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*BT* Notation - Part 1

🔗gdsecor <gdsecor@yahoo.com>

1/21/2002 10:35:51 AM

*BURIED TREASURE*
"Notation - Part 1"
From: George Secor
January 22, 2002

Patience comes to those who wait for it, and I thank you all for your
patience. Here at last is Part One of my saggital notation
presentation, and I hope you agree that it was worth waiting for.

In this first part of the presentation I will illustrate the process
by which I arrived at the 72-EDO form of the notation. Subsequent
installments will address its multi-system application, both in
native (or EDO-specific) and transcendental (or trans-system generic)
forms, leaving most of the questions and comments that have been made
regarding the more controversial aspect of the subject for the final
installment.

It is perhaps a bit of a stretch to call this buried treasure,
inasmuch as this is so new that there has barely been enough time to
get any "dust" on the paperwork (most of which is virtual, in the
form of computer files; hmmm, I do seem to notice some dust on the
monitor screen). I reasoned that the presentation would be more
widely read, especially by future members of the Tuning List, if I
put it in my Buried Treasure column.

At the beginning of the year I made a new year's resolution to
complete the development and testing of my notation, and I am sharing
it with you to elicit your comments and suggestions to make this the
very best notation possible, one that will come closest to "doing it
all" and doing it well.

So as not to keep you in further suspension, let the resolution begin!

*A Challenge I Couldn't Resist!*

Please note: The figures for this presentation are in:

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

I have always believed that the best notation is that which is
simplest. A good example of this is the Tartini fractional sharps
(shown in the right half of the top row of Figure 1), which are so
clear that they require virtually no explanation. Although these
were employed by both Ivan Vyshnegradsky (for 24-EDO) and Adriaan
Fokker (for 31-EDO), it is rather surprising that quartertone
composers never adopted these as a standard notation. Instead, they
often preferred to place arrows in front of notes, which, in
combination with sharps or flats, tend to clutter a musical
manuscript, especially when chords are notated on a single staff.

Existing methods of notating 72-EDO have also used this approach, and
the diversity of symbols used somewhat arbitrarily (and not always
logically) to designate three different amounts of alteration in
pitch strikes me as a conglomeration of add-ons or do-dads intended
to supplement traditional notational practice. However, I did not
see these 72-EDO notations (including the one devised by Ezra Sims)
until several months after I had produced the initial (expanded)
version of my saggital notation, so they had absolutely no influence
in its development. To be completely honest, once I did see them, I
was appalled. I later learned that the symbols that were proposed by
those on the Alternate Tuning List were ASCII versions for
theoretical use only, not practical notation intended for use on an
actual musical manuscript, and the goal was largely to emulate the
Sims notation. Inasmuch as my goal was to arrive at the very best
notation possible, it is understandable that, immediately upon seeing
it, I found that I had absolutely no desire to emulate the Sims
notation, and it should be evident by the end of this part of the
presentation that any similarity between the Sims and the saggital
symbols is purely coincidental.

It is not an easy matter to arrive at a simple notation that would
require only a single symbol to modify the pitch of the seven
naturals notes on the staff for 72-EDO. In the first place, 24
symbols would be needed for a complete range of alteration by a whole
tone, both upward and downward. In order for this approach to be
successful, the new symbols would need to have an intuitiveness that
would enable them to be quickly and easily understood. They would
also need to be similar enough that they could be easily remembered,
yet different enough that there would be no difficulty in
distinguishing them from one another. This was a challenge that I
couldn't resist!

The solution did not come quickly, however, as it soon became evident
that this is one situation where the desired result would not be
achieved without investing a considerable amount of time and effort.
I spent hours putting all sorts of symbols, both old and new, on a
piece of paper, seeking as many ideas as possible from which to
choose. In the end I found that the best ideas were ones that had
already been successfully used in the past, and my saggital notation
integrates three of these into a unified set of symbols. These three
ideas are: 1) the use of arrows to indicate alterations in pitch up
and down, 2) the intuitiveness of the Tartini fractional sharps, and
3) the slanted lines used by Bosanquet to indicate commatic
alterations.

*Tartini Plus Arrows*

Up and down arrows can be employed to indicate clearly the direction
in which the pitch is to be altered, and it was immediately obvious
that it would be necessary to have only 12 different symbols if each
symbol of the new notation could be inverted or mirrored vertically
to symbolize equal-but-opposite amounts of alteration. This would
require discarding the traditional single and double sharp symbols
(as well as excluding the Tartini fractional sharps from
consideration), inasmuch as they look virtually the same when
inverted. A traditional flat symbol can be inverted and does
resemble a hand with a finger pointing; the problem is that it points
in the wrong direction, so I concluded that it would also need to be
discarded. Of the conventional symbols, only the "natural" symbol
would be retained.

In my first version of the sagittal notation of August 2001 (which I
now call the expanded saggital symbols), I used arrows as semisharp
and semiflat symbols, with multiple arrowheads for single, sesqui,
and double sharps and flats. These are shown in the second row of
Figure 1. The use of arrows to represent semisharps and semiflats
may seem somewhat arbitrary, inasmuch as they have been used in
different instances to represent various amounts of pitch alteration,
but I felt that their frequent use for notating quartertones was
adequate justification.

In December I realized that these symbols could be simplified by
replacing the multiple arrowheads with single arrows that are
combined with one to three vertical strokes, as in the Tartini
fractional sharps, with an "X" for the double sharp and flat, as
shown in the third row of Figure 1. The single arrowheads not only
make the symbols more compact, but they also permit a bolder print
(or font) style to be employed, which improves legibility.

If the abandonment of the conventional sharp and flat symbols seems a
bit shocking, we need to realize that, although they have served us
well since they were devised in the Middle Ages, 21st-century
microtonality will be better served by something new and better, and
I think that it is safe to say it is about time for an upgrade. We
can continue to call these sharps and flats with semi, sesqui, and
double prefixes added as appropriate, inasmuch as it is only the
symbols that are changing, not their names or meanings.

This set of 9 symbols is sufficient to notate 17, 24, and 31-EDO.
However, more symbols would be needed for 72-EDO.

*Plus Bosanquet*

The third idea to find its way into my saggital notation was the
symbol for commatic alterations in 53-EDO that Bosanquet used around
1875. These are shown in the top row of Figure 2, which illustrates
a lateral grouping for multi-comma alterations. The single degree of
72-EDO is similar in size to that of 53-EDO, with the intervals
representing just (5:4) and Pythagorean (81:64) major thirds
differing in size by this amount in each system, so the use of this
sort of symbol would not be inappropriate to indicate an alteration
of a single degree in 72-EDO. I first added a stem to the Bosanquet
symbol to form a sort of half-arrow or flag. I then stacked several
of these flags to indicate multiple-degree alterations, as in the
second row of Figure 2.

I quickly realized that the symbol that I was already using to alter
by 3 degrees differed from the 1-degree symbol by only a right half-
arrow or flag, and that it would be quite logical to represent a 2-
degree alteration with a backward 1-degree symbol. The resulting
expanded saggital symbols are shown in the third row of Figure 2.
These were subsequently simplified into the compact saggital notation
shown in the fourth row of Figure 2. Observe that each new half-
arrow (or Bosanquet flag) symbol is adjacent to a full-arrow symbol,
with the slant of the Bosanquet flag corresponding to the direction
in which the pitch symbolized by the adjacent (full-arrow) symbol
must be altered to arrive at the pitch symbolized by the Bosanquet
flag symbol: upward slope signifies alteration one degree (or comma)
up, while downward slope signifies one degree (or comma) down.

The full range of symbols is shown in Figure 3, along with some
examples on a musical staff comparing other notations with the new
saggital notation.

Both the compact and expanded versions of the saggital symbols may be
simulated with ASCII characters for e-mail messages, etc., using a
combination of the slash, backslash, pipe, and capital X characters.
One comma down is \|, semisharp is /|\, and doubleflat is \X/
(compact) or \\\\|//// (expanded). While this generally involves
more characters than with other proposed ASCII notation, it is more
intuitive, and it inconveniences the theorist rather than the
musician. (Please note that the combination of ASCII symbols has a
better appearance when a proportionally spaced font is used; my
choice is Ariel.)

The next part of this presentation will discuss how the notation may
be applied logically and consistently to other EDO's, beginning with
31 and 41, as well as the use of the 72-EDO symbols as a
transcendental notation for sets of just (or near-just) tones mapped
onto a lesser division of the octave.

Until next time, please stay tuned!

--George

Love / joy / peace / patience ...

🔗jpehrson2 <jpehrson@rcn.com>

1/21/2002 2:13:58 PM

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

/tuning/topicId_32971.html#32971

> *BURIED TREASURE*
> "Notation - Part 1"
> From: George Secor
> January 22, 2002
>
> Patience comes to those who wait for it, and I thank you all for
your patience.

*****I suppose it would be better for me to wait until George Secor
has made his *entire* presentation before I start talking.

However, I'm excited and interested in the commentary, so I may start
anyway...

The problem, basically, is that I have been rejected from the jury.
I'm a *very* biased participant, having pretty much decided to stick
with the Maneri/Sims 72-tET notation. For one thing, I am already
writing in it, am used to it, and if composers of the stature of Sims
and Maneri have been using it and training musicians to use it, it
becomes a kind of "standard" that I don't want to subvert, even if I
have minor disagreements with this and that.

So, I would urge George to pay very little attention to what I say
about his invention, since I'm incredibly, biased, prejudiced, pre-
decided, pre-determined (and probably could be called by various
epithets if this were talk radio...)

However, believe it or not, I will proceed...

> I have always believed that the best notation is that which is
> simplest. A good example of this is the Tartini fractional sharps
> (shown in the right half of the top row of Figure 1), which are so
> clear that they require virtually no explanation. Although these
> were employed by both Ivan Vyshnegradsky (for 24-EDO) and Adriaan
> Fokker (for 31-EDO), it is rather surprising that quartertone
> composers never adopted these as a standard notation.

****Quite frankly, they may be more in existence than you give
credit. The SCORE notation program uses the Tartini symbols (a
little differently than used in 31-tET of course) for quartertones.
Some people, including several contemporary notation books I have
seen, *do* consider it a "de facto" standard, at least for quarter-
tones...

> Existing methods of notating 72-EDO have also used this approach,
and
> the diversity of symbols used somewhat arbitrarily (and not always
> logically) to designate three different amounts of alteration in
> pitch strikes me as a conglomeration of add-ons or do-dads intended
> to supplement traditional notational practice. However, I did not
> see these 72-EDO notations (including the one devised by Ezra Sims)
> until several months after I had produced the initial (expanded)
> version of my saggital notation, so they had absolutely no
influence in its development. To be completely honest, once I did
see them, I was appalled.

****There are only *THREE* symbols in this system. Arrows, half-
arrows and quarter-tone symbols. It couldn't be any simpler. The
arrows and half-arrows go up and down. The quarter-tone Sims
symbols, I agree, are a bit controversial, but if one were to use the
Tartini "one-slash" sharp and "backwards flat" for quarter-tones this
system would *really* be optimal. However, as I've mentioned before,
that fact that Sims and Maneri *already* have a practicing system
means something to me, so I would rather forgo optimization to "get
with the program" so to speak... :)

I later learned that the symbols that were proposed by
> those on the Alternate Tuning List were ASCII versions for
> theoretical use only, not practical notation intended for use on an
> actual musical manuscript, and the goal was largely to emulate the
> Sims notation.

**** I believe this is partially incorrect. I believe the ASCII
notation developed on this list is *so close* to the Sims/Maneri
written notation that *either* could be easily subsituted in actual
*playable* music, not just theoretical exercises...

^ V for the arrows... very similar
< > for the half-arrows

and

[ ] for the quarter-tones... looking somewhat similar to the Sims
but, frankly, even better...

Here are the "written" symbols again, for easy comparison:

/tuning/files/Pehrson/Sims.GIF

These three
> ideas are: 1) the use of arrows to indicate alterations in pitch up
> and down,

****This is a good idea but, also, basically incorporated in the
Sims, although I do admit, not as *UNIFORMLY* as you've done it...

2) the intuitiveness of the Tartini fractional sharps, and
> 3) the slanted lines used by Bosanquet to indicate commatic
> alterations.

****Quite frankly, I've found those "compounded slashes" *very*
difficult to read.

Any musician who has to *sight read* 72-tET... and I *do* hope that
there will be more and more of them! would absolutely *hate* these
little slashes. I *guarantee* it!!!

> In my first version of the sagittal notation of August 2001 (which
I now call the expanded saggital symbols), I used arrows as semisharp
> and semiflat symbols, with multiple arrowheads for single, sesqui,
> and double sharps and flats. These are shown in the second row of
> Figure 1. The use of arrows to represent semisharps and semiflats
> may seem somewhat arbitrary, inasmuch as they have been used in
> different instances to represent various amounts of pitch
alteration,
> but I felt that their frequent use for notating quartertones was
> adequate justification.
>
> In December I realized that these symbols could be simplified by
> replacing the multiple arrowheads with single arrows that are
> combined with one to three vertical strokes, as in the Tartini
> fractional sharps, with an "X" for the double sharp and flat, as
> shown in the third row of Figure 1. The single arrowheads not only
> make the symbols more compact, but they also permit a bolder print
> (or font) style to be employed, which improves legibility.
>

****It seems that the "compact" version of your notation is *indeed*
a significant improvement. The "stacked" version of the notation has
the same problems as the Bosanquet, in *my* opinion.

Musicians can't read all those little flags in *sight reading* fast
enough.

Unless we want to say that "anything goes" in microtonality, and it
doesn't matter if they get them right in the first place... :)

(I don't think we want that...)

> If the abandonment of the conventional sharp and flat symbols seems
a bit shocking, we need to realize that, although they have served us
> well since they were devised in the Middle Ages, 21st-century
> microtonality will be better served by something new and better,
and I think that it is safe to say it is about time for an upgrade.

****Here I part company quite severely. (I think I just cut myself
in the kitchen...ouch)

Abandoning the traditional sharps and flats is unequivocal nonsense,
in *my* opinion. (Please see caveat at beginning of my
commentary... :)

One of the *best* things about 72-tET is that traditional musicians,
accustomed to 12-tET can read *at least THOSE pitches* quickly and
easily. There is *absolutely no doubt* in my mind that a quick,
convenient system for 72-equal should *of necessity* use the 12-tET
symbols that are *already* in the system.

Particularly if one only needs *one* modifying symbol to these. Now,
you're going to say there are two, but I disagree. The sharps and
flats are *so integral* that they are practically part of the notes!
They are *very* intuitive, and part of our early music-learning.
(For *most* people I should think, anyway). So, it isn't the same
effect as having two *entirely new* modifiers...

We
> can continue to call these sharps and flats with semi, sesqui, and
> double prefixes added as appropriate, inasmuch as it is only the
> symbols that are changing, not their names or meanings.
>
> This set of 9 symbols is sufficient to notate 17, 24, and 31-EDO.
> However, more symbols would be needed for 72-EDO.
>

*****Now, I will have to admit that you possibly are going for a
*universal, theoretical* notation that will describe *MANY* different
systems.

From a *THEORETICAL* point of view, your system could be optimal, but
I believe that for practicing, performing musicians who want to make
the "leap" from 12-tET to 72-tET it is not at all. And it *will* be
a "leap" for many. You can trust me on that one...

(Please see response caveat above... :)

>
> *Plus Bosanquet*
>
> The third idea to find its way into my saggital notation was the
> symbol for commatic alterations in 53-EDO that Bosanquet used
around 1875.

*****I don't even want to discuss all these slashes. I don't feel
they are a performable notation, regardless of their *theoretical*
significance in illustrating microtonality...

>
> I quickly realized that the symbol that I was already using to
alter
> by 3 degrees differed from the 1-degree symbol by only a right half-
> arrow or flag, and that it would be quite logical to represent a 2-
> degree alteration with a backward 1-degree symbol. The resulting
> expanded saggital symbols are shown in the third row of Figure 2.

*****Your development here is, indeed, *much* better than multiple
slashes.

Let's eliminate the "slashers" from music.... :)

>
> Both the compact and expanded versions of the saggital symbols may
be
> simulated with ASCII characters for e-mail messages, etc., using a
> combination of the slash, backslash, pipe, and capital X
characters.
> One comma down is \|, semisharp is /|\, and doubleflat is \X/
> (compact) or \\\\|//// (expanded). While this generally involves
> more characters than with other proposed ASCII notation, it is more
> intuitive, and it inconveniences the theorist rather than the
> musician. (Please note that the combination of ASCII symbols has a
> better appearance when a proportionally spaced font is used; my
> choice is Ariel.)

****This is *HORRIBLE* ASCII, George! How could you say that??
Compared to the simple symbols we've evolved for the Sims/Maneri on
this very list? Come on!

(Please see introductory caveat above... :)

> The next part of this presentation will discuss how the notation
may be applied logically and consistently to other EDO's, beginning
with 31 and 41, as well as the use of the 72-EDO symbols as a
> transcendental notation for sets of just (or near-just) tones
mapped onto a lesser division of the octave.
>
> Until next time, please stay tuned!
>

****I shudda waited, but just had to blather...

Thanks again for the interesting discussion!

Joseph Pehrson

🔗gdsecor <gdsecor@yahoo.com>

1/22/2002 11:31:21 AM

Joseph,

Thank you for your comments, some of which follow.

[Joseph Pehrson (#32975):]
<< *****I suppose it would be better for me to wait until George
Secor has made his *entire* presentation before I start talking.

However, I'm excited and interested in the commentary, so I may start
anyway... >>

I much prefer eliciting a lively, sincere response than being mocked
or ignored, so fire away!

<< … I would urge George to pay very little attention to what I say
about his invention, since I'm incredibly, biased, prejudiced, pre-
decided, pre-determined (and probably could be called by various
epithets if this were talk radio...) >>

On the contrary, your opinions are very important, because you are
contributing a point of view that will no doubt be held by many
others outside the Tuning List, and I believe that it is much better
initially to air these things in an in-house debate rather than with
those who do not share our outlook and/or objectives.

<< [re the slanted lines used by Bosanquet to indicate commatic
alterations] ****Quite frankly, I've found those "compounded slashes"
*very* difficult to read.

Any musician who has to *sight read* 72-tET... and I *do* hope that
there will be more and more of them! would absolutely *hate* these
little slashes. I *guarantee* it!!! >>

You're absolutely correct! This is why I went a step farther to
convert the expanded saggital symbols into the compact version, even
as you observed:

<< ****It seems that the "compact" version of your notation is
*indeed* a significant improvement. The "stacked" version of the
notation has the same problems as the Bosanquet, in *my* opinion.

Musicians can't read all those little flags in *sight reading* fast
enough. >>

Besides showing you how I got to the final version, those expanded
saggital symbols will play a temporary (illustrative) purpose in the
next part of the presentation, when I take up the subject of how
commas enter into the whole scheme of things, particularly in
determining how to go about using the symbols for other (should I
emphasize: many other!) EDO's.

<< Abandoning the traditional sharps and flats is unequivocal
nonsense, in *my* opinion. (Please see caveat at beginning of my
commentary... :)

One of the *best* things about 72-tET is that traditional musicians,
accustomed to 12-tET can read *at least THOSE pitches* quickly and
easily. There is *absolutely no doubt* in my mind that a quick,
convenient system for 72-equal should *of necessity* use the 12-tET
symbols that are *already* in the system.

Particularly if one only needs *one* modifying symbol to these. Now,
you're going to say there are two, but I disagree. The sharps and
flats are *so integral* that they are practically part of the notes!
They are *very* intuitive, and part of our early music-learning.
(For *most* people I should think, anyway). So, it isn't the same
effect as having two *entirely new* modifiers... >>

Keeping them poses a problem that I will address in the final
installment of the presentation. I intend to show that in keyboard
music, for example, it is not difficult to come up with situations in
which the Sims notation not only makes the manuscript a difficult
mess to encode and decipher, but at times completely breaks down in
ambiguity.

<< *****Now, I will have to admit that you possibly are going for a
*universal, theoretical* notation that will describe *MANY* different
systems. >>

Now you're beginning to step back and get the big picture. If we
have one notation for 72, another for 31, something else for 41,
something else for 22, etc., etc., imagine the learning curve (and
potential confusion) this poses for someone attracted to
microtonality, but who hasn't decided where to start. If you have a
well-thought-out superset of symbols (yes, I will be introducing 8
more, and that's all) that can be used for just about anything (I
hesitate to use the term "universal"), and if each and every symbol
has a meaning that doesn't change when it's used for different tonal
systems, then it will be much easier to read music written for any
other system, once you have learned it for one.

To use an analogy, someone who speaks and reads English can
readily "read" Spanish, French, or Latin; the problem, of course, is
that they won't understand it, but, upon learning to speak a second
language that uses the same alphabet, reading it is not much of a
problem. This would not be the case with a language that uses
totally different symbols, such as Hebrew, Arabic, or Chinese.

Something that I need to emphasize is that this is a *practical*
notation to be used on the staff and not just for theoretical
discourses. And I think you can readily appreciate the prospect that
such a notation would include many who would be excluded by the
considerably narrower scope of the Sims notation.

Am I making any sense?

<< Let's eliminate the "slashers" from music.... :)

> Both the compact and expanded versions of the saggital symbols may
be
> simulated with ASCII characters for e-mail messages, etc., using a
> combination of the slash, backslash, pipe, and capital X
characters.
> One comma down is \|, semisharp is /|\, and doubleflat is \X/
> (compact) or \\\\|//// (expanded). While this generally involves
> more characters than with other proposed ASCII notation, it is more
> intuitive, and it inconveniences the theorist rather than the
> musician. (Please note that the combination of ASCII symbols has a
> better appearance when a proportionally spaced font is used; my
> choice is Ariel.)

****This is *HORRIBLE* ASCII, George! How could you say that??
Compared to the simple symbols we've evolved for the Sims/Maneri on
this very list? Come on! >>

These are just a suggestion for ASCII. The essence of my proposal is
a *practical* staff notation, and whatever you might use for an ASCII
simulation is completely up to you. The ideal solution is to use a
custom character set, but in certain instances, such as e-mail, this
isn't possible, so we have to do the best we can with ASCII
characters.

If I might make a comment on the Sims notation, it looks like he
found them in a set of previously existing special characters for a
computer, which is akin to doing it backwards. The computer has
become a valuable tool for a great number of us, but it has to keep
its place as a servant, not a master that dictates to us those
symbols from which we may or may not choose for use on a musical
manuscript. Once we have made that decision, then we can decide what
symbols, ASCII or otherwise, to use on a computer to simulate those
that best serve the purpose for the manuscript.

<< ****I shudda waited, but just had to blather... >>

Hey, you gotta do what you gotta do!

<< Thanks again for the interesting discussion! >>

And thank you. Your comments have provided an opportunity for me to
clarify a few things, and if we still don't agree about some of
these, at least we now have a much better understanding of where
we're coming from, which should help all of us in deciding where to
go from here, once all is said.

And let me take this opportunity to thank everyone for making me feel
at home here on the Tuning List. One of our primary objectives
should be to encourage one another in the pursuit of the goals and
dreams that we share. In this you are doing a marvelous job, and I
count it a privilege to have had the opportunity to join you in this
great adventure.

--George

🔗paulerlich <paul@stretch-music.com>

1/22/2002 12:39:43 PM

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

> Keeping them poses a problem that I will address in the final
> installment of the presentation. I intend to show that in keyboard
> music, for example, it is not difficult to come up with situations
in
> which the Sims notation not only makes the manuscript a difficult
> mess to encode and decipher, but at times completely breaks down in
> ambiguity.
>
> << *****Now, I will have to admit that you possibly are going for
a
> *universal, theoretical* notation that will describe *MANY*
different
> systems. >>
>
> Now you're beginning to step back and get the big picture. If we
> have one notation for 72, another for 31, something else for 41,
> something else for 22, etc., etc., imagine the learning curve (and
> potential confusion) this poses for someone attracted to
> microtonality, but who hasn't decided where to start.

I'm anticipating a big problem that John deL., I, probably Margo S.,
and others may very well end up having with your final proposal:

We feel that common-practice notation came up ESSENTIALLY in a
meantone tuning, and its 5-limit approximations could NEVER be
adequately notated in a system like 41 or 72. You'll have the "comma
problem" to contend with, and precious little Western common-practice
music is unaffected by it.

From a more general mathematical viewpoint, though perhaps a bit
too "theoretical" for many, 5-limit diatonicism (i.e. common
practice) not only requires, but IS DEFINED BY, the vanishing of the
comma 81:80 -- and then the chromatic alterations represent the
remaining unison vector, 25:24. Canasta, on the other hand, IS
DEFINED BY the vanishing of 225:224, 385:384, and 441:440 (for
example), and the remaining unison vector, 81:80, which would
function as the "chromatic alteration".

I suspect Gene will have rather similar reactions once your final
notational proposal is put forward . . . but I'm only guessing as to
what it is! I know Gene feels, like I do, that there IS NO 5-limit
diatonicism in 22-tET, for example. While 19-tET sounds just great
for common-practice diatonic music, any attempt to render it (except
for very special cases) in 22-tET leaves some of us extremely queasy.

Take all this as my 2¢ only.

Cheers!

🔗monz <joemonz@yahoo.com>

1/22/2002 4:58:17 PM

Hi George,

I'll probably have a *lot* more to say about your notational
proposal after I've studied it more. But this jumped out
right away:

> From: gdsecor <gdsecor@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, January 22, 2002 11:31 AM
> Subject: [tuning] Re: *BT* Notation - Part 1
>
>
> > [Joseph Pehrson]
> > Abandoning the traditional sharps and flats is unequivocal
> > nonsense, in *my* opinion. (Please see caveat at beginning
> > of my commentary... :)
>
> One of the *best* things about 72-tET is that traditional musicians,
> accustomed to 12-tET can read *at least THOSE pitches* quickly and
> easily. There is *absolutely no doubt* in my mind that a quick,
> convenient system for 72-equal should *of necessity* use the 12-tET
> symbols that are *already* in the system.

I agree that two great things about 72-EDO (as I like to call it)
is that it preserves both the 12-EDO scale with which everyone's
already familiar, and (at least in some versions) the standard
sharp/flat notation. But please note that these are two separate
things!

Standard musical notation is *NOT* based on 12-EDO, but rather,
it was originally derived from Pythagorean and later applied
to meantone! Paul goes into greater detail on this in his
response to you.

A notation based on 12-EDO would have an unambiguous way of
specifying *12* notes -- for example, the integer [0...11] notation
used by many authors exploring 12-EDO serialism in _Journal of
Music Theory_, etc.

Our standard notation is based on a *7*-tone diatonic set,
which is a combination of both "half-" and "whole-" steps.
Thus, I would characterize it as a "7+(7*2n)" system, where "n"
is an integer which represents how many levels of accidentals
you're using, i.e.:

- n = 0, means no accidentals, thus the regular diatonic set of
A, B, C,D, E, F, G = 7 different notes;

- n = 1, means one level of accidentals, or the basic set plus
sharps and flats = 21 different notes;

- n = 2, means all of those plus double-sharps and double-flats
= 35 different notes;

etc.

Indeed, a big part of the difficulty with notation, in today's
world where many people want to use various EDOs, is that the
traditional notation suffers from this "logical flaw" (as I've
always called it) of not indicating in the notation which of
the scale-steps are "whole-steps" and which are "half-steps".

Same thing goes for the use of the staff itself, where going
from a line to a space, or vice-versa, means traversing a
"whole-step" in 5 out of 7 cases, but a "half-step" in the
other two.

This observation led me to suggest abandoning standard 5-line
staff notation altogether. Here's one solution I devised,
where each line represents a 12-EDO pitch and the spaces
represent the quarter-tone pitches between them:

http://tonalsoft.com/enc/q/qt-staff.aspx

And about 2/3 of the way down this page, you can see how it
might be adapted for use with 72-EDO:

http://tonalsoft.com/enc/number/72edo.aspx

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗jpehrson2 <jpehrson@rcn.com>

1/22/2002 5:20:19 PM

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

/tuning/topicId_32971.html#33045
>
> Indeed, a big part of the difficulty with notation, in today's
> world where many people want to use various EDOs, is that the
> traditional notation suffers from this "logical flaw" (as I've
> always called it) of not indicating in the notation which of
> the scale-steps are "whole-steps" and which are "half-steps".
>
> Same thing goes for the use of the staff itself, where going
> from a line to a space, or vice-versa, means traversing a
> "whole-step" in 5 out of 7 cases, but a "half-step" in the
> other two.
>
> This observation led me to suggest abandoning standard 5-line
> staff notation altogether. Here's one solution I devised,
> where each line represents a 12-EDO pitch and the spaces
> represent the quarter-tone pitches between them:
>
> http://tonalsoft.com/enc/q/qt-staff.aspx
>
>
> And about 2/3 of the way down this page, you can see how it
> might be adapted for use with 72-EDO:
>
> http://tonalsoft.com/enc/number/72edo.aspx
>
>
>
> -monz
>

Well, this is very cool, Monz, and I've seen it before. It looks a
bit like Stockhausens old "graphic" notations for pieces like _Studie
II_....

However, I defy you to get "traditional" or even "open minded
experimental" performing musicians to do it!

Maybe you could prove me wrong... but I think I could make some money
on a bet like that.... :)

JP

🔗graham@microtonal.co.uk

1/23/2002 2:57:00 AM

In-Reply-To: <00ab01c1a3a9$0c4789e0$af48620c@dsl.att.net>
monz wrote:

> This observation led me to suggest abandoning standard 5-line
> staff notation altogether. Here's one solution I devised,
> where each line represents a 12-EDO pitch and the spaces
> represent the quarter-tone pitches between them:
>
> http://tonalsoft.com/enc/q/qt-staff.aspx

If you're going to replace the staff with something based around 12, I
don't think you can beat what Erv Wilson shows at
<http://www.anaphoria.com/xen3a.PDF>. Lines are black notes, spaces are
white notes. Your quarter-tone staff actually does the same thing for my
rearranged DX21.

You can also check out this link I found on Charles Lucy's site:
<http://www.speechskript.com/mnma.htm>.

Whatever, universal notation is a tall order if you expect a fixed number
of nominals. But I'm keeping my suspenders on at least until the next
installment of George Secor's notation.

Graham

🔗gdsecor <gdsecor@yahoo.com>

1/23/2002 12:56:17 PM

I'll be replying to a few things here:

[Paul Erlich (#33021):]

<< I'm anticipating a big problem that John deL., I, probably Margo
S., and others may very well end up having with your final proposal:

We feel that common-practice notation came up ESSENTIALLY in a
meantone tuning, and its 5-limit approximations could NEVER be
adequately notated in a system like 41 or 72. You'll have the "comma
problem" to contend with, and precious little Western common-practice
music is unaffected by it.

From a more general mathematical viewpoint, though perhaps a bit
too "theoretical" for many, 5-limit diatonicism (i.e. common
practice) not only requires, but IS DEFINED BY, the vanishing of the
comma 81:80 -- and then the chromatic alterations represent the
remaining unison vector, 25:24. Canasta, on the other hand, IS
DEFINED BY the vanishing of 225:224, 385:384, and 441:440 (for
example), and the remaining unison vector, 81:80, which would
function as the "chromatic alteration".

I suspect Gene will have rather similar reactions once your final
notational proposal is put forward . . . but I'm only guessing as to
what it is! I know Gene feels, like I do, that there IS NO 5-limit
diatonicism in 22-tET, for example. While 19-tET sounds just great
for common-practice diatonic music, any attempt to render it (except
for very special cases) in 22-tET leaves some of us extremely queasy.

Take all this as my 2¢ only. >>

Rest assured that I have long been familiar with the problems imposed
by commas and am under no misconceptions about what this notation (or
any other, for that matter) is or isn't capable of accomplishing,
given certain immutable acoustical facts such as you have mentioned
above.

After completing two more paragraphs of this reply, I suddenly
realized that what I was writing should be in the next installment of
the presentation. So you'll have to wait a few more days for the
rest. Sorry about that!

[Joe Monzo (#33045):]

<< I agree that two great things about 72-EDO (as I like to call it)
is that it preserves both the 12-EDO scale with which everyone's
already familiar, and (at least in some versions) the standard
sharp/flat notation. But please note that these are two separate
things!

Standard musical notation is *NOT* based on 12-EDO, but rather, it
was originally derived from Pythagorean and later applied to
meantone! Paul goes into greater detail on this in his response to
you. >>

Joe, while you go on to suggest that maybe we should consider using
something other than a seven-step (or heptatonic) notation, you have
just (inadvertently) pointed out a highly compelling reason (aside
from the learning curve involved) *not* to go that route.

Unlike the Pythagorean tuning, meantone temperament, and most of the
practical >5-limit EDO's, 72-EDO has a 12-tone *circle* rather than a
larger *series* of fifths, which removes the traditional notational
distinction between sharps and flats in a non-heptatonic notation.
Any attempt to use 72-EDO as a transcendental notation of the sort
that I am proposing (more details next installment) will utterly fail
to represent those systems by its inability to make that important
distinction.

As I said in my saggital notation proposal, I am merely replacing the
symbols for the sharp and the flat, and not their names or meanings.
I think most contemporary musicians are bright enough to be able to
remember the new sharp and flat symbols that I'm using and will
accept them, given a good enough reason for doing this. Or is there
something sacred about the present symbols that I haven't yet heard
about? (Should I start handing out bumper stickers that say "Kick
the #/b habit -- go saggital!")

[Graham Breed (#33058):]

<< If you're going to replace the staff with something based around
12, I don't think you can beat what Erv Wilson shows at
<http://www.anaphoria.com/xen3a.PDF>. Lines are black notes, spaces
are white notes. Your quarter-tone staff actually does the same
thing for my rearranged DX21. >>

Once again, when you use 72-EDO for this, you lose the sharp/flat
distinction.

<< Whatever, universal notation is a tall order if you expect a
fixed number of nominals. But I'm keeping my suspenders on at least
until the next installment of George Secor's notation. >>

Notice that I hesitated to use the term "universal." I'm just trying
to do as much as possible without using an unreasonable number of
symbols. And I'll try to cover as much ground as I can in the next
installment. (This whole subject could easily fill a book if it were
written for the novice, but the interactive nature of this
presentation should serve our purpose in far fewer words.)

Stay tuned!

--George

🔗jpehrson2 <jpehrson@rcn.com>

1/23/2002 2:02:46 PM

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

/tuning/topicId_32971.html#33068

> I think most contemporary musicians are bright enough to be able to
> remember the new sharp and flat symbols that I'm using and will
> accept them, given a good enough reason for doing this. Or is
there
> something sacred about the present symbols that I haven't yet heard
> about? (Should I start handing out bumper stickers that say "Kick
> the #/b habit -- go saggital!")
>

Hi George!

Frankly, I believe if you tell just about any *practicing* musician
that you are going to be eliminating the cherished sharps and flats
from music, you're going to get some strong opposition.... that is,
unless you start training from kindergarten or something of the
kind....

So, *yes* I believe you'll need a heavy "promotional" campaign... :)

Joseph

🔗paulerlich <paul@stretch-music.com>

1/23/2002 2:50:26 PM

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

> If I might make a comment on the Sims notation, it looks like he
> found them in a set of previously existing special characters for a
> computer, which is akin to doing it backwards. The computer has
> become a valuable tool for a great number of us, but it has to keep
> its place as a servant, not a master that dictates to us those
> symbols from which we may or may not choose for use on a musical
> manuscript. Once we have made that decision, then we can decide
what
> symbols, ASCII or otherwise, to use on a computer to simulate those
> that best serve the purpose for the manuscript.

I'll give you 100000-to-1 odds that Sims did not find the characters
on a computer (who had a computer back then?), but designed them
specifically for the manuscript.

Having said that, I don't think you'll find a lot of score-readers
who are particularly fond of Sims' "square-root sign".

Franz-Richter Herf has also designed a 72-tET notation, in use at the
International Institute for Ekmelic Music:

http://www.ekmelic-music.org/en/em/notation.htm

You might like it better.

🔗jonszanto <JSZANTO@ADNC.COM>

1/23/2002 2:54:14 PM

First off, that was a great subject line, Joe!

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> Frankly, I believe if you tell just about any *practicing* musician
> that you are going to be eliminating the cherished sharps and flats
> from music, you're going to get some strong opposition.... that is,
> unless you start training from kindergarten or something of the
> kind....

You know something? I don't think that is true _across_the_board_. I
think some of it has to do with how much the musician might be
interested in playing the music!

You mentioned that you thought various notational 'styles' that
popped up in the 50's and 60's fragmented the practice of performance
into smaller areas, but speaking as just *one* practicing musician,
I've sometimes considered the notation as another form of the
adventure of a new piece. I can't imagine Morton Feldman's "King of
Denmark" written any other way, and the variety of notations for
Partch's instruments didn't turn off players - they looked forward to
it.

This doesn't cover all musics, certainly, and for a generalized rep I
think finding some sort of 'consensus' notation is a *good* thing.
Just don't sell short a few of us out here that don't mind the
challenge!

Cheers,
Jon

🔗M. Schulter <MSCHULTER@VALUE.NET>

1/23/2002 3:08:15 PM

Hello, there, everyone, and I'd like briefly to comment on the
presentation and discussion so far on the new Secor saggital
notation.

This notation seems especially attractive to me as an interchange
standard for music conceived in a Partch-like JI or near-JI framework
of an "n-limit" type, the kind of fabric represented by a Secorian
temperament based on an ideal generator of (18/5)^(1/19), or about
116.716 cents, with likely realizations in 72/31/41-EDO.

An important point here, as often raised by Paul Erlich, is that
attempting to "translate" a piece of music between tuning systems can
have a range of consequences, vertical and melodic, calling for our
exercise of due discretion and informed common sense.

Thus using saggital notation to "map" a near-JI piece conceived in
72-EDO into 31-EDO, for example, might in many circumstances be a
useful translation. Certain intervals such as fifths and fourths will
be less accurate, and some commas will be disregarded, but the result
might well be a "reasonable facsimile" of the original composition.

Translating from a tuning system where a given comma is dispersed, to
one where it is observed, can be a considerably more problematic
endeavor. Going from 31-EDO, approached in a Renaissance meantone
style, to 72-EDO, is an obvious example.

Then, again, traditional notations also give one the means to make
translations where treading with caution might be well advised --
unless the purpose is experimentation and play for its own sake, an
invaluable means of "research and development."

Also, one might want to consider the possibility for freer
translations negotiated by flexible-pitch ensembles (e.g. strings,
brass, and of course human voices) where comma issues might be
finessed by the kind of "adaptive tuning" often discussed here.

What saggital notation provides is a new approach for complex types of
systems with a large number of steps and degrees of alteration, and
also a new outlook which can place other approaches in a richer
perspective, giving us a larger set of choices.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗paulerlich <paul@stretch-music.com>

1/23/2002 4:16:28 PM

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

> Thus using saggital notation to "map" a near-JI piece conceived in
> 72-EDO into 31-EDO, for example, might in many circumstances be a
> useful translation. Certain intervals such as fifths and fourths
will
> be less accurate, and some commas will be disregarded, but the
result
> might well be a "reasonable facsimile" of the original composition.

Or it might not be, if, say, the composer were relying on the
difference in character between the "6:5" and the "25:21" in
Blackjack, for example to evoke tension-resolution effects.

> Translating from a tuning system where a given comma is dispersed,
to
> one where it is observed, can be a considerably more problematic
> endeavor. Going from 31-EDO, approached in a Renaissance meantone
> style, to 72-EDO, is an obvious example.

While I agree with you and John deLaubenfels that this is
problematic, certain Renaissance scholars, such as Jonathan Walker,
would not agree, seeing the dispersal of the comma as a form
of "ficta" operating on a smaller level than the usual "chromatic
ficta".

Notation is a hairy subject!

🔗jpehrson2 <jpehrson@rcn.com>

1/23/2002 6:18:48 PM

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

/tuning/topicId_32971.html#33068

> I suspect Gene will have rather similar reactions once your final
> notational proposal is put forward . . . but I'm only guessing as
to what it is! I know Gene feels, like I do, that there IS NO 5-limit
> diatonicism in 22-tET, for example. While 19-tET sounds just great
> for common-practice diatonic music,

****Paul... why is this again. Is it because both 12-tET and 19-tET
are *meantones?* Help. I need to know this.

> Unlike the Pythagorean tuning, meantone temperament, and most of
the practical >5-limit EDO's, 72-EDO has a 12-tone *circle* rather
than a larger *series* of fifths,

****Paul... or somebody. I don't get this. How does this concept of
a "circle" rather than a "series" relate to the 6 "bicycle chains...?"

Help, please, again... :)

(This whole subject could easily fill a book if it were
> written for the novice, but the interactive nature of this
> presentation should serve our purpose in far fewer words.)
>

****Ummm, George. Please don't expect too much... I would hate to
get "lost..." It makes me feel very uncomfortable... :)

Joseph

🔗jpehrson2 <jpehrson@rcn.com>

1/23/2002 6:48:09 PM

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

/tuning/topicId_32971.html#33084

>
> Having said that, I don't think you'll find a lot of score-readers
> who are particularly fond of Sims' "square-root sign".
>

I almost hate to say something "good" about the square root signs
but, actually, there *is* something. They are clearly the *largest*
signs for the largest "deviating" interval...

JP

🔗paulerlich <paul@stretch-music.com>

1/23/2002 6:50:54 PM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>
> /tuning/topicId_32971.html#33068
>
> > I suspect Gene will have rather similar reactions once your final
> > notational proposal is put forward . . . but I'm only guessing as
> to what it is! I know Gene feels, like I do, that there IS NO 5-limit
> > diatonicism in 22-tET, for example. While 19-tET sounds just great
> > for common-practice diatonic music,
>
> ****Paul... why is this again. Is it because both 12-tET and 19-tET
> are *meantones?* Help. I need to know this.

You're right!
>
> > Unlike the Pythagorean tuning, meantone temperament, and most of
> the practical >5-limit EDO's, 72-EDO has a 12-tone *circle* rather
> than a larger *series* of fifths,
>
>
> ****Paul... or somebody. I don't get this. How does this concept of
> a "circle" rather than a "series" relate to the 6 "bicycle chains...?"
>
> Help, please, again... :)

Each circle is one of the bicycle chains . . . seems obvious (if you
don't mind my saying so).

🔗jpehrson2 <jpehrson@rcn.com>

1/23/2002 6:51:19 PM

--- In tuning@y..., "jonszanto" <JSZANTO@A...> wrote:

/tuning/topicId_32971.html#33086

> First off, that was a great subject line, Joe!
>
> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > Frankly, I believe if you tell just about any *practicing*
musician
> > that you are going to be eliminating the cherished sharps and
flats
> > from music, you're going to get some strong opposition.... that
is,
> > unless you start training from kindergarten or something of the
> > kind....
>
> You know something? I don't think that is true _across_the_board_.
I
> think some of it has to do with how much the musician might be
> interested in playing the music!
>
> You mentioned that you thought various notational 'styles' that
> popped up in the 50's and 60's fragmented the practice of
performance
> into smaller areas, but speaking as just *one* practicing musician,
> I've sometimes considered the notation as another form of the
> adventure of a new piece. I can't imagine Morton Feldman's "King of
> Denmark" written any other way, and the variety of notations for
> Partch's instruments didn't turn off players - they looked forward
to
> it.
>
> This doesn't cover all musics, certainly, and for a generalized rep
I
> think finding some sort of 'consensus' notation is a *good* thing.
> Just don't sell short a few of us out here that don't mind the
> challenge!
>
> Cheers,
> Jon

Absolutely, Jon! I'm all for trying different things if they work.
Now whether I'll want to use them in my *own* compositions is
another, and more "personal" matter.... :)

JP

🔗jpehrson2 <jpehrson@rcn.com>

1/23/2002 7:07:06 PM

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

/tuning/topicId_32971.html#33100

> > ****Paul... or somebody. I don't get this. How does this concept
of
> > a "circle" rather than a "series" relate to the 6 "bicycle
chains...?"
> >
> > Help, please, again... :)
>
> Each circle is one of the bicycle chains . . . seems obvious (if you
> don't mind my saying so).

****Oh sure... but what I was wondering was about the
*terminology...* What's the difference between a "circle" and
a "series" of fifths?? Is it whether they come around to the same
point...

Joseph

🔗paulerlich <paul@stretch-music.com>

1/23/2002 7:17:10 PM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> /tuning/topicId_32971.html#33100
>
> > > ****Paul... or somebody. I don't get this. How does this concept
> of
> > > a "circle" rather than a "series" relate to the 6 "bicycle
> chains...?"
> > >
> > > Help, please, again... :)
> >
> > Each circle is one of the bicycle chains . . . seems obvious (if you
> > don't mind my saying so).
>
> ****Oh sure... but what I was wondering was about the
> *terminology...* What's the difference between a "circle" and
> a "series" of fifths?? Is it whether they come around to the same
> point...
>
Yes!

🔗jpehrson2 <jpehrson@rcn.com>

1/23/2002 7:17:54 PM

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

/tuning/topicId_32971.html#33100
> >
> > ****Paul... or somebody. I don't get this. How does this concept
of
> > a "circle" rather than a "series" relate to the 6 "bicycle
chains...?"
> >
> > Help, please, again... :)
>
> Each circle is one of the bicycle chains . . . seems obvious (if you
> don't mind my saying so).

Well, I guess Paul, the other thing is that he terms Pythagorean
a "chain" not a "circle..."

And, I thought that 53-tET was *very* close to Pythagorean, and that
it *also* circled around even *closer* (logically) than 12-tET.

So why would Pythagorean be a "chain" rather than a "circle...?"

Or am I making too much of this terminology??

Joseph

🔗jpehrson2 <jpehrson@rcn.com>

1/23/2002 7:25:46 PM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

/tuning/topicId_32971.html#33104

> And, I thought that 53-tET was *very* close to Pythagorean, and
that
> it *also* circled around even *closer* (logically) than 12-tET.
>

I meant, of course, circling around closer than 12 Pythagorean
fifths... :)

JP

🔗jpehrson2 <jpehrson@rcn.com>

1/23/2002 7:39:04 PM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

/tuning/topicId_32971.html#33104

Sorry for the confusion. *Finally* I get it.

In other words, even though 53-tET is *very close* to Pythagorean
tuning, and may even *be* for practical purposes, it still "isn't"
Pythagorean.

So, 53-tET closes and is a "circle" and Pythagorean is open, hence
a "chain..." And, all *temperaments* based upon a constant generator
are "circles..."

That took a while... sorry! :)

JP

🔗paulerlich <paul@stretch-music.com>

1/23/2002 7:54:07 PM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> So, 53-tET closes and is a "circle" and Pythagorean is open, hence
> a "chain..."

Yup. But not a "bicycle chain".

> And, all *temperaments* based upon a constant generator
> are "circles..."

Nope, only equal temperaments are. There are also open *linear
temperaments", such as most forms of meantone, or MIRACLE as generated
by the "official" secor, or most of the temperaments that Dave Keenan,
Graham Breed, and Gene Ward Smith have put forward . . .

> That took a while... sorry! :)
>
> JP

Maybe the confusion was due to the fact that bicycle chains, unlike
most kind of chains, actually close on themselves and form a circle . . .

🔗jpehrson2 <jpehrson@rcn.com>

1/23/2002 8:09:49 PM

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

/tuning/topicId_32971.html#33109

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > So, 53-tET closes and is a "circle" and Pythagorean is open,
hence a "chain..."
>
> Yup. But not a "bicycle chain".

****Right... so that makes the terminology a little confusing...

>
> > And, all *temperaments* based upon a constant generator
> > are "circles..."
>
> Nope, only equal temperaments are. There are also open *linear
> temperaments", such as most forms of meantone, or MIRACLE as
generated by the "official" secor, or most of the temperaments that
Dave Keenan, Graham Breed, and Gene Ward Smith have put forward . . .

****Wow. That's an important distinction, Paul... Of course, even
thought the generator in, say, MIRACLE is constant the step sizes are
different and it *never* closes unless another 33 cent step
is "thrown in" if I understand it correctly.

> Maybe the confusion was due to the fact that bicycle chains, unlike
> most kind of chains, actually close on themselves and form a
circle . . .

Yes, that and the fact that I momentarily thought that 53-tET and
Pythagorean were synonymous... which I guess they are almost in
*sound* but not in "concept..."

Thanks!

Joseph

🔗genewardsmith <genewardsmith@juno.com>

1/23/2002 8:16:48 PM

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

> > And, all *temperaments* based upon a constant generator
> > are "circles..."
>
> Nope, only equal temperaments are.

Even then, only up to octave equivalence, of course; and the bicycle can have more than one chain.

🔗genewardsmith <genewardsmith@juno.com>

1/23/2002 8:21:34 PM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

>
> ****Wow. That's an important distinction, Paul... Of course, even
> thought the generator in, say, MIRACLE is constant the step sizes are
> different and it *never* closes unless another 33 cent step
> is "thrown in" if I understand it correctly.

If the generator is 7/72 of an octave, it will close (modulo octaves)
after 72 generator steps, when it will reach 7 octaves. It will not close before that because 7 is relatively prime to 72, meaning 7/72 is reduced to its lowest terms.

> Yes, that and the fact that I momentarily thought that 53-tET and
> Pythagorean were synonymous... which I guess they are almost in
> *sound* but not in "concept..."

You should take a look at 665-et sometime. :)

🔗jpehrson2 <jpehrson@rcn.com>

1/23/2002 8:29:54 PM

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

/tuning/topicId_32971.html#33112

> If the generator is 7/72 of an octave, it will close (modulo
octaves)
> after 72 generator steps, when it will reach 7 octaves. It will not
close before that because 7 is relatively prime to 72, meaning 7/72
is reduced to its lowest terms.
>

****Oh, Gee... I guess that makes sense, Gene, but, that would mean,
I believe that you wouldn't have "octave equivalence" for Blackjack
generated until one reached the 7th octave...

> > Yes, that and the fact that I momentarily thought that 53-tET and
> > Pythagorean were synonymous... which I guess they are almost in
> > *sound* but not in "concept..."
>
> You should take a look at 665-et sometime. :)

Yes! But I'd much rather "look" at it than play it at the
keyboard... :)

JP

🔗paulerlich <paul@stretch-music.com>

1/23/2002 8:40:35 PM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> ****Wow. That's an important distinction, Paul... Of course, even
> thought the generator in, say, MIRACLE is constant the step sizes are
> different and it *never* closes unless another 33 cent step
> is "thrown in" if I understand it correctly.

Umm . . . if the generator is exactly 116 2/3 cents, the circle *does*
close after 72 generators . . . but note that the official secor is
close to 116.7 cents . . .

🔗paulerlich <paul@stretch-music.com>

1/23/2002 8:47:26 PM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> ****Oh, Gee... I guess that makes sense, Gene, but, that would mean,
> I believe that you wouldn't have "octave equivalence" for Blackjack
> generated until one reached the 7th octave...

Well that would be true not for Blackjack itself, but it would be true
for a "non-octave" version of MIRACLE where the generator wasn't
allowed to be inverted (i.e., "wrapped" around the octave). This
version would have approximately 10 2/7 notes per octave, so would
resemble Decimal but without the "kink" . . . A "diatonic" analogy
would be a tuning built on fifths only, like C G d a e' b' f#'' c#'''
g#''' d#'''' a#'''' e#''''' . . .

🔗jpehrson2 <jpehrson@rcn.com>

1/23/2002 9:17:38 PM

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

/tuning/topicId_32971.html#33115

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > ****Oh, Gee... I guess that makes sense, Gene, but, that would
mean,
> > I believe that you wouldn't have "octave equivalence" for
Blackjack
> > generated until one reached the 7th octave...
>
> Well that would be true not for Blackjack itself, but it would be
true
> for a "non-octave" version of MIRACLE where the generator wasn't
> allowed to be inverted (i.e., "wrapped" around the octave).

***Oh... I guess that makes sense, since by "definition" Blackjack
is "octave-equivalent..."...

JP

🔗Kraig Grady <kraiggrady@anaphoria.com>

1/24/2002 12:03:25 AM

I remember it being referred to as a cycle of fifths. hence the bicycle chain?!

jpehrson2 wrote:

> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> /tuning/topicId_32971.html#33100
> > >
> > > ****Paul... or somebody. I don't get this. How does this concept
> of
> > > a "circle" rather than a "series" relate to the 6 "bicycle
> chains...?"
> > >
> > > Help, please, again... :)
> >
> > Each circle is one of the bicycle chains . . . seems obvious (if you
> > don't mind my saying so).
>
> Well, I guess Paul, the other thing is that he terms Pythagorean
> a "chain" not a "circle..."
>
> And, I thought that 53-tET was *very* close to Pythagorean, and that
> it *also* circled around even *closer* (logically) than 12-tET.
>
> So why would Pythagorean be a "chain" rather than a "circle...?"
>
> Or am I making too much of this terminology??
>
> Joseph
>
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
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-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

🔗monz <joemonz@yahoo.com>

1/24/2002 4:09:58 AM

> From: jpehrson2 <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, January 23, 2002 6:48 PM
> Subject: [tuning] Sims notation
>
>
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> /tuning/topicId_32971.html#33084
>
> >
> > Having said that, I don't think you'll find a lot of score-readers
> > who are particularly fond of Sims' "square-root sign".
> >
>
> I almost hate to say something "good" about the square root signs
> but, actually, there *is* something. They are clearly the *largest*
> signs for the largest "deviating" interval...

And George speculated that Sims possibly invented these symbols
from a computer keyboard, which Paul (correctly) said was wrong.

Sims decided to use the square-root symbol for quarter-tones
because the "ratio" of a quarter-tone is the square-root of
the "ratio" of the semitone. To "divide a semitone in half"
aurally, you take the square-root :

[ 2^(1/12) ]^(1/2) = 2^(1/24) .

This is documented either in Sims's _Computer Music Journal_
(1987) article, or the one a few years later in _Perspectives
of New Music_ ... I can't remember which.

-monz

_________________________________________________________
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Get your free @yahoo.com address at http://mail.yahoo.com

🔗jpehrson2 <jpehrson@rcn.com>

1/24/2002 5:58:33 AM

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

/tuning/topicId_32971.html#33121

>
> And George speculated that Sims possibly invented these symbols
> from a computer keyboard, which Paul (correctly) said was wrong.
>
>
> Sims decided to use the square-root symbol for quarter-tones
> because the "ratio" of a quarter-tone is the square-root of
> the "ratio" of the semitone. To "divide a semitone in half"
> aurally, you take the square-root :
>
> [ 2^(1/12) ]^(1/2) = 2^(1/24) .
>
>
> This is documented either in Sims's _Computer Music Journal_
> (1987) article, or the one a few years later in _Perspectives
> of New Music_ ... I can't remember which.
>
>
***I believe this is also in an early issue of Xenharmonicon... I'll
find the citation later...

JP

🔗robert_wendell <rwendell@cangelic.org>

1/24/2002 8:37:16 AM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> /tuning/topicId_32971.html#33109
>
> > --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> >
> > > So, 53-tET closes and is a "circle" and Pythagorean is open,
> hence a "chain..."
> >
> > Yup. But not a "bicycle chain".
>
> ****Right... so that makes the terminology a little confusing...
>
> >
> > > And, all *temperaments* based upon a constant generator
> > > are "circles..."
> >
> > Nope, only equal temperaments are. There are also open *linear
> > temperaments", such as most forms of meantone, or MIRACLE as
> generated by the "official" secor, or most of the temperaments that
> Dave Keenan, Graham Breed, and Gene Ward Smith have put
forward . . .
>
> ****Wow. That's an important distinction, Paul... Of course, even
> thought the generator in, say, MIRACLE is constant the step sizes
are
> different and it *never* closes unless another 33 cent step
> is "thrown in" if I understand it correctly.
>
>
> > Maybe the confusion was due to the fact that bicycle chains,
unlike
> > most kind of chains, actually close on themselves and form a
> circle . . .
>
> Yes, that and the fact that I momentarily thought that 53-tET and
> Pythagorean were synonymous... which I guess they are almost in
> *sound* but not in "concept..."
>
> Thanks!
>
> Joseph

Bob W.:
Pythagorean and 53-EDO almost the same in sound? How so? 53-EDO is
essentially JI within a couple of cents for any 5-limit interval.
The major thirds in Pythagorean are 21.5 cents sharp! How close is
that?!

🔗gdsecor <gdsecor@yahoo.com>

1/24/2002 10:28:37 AM

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> And George speculated that Sims possibly invented these symbols
> from a computer keyboard, which Paul (correctly) said was wrong.
>
> Sims decided to use the square-root symbol for quarter-tones
> because the "ratio" of a quarter-tone is the square-root of
> the "ratio" of the semitone. To "divide a semitone in half"
> aurally, you take the square-root :
>
> [ 2^(1/12) ]^(1/2) = 2^(1/24) .
>
> This is documented either in Sims's _Computer Music Journal_
> (1987) article, or the one a few years later in _Perspectives
> of New Music_ ... I can't remember which.
>
> -monz

Hi, Joe!

I surmised that the computer got in on the act, if not initally, then
somewhere along the line, supposing that the symbols might have gone
through some sort of subsequent, if slight, modification. Having,
seen this:

https://www.mindeartheart.org/micro.html

in which I noticed that the 1/4-up symbol is not quite the same as an
inverted 1/4-down (square root) symbol, I wondered why, concluding
that the most likely explanation was that both of these occurred in
an existing character set of some sort.

Now that the not-so-obvious meaning of the 1/4-down symbol has been
revealed, would you (or anyone else) happen to know anything about
the origin of the 1/4-up symbol?

--George

🔗jpehrson2 <jpehrson@rcn.com>

1/24/2002 11:05:46 AM

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:

/tuning/topicId_32971.html#33127

>
> Bob W.:
> Pythagorean and 53-EDO almost the same in sound? How so? 53-EDO is
> essentially JI within a couple of cents for any 5-limit interval.
> The major thirds in Pythagorean are 21.5 cents sharp! How close is
> that?!

Hi Bob!

There must be something I'm seriously misunderstanding here... I
thought that 53-tET was *very* similar to Pythagorean tuning.

So how is it possible to have Just 5-limit intervals??

I'm not getting that...

Help, Bob or somebody!

JP

🔗jpehrson2 <jpehrson@rcn.com>

1/24/2002 11:09:38 AM

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

/tuning/topicId_32971.html#33129

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > And George speculated that Sims possibly invented these symbols
> > from a computer keyboard, which Paul (correctly) said was wrong.
> >
> > Sims decided to use the square-root symbol for quarter-tones
> > because the "ratio" of a quarter-tone is the square-root of
> > the "ratio" of the semitone. To "divide a semitone in half"
> > aurally, you take the square-root :
> >
> > [ 2^(1/12) ]^(1/2) = 2^(1/24) .
> >
> > This is documented either in Sims's _Computer Music Journal_
> > (1987) article, or the one a few years later in _Perspectives
> > of New Music_ ... I can't remember which.
> >
> > -monz
>
> Hi, Joe!
>
> I surmised that the computer got in on the act, if not initally,
then
> somewhere along the line, supposing that the symbols might have
gone
> through some sort of subsequent, if slight, modification. Having,
> seen this:
>
> https://www.mindeartheart.org/micro.html
>
> in which I noticed that the 1/4-up symbol is not quite the same as
an
> inverted 1/4-down (square root) symbol, I wondered why, concluding
> that the most likely explanation was that both of these occurred in
> an existing character set of some sort.
>
> Now that the not-so-obvious meaning of the 1/4-down symbol has been
> revealed, would you (or anyone else) happen to know anything about
> the origin of the 1/4-up symbol?
>
> --George

Hello George...

Personally I think it *was* an attempt to "invert" the 1/4 tone
symbol. Just the flag at the top goes the other way. Perhaps it was
to make the most of the difference in symmetry that is shown even
better in the ASCII symbols we have adopted for that:

[ = quarter tone down
] = quarter tone up

That's still a conjecture. Maybe somebody can contact Sims who knows
him... Monzo??

JP

🔗Kraig Grady <kraiggrady@anaphoria.com>

1/24/2002 11:14:26 AM

The 5 limit interval and the pythagorean are two different sizes c-Fb as opposed to c-e i
believe?!

jpehrson2 wrote:

> --- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
>
> /tuning/topicId_32971.html#33127
>
> >
> > Bob W.:
> > Pythagorean and 53-EDO almost the same in sound? How so? 53-EDO is
> > essentially JI within a couple of cents for any 5-limit interval.
> > The major thirds in Pythagorean are 21.5 cents sharp! How close is
> > that?!
>
> Hi Bob!
>
> There must be something I'm seriously misunderstanding here... I
> thought that 53-tET was *very* similar to Pythagorean tuning.
>
> So how is it possible to have Just 5-limit intervals??
>
> I'm not getting that...
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

🔗genewardsmith <genewardsmith@juno.com>

1/24/2002 11:51:10 AM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> There must be something I'm seriously misunderstanding here... I
> thought that 53-tET was *very* similar to Pythagorean tuning.

It is.

> So how is it possible to have Just 5-limit intervals??

They are the schismic temperament intervals--instead of going up two tones to get an approximate 5/4, go down four to get an approximate
5/8.

🔗alternativetuning <alternativetuning@yahoo.com>

1/24/2002 11:54:07 AM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> >
> ***I believe this is also in an early issue of Xenharmonicon...
I'll
> find the citation later...
>
> JP

That's Xenharmonikon with a "k".

🔗jpehrson2 <jpehrson@rcn.com>

1/24/2002 11:58:28 AM

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:

/tuning/topicId_32971.html#33133

> The 5 limit interval and the pythagorean are two different sizes c-
Fb as opposed to c-e i
> believe?!
>

Oh... Thanks, Kraig!

I remember something about that now.

But wouldn't *both* Pythagorean *and* 53-tET have that feature??

signed,

Confused...

JP

🔗jpehrson2 <jpehrson@rcn.com>

1/24/2002 11:59:43 AM

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

/tuning/topicId_32971.html#33137

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > >
> > ***I believe this is also in an early issue of Xenharmonicon...
> I'll
> > find the citation later...
> >
> > JP
>
> That's Xenharmonikon with a "k".

Thanks, man... only in Amerika... :)

By the way, who *is* you??

JP

🔗monz <joemonz@yahoo.com>

1/24/2002 12:05:01 PM

Hi Bob,

> From: robert_wendell <rwendell@cangelic.org>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, January 24, 2002 8:37 AM
> Subject: [tuning] Re: finally, I get it "circle" vs. "chain" for us
"squares"
>
>

> Bob W.:
> Pythagorean and 53-EDO almost the same in sound? How so? 53-EDO is
> essentially JI within a couple of cents for any 5-limit interval.

This is true ... but ...

> The major thirds in Pythagorean are 21.5 cents sharp! How close is
> that?!

But you're looking at it the wrong way. 53-EDO maps JI as a
*schismic* temperament, where the "major 3rd" is derived from
8192/6561 [== 3^-8, i.e., the Pythagorean "diminished 4th"]
and not [== 3^4, the Pythagorean "major 3rd"].

The reason why 53-EDO emulates *both* Pythagorean and JI
nearly equally as well, is because first of all the Pythagorean
and syntonic commas are nearly the same size, and secondly
53-EDO's step-size is nearly midway between both of those:

Pythagorean comma = ~23.46001038 cents
> ~0.818500951 cent
2^(1/53) = ~22.64150943 cents
> ~1.135219837 cents
syntonic comma = ~21.5062896 cents

As for the two different mappings of "major 3rds" in 53-EDO:

prime-factor ratio ~cents

JI/schismic: 3^-8 = 8192/6561 = 384.3599931
53-EDO JI: 2^(17/53) = 384.9056604

Pythagorean: 3^4 = 81/64 = 407.8200035
53-EDO Pyth.: 2^(18/53) = 407.5471698

Pretty darn close to both.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗monz <joemonz@yahoo.com>

1/24/2002 12:10:24 PM

Hi George,

> From: gdsecor <gdsecor@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, January 24, 2002 10:28 AM
> Subject: [tuning] Re: Sims notation
>
>

> -- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > Sims decided to use the square-root symbol for quarter-tones
> > because the "ratio" of a quarter-tone is the square-root of
> > the "ratio" of the semitone. To "divide a semitone in half"
> > aurally, you take the square-root :
> >
> > [ 2^(1/12) ]^(1/2) = 2^(1/24) .
>
>
> Now that the not-so-obvious meaning of the 1/4-down symbol has been
> revealed, would you (or anyone else) happen to know anything about
> the origin of the 1/4-up symbol?

He simply made a modification of the square-root (radical) sign
so that it would point upward for the opposite modification.

That's all it was. This is documented in Sims's own article as well.

I posted something here last Spring about the process by
which Sims derived his tuning :
/tuning/topicId_22968.html#23086?expand=1
> 23086 From: monz <joemonz@y...>
> Date: Fri May 18, 2001 2:33am
> Subject: Re: Miracle/Canasta 72-tet

There's info there about Franz Richter Herf's theories too.
His name has also come up recently in connection with 72-EDO.

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗jpehrson2 <jpehrson@rcn.com>

1/24/2002 12:37:14 PM

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

/tuning/topicId_32971.html#33140

> >
>
> > Bob W.:
> > Pythagorean and 53-EDO almost the same in sound? How so? 53-EDO is
> > essentially JI within a couple of cents for any 5-limit interval.
>
> This is true ... but ...
>
>
> > The major thirds in Pythagorean are 21.5 cents sharp! How close is
> > that?!
>
>
> But you're looking at it the wrong way. 53-EDO maps JI as a
> *schismic* temperament, where the "major 3rd" is derived from
> 8192/6561 [== 3^-8, i.e., the Pythagorean "diminished 4th"]
> and not [== 3^4, the Pythagorean "major 3rd"].
>
>
> The reason why 53-EDO emulates *both* Pythagorean and JI
> nearly equally as well, is because first of all the Pythagorean
> and syntonic commas are nearly the same size, and secondly
> 53-EDO's step-size is nearly midway between both of those:
>
> Pythagorean comma = ~23.46001038 cents
> > ~0.818500951 cent
> 2^(1/53) = ~22.64150943 cents
> > ~1.135219837 cents
> syntonic comma = ~21.5062896 cents
>
>
>
> As for the two different mappings of "major 3rds" in 53-EDO:
>
> prime-factor ratio ~cents
>
> JI/schismic: 3^-8 = 8192/6561 = 384.3599931
> 53-EDO JI: 2^(17/53) = 384.9056604
>
> Pythagorean: 3^4 = 81/64 = 407.8200035
> 53-EDO Pyth.: 2^(18/53) = 407.5471698
>
>
> Pretty darn close to both.
>

****Thanks, Monz... I think this pretty much answers my questions.

This is fascinating. Might be worth a 53-tET webpage *all its
own*... I was looking for that over at "your place" and didn't see
anything like that...

Joe

🔗jpehrson2 <jpehrson@rcn.com>

1/24/2002 12:42:59 PM

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

/tuning/topicId_32971.html#33140

So, I have yet *another* question on the "distinction" between 53-tET
and Pythagorean.

Apparently, because of the 53-tET step size and finding this
additional not Fb we get pretty accurate 5-limit major thirds.

HOWEVER, didn't Margo Schulter describe a process where the
schismatic thirds were found and utilized in *pure* Pythagorean??

In that case, then, there would really be *still* no difference
between Pythagorean and 53-tET... ??

Where did I get off??

JP

🔗clumma <carl@lumma.org>

1/24/2002 1:14:03 PM

>>While 19-tET sounds just great for common-practice diatonic music,
>
> ****Paul... why is this again. Is it because both 12-tET and 19-
>tET are *meantones?* Help. I need to know this.

The best way to get to know it is to try composing diatonic music
in these two tunings!

-C.

🔗monz <joemonz@yahoo.com>

1/24/2002 1:27:19 PM

> From: jpehrson2 <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, January 24, 2002 12:42 PM
> Subject: [tuning] yet another question [Pythag vs. 53-tET]
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> /tuning/topicId_32971.html#33140
>
> So, I have yet *another* question on the "distinction" between 53-tET
> and Pythagorean.
>
> Apparently, because of the 53-tET step size and finding this
> additional not Fb we get pretty accurate 5-limit major thirds.
>
> HOWEVER, didn't Margo Schulter describe a process where the
> schismatic thirds were found and utilized in *pure* Pythagorean??

Yes. According to Mark Lindley (_Mathematical Models of Musical
Scales: A New Approach_) there is evidence that Pythagorean
tuning was used in the 1300-1400s (Margo can give more accurate
dates) in such a way that the "diminished 4th" ("Fb" if 1/1="C")
served as the "major 3rd", creating in effect a schismic temperament.

As you can see from the data I posted:

> JI/schismic: 3^-8 = 8192/6561 = 384.3599931
> 53-EDO JI: 2^(17/53) = 384.9056604

both the Pythgorean "diminished 4th" *and* 53-EDO are quite close
to the 5-limit "major 3rd" with ratio 5:4 = ~386.3137139 cents.
The difference between 5/4 and 8192/6561 is the *skhisma*

http://www.ixpres.com/interval/dict/schisma.htm
(definition #2)

hence the name of the temperament. (Ellis deliberately spelled
it "skhisma", but "schisma" is more common.)

Also note what I have to say about 12-EDO here:

http://www.ixpres.com/interval/dict/schismic.htm

>> The standard 12-EDO tuning is unique in that it is both a
>> schismic (-1 schisma) and a meantone (-1/11-comma) temperament,
>> meaning that it equates 5:4 [== 5^1] with both 81:64 [== 3^4]
>> and 8192:6561 [== 3^-8].

Understanding this is the key to understanding what Wilson
is talking about here:

"On the Development of Intonational Systems by Extended Linear Mapping"
http://www.anaphoria.com/xen3b.PDF

Wilson prefers to keep his theory generalized so that it may work
with temperaments as well as Pythagorean/JI/extended-JI, so I'll
raise "gen" (for "generator") to various powers here, instead of
using 3 as the base.

Wilson is saying that *because* of the fact that Western culture
has been using 12-EDO, which can map the "major 3rd" to *either*
gen^4 or gen^-8, we're *used* to the "major 3rd" being interpreted
as either of those linear mappings.

The standard paradigm for several centuries now has been to map
the "major 3rd" to gen^4, but Wilson sees it as advantageous that
we already use a scale that may also map it to gen^-8, thus
leaving the door open for accepting other tunings which also
do this.

If I've misinterpreted Wilson here (entirely possible), then
someone who knows better should please speak up!
(Kriag?, Paul?, Carl?, John Chalmers?)

See also Graham's "schismic temperament" page:
http://x31eq.com/schismic.htm

> In that case, then, there would really be *still* no difference
> between Pythagorean and 53-tET... ??
>
> Where did I get off??

Joe, be careful. There *is* a difference between 53-EDO and
Pythagorean, albeit an extremely tiny one.

53-EDO is calculated as roots of prime-factor 2, and Pythagorean
is combinations of powers of prime-factors 2 and 3. Because of
the Fundamental Theorem of Arithmetic

http://www.ixpres.com/interval/dict/fundo.htm

http://mathworld.wolfram.com/FundamentalTheoremofArithmetic.html

we know that primes are incommensurate: no non-zero power of any
prime will ever exactly equal any non-zero power of any other prime.
(Zero powers always equal one.)

So there is no way that 53-EDO and Pythagorean can ever be *exactly*
the same. But they have many intervals that are extremely close in
size. You're getting confused because you're ignoring those tiny
differences.

As for your previous post,

> This is fascinating. Might be worth a 53-tET webpage *all its
> own*... I was looking for that over at "your place" and didn't
> see anything like that...

Absolutely, 53-EDO definitely deserves a page of its own!

I was more interested in 53-EDO back when I was more heavily
into JI ... since getting on the tuning list and becoming
more interested in temperaments, it's been on the back burner.

I have a heavy interest in the "common-practice" repertoire,
specifically Mozart, Beethoven, and Mahler. You have to base
this repertoire on meantone and not Pythag/JI, so these days
I'm more interested in 19-, 31- and 55-EDO.

Hopefully I'll get around to making the 53-EDO page.
Actually, this post and my last one are probably already
a good start! When I have time I'll turn them into a webpage.

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗paulerlich <paul@stretch-music.com>

1/24/2002 2:31:05 PM

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

> Yes. According to Mark Lindley (_Mathematical Models of Musical
> Scales: A New Approach_) there is evidence that Pythagorean
> tuning was used in the 1300-1400s (Margo can give more accurate
> dates) in such a way that the "diminished 4th" ("Fb" if 1/1="C")
> served as the "major 3rd", creating in effect a schismic
temperament.

I believe the correct dates are more like 1420-1480.

🔗dkeenanuqnetau <d.keenan@uq.net.au>

1/24/2002 3:58:23 PM

> > > --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > >
> > > > So, 53-tET closes and is a "circle" and Pythagorean is open,
> > hence a "chain..."
> > >
> > > Yup. But not a "bicycle chain".
> >
> > ****Right... so that makes the terminology a little confusing...

I guess it does. We also have daisy-chains. I think they are usually
closed. But then we have chains of events. Those are definitely not
closed. I understand this usage of "chain" (open) is standard in graph
theory. But graph theory isn't music theory.

Maybe, since they are called "linear" temperaments we should call them
"lines" of generators. What else has been used in the literature. What
does Erv Wilson call them?

🔗Kraig Grady <kraiggrady@anaphoria.com>

1/24/2002 3:58:38 PM

joepehr!
I think it was helmholtz who noticed and as G smith pointed out that if you go down to Fb in
3/2 it is a almost a 5/4. a schisma difference. The persians it appears were already aware of this
hence also the north indians

jpehrson2 wrote:

>
> Oh... Thanks, Kraig!
>
> I remember something about that now.
>
> But wouldn't *both* Pythagorean *and* 53-tET have that feature??
>
> signed,
>
> Confused...
>
> JP
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

🔗jpehrson2 <jpehrson@rcn.com>

1/24/2002 5:31:20 PM

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

/tuning/topicId_32971.html#33155

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > Yes. According to Mark Lindley (_Mathematical Models of Musical
> > Scales: A New Approach_) there is evidence that Pythagorean
> > tuning was used in the 1300-1400s (Margo can give more accurate
> > dates) in such a way that the "diminished 4th" ("Fb" if 1/1="C")
> > served as the "major 3rd", creating in effect a schismic
> temperament.
>
> I believe the correct dates are more like 1420-1480.

OK...

So now Monz and others have shown that *both* Pythagorean *and* 53-
tET can make very close to pure 3-limit and 5-limit intervals.

So, then why does Bob Wendell say they are so different...???

I hope he's not smokin' somethin' or somethin' :)

JP

🔗paulerlich <paul@stretch-music.com>

1/24/2002 5:39:18 PM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> OK...
>
> So now Monz and others have shown that *both* Pythagorean *and* 53-
> tET can make very close to pure 3-limit and 5-limit intervals.
>
> So, then why does Bob Wendell say they are so different...???
>
> I hope he's not smokin' somethin' or somethin' :)
>
> JP

I hope he is, and is enjoying it! :)

Seriously, Bob may have been thinking 53 in the "5-limit JI" sense,
that is, following the example of 53-tET theorists such as Ellis and
Tanaka, but not of Mercator and Philolaus, while he was thinking of
Pythagorean in the "diatonic" sense, i.e., the Western sense from 800
or earlier through 1420, but not the 1420-1480 sense, nor the
Medieval Arabic sense, in which schisma seems to have been exploited.
Diatonic triads in Pythagorean are rather harsh, having 408-cent
major thirds, while schismic triads in Pythagorean are quite smooth.
Scale theory suggest scales such as 7-tone ones for the diatonic
(Pythagorean or otherwise) case, while 17-tone scales, for example,
look good in the schismic case (as Wilson seems to have more than
picked up on). More on this can be directed to tuning-
math@yahoogroups.com

🔗jpehrson2 <jpehrson@rcn.com>

1/24/2002 5:47:33 PM

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

/tuning/topicId_32971.html#33160

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > OK...
> >
> > So now Monz and others have shown that *both* Pythagorean *and*
53-
> > tET can make very close to pure 3-limit and 5-limit intervals.
> >
> > So, then why does Bob Wendell say they are so different...???
> >
> > I hope he's not smokin' somethin' or somethin' :)
> >
> > JP
>
> I hope he is, and is enjoying it! :)
>
> Seriously, Bob may have been thinking 53 in the "5-limit JI" sense,
> that is, following the example of 53-tET theorists such as Ellis
and
> Tanaka, but not of Mercator and Philolaus, while he was thinking of
> Pythagorean in the "diatonic" sense, i.e., the Western sense from
800
> or earlier through 1420, but not the 1420-1480 sense, nor the
> Medieval Arabic sense, in which schisma seems to have been
exploited.
> Diatonic triads in Pythagorean are rather harsh, having 408-cent
> major thirds, while schismic triads in Pythagorean are quite
smooth.
> Scale theory suggest scales such as 7-tone ones for the diatonic
> (Pythagorean or otherwise) case, while 17-tone scales, for example,
> look good in the schismic case (as Wilson seems to have more than
> picked up on). More on this can be directed to tuning-
> math@y...

Hi Paul. Got it.

That must be the answer to the puzzle! We're all smokin' now! :)

JP

🔗Kraig Grady <kraiggrady@anaphoria.com>

1/24/2002 6:29:47 PM

-But the man is dead!
-Continue with the lashes, Mr. Christian!

>
>
> I hope he is, and is enjoying it! :)

🔗jpehrson2 <jpehrson@rcn.com>

1/24/2002 6:58:59 PM

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:

/tuning/topicId_32971.html#33163

> -But the man is dead!
> -Continue with the lashes, Mr. Christian!
>
> >
> >
> > I hope he is, and is enjoying it! :)

ohmygod... a film that I actually *saw!*

That almost *never* happens...

JP

🔗D.Stearns <STEARNS@CAPECOD.NET>

1/24/2002 11:14:19 PM

There's evidence that indicates that some just sonorities via
schismatic spellings were edited out of Ives.

Here's a fascinating bit from an exchange I had with David Porter of
the Ives Society:

"Last year I spent several months going over the "new" (22 years old)
critical editions of Tone Roads Nos. 1 & 3 and Halloween. In TR #1
and Halloween, John Kirkpatrick had respelled many of Ives's notes to
make them more "practical" and more "correct" harmonically. The
reason I spent so much time undoing all that foolishness wa to make
these Old editions in line with our new Ives Society guidelines. (We
now preseve Ives's spelling, rhythms, and have a consistent format for
describing details of the editing.)

Ives himself states his thinking in an Appendix to his "Memos"--flats
represent repose and rest, sharps represent activity. I've seen
several places where a first draft uses flats, but the later and final
drafts use sharps. C# is nearer to D than to C, etc., Db closer to C
than to D. Hence that descending "triad" in "West London" b-ab-e.
The "3rd" ab should be flatter than a "proper" g#."

This makes an interesting addendum to Johnny Reinhard's extended
Pythagorean interpretation for Ives' spelling-tuning template, and for
anyone wanting to realize these implications in Ives, these recent
critical editions (that preserve Ives' original spellings, et al) will
be an invaluable addition to an oeuvre that still has some
surprisingly fresh tricks up its sleeve.

--Dan Stearns

----- Original Message -----
From: "monz" <joemonz@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Thursday, January 24, 2002 1:27 PM
Subject: Re: [tuning] yet another question [Pythag vs. 53-tET]

>
> > From: jpehrson2 <jpehrson@rcn.com>
> > To: <tuning@yahoogroups.com>
> > Sent: Thursday, January 24, 2002 12:42 PM
> > Subject: [tuning] yet another question [Pythag vs. 53-tET]
> >
> >
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > /tuning/topicId_32971.html#33140
> >
> > So, I have yet *another* question on the "distinction" between
53-tET
> > and Pythagorean.
> >
> > Apparently, because of the 53-tET step size and finding this
> > additional not Fb we get pretty accurate 5-limit major thirds.
> >
> > HOWEVER, didn't Margo Schulter describe a process where the
> > schismatic thirds were found and utilized in *pure* Pythagorean??
>
>
> Yes. According to Mark Lindley (_Mathematical Models of Musical
> Scales: A New Approach_) there is evidence that Pythagorean
> tuning was used in the 1300-1400s (Margo can give more accurate
> dates) in such a way that the "diminished 4th" ("Fb" if 1/1="C")
> served as the "major 3rd", creating in effect a schismic
temperament.
>
>
> As you can see from the data I posted:
>
> > JI/schismic: 3^-8 = 8192/6561 = 384.3599931
> > 53-EDO JI: 2^(17/53) = 384.9056604
>
> both the Pythgorean "diminished 4th" *and* 53-EDO are quite close
> to the 5-limit "major 3rd" with ratio 5:4 = ~386.3137139 cents.
> The difference between 5/4 and 8192/6561 is the *skhisma*
>
> http://www.ixpres.com/interval/dict/schisma.htm
> (definition #2)
>
> hence the name of the temperament. (Ellis deliberately spelled
> it "skhisma", but "schisma" is more common.)
>
>
> Also note what I have to say about 12-EDO here:
>
> http://www.ixpres.com/interval/dict/schismic.htm
>
> >> The standard 12-EDO tuning is unique in that it is both a
> >> schismic (-1 schisma) and a meantone (-1/11-comma) temperament,
> >> meaning that it equates 5:4 [== 5^1] with both 81:64 [== 3^4]
> >> and 8192:6561 [== 3^-8].
>
>
> Understanding this is the key to understanding what Wilson
> is talking about here:
>
> "On the Development of Intonational Systems by Extended Linear
Mapping"
> http://www.anaphoria.com/xen3b.PDF
>
>
> Wilson prefers to keep his theory generalized so that it may work
> with temperaments as well as Pythagorean/JI/extended-JI, so I'll
> raise "gen" (for "generator") to various powers here, instead of
> using 3 as the base.
>
> Wilson is saying that *because* of the fact that Western culture
> has been using 12-EDO, which can map the "major 3rd" to *either*
> gen^4 or gen^-8, we're *used* to the "major 3rd" being interpreted
> as either of those linear mappings.
>
> The standard paradigm for several centuries now has been to map
> the "major 3rd" to gen^4, but Wilson sees it as advantageous that
> we already use a scale that may also map it to gen^-8, thus
> leaving the door open for accepting other tunings which also
> do this.
>
>
> If I've misinterpreted Wilson here (entirely possible), then
> someone who knows better should please speak up!
> (Kriag?, Paul?, Carl?, John Chalmers?)
>
>
> See also Graham's "schismic temperament" page:
> http://x31eq.com/schismic.htm
>
>
>
> > In that case, then, there would really be *still* no difference
> > between Pythagorean and 53-tET... ??
> >
> > Where did I get off??
>
>
> Joe, be careful. There *is* a difference between 53-EDO and
> Pythagorean, albeit an extremely tiny one.
>
> 53-EDO is calculated as roots of prime-factor 2, and Pythagorean
> is combinations of powers of prime-factors 2 and 3. Because of
> the Fundamental Theorem of Arithmetic
>
> http://www.ixpres.com/interval/dict/fundo.htm
>
> http://mathworld.wolfram.com/FundamentalTheoremofArithmetic.html
>
> we know that primes are incommensurate: no non-zero power of any
> prime will ever exactly equal any non-zero power of any other prime.
> (Zero powers always equal one.)
>
> So there is no way that 53-EDO and Pythagorean can ever be *exactly*
> the same. But they have many intervals that are extremely close in
> size. You're getting confused because you're ignoring those tiny
> differences.
>
>
> As for your previous post,
>
> > This is fascinating. Might be worth a 53-tET webpage *all its
> > own*... I was looking for that over at "your place" and didn't
> > see anything like that...
>
> Absolutely, 53-EDO definitely deserves a page of its own!
>
> I was more interested in 53-EDO back when I was more heavily
> into JI ... since getting on the tuning list and becoming
> more interested in temperaments, it's been on the back burner.
>
> I have a heavy interest in the "common-practice" repertoire,
> specifically Mozart, Beethoven, and Mahler. You have to base
> this repertoire on meantone and not Pythag/JI, so these days
> I'm more interested in 19-, 31- and 55-EDO.
>
> Hopefully I'll get around to making the 53-EDO page.
> Actually, this post and my last one are probably already
> a good start! When I have time I'll turn them into a webpage.
>
>
>
> -monz
>
>
>
>
>
>
>
> _________________________________________________________
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> Get your free @yahoo.com address at http://mail.yahoo.com
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>
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🔗monz <joemonz@yahoo.com>

1/24/2002 8:32:13 PM

Hi Dan,

Thanks for a FANTASTIC post!

More below ...

> From: D.Stearns <STEARNS@CAPECOD.NET>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, January 24, 2002 11:14 PM
> Subject: Re: [tuning] yet another question [Pythag vs. 53-tET]
>
>
> There's evidence that indicates that some just sonorities via
> schismatic spellings were edited out of Ives.
>
> Here's a fascinating bit from an exchange I had with David Porter of
> the Ives Society:
>
>
> "Last year I spent several months going over the "new" (22 years old)
> critical editions of Tone Roads Nos. 1 & 3 and Halloween. In TR #1
> and Halloween, John Kirkpatrick had respelled many of Ives's notes to
> make them more "practical" and more "correct" harmonically. The
> reason I spent so much time undoing all that foolishness wa to make
> these Old editions in line with our new Ives Society guidelines. (We
> now preseve Ives's spelling, rhythms, and have a consistent format for
> describing details of the editing.)

This is really terrific. I'm glad that someone is re-correcting
the spelling of Ives's music! I've mentioned that the "Critical Edition"
of Mahler scores has done the same thing with his music too --
mainly changing double-sharps to their equivalent 12-EDO "naturals".
This is a sacrilege, given that I now know that Mahler lamented
the demise of meantone. Those double-sharps have a specific
meaning which is completely obscured by this modern respelling.

> Ives himself states his thinking in an Appendix to his "Memos"--flats
> represent repose and rest, sharps represent activity.

Wow -- to me, this *really* seems to support Johnny Reinhard's
belief that Ives had a "preferred tuning" and that it was Pythagorean.

Certainly, Dan, I agree with you that Ives's music is capable
of encompassing several different kinds of tuning systems simultaneously.
But this comment by the man himself shows that his general intonational
thinking leaned towards Pythagorean. In any case, it certainly
argues *against* 12-EDO!

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗genewardsmith <genewardsmith@juno.com>

1/24/2002 8:34:16 PM

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

> Hopefully I'll get around to making the 53-EDO page.
> Actually, this post and my last one are probably already
> a good start! When I have time I'll turn them into a webpage.

You could link to my Orwell piece also if you like. An exploration of 53-et should include its more important linear temperaments, BTW.

🔗M. Schulter <MSCHULTER@VALUE.NET>

1/24/2002 9:17:51 PM

Hello, there, Paul, and I wanted quickly to respond to a couple of
points in your last response.

First, your point about Jonathan Walker and the management of commas
by a vocal ensemble or the like as a form of _musica ficta_, in
effect, might tie in with the comment in my post about flexible pitch
ensembles where a tuning that observes a comma dispersed in the
composer's intended framework might raise less of a problem. We're
likely talking about adaptive tuning, which I much agree could be
relevant either for classic JI (in practice), or for a system like
72-EDO.

Secondly, thank you for mentioning 25:21, a ratio I hadn't considered,
and had a bit of fun trying to guesstimate as 5:4 less 21:20, giving
something close to 300 cents, or exactly that in a 72-EDO version of
the secorian temperament, as I call it (in lower case, since there are
a number of Secorian temperaments, but not all generated by secors).

An interesting thing about the 6:5 vs. 25:21 question is that while
6:5 is the simpler ratio, 21:25 if presented as part of 42:50:63 would
approximate 16:19 in 16:19:24, possibly actually having a more
"rooted" or conclusive effect than 10:12:15.

Anyway, I agree that while mapping both the secorian types we're
considering to a 310-cent interval in 1/4-comma meantone (or something
very slightly smaller, I guess, in 31-EDO, actually still a rounded
310 according to Scala) could be "tolerable," it wouldn't necessarily
be optimal for a composer who wants either a somewhat more "impure"
minor third, or possible a more "rooted" one in connection with the
fifth (the near-16:19:24 effect).

This nuance reminds me a bit about my own liking for the alternative
or supraminor third in 29-EDO at 331 cents, very close to 23:19.
Mapping it to either 310 cents or 348 cents in 31-EDO seems to me not
the most accurate representation; in 72-EDO, or 36-EDO for that
matter, I'd be happy with 333.33 cents.

A curiosity: 25:21 is the mediant of 6:5 and 19:16, while 23:19 is the
mediant of 6:5 and 17:14.

Anyway, thank you for your comments, to which I hope this is a fair
response.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗graham@microtonal.co.uk

1/25/2002 3:11:00 AM

In-Reply-To: <3C509FAD.5F1A470A@anaphoria.com>
Kraig Grady wrote:

> I think it was helmholtz who noticed and as G smith pointed out
> that if you go down to Fb in
> 3/2 it is a almost a 5/4. a schisma difference. The persians it appears
> were already aware of this
> hence also the north indians

Helmholtz/Ellis does describe 5-limit schismic temperament. I think
that's in Helmholtz's bit.

I currently have Wright's "The Modal System of Arab and Persian Music A D
1250-1300" on interlibrary loan. It gives more evidence for schismic
temperament. Both Safi ud-Din and Qutb al-Din gave string ratios along
with positions on the Pythagorean chain/string/whatever. The scales we
suspected of being schismic are written as 5-limit. However, not all
scales are schismically consistent.

Safi ud-Din does give the octave-based schismic scales as a single
category, however. In fact, these are 7-limit schismic. He divides all
scales into two groups, only the first of which are described on the lute
('oud, whatever). These are then divided into three, which we can
describe as

1) Pythagorean diatonics

2) Schismic scales.

3) Miscellaneous scales defined on a vague 17 note scale.

The highest tones on some of (2) are divided 21:20 and 15:14. The status
of the division point in the Pythagorean system isn't clear, but seems to
agree with the usual schismic interpretation. All of (2) are octave based
scales, but an additional scale covering a seventh is considered "related"
to it. This scale is not schismically correct as described.

A lot of (3) contain intervals from 11 and higher limits. There's some
consistency of spelling, but lots of exceptions. In fact, they aren't
even consistent as 17 equal steps. Sometimes an interval is stretched ad
hoc to avoid a comma appearing in the scale. They are consistent in that
each interval is only describe to ways, which are either the same number
of steps from 17, or differ by a comma. One strange detail is 8:5 and 5:3
are usually written the same when the scale encompasses one of them.

A couple of (3) are schismically-incorrect 5-limit scales. Because they
begin with a 16:15 semitone, and all scales are described with the same
tonic, it's impossible to describe these scales with in 17 note
Pythagorean system.

So that's the theory. It appears that in practice the scales described as
5-limit wouldn't have been tuned that way. There's a passage from Safi
ud-Din that's supposed to support this. 4:5:6, being an arithmetical
division of the fifth, is taken to define neutral thirds. Make of that
what you will.

I still haven't got hold of Touma's essay that specifically deals with
this question.

Graham

🔗clumma <carl@lumma.org>

1/25/2002 12:58:03 PM

Dave Keenan wrote...
>What does Erv Wilson call them?

"Linear".

-Carl

🔗Guiseppi Mendoza <guiseppi@mendozadil.freeserve.co.uk>

1/25/2002 7:12:56 PM

On Fri, 25 Jan 2002 02:58:59 -0000, SOMEONE articulated, I'd better
not say who :

>> -But the man is dead!
>> -Continue with the lashes, Mr. Christian!
>> > I hope he is, and is enjoying it! :)
>
>ohmygod... a film that I actually *saw!*
>That almost *never* happens...

I actually almost see every film I ever want to, but I don't like them
all. My lashes are quite copious !

Touche' (in no certain terms)

"If I had twice the time, half the work done"

http://www.ampcast.com/guiseppimendoza

🔗Guiseppi Mendoza <guiseppi@mendozadil.freeserve.co.uk>

1/25/2002 7:17:33 PM

On Fri, 25 Jan 2002 20:58:03 -0000, SOMEONE articulated, I'd better
not say who :

>Dave Keenan wrote...
>>What does Erv Wilson call them?
>
>"Linear".

Straight..Simple..

"If I had twice the time, half the work done"

http://www.ampcast.com/guiseppimendoza

🔗dkeenanuqnetau <d.keenan@uq.net.au>

1/26/2002 5:07:34 PM

--- In tuning@y..., "clumma" <carl@l...> wrote:
> Dave Keenan wrote...
> >What does Erv Wilson call them?
>
> "Linear".

Carl and Guiseppi, I'm sorry my question wasn't clear. I know Erv
calls them linear temperaments. I wouldn't suggest changing that.

My question was, does Erv Wilson use any word other than "chain" in,
for example, "The common diatonic scale can be considered as a <chain>
of six meantone-fifth generators."

The problem is, as the title of this thread suggests, that some folk
tend to think that a "chain" is necessarily looped, like a bicycle
chain.

🔗robert_wendell <rwendell@cangelic.org>

1/27/2002 3:14:09 PM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
>
> /tuning/topicId_32971.html#33127
>
> >
> > Bob W.:
> > Pythagorean and 53-EDO almost the same in sound? How so? 53-EDO
is
> > essentially JI within a couple of cents for any 5-limit interval.
> > The major thirds in Pythagorean are 21.5 cents sharp! How close
is
> > that?!
>
> Hi Bob!
>
> There must be something I'm seriously misunderstanding here... I
> thought that 53-tET was *very* similar to Pythagorean tuning.
>
> So how is it possible to have Just 5-limit intervals??
>
> I'm not getting that...
>
> Help, Bob or somebody!
>
>
> JP

BobW.:
The most telling argument against 53-EDO for 5-limit approximations,
Joe, is that it's not a meantone, so you have to deal with "comma
drift".

🔗robert_wendell <rwendell@cangelic.org>

1/27/2002 3:08:54 PM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
>
> /tuning/topicId_32971.html#33127
>
> >
> > Bob W.:
> > Pythagorean and 53-EDO almost the same in sound? How so? 53-EDO
is
> > essentially JI within a couple of cents for any 5-limit interval.
> > The major thirds in Pythagorean are 21.5 cents sharp! How close
is
> > that?!
>
> Hi Bob!
>
> There must be something I'm seriously misunderstanding here... I
> thought that 53-tET was *very* similar to Pythagorean tuning.
>
> So how is it possible to have Just 5-limit intervals??
>
> I'm not getting that...
>
> Help, Bob or somebody!
>
>
> JP

Bob W.:
The 53-EDO is sometimes called the "scale of commas", since its
individual steps are very close to the average of the two most
prominent commas (i.e., the Pythagorean and syntonic commas). This
means that even though a cycle of 53 perfect fifths comes out
very close to closing, which probably implied to you that it is
almost the same as Pythagorean, all you have to do to get a major
third that is almost perfect after climbing four perfect fifths is to
drop back one step.

All the 5-limit intervals are approximated to an accuracy of + - 3
cents as I recall (shooting from seat-of-the-pants memory. The
perfect fifth, major third, and minor third, and their inversions by
implication, are well within this tolerance, maybe even less than two
cents off. Don't have my calculator handy right now.

🔗paulerlich <paul@stretch-music.com>

1/27/2002 9:00:40 PM

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:

> BobW.:
> The most telling argument against 53-EDO for 5-limit
approximations,
> Joe, is that it's not a meantone, so you have to deal with "comma
> drift".

That would then be an argument against 5-limit JI for 5-limit
approximations. Take from that what you will.

🔗Kraig Grady <kraiggrady@anaphoria.com>

1/27/2002 10:31:17 PM

Graham!
I find the first chapter on the Systematist scale quite enlightening also and hope to put it
up. Wish i had chapter 2

graham@microtonal.co.uk wrote:

> In-Reply-To: <3C509FAD.5F1A470A@anaphoria.com>
> Kraig Grady wrote:
>
> > I think it was helmholtz who noticed and as G smith pointed out
> > that if you go down to Fb in
> > 3/2 it is a almost a 5/4. a schisma difference. The persians it appears
> > were already aware of this
> > hence also the north indians
>
> Helmholtz/Ellis does describe 5-limit schismic temperament. I think
> that's in Helmholtz's bit.
>
> I currently have Wright's "The Modal System of Arab and Persian Music A D
> 1250-1300" on interlibrary loan. It gives more evidence for schismic
> temperament. Both Safi ud-Din and Qutb al-Din gave string ratios along
> with positions on the Pythagorean chain/string/whatever. The scales we
> suspected of being schismic are written as 5-limit. However, not all
> scales are schismically consistent.
>
> Safi ud-Din does give the octave-based schismic scales as a single
> category, however. In fact, these are 7-limit schismic. He divides all
> scales into two groups, only the first of which are described on the lute
> ('oud, whatever). These are then divided into three, which we can
> describe as
>
> 1) Pythagorean diatonics
>
> 2) Schismic scales.
>
> 3) Miscellaneous scales defined on a vague 17 note scale.
>
> The highest tones on some of (2) are divided 21:20 and 15:14. The status
> of the division point in the Pythagorean system isn't clear, but seems to
> agree with the usual schismic interpretation. All of (2) are octave based
> scales, but an additional scale covering a seventh is considered "related"
> to it. This scale is not schismically correct as described.
>
> A lot of (3) contain intervals from 11 and higher limits. There's some
> consistency of spelling, but lots of exceptions. In fact, they aren't
> even consistent as 17 equal steps. Sometimes an interval is stretched ad
> hoc to avoid a comma appearing in the scale. They are consistent in that
> each interval is only describe to ways, which are either the same number
> of steps from 17, or differ by a comma. One strange detail is 8:5 and 5:3
> are usually written the same when the scale encompasses one of them.
>
> A couple of (3) are schismically-incorrect 5-limit scales. Because they
> begin with a 16:15 semitone, and all scales are described with the same
> tonic, it's impossible to describe these scales with in 17 note
> Pythagorean system.
>
> So that's the theory. It appears that in practice the scales described as
> 5-limit wouldn't have been tuned that way. There's a passage from Safi
> ud-Din that's supposed to support this. 4:5:6, being an arithmetical
> division of the fifth, is taken to define neutral thirds. Make of that
> what you will.
>
> I still haven't got hold of Touma's essay that specifically deals with
> this question.
>
> Graham
>
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

🔗robert_wendell <rwendell@cangelic.org>

1/27/2002 3:13:40 PM

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
>
> /tuning/topicId_32971.html#33127
>
> >
> > Bob W.:
> > Pythagorean and 53-EDO almost the same in sound? How so? 53-EDO
is
> > essentially JI within a couple of cents for any 5-limit interval.
> > The major thirds in Pythagorean are 21.5 cents sharp! How close
is
> > that?!
>
> Hi Bob!
>
> There must be something I'm seriously misunderstanding here... I
> thought that 53-tET was *very* similar to Pythagorean tuning.
>
> So how is it possible to have Just 5-limit intervals??
>
> I'm not getting that...
>
> Help, Bob or somebody!
>
>
> JP

BobW.:
The most telling argument against 53-EDO for 5-limit approximations,
Joe, is that it's not a meantone, so you have to deal with "comma
drift".

🔗genewardsmith <genewardsmith@juno.com>

1/27/2002 9:16:47 PM

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:

> The most telling argument against 53-EDO for 5-limit approximations,
> Joe, is that it's not a meantone, so you have to deal with "comma
> drift".

Only if you insist on treating it as if it *was* meantone.

🔗bps1572ya <bps1572@mail.be>

1/28/2002 12:27:46 AM

Well,

It is very simple.

When 53 EDO is seen as an approximation to Pythagorean,
a tone is 9 steps, a chromatic semitone 5 ans a diatonic semitone 4.
The major third is two times 9 = 18 steps.

53 EDO can be used as an approximation to 5-limit JI.
In that case, a major third of 17 steps is a rather good
approximation to
a just major third.

If you don't believe it, anyone can always calculate this for himself.
I don't have time to do that now, but if anyone has trouble with it,
you can always ask me.

(You don't know me ?
That's right ! This is my first post to the list.
I am working for some weeks on EDOs and their releationship to
meantones.
When I have enough free time, maybe I can write things down and post
them here.
But that can take quite a lot of time ...)

Bart Pauwels

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
>
> /tuning/topicId_32971.html#33127
>
> >
> > Bob W.:
> > Pythagorean and 53-EDO almost the same in sound? How so? 53-EDO
is
> > essentially JI within a couple of cents for any 5-limit interval.
> > The major thirds in Pythagorean are 21.5 cents sharp! How close
is
> > that?!
>
> Hi Bob!
>
> There must be something I'm seriously misunderstanding here... I
> thought that 53-tET was *very* similar to Pythagorean tuning.
>
> So how is it possible to have Just 5-limit intervals??
>
> I'm not getting that...
>
> Help, Bob or somebody!
>
>
> JP

🔗genewardsmith <genewardsmith@juno.com>

1/28/2002 2:27:53 AM

--- In tuning@y..., "bps1572ya" <bps1572@m...> wrote:

> (You don't know me ?

Hi Bart. :)

> That's right ! This is my first post to the list.
> I am working for some weeks on EDOs and their releationship to
> meantones.
> When I have enough free time, maybe I can write things down and post
> them here.

You might post a synopsis--it's likely some of what you want to say is already well known here.

🔗graham@microtonal.co.uk

1/28/2002 4:12:00 AM

In-Reply-To: <3C54F035.BB86D40D@anaphoria.com>
Kraig Grady wrote (on the Wright book):

> I find the first chapter on the Systematist scale quite
> enlightening also and hope to put it
> up. Wish i had chapter 2

Does that mean you have chapter 1 but not the rest of the book? Would you
like a photocopy of chapter 2? If so, I'm afraid you'll want the rest of
the book as well. Later chapters include transcriptions of the musical
examples. I haven't got that far, and it has to be returned in a few
days.

Ah, perhaps it's that you have chapter 1 scanned in.

Qutb al-Din did give ratios of some kind in addition to string lengths,
and in one of the examples they disagree. But Wright only gives one
rationalisation in his list, I think the string lengths. So there is even
more information if you can find a more comprehensive source.

Do you have the Touma essay specifically mentioning the schismic
interpretation (it's in Manuel's bibliography)?

Graham

🔗paulerlich <paul@stretch-music.com>

1/28/2002 10:39:14 AM

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:

> BobW.:
> The most telling argument against 53-EDO for 5-limit
approximations,
> Joe, is that it's not a meantone, so you have to deal with "comma
> drift".

??? This would also argue against 5-limit JI for 5-limit
approximations. Which seems rather odd. "Comma drift" is already
recognized as a problem with 5-limit JI for common-practice Western
music. So the identification of 53-tET with 5-limit JI seems quite
secure.

🔗Kraig Grady <kraiggrady@anaphoria.com>

1/28/2002 4:24:46 PM

Graham

graham@microtonal.co.uk wrote:

> Does that mean you have chapter 1 but not the rest of the book?

yes that is all i copied or had time to copy. It is an extensive list of
scales in JI (if you
translate it out of Cents into ratios) some of which use 13 as a mean
subdivision of the 7/6.
Since i have instruments in Just and none in 53 you might understand my
bias when it came time to
xerox.

> Would you
> like a photocopy of chapter 2?

i will go back to UCLA and take a closer look at this. but thanks for
the offer

> If so, I'm afraid you'll want the rest of
> the book as well. Later chapters include transcriptions of the musical
> examples. I haven't got that far, and it has to be returned in a few
> days.

Fortunately i got about a hundred pages transcriptions out of Erlander
book. There is also quite a
bit on their rhythmic patterns. Some going up to 128. Makes my 79 i used
once as, well, within the
mainstream:) - not really

>
>
> Qutb al-Din did give ratios of some kind in addition to string lengths,
> and in one of the examples they disagree. But Wright only gives one
> rationalisation in his list, I think the string lengths. So there is even
> more information if you can find a more comprehensive source.
>
> Do you have the Touma essay specifically mentioning the schismic
> interpretation (it's in Manuel's bibliography)?

I have Touma's book. Is there a specific essay?

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

🔗robert_wendell <rwendell@cangelic.org>

1/28/2002 7:50:09 PM

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
>
> > BobW.:
> > The most telling argument against 53-EDO for 5-limit
> approximations,
> > Joe, is that it's not a meantone, so you have to deal with "comma
> > drift".
>
> ??? This would also argue against 5-limit JI for 5-limit
> approximations. Which seems rather odd. "Comma drift" is already
> recognized as a problem with 5-limit JI for common-practice Western
> music. So the identification of 53-tET with 5-limit JI seems quite
> secure.

Bob W.:
Yes, I realized after I posted this that it was kind of redundant and
contained an implied contradiction as well. No meantone can ever be
a very close approximation of JI and vice versa. What I was actually
trying to say was that 53-EDO's principal drawback is that, like any
other close approximation of JI, 5-limit or not (72-EDO, for
example), it presents problems with comma drift.

🔗paulerlich <paul@stretch-music.com>

1/28/2002 9:56:20 PM

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:

> What I was actually
> trying to say was that 53-EDO's principal drawback is that, like
any
> other close approximation of JI, 5-limit or not (72-EDO, for
> example), it presents problems with comma drift.

Agreed. But some people may consider adaptive JI, with melodic shifts
of no more than 6 cents, to be a sort of "close approximation of JI".
See how hairy the terminology is?

🔗graham@microtonal.co.uk

1/29/2002 2:37:00 AM

In-Reply-To: <3C55EBCD.65D869C2@anaphoria.com>
Kraig Grady wrote:

> Fortunately i got about a hundred pages transcriptions out of Erlander
> book. There is also quite a
> bit on their rhythmic patterns. Some going up to 128. Makes my 79 i used
> once as, well, within the
> mainstream:) - not really

Do you mean d'Erlanger? How understandable is that? Wright does
transcribe the notation examples into staff notation, I didn't think
d'Erlanger did. But presumably you can get all the scales out of it.

And yes, there's rhythm as well.

> I have Touma's book. Is there a specific essay?

Touma, Habib Hassan. "Basics of Ratio in Arab Music. The schismatic
Permutation of Safiyyuddin al-Urmawi", Lux Oriente. Festschrift Robert
G�nther zum 65. Geburtstag, Gustav Bosse-Verlag, Kassel, 1995.

Graham

🔗robert_wendell <rwendell@cangelic.org>

1/29/2002 10:10:06 AM

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
>
> > What I was actually
> > trying to say was that 53-EDO's principal drawback is that, like
> any
> > other close approximation of JI, 5-limit or not (72-EDO, for
> > example), it presents problems with comma drift.
>
> Agreed. But some people may consider adaptive JI, with melodic
shifts
> of no more than 6 cents, to be a sort of "close approximation of
JI".
> See how hairy the terminology is?

Bob W.:
Indeed, Paul! I was writing in the context of fixed scales, however,
which would not include adaptive JI. Of course, as I'm sure you're
well aware by now, I am a strong advocate of adaptive JI in a
cappella performance of choral literature, especially renaissance,
but we also do baroque a cappella this way, apparently over the
objections of some here of its being not stylistically appropriate.
We performed Franz Biebl's Ave Maria (1930ish, I think) using the
same approach, and it works so beautifully that it moved some people
to tears.

🔗paulerlich <paul@stretch-music.com>

1/29/2002 12:39:02 PM

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:

> Bob W.:
> Indeed, Paul! I was writing in the context of fixed scales,
however,
> which would not include adaptive JI.

Unless you considered Vicentino's Second Tuning of 1555, with 19+17
tones per octave, to be a fixed scale (hey, if you're allowed to have
two different D's in 53-tET and still call that a "fixed scale", why
not? Again, terminology shoots us in the foot).

> Of course, as I'm sure you're
> well aware by now, I am a strong advocate of adaptive JI in a
> cappella performance of choral literature, especially renaissance,
> but we also do baroque a cappella this way, apparently over the
> objections of some here of its being not stylistically appropriate.

You're speaking of the advocates of fixed Werckmeister for Bach?

> We performed Franz Biebl's Ave Maria (1930ish, I think) using the
> same approach, and it works so beautifully that it moved some
people
> to tears.

Isn't it great when we, as performers, can bring others so close to
our inner emotional spheres?

🔗jpehrson2 <jpehrson@rcn.com>

1/27/2002 8:31:37 PM

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:

/tuning/topicId_32971.html#33249

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
> >
> > /tuning/topicId_32971.html#33127
> >
> > >
> > > Bob W.:
> > > Pythagorean and 53-EDO almost the same in sound? How so? 53-EDO
> is
> > > essentially JI within a couple of cents for any 5-limit
interval.
> > > The major thirds in Pythagorean are 21.5 cents sharp! How close
> is
> > > that?!
> >
> > Hi Bob!
> >
> > There must be something I'm seriously misunderstanding here... I
> > thought that 53-tET was *very* similar to Pythagorean tuning.
> >
> > So how is it possible to have Just 5-limit intervals??
> >
> > I'm not getting that...
> >
> > Help, Bob or somebody!
> >
> >
> > JP
>
> Bob W.:
> The 53-EDO is sometimes called the "scale of commas", since its
> individual steps are very close to the average of the two most
> prominent commas (i.e., the Pythagorean and syntonic commas). This
> means that even though a cycle of 53 perfect fifths comes out
> very close to closing, which probably implied to you that it is
> almost the same as Pythagorean, all you have to do to get a major
> third that is almost perfect after climbing four perfect fifths is
to
> drop back one step.
>
> All the 5-limit intervals are approximated to an accuracy of + - 3
> cents as I recall (shooting from seat-of-the-pants memory. The
> perfect fifth, major third, and minor third, and their inversions
by
> implication, are well within this tolerance, maybe even less than
two
> cents off. Don't have my calculator handy right now.

Thanks, Bob, for the clarification!

JP