Yahoo Tuning Groups Ultimate Backup tuning Sagittal font: accumulation of comma symbols

hi Torsten,

--- In tuning@yahoogroups.com, Torsten Anders <t.anders@q...> wrote:

> I am looking for a notation which shows the intended
> analytic 'meaning' in a as much as possible comprehensible
> way. I would like to use the conventional accidentals (#, b)
> to denote the untempered circle of fifth and use Sagittal
> symbols only to represent single commas which can be
> accumulated in a row. I contact you to check whether
> you would consider my intended use of Sagittal approvable
> or whether it violates some standard ;-)
>
> For example (C=1/1), I want to notate the chord
> 5/6, 25/24, 5/4, 35/24
> (i.e. the dominant seventh over 5/6). I would notate (the comma ','
> separates accidentals):
>
> A\! C#,\!,\! E\! G\!,!)
>
>
> Here, the disadvantage is certainly that I easily end up
> with more then 3 accidentals in a row. The advantage I see,
> however, is that I can easily read, e.g., a dominant seventh
> lowered by a syntonic comma.
> Alternatively, e.g. for a 72-EDO I could write
>
> A\! C||) E\! G\|/
>
>
> With this notation (apart from being less accurate) it is
> harder to recognise the familiar dominant seventh accord.
> Thus, I would prefer the former notation.
>
> Now, Sagittal certainly allows to contract G\!,!) to G\!).
> However, so far I did not found out how I could notate C#,\!,\!
> (or even C#, \!,\!,\!) in a more concise way such that the
> accumulation of syntonic commas is obvious.
>
> Any remarks?

while i do support sagittal as an emerging musical notation,
i also encourage to take a look at the two versions, just
and 72-edo, of my HEWM notation.

if C = 1/1, your "dominant 7th chord over 5/6" is:

in JI-HEWM: A- C#-- E- G<-

in 72-edo HEWM: A- C#< E- Gv

in the JI version, the minus sign represents the
syntonic comma and the less-than sign represents the
septimal comma.

the beautiful thing about 72-edo is that it does a decent
job of representing 11-limit JI while greatly simplifying
the notation of accidentals. the JI-HEWM double-minus "--"
always contracts to 72-edo-HEWM less-than "<", and the
JI-HEWM "<-" always contracts to 72-edo-HEWM "v", so
once you become familiar with how 72-edo-HEWM represents JI,
it becomes fairly transparent.

but even the JI-HEWM is more compact than sagittal.

the big advantage of sagittal, of course, is that the
same symbols are available to notate a wide variety of
other tunings. HEWM is not as portable. but for your
stated goals, HEWM seems to be a good solution.

-monz

🔗monz <monz@tonalsoft.com>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, Torsten Anders <t.anders@q...> wrote:
> I just read about the Sagittal notation in the Xenharmonikon article
> and I like its approach. However, as many others I am also
> overwhelmed by the sheer mass of supported symbols ;-)

Hi Torsten. Welcome to the list.

Carl's remark about Xenharmonikon refers to the fact that while the
Sagittal article claims to have been reprinted from Xenharmonikon 18
(2004), in fact this issue of the journal does not exist, and now can
never exist. Although the editor, John Chalmers, assures us that #18
will be published any year now. And as we all know, the sooner you
fall behind, the more time you'll have to catch up. (Sorry John, I
couldn't resist. :-)

Yes. I think perhaps we made a mistake in publishing that superset in
Table 3 of
http://dkeenan.com/sagittal/Sagittal.pdf
You could be forgiven for running screaming. And in fact George and I
have just recently agreed that we don't need them! We only need the
union of the "Athenian" and "Trojan" sets (most of which are common to
both) and a very few others. The addition of accent marks to these
symbols will take care of everything else. Two recent innovations, not
described on the website yet, have led to this simplification - right
accents and "smart defaults".

I suppose that even the Athenian JI notation can look fairly daunting
at first. But once you get used to the way the symbols are composed of
a small number of graphical elements combined in logical ways, I don't
think it's too bad. The accurate notation of extended-JI is of course
the most demanding task for any notation system.

But that's one reason why we recommend that readers start with the
mythology.
http://dkeenan.com/sagittal/gift/GiftOfTheGods.htm
It's a far gentler introduction, although as yet incomplete.

That is certainly standard Sagittal for JI, although I'd call it a
chain or spiral of fifths rather than a circle.

> and use Sagittal symbols only to represent single commas which can be
> accumulated in a row. I contact you to check whether you would consider
> my intended use of Sagittal approvable or whether it violates some
> standard ;-)

Please feel free to use them that way, if that's what works for you.
But it is not the way preferred by the authors. We prefer to use one
of two systems (a) "pure sagittal" a single sagittal accidental per
note or (b) "mixed sagittal" a maximum of two accidentals where one is
a conventional sharp or flat or double thereof and the other is a
single-shaft sagittal.

However, an important stage in the evolution of sagittal (like several
earlier systems) was the idea of having one symbol per prime number.
Those symbols still exist in Sagittal. Their prime comma meanings are
shown in bold in Table 1, although some are now only secondary
meanings for their symbols (13, 29 and greater), and for convenience a
few primes have more than one comma symbol (5, 17, 19).

If you wish to use more than three accidentals per note then we
recommend using only these single-shaft prime-comma symbols, with
conventional sharps and flats representing the prime number 3. And we
suggest they should always be listed in order of prime number as you
go away from the letter nominal or notehead (left to right in text and
right to left on the staff).

But we hope that once you get to know sagittal better you won't see
the need for this, or perhaps only rarely.

> For example (C=1/1), I want to notate the chord 5/6, 25/24, 5/4, 35/24
> (i.e. the dominant seventh over 5/6). I would notate (the comma ','
> separates accidentals):
>
> A\! C#,\!,\! E\! G\!,!)
>
> Here, the disadvantage is certainly that I easily end up with more then
> 3 accidentals in a row. The advantage I see, however, is that I can
> easily read, e.g., a dominant seventh lowered by a syntonic comma.
> Alternatively, e.g. for a 72-EDO I could write
>
> A\! C||) E\! G\|/
>
> With this notation (apart from being less accurate) it is harder to
> recognise the familiar dominant seventh accord. Thus, I would prefer
> the former notation.

This certainly _is_ less accurate! I'm not sure why you think we would
suggest it at all. See Figures 5 and 7.

In mixed sagittal we would write

A\! C#\\! E\! G\!)

and in pure sagittal

A\! C)||( E\! G\!)

although personally I'd call this a subminor seventh chord or harmonic
seventh chord since I think of a dominant seventh chord as a 5-limit
thing.

I'm sure you will agree that the seventh chord is easy enough to read
in the mixed notation above. (I'll assume you are not interested in
the pure sagittal from here on.)

I agree there are other examples of chords shifted by multiple commas,
or by commas in the opposite direction, where even the standard mixed
sagittal might obscure the chord structure.

We were talking with Robert Walker (of Fractal Tune Smithy fame)
recently about this very thing and we agreed that it made sense in
this case to indicate the common comma offset with one sagittal nearer
to the nominal or notehead and the indidual commas further away. Any
conventional sharps or flats would still be closest to the nominal or
notehead. But the aim here was to limit it to no more than 3
accidentals per note, so multiple comma shifts must still be contracted.

The result of this would be exactly as you first wrote it above
(although the separator after "#" isn't necessary).

A\! C#\!,\! E\! G\!,!)

Your choice of the comma ',' as a separator is a good one as this
character is not used in any ASCII representation of any sagittal.

> Now, Sagittal certainly allows to contract G\!,!) to G\!). However,
> so far I did not found out how I could notate C#,\!,\!

Just as \!,!) = \!)

we have \!,\! = \\!

See degree 8 of Figure 5, or the fifth symbol from the bottom in Table 1.

We also have the not so obvious contraction where a 5-comma up
combined with a 7-comma down results in a 5:7-kleisma down.

/|,!) = !(

It's not so obvious graphically because it involves a subtraction and
the result is smaller than either of the two operands. But you can
sort of imagine that the straight barb makes the convex arc flip about
a 45 degree line and become a concave scroll.

> (or even C#,\!,\!,\!) in a more concise way such that the
accumulation of syntonic
> commas is obvious.

I'm afraid that once you go beyond an accumulation of two commas, the
symbols are not so graphically obvious, and yet they are still logical
in a different way.

What we find when we accumulate three 5-commas (to obtain a
125-large-diesis) is that we are only 0.4 cents away from the 35-large
diesis whose down symbol is (!/ [See the bottom of Table 1]. So it
doesn't make sense to have a completely different symbol for the
125-L-diesis (triple 5-comma). If you really need to distinguish these
two large dieses (e.g. when abstracted from any actual notes) you can
add a right accent mark in the opposite direction to the symbol to
subtract that 0.4 cents. So a triple 5-comma down is written in ASCII
as (!/'

The upward accent is shown in ASCII as an apostrophe or single-quote '
and the downward is shown as a period or full-stop . so we have to be
careful when ASCII sagittals appear at the end of a sentence. The
usual thing is to leave a space or two after the sagittal in that
case, or to omit the full-stop altogether and start a new line.

Another lexical problem in discussing these symbols in ASCII is that
parenthetic remarks [like this] are sometimes better put in square
brackets to avoid confusion with the round brackets used in ASCII
sagittals.

Sagittal accents can often be omitted (or ignored) in actual music,
due to the system of "smart defaults" mentioned earlier. This is based
on the most common usage and uses the nominal that the accidental is
applied to (and any sharps or flats), to determine which comma is
intended. It takes into account the choice of 1/1.

A zipped Excel spreadsheet giving this information is at
http://dkeenan.com/sagittal/AthenianSmartDefaults.zip

But note that this table is really only needed for machine
interpretation of the Athenian JI notation. Humans can do the same
thing by ear, and in any case the worst possible error is only about 2
cents.

You don't need to use this table to convert ratios to Athenian
sagittal either. That is a much simpler process that involves simply
converting the comma to cents and looking it up on the number line at
the bottom of Figure 5. The exact boundaries are given in Scala's
sag_ji1.par. And of course Scala can do the job for you automatically
if you SET NOTATION SAJI1 and SET SAGITTAL MIXED. Scala can be found
at http://www.xs4all.nl/~huygensf/scala/

Unfortunately we are still working on the higher precision JI
notations that will automatically include the correct accent marks.

Now to your specific example of C#\!,\!,\!)

If this is part of an entire chord that has been shifted by two
5-commas we could write it C#\\!,\!) . If the whole chord is shifted
by three 5-commas we could write it C#(!/',!) and the accent might be
able to be unambiguously omitted to give C#(!/,!) .

In standard mixed sagittal we would calculate the sum of the commas in
cents = 3*-21.51 + -27.26 = -91.79 cents. This has an absolute value
too big for a single-shaft sagittal so we would add in (and thereby
partly cancel) the apotome represented by the sharp.
113.69 - 91.79 = 21.9 cents.

This is of course only 0.4 cents larger than a 5-comma up and it would
be written as C/|' or simply C/| .

A performer is certainly going to have a much easier time figuring out
how to play or sing a C/| or a C/|' than trying to figure out what
they are supposed to do with C#\!,\!,\!) . But of course the composer
may understand it better when written in the long form.

> PS: I am new to this list. Sorry if I ask something you already
> discussed before in length (there are so much mails on the Sagittal
> notation in this list, I did not read everything before :-P)

We certainly don't expect anyone to read all those before asking
questions about Sagittal. :-)

I hope this has been of help, and please ask more questions. I'm sure
there are many others on the list who have been wondering about
similar things.

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Torsten Anders <t.anders@qub.ac.uk>

🔗monz <monz@tonalsoft.com>

hi Torsten,

--- In tuning@yahoogroups.com, Torsten Anders <t.anders@q...> wrote:

> I read your article: the concept of your just HEWM notation
> (Pythagorean basis plus further accidental symbols for commas)
> is precisely what I am after. I only tend to prefer the
> Sagittal symbol shapes: these symbols can be assembled and
> and always clearly mark a direction, even assembled
> symbols [ as in /|) ].
>
> <snip>
>
> Actually, I have used another version of 72-edo notation for
> some time (Richter Herf). However, I had always a hard time
> deciphering the harmonic meaning later. Perhaps I did it not
> work with it long enough to really get used to it. However,
> this simplification always felt to me like simplifying
> classical music by notating it with only only a single
> accidental (e.g. discarding all sharps). With a grain
> of salt: it appears to me, notating A- C#-- as A- C#<
> (where < means either two syntonic commas or a septimal comma)
> is a bit like (in classical notation) substituting A C#
> by A Db (where C# and Db are the same key on the piano).
> I should add that I am talking here mainly about a notation
> which allows me to better understand the analytical harmonic
> 'meaning' of the music -- I may rewrite the whole thing for
> performers..

yes, well now i see clearly that the JI-HEWM concept is
the one you want. in fact, i believe that sagittal was
based on essentially the same concepts. as i said before,
the big difference between sagittal and my version of HEWM
is that HEWM is far more limited, only defined up to 11-limit JI.
the proposal by Daniel Wolf (quoted on my HEWM webpage) goes
up to 23-limit. but if 11-limit is good enough, then my HEWM
a provides a nice compact symbol set.

years before i came up with the HEWM symbols, i was struggling
with the same problem, and came up with the idea of using
the vector of the prime-factor exponents (which Gene Ward Smith
named a "monzo" after you know who) *as* an accidental. so,
to use you example chord again, if C = 1/1 = [0 0, 0 0> , then

5/6 25/24 5/4 35/24 =
A [-1 -1, 1 0>, C# [-3 -1, 2 0>, E [-2 0, 1 0>, G [-3 -1, 1 1>

much more long-winded, i know. but staff-notation shows
the pitch-height of the note, and so the exponent of 2 can
be omitted. so we get:

A [-1, 1 0>, C# [-1, 2 0>, E [0, 1 0>, G [-1, 1 1>

the brackets and commas were not part of my original proposal,
and have only been made part of the standard monzo notation
recently. in my original version, it would look like this:

A -1 1 0, C# -1 2 0, E 0 1 0, G -1 1 1

i realize that this is rather difficult for performers to
read on-the-fly, but it has the advantage of using a very
small symbol set -- only the digits 0...9 and the negative sign.
(and the sharps and flats, which in my proposal are really
part of the nominal and not part of the accidental).

-monz

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, Torsten Anders <t.anders@q...> wrote:
>
> Dear list,
>
> I just read about the Sagittal notation in the Xenharmonikon
article and
> I like its approach. However, as many others I am also overwhelmed
by
> the sheer mass of supported symbols ;-)

Hi Torsten,

I started my reply before Dave Keenan posted his, but was not able to
finish it until later, when I did not have access to the Internet.
While I duplicated part of what he said, I also make a couple of
other observations, coming from the viewpoint that Sagittal is
intended to excel as a *performance* notation, capable of being read
in real time, in which context my statements regarding restrictions
for the stacking of flags should be interpreted.

Sagittal is still very new to everyone, and not many have seen it
yet. The "reprint" of the XH article has turned out to be
a "preprint," since we are still awaiting release of Xenharmonikon 18.

We haven't yet prepared a FAQ document for the Sagittal website, but
when we do your question will definitely be included.

While the many symbols were devised to accommodate the anticipated
needs of users having various specialized requirements, we have
concluded that for general use the athenian symbol set would be
adequate for music in just intonation and smaller subsets of that for
most temperaments. It's not all that complicated if you learn them a
few at a time.

> I am looking for a notation which shows the intended
analytic 'meaning'
> in a as much as possible comprehensible way. I would like to use the
> conventional accidentals (#, b) to denote the untempered circle of
fifth
> and use Sagittal symbols only to represent single commas which can
be
> accumulated in a row. I contact you to check whether you would
consider
> my intended use of Sagittal approvable or whether it violates some
> standard ;-)

There are two principles that we have followed in devising the
Sagittal symbols that are relevant here:

1) There should not be more than one Sagittal accidental modifying a
single notehead.

2) No Sagittal symbol should contain more than two flags.

This would prohibit stacking more than two 5-comma flags at a time.
putting you in much the same situation as with Roman numerals, where
the restriction is to have no more than three like characters in a
row, making "III" okay, but "IIII" becomes "IV".

The analogy can be taken a step farther with the observation that,
just as Roman numerals proved too cumbersome for many calculations,
so would stacked symbol-elements (or flags) be too complicated for
reading pitches and intervals. Just as learning the meaning of
Arabic numerals enables calculations to be done more efficiently, so
does learning the athenian-level sequence of Sagittal symbols make
the calculation of compound commas a relatively simple matter.

> For example (C=1/1), I want to notate the chord 5/6, 25/24, 5/4,
35/24
> (i.e. the dominant seventh over 5/6). I would notate (the comma ','
> separates accidentals):
>
> A\! C#,\!,\! E\! G\!,!)

In athenian-level JI this would be notated:

A\! C#\\! E\! G\!)

I think the meaning of the symbols is about as transparent as it
could possibly be.

In pure Sagittal this would be: A\! C)||( E\! G\!)
Here the notation for the third of the chord is not so obvious, but
I'll explain more about this below.

If you're running Windows and want your computer to display the
Sagittal notation for just ratios, I recommend downloading Scala
(freeware written by Manuel Op de Coul):

http://www.xs4all.nl/~huygensf/scala/

Once you have it up and running, enter the commands:

set notation saji1
set sagittal mixed

Then go to the "Tools" menu and click on "Ratio vector calculator"
(or use the shortcut key <shift>+<alt>+V). You can then enter a
ratio (relative to C) in the "Pitch and Names" box, e.g., 25/24 (as
5/4 of A\!) and see one or two possible spellings (the SAJI1 names)
in both ASCII text and with the actual symbols displayed next to a
notehead. (It's not necessary to select a tuning or load an .scl
file to do this.) As an alternative, you can enter the prime factors
in the boxes above and let Scala calculate the ratio for you.

> Here, the disadvantage is certainly that I easily end up with more
then
> 3 accidentals in a row.
>
> The advantage I see, however, is that I can
> easily read, e.g., a dominant seventh lowered by a syntonic comma.
> Alternatively, e.g. for a 72-EDO I could write
>
> A\! C||) E\! G\|/
>
> With this notation (apart from being less accurate) it is harder to
> recognise the familiar dominant seventh accord. Thus, I would
prefer the
> former notation.

Yes, with the 72-mapping you need far fewer symbols, but you no
longer are able to make as many distinctions of pitch. With athenian-
level (medium-precision) JI you have the ability to notate all of the
11-limit consonances *exactly* and all of the 15-limit consonances to
within less than a cent. This is much better accuracy than would be
expected by the ~5.4-cent resolution that results by dividing the
apotome into 21 steps, which we have found to provide a good balance
between simplicity and precision.

> Now, Sagittal certainly allows to contract G\!,!) to G\!). However,
so
> far I did not found out how I could notate C#,\!,\!

As I indicated above, this is simplified as C#\\!, for which the
meaning should be obvious.

> or even C#,
> \!,\!,\! in a more concise way such that the accumulation of
syntonic
> commas is obvious.

[Note: I removed the parentheses from the last part of your
statement in order to clarify that you're discussing "\!,\!,\!",
not "\!,\!,\!)".]

Immediately obvious? No, but I think we can come quite close to that
with something that's fairly easy to remember and very definitely
clear and concise -- something akin to counting by fours with Arabic
numerals.

If you look at Figure 5 on page 10 of our XH article you will see the
sequence of athenian Sagittal symbols, which is not difficult to
learn. You will observe that accumulated 5-commas are notated by
every 4th step in the sequence. I already pointed out that the
meaning of the symbol for two 5-commas down \\! is clear. Follow the
sequence down into the second row (and leftward) to the symbol (|\
occupying the 12th step, for which the exact definition (8192:8505)
differs from three 5-commas by only ~0.4 cents. Its downward
counterpart (!/ is the same as if one had written \\\!, so "A" 3
commas down would therefore be notated A(!/ . However, C# lowered by
3 commas is not written C#(!/, but rather in the simpler form C/|),
since we discourage employment of a sharp or flat in combination with
a Sagittal accidental larger than 40% of an apotome (~46 cents)
altering in the opposite direction.

How did I make the conversion from C#(!/ to C/|)? Since /|) and (|\
are apotome-complements, we can write the equation:
# = /|) + (|\
which can then be rewritten:
# - (|\ = /|)
# + (!/ = /|)
Thus C#(!/ = C/|)

But, wait! We're just getting started. This, in turn, makes it easy
to deduce the notation for C# four commas down: simply remove the
left (5-comma-up) flag from C/|) to get C|); yes, this looks like a C
raised by a 7-comma. If you want to get technical, it's only ~0.4
cents higher than that, so for all practical purposes we can consider
the two to be the same pitch.

Now let's go one more! To get C# five commas down all we need to do
is replace the 7-comma-up symbol with a 5:7-kleisma-up symbol (i.e.,
5103:5120, which is the *difference* between a 7-comma and a 5-comma)
giving us C|( -- a C raised by ~6 cents. Another way of arriving at
the notation for these last two pitches is to look at the sequence
C/|) C|) C|( and see that the accidentals are 4 steps apart in the
athenian-level JI diagram.

If I were reading music that called for C# to be lowered by 5 commas,
I think I would have a much better idea of the proper pitch if I saw
C|( than if I had to read something like C#\\\\\!

So you see that this is indeed as easy as performing arithmetic using
Arabic numerals -- once you have taken the time to memorize the
athenian symbol sequence (which I think you will agree is more
logical than the sequence of Arabic numerals, which, except for "1",
provides no clue as to meaning.).

I said above that I would show how the pure Sagittal could be made a
little more obvious. Referring to Figure 5 again, observe (just
below the natural sign) that C# is written C/||\ in pure Sagittal.
From there move 4 steps to the right to a symbol that is the apotome
symbol without the right (5-comma) flag, giving C||\ as the
equivalent of C#\! . Four more steps to the right brings us to C)||(
as the equivalent of C#\\! . Four more steps (going upward around
the end and up into the first row) brings us to C/|), which is C#
lowered by three 5-commas, at which point we continue in the same
manner as above. So interval calculation becomes fairly simple once
the sequence of athenian-level symbols is learned, just as numerical
calculation is simplified by learning the sequence of Arabic numerals.

I hope this has addressed your questions satisfactorily.

Best,

--George Secor

🔗Torsten Anders <t.anders@qub.ac.uk>

Dear Dave,

Thank you very much for your most detailed response.

On Tue, 2005-02-22 at 03:24, Dave Keenan wrote:
> However, an important stage in the evolution of sagittal (like several
> earlier systems) was the idea of having one symbol per prime number.
> Those symbols still exist in Sagittal. Their prime comma meanings are
> shown in bold in Table 1, although some are now only secondary
> meanings for their symbols (13, 29 and greater), and for convenience
> a few primes have more than one comma symbol (5, 17, 19).

In Table 1, I found symbols for the following prime
commas/schismas/diesis

(5) \! 80/81 = 5/4 : (2^-6 * 3^4) (-21.5¢)

(7) !) 63/64 = 7/4 : 16/9 (-27.3¢)

(11) /|\ 33/32 = 11/8 : 4/3 (53.3¢)

(13) /|) 1053/1024 = 13/8 : 2^7 * 3^-4 (48.3¢)

(17) ~|( 4131/4096 = 17/16 : 2^8 * 3^-5 (14.7¢)

(19) )| 513/512 = 19/16 : 32/27 (3.4¢)

(23) |~ 736/729 = 23/16 : 2^-9 * 3^6 (16.5¢)

(29) (| 261/256 = 29/16 : 16/9 (33.5¢)

However, so far I was unable to find a symbol for the following

(31) 248/243 = 31/16 : 2^-7 * 3^5 (35.3 ¢)

Is prime 31 perhaps substituted by something close?

(I did not try to find symbols for higher prime numbers.)

>> to denote the untempered circle of fifth
> I'd call it a chain or spiral of fifths rather than a circle.

Sure, cough..

> If you wish to use more than three accidentals per note then we
> recommend using only these single-shaft prime-comma symbols, with
> conventional sharps and flats representing the prime number 3. And we
> suggest they should always be listed in order of prime number as you
> go away from the letter nominal or notehead (left to right in text and
> right to left on the staff).

> But we hope that once you get to know sagittal better you won't see
> the need for this, or perhaps only rarely.

>> A\! C#,\!,\! E\! G\!,!)
>
> In mixed sagittal we would write
>
> A\! C#\\! E\! G\!)
>
> and in pure sagittal
>
> A\! C)||( E\! G\!)

I see. The mixed Sagittal is nicely readable for me. I haven't realised
the \\! before. Is there also !)) or even \\!// etc allowed? Or would
there again be some 'Sagittal math' involved as explained by George D.
Secor in his answer to my first email?

> I'd call this a subminor seventh chord or harmonic seventh chord
> since I think of a dominant seventh chord as a 5-limit thing.

Like with the 'circle of fifth' you caught me: my terminology still
stucks in traditional music theory, sigh.

> I agree there are other examples of chords shifted by multiple commas,
> or by commas in the opposite direction, where even the standard mixed
> sagittal might obscure the chord structure.

> We were talking with Robert Walker (of Fractal Tune Smithy fame)
> recently about this very thing and we agreed that it made sense in
> this case to indicate the common comma offset with one sagittal nearer
> to the nominal or notehead and the indidual commas further away. Any
> conventional sharps or flats would still be closest to the nominal or
> notehead. But the aim here was to limit it to no more than 3
> accidentals per note, so multiple comma shifts must still be
> contracted.

I see. For whole chords (each note with up to two mixed Sagittal
accidentals) shifted by some commas I would thus need 3 accidentals
per note at maximum (in mixed Sagittal).

Thus, only in case I would construct a chord such that I wish to have
already more then two accidentals for some note in the unshifted chord,
e.g., by stacking three and more equal intervals like

C E\! G#\\! B#\\!,\!

and then shifting this whole chord by some comma I would need more then
3 accidentals per note like (if I not contract the symbols to some other
symbols):

C/|) E/|),\! G#/|),\\! B#/|),\\!,\!

I take it that you would suggest that I limited myself to 3 accidentals
and substitute \\!,\! by (!/.

Needless to say that such a notation (accidentals in opposite
directions) would only be for my own sake as composer/analyst and
probably would need rewriting for performers ;-)

Thanks again!

Best,
Torsten

--
Torsten Anders
Sonic Arts Research Centre
Queen's University Belfast
Tel: +44 28 9097 4761 (office)
+44 28 9066 7439 (private)
www.torsten-anders.de

🔗Torsten Anders <t.anders@qub.ac.uk>

Dear Joe,

On Tue, 2005-02-22 at 15:31, monz wrote:
> years before i came up with the HEWM symbols, i was struggling
> with the same problem, and came up with the idea of using
> the vector of the prime-factor exponents (which Gene Ward Smith
> named a "monzo" after you know who) *as* an accidental. so,
> to use you example chord again, if C = 1/1 = [0 0, 0 0> , then
>
> 5/6 25/24 5/4 35/24 =
> A [-1 -1, 1 0>, C# [-3 -1, 2 0>, E [-2 0, 1 0>, G [-3 -1, 1 1>
>
> much more long-winded, i know. but staff-notation shows
> the pitch-height of the note, and so the exponent of 2 can
> be omitted. so we get:
>
> A [-1, 1 0>, C# [-1, 2 0>, E [0, 1 0>, G [-1, 1 1>

Thank you very much! This notation looks highly suitable for my
purposes. I understand that these 'monzo accidentals' fully denote the
pitch or pitch class itself instead of being some comma notation like

A [4 -4, 1 0> C# [8 -8, 2 0> E [-4 4, -1 0> G [-6 2, 0 -1>

> the brackets and commas were not part of my original proposal,
> and have only been made part of the standard monzo notation
> recently.

Does the comma mark the prim factor of the 3? To be honest, I don't
understand the purpose of the irregular brackets [...>.

I must say that for deciphering the harmonic meaning of my own scores I
really like this pitch representation. Besides, it can be easily
extended for arbitrary higher prim numbers. So, I am currently thinking
about a notation which combines some approximate pitch notation (e.g.
some 72-EDO notation) with these 'monzo accidentals'. However, for a
more comprehensive, generic and concise representation I guess I prefer
plain numeric expressions:

A- (3^-1 * 5), C#< (3^-1 * 5^2), E- (5), Gv (3^-1 * 5 * 7)

As proposed by Dave Keenan for Sagittal, I could also isolate the pitch
'offset' of a chord (in this case not some commas but the root):

A- (3^-1*5), C#< ((3^-1*5) * 5), E- ((3^-1*5) * 3), Gv ((3^-1*5) * 7)

If this is written in conventional math notation (i.e. not ASCII), e.g.,
using TeX in Lilypond it should look better readable..

Thanks again!

Best,
Torsten

--
Torsten Anders
Sonic Arts Research Centre
Queen's University Belfast
Tel: +44 28 9097 4761 (office)
+44 28 9066 7439 (private)
www.torsten-anders.de

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, Torsten Anders <t.anders@q...> wrote:
> ...
> On Tue, 2005-02-22 at 03:24, Dave Keenan wrote:
> > However, an important stage in the evolution of sagittal (like
several
> > earlier systems) was the idea of having one symbol per prime
number.
> > Those symbols still exist in Sagittal. Their prime comma meanings
are
> > shown in bold in Table 1, although some are now only secondary
> > meanings for their symbols (13, 29 and greater), and for
convenience
> > a few primes have more than one comma symbol (5, 17, 19).
>
> In Table 1, I found symbols for the following prime
> commas/schismas/diesis
>
> (5) \! 80/81 = 5/4 : (2^-6 * 3^4) (-21.5Â¢)
>
> (7) !) 63/64 = 7/4 : 16/9 (-27.3Â¢)
>
> (11) /|\ 33/32 = 11/8 : 4/3 (53.3Â¢)
>
> (13) /|) 1053/1024 = 13/8 : 2^7 * 3^-4 (48.3Â¢)
>
> (17) ~|( 4131/4096 = 17/16 : 2^8 * 3^-5 (14.7Â¢)
>
> (19) )| 513/512 = 19/16 : 32/27 (3.4Â¢)
>
> (23) |~ 736/729 = 23/16 : 2^-9 * 3^6 (16.5Â¢)
>
> (29) (| 261/256 = 29/16 : 16/9 (33.5Â¢)
>
> However, so far I was unable to find a symbol for the following
>
> (31) 248/243 = 31/16 : 2^-7 * 3^5 (35.3 Â¢)
>
> Is prime 31 perhaps substituted by something close?

It is rather roughly approximated (within ~2.1 cents) in athenian-
level (medium-resolution) JI by (| , in high-resolution JI (within
0.47 cents) by '(| , and in extreme-resolution JI (within 0.05 cents)
by '(|' .

In addition to that, there is also:

31:32, ~54.964 cents
approximated (within ~1.7 cents) in athenian-level JI by /|\ , in in
high-resolution JI (within 0.44 cents) by (/| , and in extreme-
resolution JI (within 0.16 cents) by (/|' ;

and

65536:67797, ~58.721 cents, the apotome-complement of 31:32
approximated (within ~1.7 cents) in athenian-level JI by (|) , in in
high-resolution JI (within 0.44 cents) by |\) , and in extreme-
resolution JI (within 0.16 cents) by |\). -- note that the period
character terminating that last symbol is part of the symbol,
representing a downward right-accent mark of ~0.4 cents.

I will explain the meaning of the symbols (/| and |\) below.

With C=1/1, Dave's athenian-level smart-defaults algorithm will
interpret B(| as 31/16 and Db(! as 32/31, but it does not interpret
C/|\, C\!/, C(|), or C(!) as ratios of 31.

> (I did not try to find symbols for higher prime numbers.)

Primes above 31 are not found by athenian smart defaults, and I
expect that they will be notated only as approximations (by re-using
symbols) at higher resolution levels (unless a smart-defaults
algorithm is also implemented at those levels, which we expect to do
once we've finalized the details of high- and extreme-resolution JI).

> ...
> >> A\! C#,\!,\! E\! G\!,!)
> >
> > In mixed sagittal we would write
> >
> > A\! C#\\! E\! G\!)
> >
> > and in pure sagittal
> >
> > A\! C)||( E\! G\!)
>
> I see. The mixed Sagittal is nicely readable for me. I haven't
realised
> the \\! before. Is there also !)) or even \\!// etc allowed?

We considered !)) for the 49M-diesis (3969:4096, ~54.528c) above the
athenian level, but ended up using the symbol pair (/| and (\! for
that instead (sum of flags ~54.654c), because it provided a much more
obvious and convenient apotome-complement pair |\) and !/) for the
49L-diesis. These are the same symbols that I gave above to
approximate the 31M (~54.964c) and 31L dieses in high-resolution JI.

Something like \\!// or //|\\ is a bit of stretch (the latter symbol
would be 11^2), but I suppose you could use something like that in a
theorical context, since it's really no worse than \\\\\! for five 5-
commas down.

> Or would
> there again be some 'Sagittal math' involved as explained by George
D.
> Secor in his answer to my first email?

We call it "flag arithmetic," and at the athenian level the flags
correspond to the following numbers of steps:

1: )| and |(
2: ~|
3: |~
4: /|
5: |)
6: |\ and (|
21: # (apotome)

In 13-limit JI the prime factors would involve only barbs and arcs:

5 \!
7 !)
11 /|\ or (!)
13 (!/ or /|)

It appears that you can add these flags directly, e.g., a ratio such
as 55/54 would combine \! with /|\ , with the up and down left flags
canceling out to give |\ .

and their differences require only the left scroll in addition to
that:

5-7 or 11-13 |(
7-5 or 13-11 !(
7-11 (|
11-7 (!
5-11 or 7-13 (|(
11-5 or 13-7 (!(

When you try to take differences, something like 11/10 doesn't come
out as a combination of /|\ and /|, or //|\ ; it's instead a
combination of (!) and /| that gives D(!( as the final result --
recall that !) + /| = !(; so it helps to know the combinations of
flags that correspond to the differences of two primes. You'll have
to decide whether or not this is too complicated for your purposes.

> I take it that you would suggest that I limited myself to 3
accidentals
> and substitute \\!,\! by (!/.

It's easy enough to learn, and you would immediately see that it's
almost the same size as the 13L diesis.

Best,

--George

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, Torsten Anders <t.anders@q...> wrote:
> In Table 1, I found symbols for the following prime
> commas/schismas/diesis
>
>
> (5) \! 80/81 = 5/4 : (2^-6 * 3^4) (-21.5Â¢)
>
> (7) !) 63/64 = 7/4 : 16/9 (-27.3Â¢)
>
> (11) /|\ 33/32 = 11/8 : 4/3 (53.3Â¢)
>
> (13) /|) 1053/1024 = 13/8 : 2^7 * 3^-4 (48.3Â¢)
>
> (17) ~|( 4131/4096 = 17/16 : 2^8 * 3^-5 (14.7Â¢)
>
> (19) )| 513/512 = 19/16 : 32/27 (3.4Â¢)
>
> (23) |~ 736/729 = 23/16 : 2^-9 * 3^6 (16.5Â¢)
>
> (29) (| 261/256 = 29/16 : 16/9 (33.5Â¢)

That's correct. And in table 1 you will also see

(17) ~! 2187/2176 = 17/16 : 2^? * 3^7 (-8.7 c)

(19) )!~ 19683/19456 = 19/16 : 2^? * 3^9 (-20.1 c)

> However, so far I was unable to find a symbol for the following
>
> (31) 248/243 = 31/16 : 2^-7 * 3^5 (35.3 Â¢)

There is a simpler 31-diesis

> Is prime 31 perhaps substituted by something close?

Yes. It is a symbol that appears in Table 3 of the XH article (right
side of first row), but is not explained in the article and is not yet
in the published font.

(31) (\! 31/32 = 31/16 : 2/1 (-55.0 c)

This symbol's primary meaning is as the 49-M-diesis (double 7-comma)
which is 0.4 c smaller. i.e. !),!) = (\!

If you need to distinguish the 31-M-diesis from the 49-M-diesis you
would add a right accent in the same direction as the arrow.

(\!. [That dot is the down accent or grave, in ASCII]

You might ask why we didn't we make a symbol !)) for the 49-M-diesis.
The main reason is that (\! gives us an obvious pairing with the
symbol !/) for its apotome-complement the 49-L-diesis, while !)) does not.

> (I did not try to find symbols for higher prime numbers.)

They would all have to adopt symbols whose primary meaning is at a
lower-prime-limit, as is already the case with the 13, 29 and 31
symbols. And you would need to add accent marks to such symbols if the
tiny errors mattered.

> I see. The mixed Sagittal is nicely readable for me. I haven't realised
> the \\! before. Is there also !)) or even \\!// etc allowed?

No. We decided to limit our symbols to a maximum of two-flags per
symbol. The values of the flags were painstakingly chosen so that this
would get us very close to anything while keeping the graphical
clutter to a minimum.

The equivalent of !)) is precisely (\! as described above, but in the
Athenian (i.e. medium precision) notation it is approximated by \!/

The equivalent of \\!// is approximated by b|( in the Athenian JI
notation. A higher precision version is as yet undecided.

> Or would
> there again be some 'Sagittal math' involved as explained by George D.
> Secor in his answer to my first email?

Yes.

> I see. For whole chords (each note with up to two mixed Sagittal
> accidentals) shifted by some commas I would thus need 3 accidentals
> per note at maximum (in mixed Sagittal).

Yes.

> Thus, only in case I would construct a chord such that I wish to have
> already more then two accidentals for some note in the unshifted chord,
> e.g., by stacking three and more equal intervals like
>
> C E\! G#\\! B#\\!,\!
>
> and then shifting this whole chord by some comma I would need more then
> 3 accidentals per note like (if I not contract the symbols to some other
> symbols):
>
> C/|) E/|),\! G#/|),\\! B#/|),\\!,\!
>
> I take it that you would suggest that I limited myself to 3 accidentals
> and substitute \\!,\! by (!/.

Yes.

> Needless to say that such a notation (accidentals in opposite
> directions) would only be for my own sake as composer/analyst and
> probably would need rewriting for performers ;-)

In that case, use as many accidentals as you like. :-)

But I think you will quickly learn the arithmetic of Athenian degree
numbers as described by George Secor and shown in figure 5.

Regards,
-- Dave Keenan

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, Torsten Anders <t.anders@q...> wrote:

> Dear Joe,
>
> On Tue, 2005-02-22 at 15:31, monz wrote:
> >
> > years before i came up with the HEWM symbols, i was struggling
> > with the same problem, and came up with the idea of using
> > the vector of the prime-factor exponents (which Gene Ward Smith
> > named a "monzo" after you know who) *as* an accidental. so,
> > to use you example chord again, if C = 1/1 = [0 0, 0 0> , then
> >
> > 5/6 25/24 5/4 35/24 =
> > A [-1 -1, 1 0>, C# [-3 -1, 2 0>, E [-2 0, 1 0>, G [-3 -1, 1 1>
> >
> > much more long-winded, i know. but staff-notation shows
> > the pitch-height of the note, and so the exponent of 2 can
> > be omitted. so we get:
> >
> > A [-1, 1 0>, C# [-1, 2 0>, E [0, 1 0>, G [-1, 1 1>
>
> Thank you very much! This notation looks highly suitable
> for my purposes.

i thought that it would!
you're quite welcome.

> I understand that these 'monzo accidentals' fully denote
> the pitch or pitch class itself instead of being some
> comma notation like
>
> A [4 -4, 1 0> C# [8 -8, 2 0> E [-4 4, -1 0> G [-6 2, 0 -1>

i'm not really clear on what you're notating here,
but yes, using the monzo as an accidental pinpoints
precisely where a just-intonation pitch-class lies
in octave-equivalent ratio-space. the monzos are like
a set of driving directions showing routes arond the
lattice.

> > the brackets and commas were not part of my original proposal,
> > and have only been made part of the standard monzo notation
> > recently.
>
> Does the comma mark the prim factor of the 3?

yes. i convention that was proposed and which i
enthusatically support is to put commas after the
exponent of 3, and then subsequently after every
third prime. it just helps to make it easier to
see two things:

1) whether 2 is being used or ignored, and

2) which exponent goes with which prime, especially
for ratios which have lots of prime-factors.

> To be honest, I don't understand the purpose of the
> irregular brackets [...>.

neither do i ... but Gene Ward Smith is a mathematician
and he and Paul Erlich have found it useful to use the
"bra-ket" notation which uses one square and one angle bracket.
i have links to some definitions of this notation near
the bottom of the Encyclopedia "monzo" page:

http://tonalsoft.com/enc/monzo.htm

the monzo [...> notation is the "ket" notation, and
its corresponding opposite is the "val", which uses
the "bra" notation:

http://tonalsoft.com/enc/val.htm

the post from Gene quoted on the "monzo" page (the section
where the definition links are located) explains how the
two go together.

> I must say that for deciphering the harmonic meaning of
> my own scores I really like this pitch representation.
> Besides, it can be easily extended for arbitrary higher
> prim numbers.

yes, absolutely true. that's exactly why i thought it up
in the first place. when i first began writing my book
(about ten years ago), one purpose was to examine a large
number of rational (i.e, just-intonation) tuning theories
that had been proposed thru-out history, and to compare
them all to each other by making them all subsets of a
"universal tuning". this was the way to do it.

> So, I am currently thinking about a notation which combines
> some approximate pitch notation (e.g. some 72-EDO notation)
> with these 'monzo accidentals'.

yes, i would suggest combining 72-edo HEWM with monzos.
the 72-edo HEWM is very simple and gets you within a few
cents of the just-intonation pitch, then the monzo specifies
precisely the rational mathematics.

> However, for a more comprehensive, generic and concise
> representation I guess I prefer plain numeric expressions:
>
> A- (3^-1 * 5), C#< (3^-1 * 5^2), E- (5), Gv (3^-1 * 5 * 7)

that was my other proposal back in my first paper, and which
i used all thru my book. it actually works quite well, as
long as there are only 2 or 3 prime-factors. with more
than 3 factors, the monzo notation is better.

> As proposed by Dave Keenan for Sagittal, I could also isolate
> the pitch 'offset' of a chord (in this case not some commas
> but the root):
>
> A- (3^-1*5), C#< ((3^-1*5) * 5), E- ((3^-1*5) * 3),
> Gv ((3^-1*5) * 7)

this "relative" notation is good for harmonic analysis ...
but the "absolute" notation we discussed above is better
for actually notating a composition.

> If this is written in conventional math notation
> (i.e. not ASCII), e.g., using TeX in Lilypond it should
> look better readable..

yes, that's how i did it in my book (with superscript
exponents).

-monz

Sagittal font: accumulation of comma symbols

🔗Torsten Anders <t.anders@qub.ac.uk>

🔗Carl Lumma <ekin@lumma.org>

🔗Torsten Anders <t.anders@qub.ac.uk>

🔗Carl Lumma <ekin@lumma.org>

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Torsten Anders <t.anders@qub.ac.uk>

🔗monz <monz@tonalsoft.com>

🔗George D. Secor <gdsecor@yahoo.com>

🔗Torsten Anders <t.anders@qub.ac.uk>

🔗Torsten Anders <t.anders@qub.ac.uk>

🔗George D. Secor <gdsecor@yahoo.com>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗monz <monz@tonalsoft.com>