nature of the
keyboard technique and my modest accomplishments therein), and some
more cadences with descending semitones (quasi-steps which could be
called small thirdtones or possibly large quartertones) as in the
opening phrase:

4G ----- 4B\!!/ 4A 4G 4F 4G
4D ----- 4F 4E 4D 4C 4D
3G 3A/|\ 4C/|\ 3B/|\ 3A/|\ 3G/|\ 3G

4G 4A 5C 5C/|\
4D 4E 4G 4G/|\
3A/|\ 3B/|\ 4D/|\ 4C/|\

4B\!!/ 4A 4G 4F 4F/|\
4F 4E 4D 4C 4C/|\
4C/|\ 3B/|\ 3A/|\ 3G/|\ 3F/|\

4F 4G 4A 4A/|\
4C 4D 4E 4E/|\
3G/|\ 3A/|\ 3B/|\ 3A/|\

4G 4F 4E\!!/ 4D 4D/|\
4D 4C 3B\!!/ 3A 3A/|\
3A/|\ 3G/|\ 3F/|\ 3E/|\ 3D/|\

Again, Happy Holidays to all!

Most appreciatively,

Margo
mschulter@value.net

--- In tuning-math@yahoogroups.com, "M. Schulter" <mschulter@m...>
wrote:
> ... Now for the arguable glitch in 121-ET, which I raise in order
to seek
> advice from more experienced hands as to whether it is actually a
> problem as long as the 121-symbols show what is happening in
> Peppermint itself.
>
> One feature of Peppermint is that a fourth less the Archytas comma
or
> 7 comma (2deg121) -- e.g. 4F-4A/|\ -- yields an approximation of
16:21
> at ~475.062 cents, or ~4.282 cents narrow, the same absolute
deviation
> as with 8:9 in the wide direction. This could alternately be spelled
> as 4F-4B!!!) using your notational equivalents, George.
>
> In a 34-note version of Peppermint with two chains of 17 notes, the
> next MOS size for Wilson/Pepper after 12, we would also have an
> approximation for 13:17 -- G\!!/-A/||\ or G(!)-A/|||\ -- formed
from a
> chain of 16 regular fifths up at ~465.530 cents, or ~1.102 cents
wide.
>
> In 121-ET, the ~16:21 of Peppermint would map to 48 steps at
~476.033
> cents, and the ~13:17 to 47 steps at ~466.116 cents. We can derive
the
> first interval as regular major third plus quasi-diesis (42 + 6) or
> fourth less 7 comma (50 - 2); and the second as a major third plus
the
> regular diesis or "Pythagorean-like comma" of 5 steps (42 + 5).
>
> Unfortunately, in 121-ET as opposed to Peppermint, the Archytan
comma
> or 7 comma at 2 steps is only ~19.835 cents rather than ~20.842
cents,
> so that the 48deg121 interval of ~476.033 cents is ~5.252 cents
wide,
> actually further from 16:21 than 47deg121 (~4.665 cents narrow).
>
> One way of putting the problem is to say that in 121-ET, in contrast
> to Peppermint, the best approximation of 4:7 less the best
> approximation of 3:4 does not yield the best approximation of 16:21.

A concise way of stating this is to say that 121-ET is not 1,3,7,21-
consistent.

> Another approach is to say that in 121-ET, the 272:273 comma (~6.353
> cents) defining the distinction between 16:21 and 13:17 is tempered
> out, while in Peppermint this distinction is observed.

Since there are two ways of arriving at 16:21 in 121-ET, I wouldn't
put it that way, especially since the one you're interested in
maintains the distinction.

> Now for the practical issue: given that the 121-notation seems
neatly
> to fit what actually happens in Peppermint, where F-A/|\ or F-B!!!)
is
> indeed the best approximation of 16:21, and G\!!/-A/||\ of 13:17, is
> the "inconsistency" or "nonuniqueness" of 121-ET really relevant
here?

Not as far as I can tell. The place where you might start to
experience a problem is if a tone at one end of one chain of fifths
is the same number of degrees as a tone at the other end of the other
chain of fifths, and you've reached the point where you've gotten
very close -- the 34th tone in a downward chain of fifths is 6deg121
(or go 17 fifths in opposite directions from the middle of each
chain). But you're out of the woods if you happen to use different
nominals (or letter names) to spell the two tones.

> ... If we really want an ET yet
> more closely modelling Peppermint, then I might look at 513-ET.

That's possible; 513 isn't 13-limit consistent, but you could get
around that by taking 13 as 360 instead of 359 degrees -- you just
wouldn't be using the best representation of 13.

> Now to some points in your responses, Dave and George. First, Dave,
> you gave this alternative ASCII version of a 121-notation:
>
> > The lower chain of 704.1 cent fifths is notated
> > F C G D A E B F# C# G# D# A#
> > The upper chain, a 58.7 cent quasi-diesis above it, is notated
> > F^ C^ G^ D^ A^ FL CL GL DL AL EL BL
> > with FL and CL optionally being notated as
> > E^ B^
>
> Here I see that ^ saves some typing in comparison with /|\, although
> I like the sagittal style of the latter -- a bias possibly related
to
> my custom of using ^ in nonsagittal notation to show a note raised
by
> about a Pythagorean comma or Archytan (63:64) comma, as George has
> aptly named the second ratio.
>
> > Upon examining 121, I found that ratios of 13 aren't compatible
with
> > ratios of 7 for the notation (as often occurs with ETs). For
> > Margo's benefit I will explain: /| is 3deg and |) is 2deg, so /|)
> > would be 5deg by adding the flags, but the 13 comma should be
> > 6deg121.
>
> Let's be sure that I'm following this. In Peppermint, or generally
in
> 121-ET, the Archytan comma |) maps to 2deg121, and the Didymic comma
> /| to 3deg121, with the 13 comma usually the sum of these or /|) --
> with a schisma in JI of 4095:4096. Here, however, it's actually
equal
> to the quasi-diesis of 6deg121, the difference between a regular
minor
> sixth and a ~8:13, e.g. 3B-4G and 3B-4G/|\ (~7:11 and ~8:13).

You've got it.

> > So one must make a choice between the 7 and 13-comma symbols for
the
> > symbol set. Even though 13 has less error in 121-ET than either 7
> > or 11, 7 is preferred over 13 because 1) it's a lower prime, and
2)
> > the 11 and 11' commas are the same number of degrees as the 13 and
> > 13' commas, respectively, so the symbols for a lower prime, 11,
can
> > be used for both 11 and 13 (which is appropriate, since 351:352
> > vanishes in both 121-ET and Peppermint).
>
> Why don't I first confirm my own intution that the 7-comma symbols
are
> indeed best, and offer the additional argument that the one just
> interval in Peppermint other than the 1:2 octave is the 6:7 or 7:12.
> An interesting point is that while "11" or "13" often suggests to
me a
> family of intervals sharing a factor (e.g. 11:14, 11:12, 9:11; or
> 11:13, 8:13, 7:13, 9:13), here we are referring to a single ratio or
> comma, thus:
>
> 3rd harmonic 2:3 +~2.141 cents
> 7th harmonic 4:7 +~2.141 cents
> 9th harmonic 8:9 +~4.282 cents
> 11th harmonic 8:11 +~3.266 cents
> 13th harmonic 8:13 +~1.770 cents
>
> As discussed in the paper, this approach permits comparing these
> deviations to calculate the accuracy of other ratios -- for example
> 9:11 at (~3.266c - ~4.282c) or ~1.015 cents narrow in Peppermint 24.

Correct.

> Also, I get the point that the 11-diesis (regular fourth vs. ~8:11)
is
> identical to the 13-diesis for ~8:13, thus 4D-4G/|\ and 4E-5C/|\
> respectively. Thus 351:352 is tempered out -- as is also 891:896,
with
> the 11-diesis or ~32:33 also representing ~27:28, the difference
> between a regular major second and a 6:7 third, e.g. 3B\!!/-4C/|\ or
> 3B\!!/-4D!!!).

Since you need the symbol for the 7 comma, but since the difference
between the 11 and 13 commas vanishes in 121, you choose the 11-comma
symbol, because the 13-comma symbol is inconsistent with the 7-comma
symbol in 121-ET.

> > The one thing that differs from what I suggested is this: Should
> > Margo would want to respell E (~14/11 relative to C) as F-
something,

Forget about this.

--George