*BT* Notation - Part 2

*BURIED TREASURE*
"Notation - Part 2"
From: George Secor
January 28, 2002
(Prerequisite: "Notation - Part 1", #32971)

*Background Note*

It had not been more than a week or two after devising the expanded
version of the saggital symbols that Margo Schulter contacted me
about my 17-tone well temperament, and in the course of our
correspondence we frequently mentioned the occurrence of commas in
various tonal systems that are realizations of her neo-medieval
approach to microtonality. In the course of our discussion, I made a
suggestion concerning the nomenclature of commas that we adopted.
This will be included in the 17-tone article that will be appearing
in the next issue of Xenharmonikon (#18), and is as follows:

<< The two different sizes of whole tone [9:8 and 8:7 in Archytas'
diatonic scale] differ in size by 63:64 (27 cents), which Alexander
Ellis referred to as the septimal comma. Ptolemy listed Archytas'
diatonic tuning as the diatonic toniaion, from which we might be a
little hesitant to coin the term toniaic comma. Instead, I believe
it would be fitting to honor the originator of this scale by calling
this Archytas' comma. (For many years I have felt that the use of
the names Pythagoras and Didymus in association with their respective
commas is a clearer and more memorable way of identifying them than
the adjectives ditonic and syntonic, which only a scholar could
love. Confusion between these two terms can happen to the best of
us: Even as knowledgeable an authority as J. Murray Barbour slipped
up in this regard in the beginning of the first chapter of his book,
Tuning and Temperament. >>

With our discussion of commas fresh in my mind, I happened to come
across a couple of papers on which I had sketched out the expanded
saggital symbols, and I began to see a connection between the two
that led to the idea of using the saggital symbols, not only for
systems other than 72-EDO, but also as a transcendental notation.

*Didymus and Archytas*

In the first part of the presentation I mentioned a distinction
between *native* (or EDO-specific) and *transcendental* (or EDO-trans-
generic) applications of the saggital notation. As a general
principle, a composition of even moderate complexity written for a
specific EDO could not be expected to be directly transferable into
another EDO, and it would be appropriate to employ a native (or
system-specific) notation for this purpose. However, it would be
highly desirable for the symbols used in a system-specific notation
to be selected from a master superset of symbols. An analogy from
written language would help to illustrate this point.

It is more difficult for those whose native language is English to
learn to read a language that uses a different alphabet or set of
symbols different from the Roman alphabet (e.g., Hebrew, Arabic,
Chinese, or Japanese) than one that uses essentially the same
alphabet (e.g., Spanish, French, or German). For the same reason it
is better, if at all possible, to use a single unified set of symbols
for many EDO's, rather than different symbols for different EDO's –
provided that each symbol were to have the same meaning across all
(or at least most) of those EDO's, just as specific vowels and
consonants in the Roman alphabet represent similar (but not
identical) sounds in the various European languages that employ them.

The fundamental principle that sets the saggital notation apart from
other systems is as follows:

Whereas other systems of notation use symbols to represent
alteration in pitch in terms of specific numbers of system degrees or
specific fractions of a sharp or flat, the saggital symbols
represents alterations in pitch by using symbols to represent
alterations *by approximations of certain superparticular ratios*,
namely Didymus' comma (81:80), Archytas' comma (64:63), and the
unidecimal diesis (33:32).

As before, the figures for this presentation are in:

/tuning/files/secor/notation/figures.bmp

Reference to the expanded saggital symbols in the third row of Figure
3 will serve to illustrate this fundamental principle. Immediately
to the left of the natural sign is a symbol with a stem with a left
flag, \|. In 72-EDO this would represent one degree down, but in the
saggital notation this actually represents a Didymus-comma-down
(80:81), which just happens to be one degree of 72-EDO. The next
symbol to the left is a stem with a right flag, |/. Specific to 72-
EDO this would represent two degrees down, but in the saggital
notation this represents an Archytus-comma-down (63:64). The next
symbol to the left combines these in a single-headed arrow with both
left and right flags, \|/, symbolizing a unidecimal-diesis-down
(32:33). For those interested in the numbers, observe the following
interval arithmetic (in which it should be understood that plus and
minus symbols actually indicate multiplication and division of the
numerical ratios):

Unidecimal diesis – (Archytus' comma + Didymus' comma) = 4.503 cents

33/32 – (64/63 + 81/80) = 385/384, or 4.503 cents

Those in the tuning-math group will immediately recognize this as a
zero vector encountered in their exploration of the Miracle tuning,
perhaps more recognizable as the difference between just intervals
represented by certain basic intervals in that tuning:

(3 secors) – (2 secors + 1 secor) = a zero vector

11/9 – {8/7 + 16/15) = 385/384, or 4.503 cents

This serves to demonstrate not only the close relationship of the
saggital notation to the Miracle tuning geometry, but also to point
out that the power and versatility of the saggital symbols is drawn
from the very same geometry that gives the Miracle tuning its
versatility and efficiency, so it would not be inappropriate to call
this a Miracle notation. But first let's see if it accomplishes any
miracles.

As we noted, in 72-EDO Didymus' comma is 1 degree and Archytus' comma
is two degrees. In 41-EDO, these are each one degree, while in 31-
EDO they are zero degrees and one degree, respectively. Now look at
any saggital symbol in the third of figure 3; the number of degrees
of alteration accomplished by that symbol in a given EDO is found by
totaling the number of degrees represented by Didymus (left) and
Archytas (right) flags in the symbol for that EDO. That's how it's
done, plain and simple! And this principle can also be used to
notate many other EDO's as well.

Just as a certain vowel symbolized by a letter of the alphabet is
pronounced somewhat differently from one spoken language to another,
so does a Didymus-comma-down symbol, for example, lower the pitch by
different amounts in different EDO's. In the case of systems that
disperse Didymus' comma, such as 24 or 31-EDO, the Didymus-comma-down
symbol will indicate an alteration of zero degrees, so that the left
flags in the expanded saggital symbols are disregarded in these
systems.

It is obvious that nobody is going to take the time or trouble to
count flags, which is why the compact saggital symbols were
developed. These contain all of the information conveyed by the
expanded symbols in a simpler form. Each of the compact symbols can
be readily identified as having a Didymus (left-handed), Archytas
(right-handed), or diesis (laterally symmetrical) appearance and
function.

*Native vs. Transcendental*

It was noted above that compositions written for one EDO are not
generally transferable into another EDO. However, there are
circumstances under which such transference is not only possible, but
highly desirable and practical. For example, music written in the
Blackjack or Canasta scales (~72-EDO) could easily be played on wind
instruments (with relatively small amounts of pitch-bending)
specifically built for 31 or 41-EDO, and the result would be far
better than using 12-EDO instruments with extended (or extraordinary)
techniques. (I believe that 72-EDO wind instruments are a bit out of
the question for the foreseeable future, if ever.) Another example
would be just intonation (e.g., the 43-tone Partch set) mapped onto
72 and played on 31 or (for Partch, preferably) 41-EDO instruments.
(I have to assume that, should microtonality go mainstream, we would
no longer want to suffer the limitations of 12-EDO instruments, and
whatever notation we might use should take this into account. It
never hurts to plan ahead!) In these instances the 72-tone notation
could be employed as a transcendental notation that could be read
directly into 31 or 41, making it unnecessary to have separate parts
for instruments in each of those systems. This would remove a
considerable burden from the composer, who might not know in advance
what instruments would eventually be used to perform a given
composition.

The saggital notation accomplishes this by a principle similar to the
way decimal numbers are rounded off to whole numbers, as illustrated
in Figure 4. In native 31-EDO notation, all symbols are multiples of
a single diesis. (As it happens, in all three EDO's – 31, 41, and
72 – three 5:4's fall short of an octave by the number of degrees
representing 33:32 in each of those systems, so the term "diesis" can
be used here in a broader sense to refer to either the unidecimal or
meantone diesis.) Each left or right-handed symbol in the
transcendental notation may be regarded as an alteration of the
neighboring symmetical symbol by a Didymus flag that represents
81:80, which is a zero vector in 31-EDO. (The left-handed symbols
are thus interpreted as a symmetrical symbol plus Didymus flag, while
right-handed symbols are interpreted as a symmetrical symbol minus a
Didymus flag.) The small arrows in the figure show how
mental "rounding" to the nearest diesis can be accomplished.

In the 41-EDO native notation some new symbols are required to
indicate odd-number multiples of a half-diesis (i.e., quarter-sharp
and quarter-flat) alteration. It should be evident that
mental "rounding" (to the nearest half-diesis) is even simpler than
what is required in 31-EDO.

This process of mental rounding works in this fashion with almost all
simple 11-limit ratios, and the exceptions are easy to remember: the
few ratios having a 7 factor in the numerator and a 5 factor in the
denominator (or vice versa) – 7/5, 10/7, 21/20, 40/21 – must be duly
noted and memorized. As far as I have been able to tell, most
everything else works like a charm!

For your reference, there is a one-octave diagram of 72-EDO (with
numbered system degrees and saggital symbols) compared with 19-limit
just intonation at:

/tuning/files/secor/notation/72vsJI.bmp

The next part of this presentation will continue with application of
the saggital notation to some other EDO's.

And your questions and comments will be appreciated.

Until next time, please stay tuned!

--George

Love / joy / peace / patience …

Over on tuning-math, Paul has told us that my curious and technical definition of what I call a "notation" is unlike any notation anyone has ever actually seen or spoken of, and since it is abstract there's some truth in that. However, I see that saggital notation is also a "notation" in my sense--specifically, it seems to be a notation for the 11-limit temperament with the single comma 225/224, and hence can be used, if not for absolutely everthing, then certainly for any system in which 225/224 is a unison and no primes higher than 11 are represented, and that covers a lot of ground.

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

> Whereas other systems of notation use symbols to represent
> alteration in pitch in terms of specific numbers of system degrees or
> specific fractions of a sharp or flat, the saggital symbols
> represents alterations in pitch by using symbols to represent
> alterations *by approximations of certain superparticular ratios*,
> namely Didymus' comma (81:80), Archytas' comma (64:63), and the
> unidecimal diesis (33:32).

Implicit in all of this, if I understand correctly, is also a count of semitones. Hence I presume 16/15 and 15/14 should be added to the mix, and the two equated. We could then expand saggital by adding a symbol for 225/224, making it into a complete 11-limit notation. We then have:

<81/80, 64/63, 33/32, 16/15, 15/14>^(-1) =

[-h2, -g5, v11, h7, h5]

Here h2, h5 and h7 are the standard 2, 5, and 7-et maps, g5 is a nonstandard 5-et map [5,8,12,15,17], and v11 is the 11-adic valuation,
meaning the map which sends 11 to 1 and everything else to 0. This amounts to a proof saggital notation works under the conditions I stated, and a recipe for calculating how any particular 11-limit
interval should be notated. Note that if 225/224~1 we can conflate
h5 and h7 into h5+h7=g12. I also would like to draw Joe Monzo's attention to the similarity between saggital and his analysis of Schoenberg.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

>Over on tuning-math, Paul has told us that my curious and technical
>definition of what I call a "notation" is unlike any notation anyone
>has ever actually seen or spoken of, and since it is abstract
>there's some truth in that.

Probably my understanding was limited to the case of Monz's
Schoenberg-PB query.

>However, I see that saggital notation is >also a "notation" in my
>sense--specifically, it seems to be a >notation for the 11-limit
>temperament with the single comma 225/224, >and hence can be used,
>if not for absolutely everthing, then >certainly for any system in
>which 225/224 is a unison and no primes >higher than 11 are
>represented, and that covers a lot of ground.

Rami Vitale was very unhappy with the idea of 225:224 vanishing, for
his JI presentation of Byzantine scales. But most of the rest of us
who have spoken up on this seem to be able to live with it -- even
Justin White seemed so disposed last we heard from him.

225:224 vanishes in paultone. So how do we notate paultone using this
system? Will it improve over the notation Alison Monteith is
currently using?

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

> This will be included in the 17-tone article that will be appearing
> in the next issue of Xenharmonikon (#18),

No offense, George, but John Chalmers has informed me that he has not
yet accepted your or any other articles for publication in XH 18. Out
of respect for John, perhaps you should put this a bit differently?

> Whereas other systems of notation use symbols to represent
> alteration in pitch in terms of specific numbers of system degrees
or
> specific fractions of a sharp or flat, the saggital symbols
> represents alterations in pitch by using symbols to represent
> alterations *by approximations of certain superparticular ratios*,
> namely Didymus' comma (81:80), Archytas' comma (64:63), and the
> unidecimal diesis (33:32).

And the naturals represent Pythagorean? Please remind me.

> *Native vs. Transcendental*
>
> It was noted above that compositions written for one EDO are not
> generally transferable into another EDO. However, there are
> circumstances under which such transference is not only possible,
but
> highly desirable and practical. For example, music written in the
> Blackjack or Canasta scales (~72-EDO) could easily be played on
wind
> instruments (with relatively small amounts of pitch-bending)
> specifically built for 31 or 41-EDO, and the result would be far
> better than using 12-EDO instruments with extended (or
extraordinary)
> techniques.

Hmm . . . a devil's advocate response: when Johnny Reinhard and his
cohorts play on conventional "12-tET" instruments to an accuracy of
one cent, you'd prefer to allow 16-cent differences in Blackjack
intervals to be totally disregarded, and triple the errors in the
consonances, via some hypothetical 31-EDO instruments that virtually
no one has built yet?

Also, MIRACLE does not look so special once you've committed to 31-
tET or even 41-tET -- given one of these ETs, other similarly
constructed systems get you the interesting ratios with less
complexity than, and of course the same (inflated relative to 72-tET)
errors as, MIRACLE.

> (I believe that 72-EDO wind instruments are a bit out of
> the question for the foreseeable future, if ever.) Another example
> would be just intonation (e.g., the 43-tone Partch set) mapped onto
> 72 and played on 31 or (for Partch, preferably) 41-EDO
instruments.
> (I have to assume that, should microtonality go mainstream, we
would
> no longer want to suffer the limitations of 12-EDO instruments, and
> whatever notation we might use should take this into account. It
> never hurts to plan ahead!) In these instances the 72-tone
notation
> could be employed as a transcendental notation that could be read
> directly into 31 or 41, making it unnecessary to have separate
parts
> for instruments in each of those systems. This would remove a
> considerable burden from the composer, who might not know in
advance
> what instruments would eventually be used to perform a given
> composition.
>
> The saggital notation accomplishes this by a principle similar to
the
> way decimal numbers are rounded off to whole numbers, as
illustrated
> in Figure 4. In native 31-EDO notation, all symbols are multiples
of
> a single diesis. (As it happens, in all three EDO's – 31, 41, and
> 72 – three 5:4's fall short of an octave by the number of degrees
> representing 33:32 in each of those systems, so the term "diesis"
can
> be used here in a broader sense to refer to either the unidecimal
or
> meantone diesis.) Each left or right-handed symbol in the
> transcendental notation may be regarded as an alteration of the
> neighboring symmetical symbol by a Didymus flag that represents
> 81:80, which is a zero vector in 31-EDO. (The left-handed symbols
> are thus interpreted as a symmetrical symbol plus Didymus flag,
while
> right-handed symbols are interpreted as a symmetrical symbol minus
a
> Didymus flag.) The small arrows in the figure show how
> mental "rounding" to the nearest diesis can be accomplished.
>
> In the 41-EDO native notation some new symbols are required to
> indicate odd-number multiples of a half-diesis (i.e., quarter-sharp
> and quarter-flat) alteration. It should be evident that
> mental "rounding" (to the nearest half-diesis) is even simpler than
> what is required in 31-EDO.
>
> This process of mental rounding works in this fashion with almost
all
> simple 11-limit ratios, and the exceptions are easy to remember:
the
> few ratios having a 7 factor in the numerator and a 5 factor in the
> denominator (or vice versa) – 7/5, 10/7, 21/20, 40/21 – must be
duly
> noted and memorized. As far as I have been able to tell, most
> everything else works like a charm!

Can Gene get into the math behind these observations?

> Until next time, please stay tuned!
>
> --George

My mouth is watering. Your notation system is very beautiful looking!

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

/tuning/topicId_33306.html#33306
>
> For your reference, there is a one-octave diagram of 72-EDO (with
> numbered system degrees and saggital symbols) compared with 19-
limit just intonation at:
>
> /tuning/files/secor/notation/72vsJI.bmp
>

Hello George!

Well, this complete octave mapping was *very* helpful for me in
understanding your system.

And, the more I look at it, the more it appeals to me! There
certainly is much logic in it.

I'm particularly impressed by the fact that if you *have* to go to
the drastic step of eliminating sharps and flats from music
(gotterdammerung!), you *at least* have tidy full arrows (in the
compact saggital notation) to do the job.

Those are something that somebody could *hold on to,* so to speak.

Likewise, the notation for quarter-tones and three-quarter tones is
also very easy to get used to, since these are both related to the
semi-tone arrow and also to the traditional Tartini, in a way.

So... pretty cool.

I guess my only "objection" is the fact that there has to be a set of
12 different symbols for each whole tone. Of course, using sharps
and flats, the Sims notation uses only *three.* (Well, 6, if you
consider the "upside-down" inversions, but I'm not going to... :) )

However, admittedly, Sims (and actually *my* humble self) was not
going for the kind of "universality" that you're trying for in this
system... only trying to write something easily in 72-EDO.

So, bravo!

Joseph Pehrson

P.S. Oh... I guess somebody *did* ask me for my opinion... why, it
was George himself! :)

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

> 225:224 vanishes in paultone.

It's gone back to being paultone?

So how do we notate paultone using this
> system?

That's a little like asking how you would notate it using JI notation--there are many ways to do so, such as your "standard" version. However, there is only one way to go the other way, so one approach would be to start from your standard scale.

My understanding of what George is up to is that (given the commas he presented) there should be a way of going between his symbolism and the mapping [g12, -h2, -g5, v11] where g12 is the variant 12-et (counting numbers of semitones) given by [12,19,28,34,41], g5 is the variant 5-et given by [5,8,12,15,17], h2 is the "standard" 2-et, and
v11 is the 11-adic valuation. If we apply this map to your standard
JI version of decatonic, which is

1--21/20--8/7--6/5--4/3--7/5--3/2--8/5--12/7--9/5

we get

[0, 0, 0, 0]
[1, 0, -1, 0]
[2, 0, 0, 0]
[3, 0, -1, 0]
[5, -1, -2, 0]
[6, -1, -3, 0]
[7, -1, -3, 0]
[8, -1, -3, 0]
[9, -1, -3, 0]
[10, -1, -4, 0]

This ought to be pretty directly translatable to and from saggital; however, it is not unique, and the best way to notate things in this and many other cases would have to be decided on. The first degree is of course 1, the second, of 21/20, is (16/15)(64/63)^(-1), and so one
64/63 down from a semitone, the third is 8/7 = (16/15)(15/14) and so forth; you can see the correspondence with the saggital mapping I gave, I hope.

Will it improve over the notation Alison Monteith is
> currently using?

I have no idea.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > 225:224 vanishes in paultone.
>
> It's gone back to being paultone?

OK, twintone (but what do we call double-diatonic, as in 26-tET,
where 50:49 and 81:80 are tempered out, but 64:63 isn't? That's sort
of a "twinny" system too . . .)

> > So how do we notate paultone using this
> > system?
>
>That's a little like asking how you would notate it using JI
>notation--there are many ways to do so,

I realize that . . . I'm just wondering if saggital makes it
something more immediately meaningful on the 5-line staff than the
notation Alison is currently using, which is based on 22-
tET "Pythagorean", with "Pythagorean" sharps and flats, and "/"
and "\" signifying 2/22 oct. alterations. It doesn't appear likely.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

> My understanding of what George is up to is that (given the commas
>he presented) there should be a way of going between his symbolism
>and the mapping [g12, -h2, -g5, v11] where g12 is the variant 12-et >
(counting numbers of semitones) given by [12,19,28,34,41], g5 is the
>variant 5-et given by [5,8,12,15,17], h2 is the "standard" 2-et, and
> v11 is the 11-adic valuation.

What bothers me about this is that, rather than preserving 12-tET
notation, George has reverted to a 7-naturals basis supplied with a
whole new set of modifiers. So I'm not immediately seeing how the
translation would be made.

There are really two different aspects to this system that one can
take or leave independently. I'll call them semantic and syntactic (or
symbolic).

I haven't fully understood the semantics of it since so far I've only
read the messages thru once quickly, but I've failed to understand how
one decides which notes in any given system get to be the 7 naturals.

WRT the symbolic level, it is indeed ingenious and logical but I
suspect it is doomed because it is bad for rapid sight reading without
error. The ability to recognise the difference between left and right
reflections of the same symbol, and the ability to tell a triple
something from a double at a glance, are late developing in childhood
relative to other symbol recognition tasks. Some Australian aboriginal
languages have no separate word for three or any exact larger numbers.
There just hasn't been the need for these sorts of recognition
abilities except in relatively recent human cultures.

I suggest that this means that such recognition tasks are of a higher
order than others and will always take more time to be processed. This
of course argues against the Tartini/Couper accidentals too. But in
favour of the Sims, and against the ASCII-fied Sims.

So dare I suggest these rules for optimal accidental symbols for sight
reading:

One shouldn't need to be able to tell left from right.
One shouldn't need to "count" past two.

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
George,

I'm sorry I have so little time to spend understanding your notation
at this time. Paul Erlich has asked whether it is equivalent on some
level to the system of interval naming in:

http://dkeenan.com/Music/Miracle/MiracleIntervalNaming.txt

I conjecture that this system can be applied sensibly to all ETs/EDOs
with fewer than 41 notes (and some others with fewer than 72) by
simply rounding their intervals to the nearest 72-EDO interval. Paul
then suggests turning this into a notation for pitches rather than
intervals by using C as the unison.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> George,
>
> I'm sorry I have so little time to spend understanding your
notation
> at this time. Paul Erlich has asked whether it is equivalent on
some
> level to the system of interval naming in:
>
> http://dkeenan.com/Music/Miracle/MiracleIntervalNaming.txt
>
> I conjecture that this system can be applied sensibly to all
ETs/EDOs
> with fewer than 41 notes (and some others with fewer than 72) by
> simply rounding their intervals to the nearest 72-EDO interval.
Paul
> then suggests turning this into a notation for pitches rather than
> intervals by using C as the unison.

A big problem with this is that A-E will not be a consonant fifth.

I wrote,

> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> >
> > > 225:224 vanishes in paultone.
> >
> > It's gone back to being paultone?
>
> OK, twintone
> (but what do we call double-diatonic, as in 26-tET,
> where 50:49 and 81:80 are tempered out, but 64:63 isn't? That's
sort
> of a "twinny" system too . . .)

Why don't we call it "Pajara", since that was the name of the group
that performed music in it at Microthons I and II. I will upload a
table of Pajara key signatures momentarily.

In-Reply-To: <a350bv+57mt@eGroups.com>
paulerlich wrote:

> OK, twintone (but what do we call double-diatonic, as in 26-tET,
> where 50:49 and 81:80 are tempered out, but 64:63 isn't? That's sort
> of a "twinny" system too . . .)

Double negative.

Graham

--- In tuning@y..., graham@m... wrote:
> In-Reply-To: <a350bv+57mt@e...>
> paulerlich wrote:
>
> > OK, twintone (but what do we call double-diatonic, as in 26-tET,
> > where 50:49 and 81:80 are tempered out, but 64:63 isn't? That's
sort
> > of a "twinny" system too . . .)
>
> Double negative.

Excuse me if I'm wrong, but aren't there some tunings which
are "double negative" according to Bosanquet, which however don't
fall into this category?

dkeenanuqnetau schrieb:
...
> One shouldn't need to be able to tell left from right.
> One shouldn't need to "count" past two.

Maybe all those arrows could be just one with different
inclinations (and not going all around to 360°). \, |, and
/, each for a komma, and with and arrowhead indicating
whether it's up or down? - as a natural, so that the others
are offset from the horizontal (and I would opt for an
optimal just/meantone diatonic scale in a system to be
"natural", not the closest imitation of 12-EDO).

But you won't be able to avoid multiple accidentals this
way.

klaus

--- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:

> I would opt for an
> optimal just/meantone diatonic scale in a system to be
> "natural", not the closest imitation of 12-EDO).

What choice would you make in 41-tET?

paulerlich schrieb:
>
> --- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:
>
> > I would opt for an
> > optimal just/meantone diatonic scale in a system to be
> > "natural", not the closest imitation of 12-EDO).
>
> What choice would you make in 41-tET?

Just.

If not, it could be indicated. Different strokes for
different folks, and certainly for different types of
instruments. (Also, transposing instruments have been heard
of before - the naturals could be whatever is under the
primary finger holes.)

klaus

In-Reply-To: <a34cso+d05@eGroups.com>
gdsecor wrote:

> As before, the figures for this presentation are in:
>
> /tuning/files/secor/notation/figures.bmp

>From this, it's clear that the notation is based on 7 nominals. And those
7 notes are a chain of fifths, not a 5-limit kludge.

> As we noted, in 72-EDO Didymus' comma is 1 degree and Archytus' comma
> is two degrees. In 41-EDO, these are each one degree, while in 31-
> EDO they are zero degrees and one degree, respectively. Now look at
> any saggital symbol in the third of figure 3; the number of degrees
> of alteration accomplished by that symbol in a given EDO is found by
> totaling the number of degrees represented by Didymus (left) and
> Archytas (right) flags in the symbol for that EDO. That's how it's
> done, plain and simple! And this principle can also be used to
> notate many other EDO's as well.

That means Archytas' comma is a miracle "quomma" or the difference between
10 secors and an octave. Didymus' comma is the difference between 3
quommas and a secor. I think George is suggesting that these be
differentiated in a 72-equal notation, to make it miracle-unique.

That's like adding "diesis" symbols to meantone, and then differentiating
31-equal equivalences. So C-Bbv is 7:4 but not A#. Hence the one-to-one
mapping between saggital and Couper/Tartini notation with added commas.
Or something like that.

> For example, music written in the
> Blackjack or Canasta scales (~72-EDO) could easily be played on wind
> instruments (with relatively small amounts of pitch-bending)
> specifically built for 31 or 41-EDO, and the result would be far
> better than using 12-EDO instruments with extended (or extraordinary)
> techniques.

Note the bit about pitch bending there. Some commentators seem to have
missed it.

> In the 41-EDO native notation some new symbols are required to
> indicate odd-number multiples of a half-diesis (i.e., quarter-sharp
> and quarter-flat) alteration. It should be evident that
> mental "rounding" (to the nearest half-diesis) is even simpler than
> what is required in 31-EDO.

Oh. So it isn't all done by combining quommas and commas.

Graham

In-Reply-To: <a35ufp+6iia@eGroups.com>
paulerlich wrote:

> Excuse me if I'm wrong, but aren't there some tunings which
> are "double negative" according to Bosanquet, which however don't
> fall into this category?

Hmm. It isn't consistent with 50, which means my catalog is wrong.

Graham

In-Reply-To: <a34iv0+s97s@eGroups.com>
paulerlich wrote:

> Also, MIRACLE does not look so special once you've committed to 31-
> tET or even 41-tET -- given one of these ETs, other similarly
> constructed systems get you the interesting ratios with less
> complexity than, and of course the same (inflated relative to 72-tET)
> errors as, MIRACLE.

That all depends on which intervals you're interested in. I can't improve
on Miracle for the 7-limit plus neutral thirds.

One side-effect of the excellent optimisation is that the most complex
Miracle intervals are also the most out of tune in 31- and 41-equal. If
you use 11-limit chords that are easy to write in Miracle notation, you'll
tend to have more of those in-tune intervals. Such music will therefore
tend to require small pitch bends to play as 11-limit JI on an instrument
built for 31- or 41-equal.

Graham

In-Reply-To: <memo.362880@cix.compulink.co.uk>
I wrote:

> /tuning/files/secor/notation/figures.bmp
>
> From this, it's clear that the notation is based on 7 nominals. And
> those 7 notes are a chain of fifths, not a 5-limit kludge.

Well, I wasn't looking closely enough. The second 31= chord includes a
16:10, or 8:5, which is written with naturals. I was also wrong in saying
there's a one-to-one mapping between Tartini/Couper and saggital. Unless
I meant that you wouldn't use the rest of saggital in 31= or something
like that.

Furthermore, and I'm still dazed by this, the mapping to Tartini/Couper is
*not* Miracle consistent! The half-sharp is written as 3/72 which is not
a special Miracle interval. If you interpret the 31-equal notation as
72-equal, you get 24-equal, not Canasta. So it's really a new set of
symbols for a meantone notation. You can't write a piece in Miracle
temperament and have it automatically work across 31, 41 and 72 with this
notation.

Graham

--- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:
>
>
> paulerlich schrieb:
> >
> > --- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:
> >
> > > I would opt for an
> > > optimal just/meantone diatonic scale in a system to be
> > > "natural", not the closest imitation of 12-EDO).
> >
> > What choice would you make in 41-tET?
>
> Just.

Very ugly. Consider what happens to the key signature when you
modulate up a two fifths, for example.

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> The fundamental principle that sets the saggital notation apart from
> other systems is as follows:
>
> Whereas other systems of notation use symbols to represent
> alteration in pitch in terms of specific numbers of system degrees
or
> specific fractions of a sharp or flat, the saggital symbols
> represents alterations in pitch by using symbols to represent
> alterations *by approximations of certain superparticular ratios*,
> namely Didymus' comma (81:80), Archytas' comma (64:63), and the
> unidecimal diesis (33:32).

This is the approach taken by Paul Rapoport in his 1995 Xenharmonikon
article 'The Notation of Equal Temperaments' (sorry I don't have the
issue number). However he chooses a slightly different set of commas,
all 5-limit, in order of importance:

syntonic comma 80:81 21.5 c
diesis 125:128 41.1 c
diaschisma 2025:2048 19.6 c
schisma 32768:32805 2.0 c

He notes that this choice is not necessarily the best for JI beyond
5-limit and suggests that further study may challenge it.

They both have the syntonic comma, but which of the other commas are
equivalent when cast in Miracle temperament?

--- In tuning@y..., graham@m... wrote:
> In-Reply-To: <memo.362880@c...>
> I wrote:
>
> >
/tuning/files/secor/notation/figures.bmp
> >
> > From this, it's clear that the notation is based on 7 nominals.
And
> > those 7 notes are a chain of fifths, not a 5-limit kludge.
>
> Well, I wasn't looking closely enough. The second 31= chord
includes a
> 16:10, or 8:5, which is written with naturals.

That's because 31 is a meantone!

>You can't write a piece in Miracle
> temperament and have it automatically work across 31, 41 and 72
with this
> notation.

With the Tartini/Couper notation?

I think the term "twintone" would be better used to mean "twin
meantone", (which I assume is what Paul meant) not "twin
super-pythagorean".

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> I think the term "twintone" would be better used to mean "twin
> meantone", (which I assume is what Paul meant) not "twin
> super-pythagorean".

Well, unfortunately "twin meantone" could also be interpreted as
implying the Vicentino Enharmonic as Graham Breed has analyzed it.
What I was referring to, and I assume you are by "twin meantone", is
the tuning with fifths near 694 cents that you put your finger on
here:

http://www.uq.net.au/~zzdkeena/Music/2ChainOfFifthsTunings.htm

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

> > http://dkeenan.com/Music/Miracle/MiracleIntervalNaming.txt
> >
> > I conjecture that this system can be applied sensibly to all
> ETs/EDOs
> > with fewer than 41 notes (and some others with fewer than 72) by
> > simply rounding their intervals to the nearest 72-EDO interval.
> Paul
> > then suggests turning this into a notation for pitches rather than
> > intervals by using C as the unison.
>
> A big problem with this is that A-E will not be a consonant fifth.

Yeah George, forget it. While it's fine for naming intervals, it will
never work for pitches because its "naturals" do not form a chain of
fifths. I think I knew that before and just forgot momentarily. Thanks
Paul.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > I think the term "twintone" would be better used to mean "twin
> > meantone", (which I assume is what Paul meant) not "twin
> > super-pythagorean".
>
> Well, unfortunately "twin meantone" could also be interpreted as
> implying the Vicentino Enharmonic as Graham Breed has analyzed it.

True!

> What I was referring to, and I assume you are by "twin meantone", is
> the tuning with fifths near 694 cents that you put your finger on
> here:
>
> http://www.uq.net.au/~zzdkeena/Music/2ChainOfFifthsTunings.htm

Yes.

Ok so there are _two_ other tuning systems that "twintone" could apply
to. I'm mainly just arguing _against_ its use for Paul Erlich's
7-limit linear temperament where two perfect fourths give the 4:7 and
a half-octave gives the 5:7.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
http://dkeenan.com/Music/Miracle/MiracleIntervalNaming.txt

> Yeah George, forget it. While it's fine for naming intervals, it
will
> never work for pitches because its "naturals" do not form a chain of
> fifths. I think I knew that before and just forgot momentarily.
Thanks
> Paul.

The "naturals" of the interval naming scheme can be considered to be
the neutral and perfect intervals. Considered as pitches,
these actually form an open chain of 3-secor neutral thirds (i.e. a
Mohajira scale) as follows:

N2 P4 N6 P8/P1 N3 P5 N7

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> http://dkeenan.com/Music/Miracle/MiracleIntervalNaming.txt
>
> > Yeah George, forget it. While it's fine for naming intervals, it
> will
> > never work for pitches because its "naturals" do not form a chain
of
> > fifths. I think I knew that before and just forgot momentarily.
> Thanks
> > Paul.
>
> The "naturals" of the interval naming scheme can be considered to
be
> the neutral and perfect intervals. Considered as pitches,
> these actually form an open chain of 3-secor neutral thirds (i.e. a
> Mohajira scale) as follows:
>
> N2 P4 N6 P8/P1 N3 P5 N7

This argues for an entirely new system of 7-nominal notation for
MIRACLE!

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

/tuning/topicId_33306.html#33346

> I suggest that this means that such recognition tasks are of a
higher order than others and will always take more time to be
processed. This of course argues against the Tartini/Couper
accidentals too. But in favour of the Sims, and against the ASCII-
fied Sims.
>

****Yes, I believe without a doubt that the *original* Sims is easier
to sight read than the Ascii. Although it's a bit bizarre, the large
shapes and the up and down indications are fast to intuit. The
*size* also makes a difference, with the "bigger" symbols being the
larger intervals... That helps in sight reading, too...

JP

Hello everyone! As before, I will be addressing a bunch of questions
& comments in one message.

[Paul Erlich (#3337):]
<< Subject: What does saggital mean? >>

"Saggita" is Latin for "arrow", and the notation is so named because
each symbol is a arrow of some sort.

> [GS:]
> This will be included in the 17-tone article that will be appearing
> in the next issue of Xenharmonikon (#18),

[Paul Erlich (#33311):]
<< No offense, George, but John Chalmers has informed me that he has
not yet accepted your or any other articles for publication in XH 18.
Out of respect for John, perhaps you should put this a bit
differently? >>

I stand corrected. Although I have good reason to believe from my
private communication with John Chalmers that I would have no problem
getting my article into XH 18, he hasn't accepted it yet because I
haven't sent it yet, since I will be sending another article along
with it that hasn't yet been completed. So I should change "will be
appearing" to "will probably be appearing". (My apologies, John, if
you happen to see this.)

> [GS:]
> Whereas other systems of notation use symbols to represent
> alteration in pitch in terms of specific numbers of system degrees
or
> specific fractions of a sharp or flat, the saggital symbols
> represents alterations in pitch by using symbols to represent
> alterations *by approximations of certain superparticular ratios*,
> namely Didymus' comma (81:80), Archytas' comma (64:63), and the
> unidecimal diesis (33:32).

[PE:]
<< And the naturals represent Pythagorean? Please remind me. >>

The naturals represent tones in a series of fifths of indefinite
size, which means that they can be used to represent whatever you
want them to (within reason), just as our conventional notation can
represent Pythagorean, meantone, or 12-EDO. (But the tuning-math
group, having a different purpose in mind, may be looking for a
different answer to this question, which I will address later in this
message.)

There also seems to have been some confusion regarding how many
nominals are in this notation: 7 or 12 (there are 7), and whether a
chain of fifths forms a 12-tone circle or a series. Again, as with
conventional notation, you can have it as either a circle (of 12 or
some other number) or a series, just as conventional notation can be
used for a circle of 12 (equal or unequal), a circle of 19 (equal or
unequal), or a meantone or Pythagorean series. So you can have 12-
tone enharmonic equivalence if you want it, or you can differentiate
the (saggital) sharps and flats if you want that.

[PE:]
<< My mouth is watering. Your notation system is very beautiful
looking! >>

Thank you; I appreciate that.

[Joseph Pehrson (#33312):]
<< --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

/tuning/topicId_33306.html#33306
>
> For your reference, there is a one-octave diagram of 72-EDO (with
> numbered system degrees and saggital symbols) compared with 19-
limit just intonation at:
>
> /tuning/files/secor/notation/72vsJI.bmp
>

Hello George!

Well, this complete octave mapping was *very* helpful for me in
understanding your system.

And, the more I look at it, the more it appeals to me! There
certainly is much logic in it.

I'm particularly impressed by the fact that if you *have* to go to
the drastic step of eliminating sharps and flats from music
(gotterdammerung!), you *at least* have tidy full arrows (in the
compact saggital notation) to do the job.

Those are something that somebody could *hold on to,* so to speak.

Likewise, the notation for quarter-tones and three-quarter tones is
also very easy to get used to, since these are both related to the
semi-tone arrow and also to the traditional Tartini, in a way.

So... pretty cool.

I guess my only "objection" is the fact that there has to be a set of
12 different symbols for each whole tone. Of course, using sharps
and flats, the Sims notation uses only *three.* (Well, 6, if you
consider the "upside-down" inversions, but I'm not going to... :) )

However, admittedly, Sims (and actually *my* humble self) was not
going for the kind of "universality" that you're trying for in this
system... only trying to write something easily in 72-EDO.

So, bravo!

Joseph Pehrson >>

Thanks for your kind words, Joseph! The symbols have a lot of
information packed into them that can be interpreted in several
different ways, and they can be used for a lot of different tunings,
so it will take some time to get comfortable with them. Remember
that I said that there are some things about this notation that took
me months to see, even though I'm the one who came up with it.
That's why I'm more inclined to use the word "discovered"
than "devised" for what I did. And I expect that I will be learning
even more about it in the next several months, inasmuch as Margo and
I have recently been giving it a test-drive over some less-frequented
EDO's to see how well it handles off the beaten path. (So far I
haven't been disappointed.)

As you noted, there are more symbols to learn than with the Sims
notation, so there is no question in my mind that you are going to
get quicker results with that than with the saggital notation. On
the other hand, since a multi-system notation is capable of so much
more than one for a single system, the saggital notation would, in
the long run, provide a greater return on investment of one's time
and effort.

By the way, in case you want to print out that full-octave diagram of
72-EDO on a single 8.5 x 11 page, go to the following and print it.
(It looked awful on my screen but printed very nicely when I tried
it.)

/tuning/files/secor/notation/72vsJI.doc

[Re: Saggital notation as a "notation" (several postings by Gene
Ward Smith beginning with #33307, including replies by Paul Erlich)]

Just one comment: I did notice a 22-tone system mentioned (#33320).
This is one octave division with which I am very familiar
(particularly the EDO), and I intend to cover the application of the
saggital notation to 22 in some detail in the next part of the
presentation.

[Re: 13-limit Saggital? (several postings by Gene Ward Smith
beginning with #33330, including replies by Paul Erlich)]

Up to this point I have been too busy with other things to be seduced
into a love affair with the math, but it looks as if you're doing
your best to lead me into temptation, so let me throw my two cents in
here, for what it's worth.

For the purpose of notating rational intervals, the saggital notation
begins with 7 nominals in a Pythagorean (or 3-limit) series, i.e.,
the naturals from F to B. (This excludes 12-tone enharmonic
equivalence.) Introduction of symbols for 81/80, 64/63, and 33/32
gives you additional tones related by primes 5, 7, and 11,
respectively. (I have not delved deeply enough into all of this to
see why you need both 25/24 and 16/15 as defining intervals, but I
assume you know what you're doing and will just have to do more
reading to see why.)

In going to the 13 limit you have added 39/40 (40/39?). Considering
that each of the primes (5, 7, and 11) were introduced with ratios
containing no other primes above 3, would it not be more advisable
(or "cleaner") to use 27/26 or 13/12 instead?

Personally, I consider ratios of 13 indispensable and would be very
interested to see where this leads. My method for successfully
incorporating 13 into a tonal system is probably a little different
from what most others have tried, but I have found that the saggital
notation (as I have presented it) works extremely well for my
purposes. (My approach to 13 will need to be presented in a couple
of *Buried Treasure* articles.)

[Dave Keenan (#33346):]

<< There are really two different aspects to this system that one can
take or leave independently. I'll call them semantic and syntactic
(or symbolic).

I haven't fully understood the semantics of it since so far I've only
read the messages thru once quickly, but I've failed to understand
how one decides which notes in any given system get to be the 7
naturals. >>

Unless I don't understand the question, or there is something here
that I'm missing, I thought that it would be obvious that they were
the tones in a series of fifths from F to B, just as we presently
have. If you have a tonal system in which reasonable approximations
of perfect fifths don't exist, e.g., 8, 11, 13, or 23-EDO, then you
can notate them as subsets of 72, 22, 26, and 46-EDO, respectively.

<< WRT the symbolic level, it is indeed ingenious and logical but I
suspect it is doomed because it is bad for rapid sight reading
without error. The ability to recognise the difference between left
and right reflections of the same symbol, and the ability to tell a
triple something from a double at a glance, are late developing in
childhood relative to other symbol recognition tasks. Some Australian
aboriginal languages have no separate word for three or any exact
larger numbers. There just hasn't been the need for these sorts of
recognition abilities except in relatively recent human cultures. >>

Would rapid reading of something that's dackwarbs in this sentence
cause you to miss it? I realize that mirrored notation is not
exactly the same thing as mirrored letters in the alphabet, but my
point is that every symbol has a context, and just as it is fairly
easy for most of us to spot a wrong note in tonal music, I don't
think that it's going to be as difficult as you suggest. (I'm sure
that many of us have heard music that is so unintelligible that it
wouldn't have mattered much whether anyone played any "wrong" notes
or not, but that's another issue entirely.)

As for exposing children to microtonal notation, I don't think you're
going to expect children to handle more than 31 tones per octave (if
even that many). That being the case, I can't think of a single
example of an EDO of less than 32 tones per octave that uses any
saggital symbols that are mirrors of each other. While a couple of
them do use some left-handed and right-handed symbols (22 and 29-
EDO), none of those in the applicable subset mirror each other, so I
think this is not really an issue.

Likewise, I think that three of something is pretty easy to
distinguish from two, but when it comes to four vs. three, I found
that the difference is not perceived as easily, which is why I chose
an X for the double sharp and double flat symbols. So I am in
partial agreement with you on this point.

<< I suggest that this means that such recognition tasks are of a
higher order than others and will always take more time to be
processed. This of course argues against the Tartini/Couper
accidentals too. But in favour of the Sims, and against the ASCII-
fied Sims.

So dare I suggest these rules for optimal accidental symbols for
sight reading:

One shouldn't need to be able to tell left from right.
One shouldn't need to "count" past two. >>

All seriousness aside, you're free to suggest whatever you want, but
then dare I suggest that we ban triple meter and get rid of all the
instruments that would require music students to move both hands (or
fingers thereof) at once in different ways? (This reminds me of the
joke about the church organist: His playing displays a true
ecclesiastical spirit -- his left hand knows not what his right hand
is doing.) We can laugh at things like this, but please, let's not
dismiss something as versatile as the saggital notation without at
least a few of us trying it and giving it a fair chance, so we can
evaluate any potential problems.

As evidence that I did some homework before showing up in this forum,
I think this would be a good time to look at one of Dave Keenan's
earlier messages (#24012 of May 30, 2001), which contains some
relevant comments by Ted Mook:

<< -- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
Ted Mook has given me permission to post the following edited email
exchange.

------------
I asked Ted:
------------
With regard to your page
https://www.mindeartheart.org/micro.html

I was wondering why the Sims system is preferred over the
Tartini/Fokker system in the case of 72-equal. The Tartini Fokker
system requires only 4 new symbols rather than 6 (the sharp-and-a-
half and flat-and-a-half being transparent shorthand and not really
necessary).

See http://www.xs4all.nl/~huygensf/doc/fokkerpb.html for examples of
the symbols.

I prefer the Tartini 1/4-tone symbols to Sims' because the Tartini
symbols _look_ like half of a sharp symbol and some kind of a flat.
Can you enlighten me?

-------------------
Ted kindly replied:
-------------------
As a performer, I worked with Ratios (Partch), Ben Johnston's
lattice, and various combinations of hooks and arrows. In my
experience, Ezra's notation was the most direct and accurate
extension of "normal" notation based on the ubiquitous ET tuning we
all learn as string players in school.

None of the symbols are combined with anything other than sharps or
flats, they behave like normal accidentals, that is, they hold
through a measure unless cancelled or replaced.

There are no chains of accidentals, which is the reason for the 1/6
tone, otherwise, you could have some construction that would be 1/4 +
1/12 sharp something and double dotted to boot...it would take three
feet of paper to describe the pitch and its duration.

There is the issue of high speed reading: The Tartini single stroke
sharp, double stroke sharp and triple stroke sharp, at music stand
distance, in crummy light become indistinguishable without careful
scrutiny.
-----------------------

Regards,
-- Dave Keenan
Brisbane, Australia >>

Given that, in some instances, two symbols may need to be used to
modify a natural note, this explanation for preferring the Sims
notation to the Tartini notation with add-ons is well justified. But
given that the saggital notation never uses two symbols to modify a
natural, the observation about chains of accidentals has no relevance
here.

Considering the appearance of the saggital symbols, the point about
the symbols being too small for music stand distance and crummy
lighting is a pertinent one, but let's not throw out the baby with
the bath water. A very simple solution with the saggital symbols is
to make them larger in the individual parts so that they can be
easily read from a distance. (On the score or for keyboard music
they can be the ordinary size.) This may look a little ugly, but the
Sims notation isn't about to win any beauty contests either, and the
one-note-at-a-time (or at most two) nature of an orchestral part
isn't going to create any real clutter or confusion.

Now if the goal is to get results from the players as quickly and
easily as possible, then an approach that minimizes the learning
curve (with the fewest new symbols) is to be preferred, and the Sims
notation wins, hands down. I would consider this an example of
a "quick and dirty fix," or something that requires the least amount
of effort to do the job. In connection with this, also see Joseph
Pehrson's posting (#27626) regarding an article on the Sims notation
from Xenharmonikon XI, particularly the two paragraphs near the end
which quote Sims directly:

/tuning/topicId_27626.html#27626

If, on the other hand, we want the notation to be useful for as many
tonal systems as possible (and also allow for the possibility that
new instruments may be used), then we are talking about an entirely
different philosophy of microtonality, one which is going to take
more effort to learn, but which will pay more dividends in the end.
To get the right answers, you need to ask the right questions;
likewise, to select the proper notation, you need to identify your
objective. This, I think, is the real issue, and I believe that many
of us who are dedicated to the approach to microtonality that finds
new consonant harmonies by using higher prime numbers will affirm
that one of the great strengths of our movement has been our
inclusiveness and diversity.

Am I making any sense?

[Dave Keenan (#33362):]

<< --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
George,

I'm sorry I have so little time to spend understanding your notation
at this time. Paul Erlich has asked whether it is equivalent on some
level to the system of interval naming in:

http://dkeenan.com/Music/Miracle/MiracleIntervalNaming.txt

I conjecture that this system can be applied sensibly to all ETs/EDOs
with fewer than 41 notes (and some others with fewer than 72) by
simply rounding their intervals to the nearest 72-EDO interval. Paul
then suggests turning this into a notation for pitches rather than
intervals by using C as the unison. >>

Dave,

It appears that this system of interval naming, inasmuch as it is
related to the Miracle tuning, harmonizes very nicely with the
saggital notation.

Simple rounding of intervals of a system (e.g., 22-EDO) to the
nearest 72-EDO interval will not result in a consistent notation for
that system, so you can't turn this into a notation for pitches.
Insofar as the fifths of 22-EDO are much wider than those of 72, the
difference between F-sharp and G-flat (saggital or otherwise) in 22
is over 100 cents, whereas in 72 they are the same pitch (being 12-
tone enharmonic equivalents). However, if you wish to notate a
tritone (i.e., three whole tones) above C in both 22-EDO and 72-EDO,
it will be F/||\ (a saggital F-sharp). And if you wish to notate the
tone 5:4 above a D (which is, in turn, 9:8 above C), in both 22-EDO
and 72-EDO it will be F||\ (a saggital F-sharp minus a Didymus
flag). This works, even though the difference between the two F-
sharps is ~16.67 cents in 72 and ~54.55 cents in 22. So you can
notate some simple things with 5-limit intervals in 22-EDO and have
them come out the way you would expect them to. (I'm really getting
ahead of myself, because this is something that should be coming out
in the next installment of the presentation. Just consider this a
sneak preview!)

So little time -- so many questions and comments. I'm going to have
to stop for now, but I will be reading the rest. (Keep those cards
and letters coming, folks!)

--George

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

> [PE:]
> << And the naturals represent Pythagorean? Please remind me. >>
>
> The naturals represent tones in a series of fifths of indefinite
> size,

Well, that's really what I meant by "Pythagorean" -- just as your
accidentals won't represent the exact ratios of 64:63, 33:32, and
whatnot, yet you named them after Greek JI thinkers, the fifths won't
be exact 3:2 ratios -- but I thought the analogy would have been
clear enough.
>
> Unless I don't understand the question, or there is something here
> that I'm missing, I thought that it would be obvious that they were
> the tones in a series of fifths from F to B, just as we presently
> have.

I don't think you said so until this message. Ben Johnston defines
them differently, so it certainly isn't "obvious".

> saggital

Why do you spell it this way, and not "sagittal"?

>
>
> [Dave Keenan (#33362):]
>
> << --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> George,
>
> I'm sorry I have so little time to spend understanding your
notation
> at this time. Paul Erlich has asked whether it is equivalent on
some
> level to the system of interval naming in:
>
> http://dkeenan.com/Music/Miracle/MiracleIntervalNaming.txt
>
> I conjecture that this system can be applied sensibly to all
ETs/EDOs
> with fewer than 41 notes (and some others with fewer than 72) by
> simply rounding their intervals to the nearest 72-EDO interval.
Paul
> then suggests turning this into a notation for pitches rather than
> intervals by using C as the unison. >>

Note that Dave later retracted this conjecture. It doesn't work.

> Dave,
>
> It appears that this system of interval naming, inasmuch as it is
> related to the Miracle tuning, harmonizes very nicely with the
> saggital notation.
>
> Simple rounding of intervals of a system (e.g., 22-EDO) to the
> nearest 72-EDO interval

No one ever suggested such a thing.

Looking forward to more!

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > Simple rounding of intervals of a system (e.g., 22-EDO) to the
> > nearest 72-EDO interval
>
> No one ever suggested such a thing.

I think I did. But this was for the purpose of _naming_ the intervals,
not for playing them. George is of course right that it is useless for
naming pitches. But Paul, do you see something wrong with it for
naming intervals, or was it just that you thought we were talking
about actually retuning the notes?

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > > Simple rounding of intervals of a system (e.g., 22-EDO) to the
> > > nearest 72-EDO interval
> >
> > No one ever suggested such a thing.
>
> I think I did. But this was for the purpose of _naming_ the
intervals,
> not for playing them. George is of course right that it is useless
for
> naming pitches. But Paul, do you see something wrong with it for
> naming intervals,

Well, I guess it's nothing new that we haven't talked about already.
I just like the elegance of naming the intervals based on a sub-block
of the ET in question. For example, in 12-tET, 19-tET, 31-tET, etc.,
the two 5-limit commas defining the ET are 81:80 and something more
complex, call it x. Then, preserving the 81:80 but replacing x with
the more directly-faced (in composing) 25:24, one gets a 7-tone PB,
which forms the set of basic, unaltered categories for the naming
scheme. I'm sure you'd agree that, similarly, a Decimal-based
interval naming scheme would come out more elegant for MIRACLE, and
perhaps you'd agree that it could lead a composer to more
indigenously MIRACLEian ways of thinking and composing, since all but
one of the unison vectors are already dealt with in the notation
(yes, I'm arguing for Decimal notation here, and of course Decimal-
based interval naming to go along with it).

Unfortunately, we have the consideration that most _performers_ will
have no idea what such notation systems represent. With 72-tET we
have a unique opportunity to introduce MIRACLE-based, and other
microtonal, music in terms 12-tET players can understand. You teach
the quarter-tone inflections along with ratios of 11, the sixth-tone
inflections along with ratios of 7, and the twelfth-tone inflections
along with ratios of 5 (in this order or the reverse), and you're
done. You've ensured good harmonic intonation _and_ introduced very
little that is unfamiliar from the point of view of either notation
or interval-naming.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

/tuning/topicId_33306.html#33488
>
> Unfortunately, we have the consideration that most _performers_
will
> have no idea what such notation systems represent. With 72-tET we
> have a unique opportunity to introduce MIRACLE-based, and other
> microtonal, music in terms 12-tET players can understand. You teach
> the quarter-tone inflections along with ratios of 11, the sixth-
tone
> inflections along with ratios of 7, and the twelfth-tone
inflections
> along with ratios of 5 (in this order or the reverse), and you're
> done. You've ensured good harmonic intonation _and_ introduced very
> little that is unfamiliar from the point of view of either notation
> or interval-naming.

****Hear, hear!

It's a logical and actually *do-able* (in the "real world")
possibility!

JP

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > [PE:]
> > << And the naturals represent Pythagorean? Please remind me. >>
> >
> > The naturals represent tones in a series of fifths of indefinite
> > size,
>
> Well, that's really what I meant by "Pythagorean" -- just as your
> accidentals won't represent the exact ratios of 64:63, 33:32, and
> whatnot, yet you named them after Greek JI thinkers, the fifths
won't
> be exact 3:2 ratios -- but I thought the analogy would have been
> clear enough.
> >
> > Unless I don't understand the question, or there is something
here
> > that I'm missing, I thought that it would be obvious that they
were
> > the tones in a series of fifths from F to B, just as we presently
> > have.
>
> I don't think you said so until this message. Ben Johnston defines
> them differently, so it certainly isn't "obvious".

Hi Paul!

I have been out of touch with what has been going on in microtonality
for quite a while and still have a lot of catching up to do, so I had
(and still have) no idea what Ben Johnston has been up to for the
past fifteen years. (So you can see that there is an element of
truth in that observation I made in my self-introduction that all of
you probably considered me "as good as dead".)

Working in isolation can have both advantages and disadvantages. For
me it was not a choice for the first twelve years (until 1975), and
even after that it seemed that everyone pretty much preferred to do
their own thing, so that has been very much my style until just
recently.

The rapid (and sometimes frantic) pace of life on the tuning list is
a very new experience for me, as I frequently observe comments shot
from the hip and opinions blowing every which way in the changing
wind, whereas I try to think things through before saying anything.
I may need to make some adjustments, so I ask for your patience and
understanding.

>
> > saggital
>
> Why do you spell it this way, and not "sagittal"?
>

For the first time in my life I am having some embarrassing problems
with spelling, first with your last name, and now with the
idiosyncracies of single vs. double consonants in Latin words. (I
never took Latin, but knew that the Latin word for "arrow"
is "sagitta" -- spelled correctly this time -- because I am also
interested in astronomy, and sagitta the arrow happens to be one of
the 88 constellations, but I never had occasion to spell it before.)
It seems that the people who gave us Latin should have followed a
modification of one of Dave Keenan's rules:

----- One shouldn't need to "count" past one.

One unfortunate consequence of all of this is that a search of the
postings for the word "sagittal" is going to miss everything that has
been posted up till now, so may I take this opportunity to advise
anyone who will be doing this also to search for "saggital".

I happened to catch the misspelling last night, when I was looking at
an astronomy book (I didn't read your message until this morning),
and, as you will see, I've already corrected the spelling
of "sagittal" in the file (which kept me busy for a while last night):

/tuning/files/secor/notation/Figures.bmp

At this point I think it would be fitting to give more details about
how I arrived at the term.

Back in August, while I was looking through back issues of
Xenharmonikon for examples of notation (for ideas for my new
notation), I found a comment by Ivor Darreg in which he referred to a
sharp symbol with an arrow affixed to the upper left as "sagittarian
notation". (Xenharmonic Bulletin No. 9, Oct. 1978, "The Calmer Mood.
31 Tones/Octave", p. 15; bound into Xenharmonikon 7&8; this, by the
way, is the last issue of XH that I have; the Sims notation appeared
later.) I puzzled over the meaning for a moment and then concluded
that his term would have been more suitable for an archer than for
arrows, which made me think of the term "sagittal". Once I developed
the arrow-like expanded symbols, the term fit, so I used it.

And that (pause) is the rest of the story.

>
> >
> > [Dave Keenan (#33362):]
> >
> > << --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> > George,
> >
> > I'm sorry I have so little time to spend understanding your
> notation
> > at this time. Paul Erlich has asked whether it is equivalent on
> some
> > level to the system of interval naming in:
> >
> > http://dkeenan.com/Music/Miracle/MiracleIntervalNaming.txt
> >
> > I conjecture that this system can be applied sensibly to all
> ETs/EDOs
> > with fewer than 41 notes (and some others with fewer than 72) by
> > simply rounding their intervals to the nearest 72-EDO interval.
> Paul
> > then suggests turning this into a notation for pitches rather
than
> > intervals by using C as the unison. >>
>
> Note that Dave later retracted this conjecture. It doesn't work.
>
> > Dave,
> >
> > It appears that this system of interval naming, inasmuch as it is
> > related to the Miracle tuning, harmonizes very nicely with the
> > saggital notation.
> >
> > Simple rounding of intervals of a system (e.g., 22-EDO) to the
> > nearest 72-EDO interval
>
> No one ever suggested such a thing.
>

I'm just having trouble keeping up with the rapid pace of the
postings. I responded to your message immediately, inasmuch as I
thought it important to clear up the problem about the misspelling.
(And I haven't yet read anything subsequent to the message to which I
am replying, so bear with me.)

> Looking forward to more!

Okay! I don't know how soon the next installment of the presentation
will come out. There seems to be some confusion, miscommunication,
and/or misinformation about just what this notation can and can't do
or is and isn't supposed to do, and sometimes it seems that the more
that is said, the worse it gets. Trying to keep track of all this
has gotten difficult, so I think I may have to let the dust settle
before I say any more (which will give me time to work on the
presentation).

--George

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

> The rapid (and sometimes frantic) pace of life on the tuning list is
> a very new experience for me, as I frequently observe comments shot
> from the hip and opinions blowing every which way in the changing
> wind, whereas I try to think things through before saying anything.
> I may need to make some adjustments, so I ask for your patience and
> understanding.

My experience before and after coming to this list was exactly the same, so I can relate to this. I'm a shoot-from-the-hip person myself, and I sometimes miss. I don't see any reason you need to adopt that style if it doesn't suit you.

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> >
> > > [PE:]
> > > << And the naturals represent Pythagorean? Please remind me. >>
> > >
> > > The naturals represent tones in a series of fifths of
indefinite
> > > size,
> >
> > Well, that's really what I meant by "Pythagorean" -- just as your
> > accidentals won't represent the exact ratios of 64:63, 33:32, and
> > whatnot, yet you named them after Greek JI thinkers, the fifths
> won't
> > be exact 3:2 ratios -- but I thought the analogy would have been
> > clear enough.
> > >
> > > Unless I don't understand the question, or there is something
> here
> > > that I'm missing, I thought that it would be obvious that they
> were
> > > the tones in a series of fifths from F to B, just as we
presently
> > > have.
> >
> > I don't think you said so until this message. Ben Johnston
defines
> > them differently, so it certainly isn't "obvious".
>
> Hi Paul!
>
> I have been out of touch with what has been going on in
microtonality
> for quite a while and still have a lot of catching up to do, so I
had
> (and still have) no idea what Ben Johnston has been up to for the
> past fifteen years.

This is probably for the best. Ben Johnston does not name his 7
naturals according to a Pythagorean scheme, and thus his system is
*extremely* confusing. BTW, I'm pretty sure he's been doing this for
far more than fifteen years.

> I may need to make some adjustments, so I ask for your patience and
> understanding.

I pledge it to you!

> Trying to keep track of all this
> has gotten difficult, so I think I may have to let the dust settle
> before I say any more (which will give me time to work on the
> presentation).

Sounds like a great plan!

Hello, everyone. Lately I've been re-examining Paul Erlich's pajara
scale system, and I've become curious about the origin of the
name "pajara". A search turned up the following message:
/tuning/topicId_33306.html#33368
to which I'm replying now, because there's something in it that I
don't follow:

--- In tuning@yahoogroups.com, "paulerlich" <paul@...> wrote:
>
> I wrote,
>
> > --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > > --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > >
> > > > 225:224 vanishes in paultone.
> > >
> > > It's gone back to being paultone?
> >
> > OK, twintone
> > (but what do we call double-diatonic, as in 26-tET,
> > where 50:49 and 81:80 are tempered out, but 64:63 isn't? That's
sort
> > of a "twinny" system too . . .)
>
> Why don't we call it "Pajara", since that was the name of the group
> that performed music in it at Microthons I and II. I will upload a
> table of Pajara key signatures momentarily.

This is very confusing. As I understand it:

1) In pajara, 50:49 and 64:63 are tempered out, but 81:80 isn't.

2) In 26-ET, 50:49 and 81:80 are indeed tempered out, and 64:63
isn't; however pajara isn't possible in 26-ET.

So this doesn't make any sense -- unless the term "pajara" was at
that time used for a different scale system. However, that's
definitely not the case, because Paul posted another message directly
after that one containing a link to a file with the pajara key
signatures, showing a keyboard with 22 tones/octave:
/tuning/topicId_12590.html#33369

The only way I can make any sense out of the above is to observe that
the three lines in parentheses must be a *parenthetical question*
that Paul completely ignored in his reply -- and, if so, then he must
have been replying to Gene's answer to Paul's previous question.

Assuming that I've figured this out, then here are my next questions -
- the ones I really wanted to ask:

1) Am I correct in concluding that "Pajara" was the name of the group
in which Paul performed his 22-tone compositions (of which a couple
of mp3 files exist as examples)?

2) How did the group get its name?

I would appreciate it if someone could supply this information, just
so I have the story straight.

Thanks.

--George

George D. Secor wrote:
> This is very confusing. As I understand it:
> > 1) In pajara, 50:49 and 64:63 are tempered out, but 81:80 isn't.
> > 2) In 26-ET, 50:49 and 81:80 are indeed tempered out, and 64:63 > isn't; however pajara isn't possible in 26-ET.

"Double diatonic" was the term being used at the time for what later became known as "injera" (named after a kind of Ethiopian bread). The comment about 26-ET was referring to that, not to pajara:

>>> (but what do we call double-diatonic, as in 26-tET,
>>> where 50:49 and 81:80 are tempered out, but 64:63 isn't? That's
> sort
>>> of a "twinny" system too . . .)

"Paultone" and "twintone" were alternative names being proposed for pajara, but "twintone" also would have made sense as a name for injera (which shares a comma with meantone).

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> This is very confusing. As I understand it:
>
> 1) In pajara, 50:49 and 64:63 are tempered out, but 81:80 isn't.

Correct. Pajara is the 12&22 system, <<2 -4 -4 -11 - 12 2||.

> 2) In 26-ET, 50:49 and 81:80 are indeed tempered out, and 64:63
> isn't; however pajara isn't possible in 26-ET.

This is injera; the 12&26 system <<2 8 8 8 7 -4||.