Sagittal Notation

By Mercury dear George, your epic endeavour was so much entertaining to read. For a deific undertaking of such proportions, the grandeur is worthy of appreciation indeed.

Besides that, I thank you for finding the time to look into my humble efforts and propound your valuable criticism. However, I have some reservations as to the following issues:

1- The Sagittal Notation, meant to identify the infinitesimal nuances used in microtonal music, seems to disregards the `elasticity` that only a single sharp or flat offers to musicians at large. What's more, does the accidentals of western notation not correspond to several different intervalistic jumps depending on the temperament? And what of free chordal instruments unbounded by static tuning? Is it not safe to claim that western tones are more arbitrary than fixed?

2- Also, the level of complication involved with the full subset of sagittal symbols begin to seem incomprehensible after a certain point. I, for one, found too many variables to feel comfortable with the notation. In my opinion, deciphering a score riddled with so many `arrows` would truly be a `herculean task`, a feat that even the Olympian deities would not perform.

3- My suggestion here would be to use a maximum of 3 microtonal sharps and 3 microtonal flats (like I suggested) in addition to the regular accidentals of western notation, and specify the increments by the number of cents just scribbled over these signs (for example: quarter-tone diesis+43c). Another method would be to inscribe the permitted ratios within the scope of a microtonal sign. I have already provided some insight in this direction on page 16 of my doctorate thesis report (www.musiki.org). Of course, my proposal is meant for an educational basis only. By omitting such numeric directions in professional playback, we could save a lot of time and font characters in our compositions. Thus, we can preserve a certain level of independence that a performer of microtonal music would like to exercise. Don't you think?

4- The incompatibility between our respective terminologies regarding the microtonal accidentals I use is due to the fact that the Turkish school founded by Rauf Yekta a century ago cleaved a path that lead to our seperation from the rest of the Arab world in transcribing our maqams. The famous quarter-tone signs that are accepted elsewhere with the values you remind, poses a real problem for us, for you see, these signs are tought to all maqam musicians in Turkey the way established by Yekta's followers: specifically Arel and Ezgi. A musician hereabouts will not play quarter-tones when reading the accidentals that you and the rest of the world understand as quarter-tones. I can find no way to remedy this situation, other than show you how we strange beings operate on this side of the world.

5- The reason I have reached the double flat or double sharp on the 7th step is due to the fact that I have been trying to preserve the elasticity I mentioned above as well as the symmetry of the whole-tone division. You'll notice that 9/8 is equally divided into 2 parts, and these in turn are divided to 3 portions each. Thus the sequence of microtonal signs in a whole-tone from C to D will be like this:

C +1/8 +1/4 +1/2 X
BB -1/2 -1/4 -1/8 D

Bear in mind that the half-tone is very very elastic and comprises all the ratios between 70 to 130 cents. This is in accordance with all the western temperaments I have come to know. So, the ordinary sharp and flat are enharmonics, and correspond to the apotome of western music today. The 4-comma sharp and flat are also enharmonics and correspond to limma of maqam music. Here, there has to be a criss-cross logic, as the enharmonics of each sign is inverted due to the direction of the diatonic. You may want to consider this temperament a variation of 41-EDO.

Cordially,
Ozan Yarman

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>You may want to consider this temperament a variation of 41-EDO.

I had thought 53-et was more or less standard for Turkish theory, but
both 41 and 53 are schismic temperament systems, meaning they equate
32805/32768 to a unison. Can you expand on your suggestion about 41?

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> By Mercury dear George, your epic endeavour was so much
entertaining to read. For a deific undertaking of such proportions,
the grandeur is worthy of appreciation indeed.

Thank you for your kind words. There will be several more episodes
(but I cannot say when, since my time is now very limited), so I hope
that you (and others) will return to the Sagittal website later to
read the rest of the story. For those others who wish to read what
we have so far, here is the link to the mythology (beginning at the
introduction):

http://dkeenan.com/sagittal/gift/GiftOfTheGods.htm

> Besides that, I thank you for finding the time to look into my
humble efforts and propound your valuable criticism.

I apologize that my reply has been delayed due to time constraints
beyond my control, but perhaps this is fortunate in that it has given
me more time to reflect on the issues you brought up, since they are
deserving of a well-thought-out response, rather than one prepared in
haste.

> However, I have some reservations as to the following issues:
>
> 1- The Sagittal Notation, meant to identify the infinitesimal
nuances used in microtonal music, seems to disregards the
`elasticity` that only a single sharp or flat offers to musicians at
large. What's more, does the accidentals of western notation not
correspond to several different intervalistic jumps depending on the
temperament? And what of free chordal instruments unbounded by static
tuning? Is it not safe to claim that western tones are more arbitrary
than fixed?

Yes, although I don't know what you are referring to by "free chordal
instruments", I would agree with you on nearly all of these points,
except that I would insist that your impression that the Sagittal
notation does not allow for flexibility in the interpretation of its
symbols is incorrect. Since you have had only a day or so to read
and digest our documentation (the result of several years of work),
it is very possible that you may have misunderstood a few things.

Beginning on page 14 of our Xenharmonikon article, we explain that
the same symbols that are used to notate exact comma-ratios in just
intonation are also used to notate degrees in various equal divisions
of the octave, so that approximations of just intervals in these
divisions will be notated as much as possible in the same way as the
just intervals themselves. This implies flexible usage of symbols.
Several times after that it is clearly stated that a symbol can
represent an interval that varies in size according to the tuning:
see p. 16 (last paragraph), p. 17 (2nd paragraph), and p. 22 (first
two paragraphs). For the benefit of others who may be following our
discussion, here is a link to the article:

http://dkeenan.com/sagittal/Sagittal.pdf

> 2- Also, the level of complication involved with the full subset of
sagittal symbols begin to seem incomprehensible after a certain
point. I, for one, found too many variables to feel comfortable with
the notation. In my opinion, deciphering a score riddled with so many
`arrows` would truly be a `herculean task`, a feat that even the
Olympian deities would not perform.

Dave Keenan and I are painfully aware that your reaction is shared by
others, particularly those new to the alternate tunings group, who
were not following our progress on this group (and also in much more
detail on the tuning-math group). The last paragraph of page 7
addresses this concern by pointing out that only a small fraction of
the total Sagittal symbol set will be necessary for most
applications, and a sentence in the third paragraph of the first page
states that our goal has been to produce "a notation system that
would be both versatile and powerful, but for which the required
complexity would not make it more difficult to do the simpler things."

For my own purposes, this "small fraction of the total symbol set"
will in most instances amount to fewer symbols than what is shown in
the spartan (i.e., trimmed-down) symbol set in Figure 4 (at the top
of p. 8), and I imagine that this will also be enough for the great
majority of users. As the text points out, the spartan set will
handle more than 40 edo's (all of those listed in Figure 8, plus two
divisions in Figure 9: 130 and 142). Observe that 41-edo requires
only 3 symbols intermediate in size between the natural and sharp (or
flat), which is only 1/4 of the spartan set. This is the same number
that you have specified for your 41-tone (unequal) tuning, so what I
am proposing for that number of tones is no more complicated than
what you have set forth. The essential difference seems to be
concerned mostly with the maximum number of symbols that should be
provided to divide the apotome, which depends, in turn, on the number
of tones that one wishes to distinguish from their neighbors (as
opposed to viewing them as essentially the same tone with varying
nuances of pitch). With the Sagittal symbols one is given the option
of making this number much larger than what most others would
consider practical, or (alternatively) of limiting the symbol set to
allow the sort of flexible interpretation of pitch that you have
suggested. If your approach to notation does not accommodate those
who wish to take the former approach, then your notation will not be
usable by everyone who uses alternate tunings.

For example, if Paul Erlich wishes to notate 152-edo (which he does),
then there will be Sagittal symbols to notate all of the tones of
that division as discrete pitches, with alternate spellings for each
tone. A notation with fewer than the number of symbols required for
152-edo that would appeal to the principle of "flexibility" of
(fewer) pitches to express Paul's intentions would probably cause him
to object that his pitches would be represented ambiguously at best.
Now if you happen to think that 152-edo is impractical, then I think
it would be best to let Paul deal with that problem and to leave it
at that. My only concern is to provide a notation that is as
generally useful as possible, so I will therefore make it capable of
notating 152-edo. The complexity of the notation for that tuning is
a consequence of its complexity, and it is not expected that
something of this sort would be mastered in a very short time. To us
the reading of Chinese characters might seem a herculean task, but a
quarter of the population of the world uses them every day, and I
believe it is safe to say that Sagittal symbols are far less numerous
and less complicated than Chinese characters.

I believe that the task of committing the spartan symbol set to
memory should be fairly simple. Certainly this should be much less
difficult than it would be for someone who knew only Roman numerals
to memorize the Arabic numerals 0 through 9 (which are less
distinctive in appearance from one another than are the spartan
symbols, besides the fact that only the "1" numeral offers any clue
by its appearance as to what it is symbolizing, whereas the spartan
set correlates the appearance of the symbol with the size of its
alteration). In short, there is not a single stroke in any Sagittal
symbol that does not contribute to the meaning of that symbol, making
each and every one maximally compact, efficient, and logical.

At its highest level of precision the Sagittal notation will soon be
capable of pitch resolution comparable to 2460-edo (better than 1/2-
cent, which is required by some who produce electronic music in
rational tunings) that will exactly notate multitudes of just ratios
through at least a prime limit of 23. The symbol refinements (in the
form of supplementary accent marks) are in the process of being
engineered so that they could be included on an actual musical
staff. Performers reading these extremely precise symbols in real
time could, at their option, ignore the accent marks and would
thereby be able to read the notes at a lower precision. (Please
observe that 2460-edo is not only a multiple of 12, but it also
divides each degree of 41-edo into exactly 60 parts. Interesting,
no?)

There have been many times during the development of the notation
when Dave and I could not help reminding one another that we must be
crazy to be taking the notation to such extremes, only to find that
there were others who asked us if we could take it even farther. Our
reason for having taken the notation so far is so that no one can
complain that the notation is too limited for their purposes. To
those who believe that we have too many symbols, we say that you may
use as many or as few as you require for your purposes -- only that
you use them in a manner consistent with how we have defined them.

> 3- My suggestion here would be to use a maximum of 3 microtonal
sharps and 3 microtonal flats (like I suggested) in addition to the
regular accidentals of western notation,

The figures on pages 13 and 16 of your doctoral thesis show 5 of
each, so I'm not sure if I understand this statement. As best as I
can tell, you're referring only to the symbols smaller than a
sharp/flat. For those following our discussion, here is the link:

http://www.musiki.org/yarman_tez_rap2.pdf

Since the text is in Turkish (which I'm not able to read), my
understanding of the diagrams may not be completely accurate, so
please forgive me if I have misinterpreted your ideas.

> and specify the increments by the number of cents just scribbled
over these signs (for example: quarter-tone diesis+43c).

We briefly discussed the possibility of using cent-numbers in the XH
article (p. 20, first two full paragraphs, and p. 21, last
paragraph), concluding that they are very useful for certain
instruments and/or situations, but at best only marginally useful for
others. The place where it is most difficult to employ cent-numbers
is for instruments having a special microtonal keyboard: if you
notate chords, where do you put all of the cent-numbers?

I realize that a ready answer to this question is that cents are not
necessary for a keyboard instrument, and that it should be possible
to identify each key using the number of symbols you have proposed.
Over the past 30 years I have designed two different generalized
keyboards that will accommodate multiple microtonal tunings. One of
these (based on the Bosanquet chain-of-fifths keyboard geometry)
keyboards is on an instrument (a Motorola Scalatron built in 1975) in
my home; given enough keys it will accommodate the 12, 17, 19, 22,
26, 29, 31,41, 43, 46, and 53 divisions (and still others), with
homogeneous fingering patterns in all keys. The other keyboard,
which has never been built, is based on the Miracle tuning geometry,
and it will best accommodate 31, 41, and 72 (and also 19, 46, and 53,
in a somewhat different fashion). Now suppose that I wish to compose
a piece for one of the divisions that could be used on either
keyboard: for example, 46-edo or a subset of 72-edo -- or even a 12-
note subset of something that might be used on a conventional
keyboard (retuned). Should I notate these in different ways,
depending on what sort of keyboard will be used (as a sort of
tablature), or should I notate the pitches in such a way that they
are independent of any keyboard (or of any instrument, for that
matter). While the former approach may give quicker results short-
term, my preference is for the latter approach, in which the notation
directly conveys the meaning of the pitches.

However, with the use of cent-numbers there is also a loss of
flexibility in that one can no longer readily translate symbols for
pitches in just intonation into pitches in edo's. Sometimes I will
jot down an idea for a microtonal melody or chord progression without
desiring to commit myself to a specific tuning, or I may wish to
write a composition that is capable of being played in more than one
tuning. By using symbols that explicitly indicate the prime numbers
in the ratios for the various pitches (independent of any particular
division of the octave), I may easily translate these symbols into
degrees of whatever octave division I may wish to use for this
composition, even at the speed of sight-reading. Here is an example
of the conversions for a few basic symbols into some of the best
octave divisions:

Degrees in EDO
Symbol/meaning 19 22 31 41 53 72 130
------------------- ---------------------------
/| 5-comma (80:81) 0 1 0 1 1 1 2
|) 7-comma (63:64) 1 0 1 1 1 2 3
/|\ 11M-diesis (32:33) [1] 1 1 2 [2] 3 6
/|) 13M-diesis (1024:1053) 0 [1] [1] 2 2 [2] 5

I have put degree numbers in square brackets for those primes that
are represented with relatively large error or inconsistently.

A proper distinction between the 5-comma and 7-comma (i.e., different
numbers of degrees, with neither one vanishing to zero) is not made
until 72-edo, which uniquely notates all of the 11-limit
consonances. Since I have chosen to compose music at the 13 limit, I
need a symbol set comparable to the one required for 130-edo.

> Another method would be to inscribe the permitted ratios within the
scope of a microtonal sign. I have already provided some insight in
this direction on page 16 of my doctorate thesis report
(www.musiki.org). Of course, my proposal is meant for an educational
basis only. By omitting such numeric directions in professional
playback, we could save a lot of time and font characters in our
compositions. Thus, we can preserve a certain level of independence
that a performer of microtonal music would like to exercise. Don't
you think?

Or why not use only the sharp and flat symbols of 12-edo and give
them even more independence? (Just joking!) Or why would you not
want to use only the Tartini symbols of 31-edo, thereby giving
performers the independence of either articulating or tempering out
Didymus' comma, as they wish? The reason is that the symbols are too
few to communicate your intentions adequately.

The limited number of symbols that you're allowing may be sufficient
for you, but they will not be sufficient for everyone. In my case
they would indeed be adequate (inasmuch as I use at most only 29 to
31 tones in the octave), but I have found that the greater number of
symbols in the spartan set is able to indicate the harmonic (i.e.,
prime-number) function of tones in my 29-tone system, by
distinguishing 5 from 7 and 11 from 13 (as I indicated in the table
above). A microtonal tuning is like a new musical language, and in
order for the performer to understand that language properly, there
must be a sufficient number of symbols to convey the intended
meaning. Would you teach the Turkish language using a simplified
subset of your alphabet and then permit your students to mispronounce
words for the sake of independence? I think not.

Please don't misunderstand me. I'm not insisting that anyone use
more symbols than they want or need, but a generalized system of
notation must be able to suit widely varying requirements, and if a
particular notation is not able to do that, then it will not be
acceptable to all.

> 4- The incompatibility between our respective terminologies
regarding the microtonal accidentals I use is due to the fact that
the Turkish school founded by Rauf Yekta a century ago cleaved a path
that lead to our seperation from the rest of the Arab world in
transcribing our maqams. The famous quarter-tone signs that are
accepted elsewhere with the values you remind, poses a real problem
for us, for you see, these signs are tought to all maqam musicians in
Turkey the way established by Yekta's followers: specifically Arel
and Ezgi. A musician hereabouts will not play quarter-tones when
reading the accidentals that you and the rest of the world understand
as quarter-tones. I can find no way to remedy this situation, other
than show you how we strange beings operate on this side of the
world.

You are speaking primarily of the Tartini semisharp and sesquisharp
symbols, which you are using for alterations in the range of 20 to 40
and of 160 to 180 cents, respectively. Inasmuch as the Sagittal
notation does not use Tartini's symbols, would that not provide a
ready-made remedy? Abandonment of symbols presently in use may be a
bit radical for some, but it does have its advantages.

> 5- The reason I have reached the double flat or double sharp on the
7th step is due to the fact that I have been trying to preserve the
elasticity I mentioned above as well as the symmetry of the whole-
tone division. You'll notice that 9/8 is equally divided into 2
parts, and these in turn are divided to 3 portions each. Thus the
sequence of microtonal signs in a whole-tone from C to D will be like
this:
>
> C +1/8 +1/4 +1/2 X
> BB -1/2 -1/4 -1/8 D
>
> Bear in mind that the half-tone is very very elastic and comprises
all the ratios between 70 to 130 cents. This is in accordance with
all the western temperaments I have come to know. So, the ordinary
sharp and flat are enharmonics, and correspond to the apotome of
western music today.

I don't know if I can entirely agree with this. In your table on p.
16 you have given a range of 100 to 130 cents for the apotome and 180
to 220 cents for the double-apotome. The divisions of the octave
that are most hospitable to traditional European harmony are those
with narrow fifths, most notable 19 and 31, in which the apotome is 1
degree (~63 cents) and 2 degrees (~77 cents), respectively, and the
double-apotome twice that. These figures are far outside the ranges
you have specified. In the case of the diatonic semitone (a limma,
or minor 2nd), these figures are 2 degrees of 19 (~126 cents) and 3
degrees of 31 (~116 cents), which does fall within your ranges, but
this is functionally not the same interval as the apotome (chromatic
semitone). I am probably stating things with which you are already
familiar; if so, then please bear with me for a couple more
paragraphs, just so there is no misunderstanding.

The basis for traditional European harmony was established at a time
when the meantone temperament was the tuning in common use, and as I
stated in the introduction to the mythology, its enharmonically
related sharps and flats differ in pitch by approximately 1/5-tone
and are generally incapable of being substituted for one another.
Even in 12-edo seemingly identical intervals are not perceived in the
same fashion: e.g., a minor third is considered consonant, yet an
augmented 2nd (in the appropriate harmonic context) is perceived as
dissonant. In this link there are a couple of examples that
illustrate this:

/tuning/topicId_48499.html#49594

When employed in edo's in the meantone family (such as 31, and
especially 19), so-called enharmonic sharps and flats result in
distinctly different harmonic intervals that should not be confused
or "clouded" with one another. In 19-edo C-sharp and D-flat are as
different in size as C and C-sharp, and to consider the differences
between either of the two pairs of tones as "subtleties"
or "flexibility" in intonation would be tantamount to ignoring the
difference between a major 3rd and a minor 3rd. Thus an augmented
prime (C to C#) is quite different in function from a minor 2nd (C to
Db), even in 12-edo, and the two intervals are not generally
interchangeable in traditional harmony.

Finally, I will give an entirely different example, one quite foreign
to the Western harmonic tradition. In 22-edo the diatonic semitone
is a single degree (~55 cents), while the apotome is 3 degrees (~164
cents) and the Didymus comma is a single degree (a whopping 55
cents), none of which are in the ranges you specify.

Since I am not able to read the text of your thesis, I may have
misunderstood or misapplied what is shown in your diagrams, so please
correct me if that is the case.

> The 4-comma sharp and flat are also enharmonics and correspond to
limma of maqam music. Here, there has to be a criss-cross logic, as
the enharmonics of each sign is inverted due to the direction of the
diatonic. You may want to consider this temperament a variation of 41-
EDO.

It is refreshing and enlightening to have your viewpoint, especially
since you have found cause to differ with conventional Turkish music
theory. In this latest occurrence of "East meets West" we share the
common goal of resisting the trend toward worldwide domination of
music by 12-edo.

Best wishes,

--George Secor

Dear George, I'm sorry for my delay in replying to your message, I had a few things to sort out before I could clear my head on this mind-consuming subject. BTW, I hope nothing is wrong with your health considering your statement `...since my time is now very limited`. Now, back to the topic at hand with renewed vigor:

One particular issue about the extendible-elastic nature of temperaments performed on unbounded (free) chordal instruments such as the violin is very much apparent if one listens to a particular folk tune from Hungary and compares it with one from Turkey and then from India and so forth... The way violin players utilize the accidentals are not the same everywhere, even in the same genre of music involved. This fact is very much evident in `historical performances` where artists try to reflect the age in which a certain music was composed. Such a level of independence with the sharps and flats used throughout the world cannot be disregarded in such a way as to consign the sharp and flat to nailed down values. The boundaries I have proposed for the half-tone interval results in a chromatically-centered zone which is 60 cents wide. It is my intention to attribute the concept of `half-tone` to this zone in alignment with the flexibility seen in the usage of half-tone accidentals.

But the way I see it (forgive all my misunderstandings from this point forward), Sagittal notation (as defined on page 18) introduces 12 different symbols to quasi-equally spaced portions of this half-tone zone. Although I admit there IS a generous level of flexibility (about 4-5 cents max) with each symbol, I am not satisfied that this region should be defined by anything unresemblant to the sharp or flat as a regular musician like myself is used to.

Yes, it is possible for me to learn Sagittal notation if I commit myself to it starting from now on. But as a speaker of a particular Chinese dialect spends all his childhood memorizing the glyphs and their attributes, so will I have to face a similar task of `un-learning` and `re-learning` notation from scratch. I was entertaining the idea that `microtonality` should be viewed as a welcome addition to the conventional musical knowledge, but I never imagined that it would result in the obsolescence of the sharp and flat all together.

Yes, you have preserved the sharp and flat in their original form as a substitute to the understanding of Sagittal Notation, plus as a means to quell any opposition to the contrary. However, these have been reduced to a supplementary state rather than forming the basis of the pragmatical comprehension of `intermediate tone`. I am not sure if I can find it in my heart to abondon my `roots`. (Do forgive my criticism if it sounds too harsh. I really appreciate and congratulate your valuable efforts.)

The complete set of symbols which amounts to 53 per half-tone is a bewildering number for someone like me who has a hard time with 72-TET alone. Even the spartan symbol set with 13 symbols per half-tone is too much given their similitude with each other. The distinctions are simply not enough for a layman like me who wants to use microtones for orchestral authentic maqam music.

To the detriment of complexity, I venture to think that additional symbols after 72-TET simply are a waste of notation space (I dare risk being branded as a reactionary). Why would someone want to persist in using 152-TET on the olde staff? If one wants to notate music that no single professional musician could possibly play correctly, I suggest that other methods for electronic notation be developed aside from that which we wish to upgrade.

I am well pleased that we converge on the same number of symbols for 41-edo. I also agree that your symbols take no less time than mine to learn. However, I live in Turkey where thousands of musicians are accustomed to the usage of the sharp as limma, the slashed flat as the quarter-to-half tone and the reversed flat as the comma. In order to preserve the application of these microtones, I have to preserve the microtonal accidentals that go along with them. The solution I found was not easy, and I expect to deal with unsettling commotion from the instigators and traditionalists who uphold the Yekta-Arel-Ezgi school with zeal. But for the sake of clarity, transposability and flexibility, I can go no further in number with the sharps and flats I proposed.

Yes, you are right in making the distinction between `the number of tones one wishes to distinguish from their neighbours` as opposed to `viewing them as essentially the same tone with varying nuances of pitch`. For that, I will stick to my symbolist doctrine to the bitter end, and suggest that the half-tone should be detailed in such a way as to preserve the identity of the common sharp and flat.

I am grateful that you have reminded that my clandestine efforts will not appeal to every microtonalist in its current form. Allow me to say that you have motivated me into searching for a feasible method for differentiating all possible microtones using only a total of 7 sharps and 7 flats. Perhaps flats and sharps could be utilized with varying colors on the sheet? If my windows desktop settings do not deceive me, I can identify about 48 different colors at once. So, a sequence of 48 colors per microtone pertaining to a rainbow spectrum would provide me three times greater resolution than your Sagittal Notation. This way, I would not require any more than 4 sharps and 4 flats per half-tone (I call the ones between 130-180 cents macrotones). I'm sure Mr. Erlich would not object to the liberty of using even more colors for higher resolution. Talk about flexibility.

If this approach seems too ambiguous, accent marks of increasing complexity could be employed the way you derive for 2460-edo. (Not for the faint of heart!) As you say: `...Performers reading these extremely precise symbols in real time could, at their option, ignore the accent marks and would thereby be able to read the notes at a lower precision.`

If 2460-edo is 60 times more voluminous than 41-edo, then I need only devise 60 supplementary
(plus&minus type) accent marks for each of my accidentals. If the eye can perceive 48 different colors at once, so can it perceive a little more than that (or so I hope). And if I use a different color per accent, I can attain resolutions you may not reach in any time soon. Crazy? yep...

For those who still have no idea on what I'm drawling about, here is a copy of my doctorate thesis report:

http://www.musiki.org/yarman_tez_rap2.pdf

I would love to read more about the keyboards you designed. Yes, I agree with you about the latter approach where you clearly outlined the necessity of notating pitches regardless of where or how they are played. And I believe that my symbols can accomodate any pitch with the desired amount of flexibility that one might want to exercise. The accents could be assigned to such abbreviations as:

5S (5 schisma)
17K (17 kleisma)
7C (7 comma)
11mD (11 M-diesis)

This is for the purists of course: either the color, accent, colored accent, abbreviations, or some/all of them at once. With my approach, one might eliminate large errors and inconsistencies in temperament-transitions (such as from 19 to 72-edo). I hope that my proposal balances the articulation independence versus communication accuracy.

As to your question:

`...Would you teach the Turkish language using a simplified
subset of your alhabet and then permit your students to mispronounce
words for the sake of independence? I think not.`

I can tell you that the revolution which commenced about 80 years ago did just that. In about 10 years, the old Ottoman alphabet was discarded, a simpler set of latin characters was adopted, lots of words of Arabic and Persian origin junked, mispronounciation was unleashed and the entire Turkish language was thus revised. I'm not particularly pleased with what's been done, but I'm speaking this modified language that has little bearing to what was being spoken a century ago. Still, with some practice, I can understand Ottoman if it is converted to latin form and can pronounce the words in Ottoman and Arabic even if written in simplified latin. What the heck, I can read Arabic written in English! I do not need any special characters in latin once I understand what word I'm reading. Thus, you can say that I gained a certain level of independence with the dialect thanks to the latin alphabet. It's a pity that I was deprived of the Arabic alphabet till so late a stage in my career. The single biggest mistake of the Language Reform was the elimination of the Arabic alphabet altogether. As to the degeneration that takes place around these parts (what with the introduction of technology and the adoption of western latin due to the lack of Turkish code within the designs), us Turks can lately read and understand words written with western latin. Thus, the youth here can comprehend Turkish with only 23 characters as compared to 36 characters of Ottoman used before. Facing such a state of affairs, I have no choice but to teach Turkish using as simplified an alphabet as possible (29 in number since the Republic).

As for our terminological dispute:

I refer to the apotome based on the ditonic series starting from C to its octave (series of white keys). Accordingly, the apotome is as close a sharp to its neighbour as possible. Thus, D sharp is about 110 to 120 cents higher than D, and consequently about 90 to 80 cents far from E. Maybe I'm using out-dated concepts, so please correct me if I'm wrong. As far as I know, in western music the sharps raise and flats lower an alternate tone in such a way that they by-pass each other by 1/3 tone although they are considered to be enharmonical. This is also a necessity of the harmonic context as you describe.

No, I'm not ignoring the subleties that enharmonical intervals represent. On the contrary, by splitting the half-tone zone into two sub-regions, I believe I am able to indicate the difference between the sharp and the flat just the way it is meant to be.

But I'm rather confused as to how a semi-tone could be a quarter-tone in 22-edo, and an apotome a 3/4 tone. And how is it that the syntonic comma (80:81) can be diverted to mean 55 cents when it is 20 cents? Please forgive my ignorance in these matters.

I thank you for investing your valuable time to discipline my far-fetched ideas. Our encounter has proved to be most worthy and precious in my eyes.

Cordially,
Ozan Yarman

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George, I'm sorry for my delay in replying to your message, I
had a few things to sort out before I could clear my head on this
mind-consuming subject. BTW, I hope nothing is wrong with your health
considering your statement `...since my time is now very limited`.

Dear Ozan,

No, it is not myself, but two of my wife's close relatives have been
seriously ill over the past several months, and we have been very
busy attending to their needs. But I appreciate your concern.

> Now, back to the topic at hand with renewed vigor:
>
> One particular issue about the extendible-elastic nature of
temperaments performed on unbounded (free) chordal instruments such
as the violin is very much apparent if one listens to a particular
folk tune from Hungary and compares it with one from Turkey and then
from India and so forth... The way violin players utilize the
accidentals are not the same everywhere, even in the same genre of
music involved. This fact is very much evident in `historical
performances` where artists try to reflect the age in which a certain
music was composed. Such a level of independence with the sharps and
flats used throughout the world cannot be disregarded in such a way
as to consign the sharp and flat to nailed down values.

Yes, I agree completely. Those of us in the Western European
tradition who are knowledgeable about the history of tuning and of
the apparent conflict between the requirements of expressive melody
(which tends to prefer a smaller diatonic semitone, or limma,
resulting from a chain of pure fifths) and consonant harmony (which
prefers a larger diatonic semitone that results from fifths tempered
narrower than in 12-EDO) will readily admit that flexibility must be
allowed not only for considerations of cultural differences
(encompassing both era and geography), but also for artistic
license. A comprehensive system of notation must be versatile enough
to allow this freedom, when it is appropriate. On the other hand, it
must also be versatile enough to be more specific about pitch, when
that is required. It would be of little value for us to debate
whether it is better to allow the performer more freedom or to allow
the composer more control with pitch. With the Sagittal notation it
has been our objective to have a notation that is versatile enough to
allow either of these approaches.

> The boundaries I have proposed for the half-tone interval results
in a chromatically-centered zone which is 60 cents wide. It is my
intention to attribute the concept of `half-tone` to this zone in
alignment with the flexibility seen in the usage of half-tone
accidentals.

I think that it would be necessary that you maintain a clear
distinction between the diatonic semitone (limma) and chromatic
semitone (apotome) in your reckoning. The Sagittal notation allows
the size of both the limma and apotome to vary by an amount
significantly greater than 60 cents. I infer from your question
below about 22-EDO that an explanation will be necessary. So that
there is no misunderstanding, I will start at the very beginning.

On the musical staff we have a set of 7 nominal notes, arranged here
as an open chain of fifths, for which I have numbered the tones zero
through 6, beginning with F as the starting tone:

0 1 2 3 4 5 6
F C G D A E B

If we put all of these tones into the same octave, we have a diatonic
scale, in which we find the diatonic semitone (or limma) in two
places, between E and B and between B and C. We observe that the
upper note of the interval of a limma is found by moving 5 positions
to the left in the chain of fifths from the lower note.

The introduction of sharps and flats (and doubles) introduces more
tones at each end of the chain:

-8 –7 –6 –5 –4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G#

10 11 12 13 14 15
D# A# E# B# Fx Cx ...

Thus the upper note of the interval of an apotome is found by moving
7 positions to the right in the chain of fifths from the lower note,
and the double-apotome will be 14 positions. This will be true
regardless of the tuning (or division of the octave).

In just intonation the fifths in the chain are of a specific size (a
2:3 ratio of frequency), but in various EDO's they will only
approximate this ratio, being either wider or narrower that the just
fifth, according to the particular division. With 41-EDO they are
very slightly wide. With 19 and 22-EDO (to give opposite examples)
they will be over 7 cents narrow and wide, respectively. While 7
cents may seem to be a small amount, this difference will accumulate
as one moves along the chain of fifths. For example, if one moves 7
places to the right (an apotome), the difference from the Pythagorean
apotome will exceed 49 cents (i.e., 7 cents multiplied by 7 places).

A chain of 19-EDO fifths (~694.7 cents) will therefore result in a
limma of ~126.3 cents and an apotome of ~63.2 cents, whereas a chain
of 22-EDO fifths (~709.1 cents) will result in a limma of ~54.5 cents
and an apotome of ~163.6 cents. Between these two divisions the
limma varies more than 70 cents and the apotome more than 100 cents,
and they vary in opposite directions.

For 19-EDO I should also point out a couple of things about the
sequence of symbols:

0 +1 +2 +3
C C# Cx
Dbb Db D

No special symbols are required for 19. The double-apotome is only
2/3 of a whole tone, so we have here a clear example where the double-
apotome is not the same as a whole tone (which in this instance would
be equal to a triple apotome).

With 12-EDO we have a special case in which the fifths of 700 cents
result in a limma and apotome of the same size, 100 cents. In other
words, the difference between the limma and apotome is tempered out,
so that the distinction between them is (unfortunately) lost.
However, as I pointed out in my previous message, in traditional
Western harmony these two intervals are functionally different and
should not be confused with one another. As a corollary, it is
important to note that the whole tone is the sum of the limma and the
apotome, and this will be true regardless of the tuning (or division
of the octave). A whole tone (8:9) is therefore *not* two limmas or
two apotomes (appearances in 12-EDO notwithstanding).

The Sagittal notation takes all of these things into account and
allows the symbols to be used accordingly, with even more flexibility
than you have suggested.

> But the way I see it (forgive all my misunderstandings from this
point forward), Sagittal notation (as defined on page 18) introduces
12 different symbols to quasi-equally spaced portions of this half-
tone zone. Although I admit there IS a generous level of flexibility
(about 4-5 cents max) with each symbol, I am not satisfied that this
region should be defined by anything unresemblant to the sharp or
flat as a regular musician like myself is used to.

Ah, let me explain further! The diagram to which you refer
illustrates the allowable ranges for each of the symbols in the
trojan (12-relative, or 12R) symbol set, but these particular ranges
are applicable *only* with the 12-EDO fifth of 700.0 cents (and thus
an apotome of 100.0 cents). One could allow these intervals to vary
in size somewhat, but not such that their departures from 12-EDO
would accumulate along the chain of fifths. Observe that the 5-
comma /| is given a nominal size of 15 cents in the trojan set, since
major and minor thirds and sixths are in error by approximately that
amount in 12-EDO. The 7-comma |) is given a nominal size of 31
cents, approximately double that of the 5-comma, while the 7:11-comma
(| is set at 20 cents. As you noted, there is a small amount of
flexibility for each of these symbols in the neighborhood of each of
their nominal values.

Compare that with the range of sizes given for the athenian symbol
set in Figure 5 (p. 10) for medium-precision just intonation, in
which the fifths are in exact 2:3 ratio and the apotome is therefore
~113.7 cents. Here the "nominal" size for the 5-comma symbol is
~21.5 cents (80:81), while the 7-comma is ~27.3 cents (63:64) and the
7:11-comma is ~33.1 cents. Again there is a small amount of
flexibility for each symbol in the neighborhood of each of their
nominal values, but the most significant difference between the
athenian and the trojan symbol usage is that the nominal values for
the symbols have changed.

Having said all of this, I must now point out that the sort of
flexibility of Sagittal symbols that I have just discussed is, by and
large, *irrelevant* to your purposes! With the 41-division you are
using a fifth that is essentially just, and the tones in the chain of
fifths will therefore be close to Pythagorean. This would require
that the Sagittal symbols have nominal values at (or close to) those
given for the athenian symbol set (in Figure 5). You require both
fewer symbols and greater flexibility of pitch than what is allowed
with the athenian symbol set, and both of these may be achieved in
Sagittal by means of a 41-EDO mapping and a smaller set of symbols.

With the near-pure fifths of 41-EDO, the 5-comma and 7-comma both
have nominal sizes of 1 degree (~29 cents), so that only one of these
symbols will be necessary. In Sagittal we would choose the 5-comma
symbol, and the 11-diesis symbol (remember Apollo-11) would be used
for the half-apotome. These, combined with apotome and double-
apotome symbols, result in the following symbol sequence:

0 +1 +2 +3 +4 +5 +6 +7 +8
C C/| C/|\ C||\ C/||\ C/||| C/|||\ CX\ C/X\ pure
DY/ D\!!!/ D\!!! D\!!/ D!!/ D\!/ D\! D D/|
C C/| C/|\ C#\! C# C#/| C#/|\ Cx\! Cx mixed
Dbb/| Db\!/ Db\! Db Db/| D\!/ D\! D D/|

Please observe that in the 41 division a double-sharp (or double-
apotome) is not the same thing as a whole step (or major 2nd), as
will be the case with all divisions that are not multiples of 12. If
you are basing your notation on the 41 division, then I don't believe
that you can disregard this where you said:

< The reason I have reached the double flat or double sharp on the
7th step is due to the fact that I have been trying to preserve the
elasticity I mentioned above as well as the symmetry of the whole-
tone division. You'll notice that 9/8 is equally divided into 2
parts, and these in turn are divided to 3 portions each. Thus the
sequence of microtonal signs in a whole-tone from C to D will be like
this: [simplified version of the diagram on p. 13 of your thesis]

In your diagram I observe that the amount of flexibility is (over
8:9) smallest at the sides and increases toward the middle. The 41
and 53 (schismic) divisions are, in my opinion, the divisions that
conform most closely to the set of tones you describe. I tabulated
the sizes of the main interval-ratios in these two representative
divisions (lumping two adjacent degrees together for the apotome):

Ratio: 1:1 80:81 32:33 apotome 11:12 9:10 8:9
41 0 1 2 3 4 5 6 7 degrees
41: 0c 29c 59c 88c 117c 146c 176c 205 cents
53: 0 1 2 4 5 7 8 9 degrees
53: 0c 23c 45c 91c 100c 158c 181c 204 cents
Range: 0c 6c 11c <--29c--> 12c 5c 1c

The range of flexibility does indeed progress as desired (if you lump
together two adjacent degrees under the apotome label; however, I
have already said that I do not think that this is advisable.)
There is also symmetry without the necessity of equating the double-
apotome with 8:9.

On page 16 of your thesis you have a table showing categories of
small intervals with a range of sizes for each. You may be
interested to learn that Dave Keenan systematically derived
boundaries for categories of comma-ratios as follows (from messages
#54577, 54584):

Square of lowerbound
.. 2-exponent
...... 3-exponent
................ Lowerbound (cents)
................................. Size range name
-------------------------------------------------
.. [ 0, 0 > ....... 0 ........... schismina
.. [-84, 53 > ..... 1.807522933 . schisma
.. [ 317,-200 > ... 4.499913461 . kleisma
.. [-19, 12 > .... 11.73000519 .. comma
.. [-57, 36 > .... 35.19001558 .. small-diesis
.. [ 8, -5 > ..... 45.11249784 .. (medium-)diesis
.. [-11, 7 > ..... 56.84250303 .. large-diesis
.. [-30, 19 > .... 68.57250822 .. small-semitone
.. [ 35, -22 > ... 78.49499048 .. limma
.. [-3, 2 > ..... 101.9550009 ... large-semitone
.. [ 62, -39 > .. 111.8774831 ... apotome
.. [-106, 67 > .. 115.492529

It is interesting to reduce the number of Dave's categories (to
correspond to degrees of the 41 division) and then compare his
boundaries with yours, as I have done here:

Sagit Category DK boundaries OY boundaries
----- -------- --------------- -------------
/| comma (C) 11.73 – 35.19 20 - 40
/|\ diesis (D) 35.19 - 68.57 40 - 70
||\ limma (L) 68.57 – 101.96 70 - 100
/||\ apotome (A) 101.96 – 125.41 100 - 130
/||| 2A-L 125.41 – 158.80 130 - 160
/|||\ 2A-D 158.80 – 192.18 160 - 180
X\ 1 tone (A+L) 192.18 – 215.64 180 - 220
or 2A-C

As you see, there is considerable agreement in these numbers.

I believe that you can have both the required symmetry and
flexibility of pitch without requiring the double apotome to be
equated with (i.e., the same number of degrees as) the whole tone.

> Yes, it is possible for me to learn Sagittal notation if I commit
myself to it starting from now on. But as a speaker of a particular
Chinese dialect spends all his childhood memorizing the glyphs and
their attributes, so will I have to face a similar task of `un-
learning` and `re-learning` notation from scratch. I was entertaining
the idea that `microtonality` should be viewed as a welcome addition
to the conventional musical knowledge, but I never imagined that it
would result in the obsolescence of the sharp and flat all together.

The change is not really as radical as you (and some others) might
think it to be. In the paper I have tried to emphasize that in the
pure Sagittal system it is only a few *symbols* that are changing.
Their *names* (sharp, flat) and *meanings* remain exactly the same as
before.

> Yes, you have preserved the sharp and flat in their original form
as a substitute to the understanding of Sagittal Notation, plus as a
means to quell any opposition to the contrary. However, these have
been reduced to a supplementary state rather than forming the basis
of the pragmatical comprehension of `intermediate tone`. I am not
sure if I can find it in my heart to abondon my `roots`. (Do forgive
my criticism if it sounds too harsh. I really appreciate and
congratulate your valuable efforts.)

Thank you for your honesty. Dave Keenan and I have tried to provide
options (such as the mixed symbol version) to make the notation as
broadly useful and acceptable as possible to a great diversity of
composers and musicians, and we realize that innovations may not be
accepted as quickly as we would like. For some the mixed-symbol
option may serve in a transitional capacity.

> The complete set of symbols which amounts to 53 per half-tone is a
bewildering number for someone like me who has a hard time with 72-
TET alone. Even the spartan symbol set with 13 symbols per half-tone
is too much given their similitude with each other. The distinctions
are simply not enough for a layman like me who wants to use
microtones for orchestral authentic maqam music.

I defer to your judgment that a 41-EDO symbol set will be sufficient
for your purposes.

> To the detriment of complexity, I venture to think that additional
symbols after 72-TET simply are a waste of notation space (I dare
risk being branded as a reactionary). Why would someone want to
persist in using 152-TET on the olde staff? If one wants to notate
music that no single professional musician could possibly play
correctly, I suggest that other methods for electronic notation be
developed aside from that which we wish to upgrade.

On the other hand, it would be very convenient, if at all possible,
to have the same sort of notation for both performance and electronic
music, for that would permit the latter to be converted to a printed
score that could be studied or (when technically feasible) to a
printed part that could be played on an acoustic instrument.
Conversely, it would also simplify the process of entering electronic
pitch data, since it would not be necessary to learn a completely
different notation for that purpose. Some of the more intricate
details regarding how this could best be accomplished are currently
in the process of being worked out.

> I am well pleased that we converge on the same number of symbols
for 41-edo. I also agree that your symbols take no less time than
mine to learn. However, I live in Turkey where thousands of musicians
are accustomed to the usage of the sharp as limma,

Then perhaps you should be one to educate them that this is incorrect
and that the sharp is actually an apotome. Ignorance about these
things also abounds in my part of the world, but I think that I
should then do something to remedy the problem.

> the slashed flat as the quarter-to-half tone and the reversed flat
as the comma. In order to preserve the application of these
microtones, I have to preserve the microtonal accidentals that go
along with them. The solution I found was not easy, and I expect to
deal with unsettling commotion from the instigators and
traditionalists who uphold the Yekta-Arel-Ezgi school with zeal. But
for the sake of clarity, transposability and flexibility, I can go no
further in number with the sharps and flats I proposed.

It sounds as if you're caught in a quagmire, with the symbols given
meanings at variance with what we have recognized for centuries in
the West. I cannot avoid the conclusion that the only way our
cultures might ever be able to have a notation in common would be to
start fresh with completely new symbols about which there would be no
dispute as to their meaning. As for how difficult it might be for
anyone to be persuaded to take that step, I will not lose any sleep
over that question; but I do what I can to make a difference.

> ...
> For those who still have no idea on what I'm drawling about, here
is a copy of my doctorate thesis report:
>
> http://www.musiki.org/yarman_tez_rap2.pdf

And I encourage them to look at the figure and tables on pages 13 and
16-19.

> I would love to read more about the keyboards you designed.

Please see these messages:
/tuning/topicId_39323.html#39399
/tuning/topicId_39323.html#39407
/tuning/topicId_36964.html#37151
as well as others in these subject threads.

In order to view the actual diagrams, you would have to join the
tuning-math group:
/tuning-math/
They are located here (inasmuch as there was no longer any room in
the files section for the alternate tunings group):
/tuning-math/files/secor/kbds/
Files: KbDec72.gif and KbScal31.gif

> ...
> As to your question:
>
> `...Would you teach the Turkish language using a simplified
> subset of your alhabet and then permit your students to
mispronounce
> words for the sake of independence? I think not.`
>
> I can tell you that the revolution which commenced about 80 years
ago did just that. ... Facing such a state of affairs, I have no
choice but to teach Turkish using as simplified an alphabet as
possible (29 in number since the Republic).

Well, I guess the joke is on me! I thought I was presenting a
hypothetical situation, but it was all too true! The point I was
trying to make is that there must enough symbols in the notation to
convey the various shades of meaning.

> As for our terminological dispute:
>
> I refer to the apotome based on the ditonic series starting from C
to its octave (series of white keys). Accordingly, the apotome is as
close a sharp to its neighbour as possible. Thus, D sharp is about
110 to 120 cents higher than D, and consequently about 90 to 80 cents
far from E. Maybe I'm using out-dated concepts, so please correct me
if I'm wrong. As far as I know, in western music the sharps raise and
flats lower an alternate tone in such a way that they by-pass each
other by 1/3 tone

The difference is generally much smaller than that, more like 1/9
tone -- but your 1/3-tone figure (which corresponds to a 17-tone
octave) is not out of the question. But this is only the case when
melodic expressiveness governs the nuances of pitch, which is only
half the story.

> although they are considered to be enharmonical. This is also a
necessity of the harmonic context as you describe.

The other half of the story is that when consonant harmony is of
primary concern, then sharps will be lower in pitch than
enharmonically related flats. In 31-EDO, for example, this
difference will amount to 1/5 tone:

0 +1 +2 +3 +4 +5
C C/|\ C# C#/|\ Cx
Dbb Db\|/ Db D\|/ D

> No, I'm not ignoring the subleties that enharmonical intervals
represent. On the contrary, by splitting the half-tone zone into two
sub-regions, I believe I am able to indicate the difference between
the sharp and the flat just the way it is meant to be.

Okay, I see that you have small triangles affixed to the sharp and
flat signs to give alternate spellings. These are equivalent to the
Sagittal ||\ sharp and !!/ flat without the left flag, such that:

D to F/||\ is a Pythagorean major 3rd (64:81), whereas
D to F||\ is a just major 3rd (4:5), and
C to E\!!/ is a Pythagorean minor 3rd (27:32), wherease
C to E!!/ is a just minor 3rd (5:6).

It appears that the only significant difference of opinion we have is:

1) Regarding the particular symbols to be used:
a) Present Turkish usage of the Tartini fractional symbols is at
variance with West European usage going back centuries (which puts us
at an impasse), but
b) The pure Sagittal symbols that would resolve that impasse are
too radical a change for you (and others) to accept.

2) You equate C-double-sharp with D in your 41-tone framework (if I
understand you correctly), whereas it is generally established (to
the extent that I'm completely confident that everyone knowledgeable
in this group would agree with me) that the double-sharp in that
division overshoots the major 2nd by one degree. I insist that a
double-apotome should *always* be twice as large as an apotome --
always!

> But I'm rather confused as to how a semi-tone could be a quarter-
tone in 22-edo, and an apotome a 3/4 tone. And how is it that the
syntonic comma (80:81) can be diverted to mean 55 cents when it is 20
cents? Please forgive my ignorance in these matters.

Since the fifths of 22-EDO are so wide, a chain of four fifths (e.g.,
C to E) will result in a wide major third of ~436 cents (8 degrees of
22, which is quite close to 7:9). Taking C as 1/1, we find that the
best approximation of 5/4 is 7 degrees (1 degree lower), so we
therefore notate this tone as E\! (E lowered by a 5-comma). The 5-
comma in 22-EDO therefore corresponds to a single degree (~54.5
cents). On the other hand, the 7-comma vanishes to zero.

Here is how we notate several degrees of 22:

0 +1 +2 +3 +4 +5 +6 +7
C C/| C||\ C/||\ C/||| CX\ C/X\ pure
D\!!! D\!!/ D!!/ D\! D D/| D||\ D/||\
C C/| C#\! C# C#/| Cx\! Cx mixed
Db\! Db Db/| D\! D D/| D#\! D#

Here, also, the double-apotome is twice as many degrees as the
apotome.

> I thank you for investing your valuable time to discipline my far-
fetched ideas. Our encounter has proved to be most worthy and
precious in my eyes.

Ozan, your ideas are not at all far-fetched, and I have learned some
valuable things from our discussion. There is much that I do not
understand concerning the way that you are seeking to apply your
notation to musical cultures with which I am for all practical
purposes unfamiliar, so I can only explain what we have done with
notation from a Western perspective and hope that it may be of
benefit to you.

Best wishes,

--George Secor

Dear George,

I need some time to fully digest the science behind your concepts. It also shows that I need to keep my spontenaity in check lest I ridicule myself further among my superiors. While I begin researching the subject and prepare a reply, I ask you to take a look at my recent doodle which I named Spectral Notation. I'm certain to benefit from your edifying directions and look forward to continue our lessons.

Cordially,
Ozan

----- Original Message -----
From: George D. Secor
To: tuning@yahoogroups.com
Sent: 26 Ekim 2004 Salı 22:37
Subject: [tuning] Re: Sagittal Notation

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George, I'm sorry for my delay in replying to your message, I
had a few things to sort out before I could clear my head on this
mind-consuming subject. BTW, I hope nothing is wrong with your health
considering your statement `...since my time is now very limited`.

Dear Ozan,

No, it is not myself, but two of my wife's close relatives have been
seriously ill over the past several months, and we have been very
busy attending to their needs. But I appreciate your concern.

> Now, back to the topic at hand with renewed vigor:
>
> One particular issue about the extendible-elastic nature of
temperaments performed on unbounded (free) chordal instruments such
as the violin is very much apparent if one listens to a particular
folk tune from Hungary and compares it with one from Turkey and then
from India and so forth... The way violin players utilize the
accidentals are not the same everywhere, even in the same genre of
music involved. This fact is very much evident in `historical
performances` where artists try to reflect the age in which a certain
music was composed. Such a level of independence with the sharps and
flats used throughout the world cannot be disregarded in such a way
as to consign the sharp and flat to nailed down values.

Yes, I agree completely. Those of us in the Western European
tradition who are knowledgeable about the history of tuning and of
the apparent conflict between the requirements of expressive melody
(which tends to prefer a smaller diatonic semitone, or limma,
resulting from a chain of pure fifths) and consonant harmony (which
prefers a larger diatonic semitone that results from fifths tempered
narrower than in 12-EDO) will readily admit that flexibility must be
allowed not only for considerations of cultural differences
(encompassing both era and geography), but also for artistic
license. A comprehensive system of notation must be versatile enough
to allow this freedom, when it is appropriate. On the other hand, it
must also be versatile enough to be more specific about pitch, when
that is required. It would be of little value for us to debate
whether it is better to allow the performer more freedom or to allow
the composer more control with pitch. With the Sagittal notation it
has been our objective to have a notation that is versatile enough to
allow either of these approaches.

> The boundaries I have proposed for the half-tone interval results
in a chromatically-centered zone which is 60 cents wide. It is my
intention to attribute the concept of `half-tone` to this zone in
alignment with the flexibility seen in the usage of half-tone
accidentals.

I think that it would be necessary that you maintain a clear
distinction between the diatonic semitone (limma) and chromatic
semitone (apotome) in your reckoning. The Sagittal notation allows
the size of both the limma and apotome to vary by an amount
significantly greater than 60 cents. I infer from your question
below about 22-EDO that an explanation will be necessary. So that
there is no misunderstanding, I will start at the very beginning.

On the musical staff we have a set of 7 nominal notes, arranged here
as an open chain of fifths, for which I have numbered the tones zero
through 6, beginning with F as the starting tone:

0 1 2 3 4 5 6
F C G D A E B

If we put all of these tones into the same octave, we have a diatonic
scale, in which we find the diatonic semitone (or limma) in two
places, between E and B and between B and C. We observe that the
upper note of the interval of a limma is found by moving 5 positions
to the left in the chain of fifths from the lower note.

The introduction of sharps and flats (and doubles) introduces more
tones at each end of the chain:

-8 –7 –6 –5 –4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G#

10 11 12 13 14 15
D# A# E# B# Fx Cx ...

Thus the upper note of the interval of an apotome is found by moving
7 positions to the right in the chain of fifths from the lower note,
and the double-apotome will be 14 positions. This will be true
regardless of the tuning (or division of the octave).

In just intonation the fifths in the chain are of a specific size (a
2:3 ratio of frequency), but in various EDO's they will only
approximate this ratio, being either wider or narrower that the just
fifth, according to the particular division. With 41-EDO they are
very slightly wide. With 19 and 22-EDO (to give opposite examples)
they will be over 7 cents narrow and wide, respectively. While 7
cents may seem to be a small amount, this difference will accumulate
as one moves along the chain of fifths. For example, if one moves 7
places to the right (an apotome), the difference from the Pythagorean
apotome will exceed 49 cents (i.e., 7 cents multiplied by 7 places).

A chain of 19-EDO fifths (~694.7 cents) will therefore result in a
limma of ~126.3 cents and an apotome of ~63.2 cents, whereas a chain
of 22-EDO fifths (~709.1 cents) will result in a limma of ~54.5 cents
and an apotome of ~163.6 cents. Between these two divisions the
limma varies more than 70 cents and the apotome more than 100 cents,
and they vary in opposite directions.

For 19-EDO I should also point out a couple of things about the
sequence of symbols:

0 +1 +2 +3
C C# Cx
Dbb Db D

No special symbols are required for 19. The double-apotome is only
2/3 of a whole tone, so we have here a clear example where the double-
apotome is not the same as a whole tone (which in this instance would
be equal to a triple apotome).

With 12-EDO we have a special case in which the fifths of 700 cents
result in a limma and apotome of the same size, 100 cents. In other
words, the difference between the limma and apotome is tempered out,
so that the distinction between them is (unfortunately) lost.
However, as I pointed out in my previous message, in traditional
Western harmony these two intervals are functionally different and
should not be confused with one another. As a corollary, it is
important to note that the whole tone is the sum of the limma and the
apotome, and this will be true regardless of the tuning (or division
of the octave). A whole tone (8:9) is therefore *not* two limmas or
two apotomes (appearances in 12-EDO notwithstanding).

The Sagittal notation takes all of these things into account and
allows the symbols to be used accordingly, with even more flexibility
than you have suggested.

> But the way I see it (forgive all my misunderstandings from this
point forward), Sagittal notation (as defined on page 18) introduces
12 different symbols to quasi-equally spaced portions of this half-
tone zone. Although I admit there IS a generous level of flexibility
(about 4-5 cents max) with each symbol, I am not satisfied that this
region should be defined by anything unresemblant to the sharp or
flat as a regular musician like myself is used to.

Ah, let me explain further! The diagram to which you refer
illustrates the allowable ranges for each of the symbols in the
trojan (12-relative, or 12R) symbol set, but these particular ranges
are applicable *only* with the 12-EDO fifth of 700.0 cents (and thus
an apotome of 100.0 cents). One could allow these intervals to vary
in size somewhat, but not such that their departures from 12-EDO
would accumulate along the chain of fifths. Observe that the 5-
comma /| is given a nominal size of 15 cents in the trojan set, since
major and minor thirds and sixths are in error by approximately that
amount in 12-EDO. The 7-comma |) is given a nominal size of 31
cents, approximately double that of the 5-comma, while the 7:11-comma
(| is set at 20 cents. As you noted, there is a small amount of
flexibility for each of these symbols in the neighborhood of each of
their nominal values.

Compare that with the range of sizes given for the athenian symbol
set in Figure 5 (p. 10) for medium-precision just intonation, in
which the fifths are in exact 2:3 ratio and the apotome is therefore
~113.7 cents. Here the "nominal" size for the 5-comma symbol is
~21.5 cents (80:81), while the 7-comma is ~27.3 cents (63:64) and the
7:11-comma is ~33.1 cents. Again there is a small amount of
flexibility for each symbol in the neighborhood of each of their
nominal values, but the most significant difference between the
athenian and the trojan symbol usage is that the nominal values for
the symbols have changed.

Having said all of this, I must now point out that the sort of
flexibility of Sagittal symbols that I have just discussed is, by and
large, *irrelevant* to your purposes! With the 41-division you are
using a fifth that is essentially just, and the tones in the chain of
fifths will therefore be close to Pythagorean. This would require
that the Sagittal symbols have nominal values at (or close to) those
given for the athenian symbol set (in Figure 5). You require both
fewer symbols and greater flexibility of pitch than what is allowed
with the athenian symbol set, and both of these may be achieved in
Sagittal by means of a 41-EDO mapping and a smaller set of symbols.

With the near-pure fifths of 41-EDO, the 5-comma and 7-comma both
have nominal sizes of 1 degree (~29 cents), so that only one of these
symbols will be necessary. In Sagittal we would choose the 5-comma
symbol, and the 11-diesis symbol (remember Apollo-11) would be used
for the half-apotome. These, combined with apotome and double-
apotome symbols, result in the following symbol sequence:

0 +1 +2 +3 +4 +5 +6 +7 +8
C C/| C/|\ C||\ C/||\ C/||| C/|||\ CX\ C/X\ pure
DY/ D\!!!/ D\!!! D\!!/ D!!/ D\!/ D\! D D/|
C C/| C/|\ C#\! C# C#/| C#/|\ Cx\! Cx mixed
Dbb/| Db\!/ Db\! Db Db/| D\!/ D\! D D/|

Please observe that in the 41 division a double-sharp (or double-
apotome) is not the same thing as a whole step (or major 2nd), as
will be the case with all divisions that are not multiples of 12. If
you are basing your notation on the 41 division, then I don't believe
that you can disregard this where you said:

< The reason I have reached the double flat or double sharp on the
7th step is due to the fact that I have been trying to preserve the
elasticity I mentioned above as well as the symmetry of the whole-
tone division. You'll notice that 9/8 is equally divided into 2
parts, and these in turn are divided to 3 portions each. Thus the
sequence of microtonal signs in a whole-tone from C to D will be like
this: [simplified version of the diagram on p. 13 of your thesis]

In your diagram I observe that the amount of flexibility is (over
8:9) smallest at the sides and increases toward the middle. The 41
and 53 (schismic) divisions are, in my opinion, the divisions that
conform most closely to the set of tones you describe. I tabulated
the sizes of the main interval-ratios in these two representative
divisions (lumping two adjacent degrees together for the apotome):

Ratio: 1:1 80:81 32:33 apotome 11:12 9:10 8:9
41 0 1 2 3 4 5 6 7 degrees
41: 0c 29c 59c 88c 117c 146c 176c 205 cents
53: 0 1 2 4 5 7 8 9 degrees
53: 0c 23c 45c 91c 100c 158c 181c 204 cents
Range: 0c 6c 11c <--29c--> 12c 5c 1c

The range of flexibility does indeed progress as desired (if you lump
together two adjacent degrees under the apotome label; however, I
have already said that I do not think that this is advisable.)
There is also symmetry without the necessity of equating the double-
apotome with 8:9.

On page 16 of your thesis you have a table showing categories of
small intervals with a range of sizes for each. You may be
interested to learn that Dave Keenan systematically derived
boundaries for categories of comma-ratios as follows (from messages
#54577, 54584):

Square of lowerbound
.. 2-exponent
...... 3-exponent
................ Lowerbound (cents)
................................. Size range name
-------------------------------------------------
.. [ 0, 0 > ....... 0 ........... schismina
.. [-84, 53 > ..... 1.807522933 . schisma
.. [ 317,-200 > ... 4.499913461 . kleisma
.. [-19, 12 > .... 11.73000519 .. comma
.. [-57, 36 > .... 35.19001558 .. small-diesis
.. [ 8, -5 > ..... 45.11249784 .. (medium-)diesis
.. [-11, 7 > ..... 56.84250303 .. large-diesis
.. [-30, 19 > .... 68.57250822 .. small-semitone
.. [ 35, -22 > ... 78.49499048 .. limma
.. [-3, 2 > ..... 101.9550009 ... large-semitone
.. [ 62, -39 > .. 111.8774831 ... apotome
.. [-106, 67 > .. 115.492529

It is interesting to reduce the number of Dave's categories (to
correspond to degrees of the 41 division) and then compare his
boundaries with yours, as I have done here:

Sagit Category DK boundaries OY boundaries
----- -------- --------------- -------------
/| comma (C) 11.73 – 35.19 20 - 40
/|\ diesis (D) 35.19 - 68.57 40 - 70
||\ limma (L) 68.57 – 101.96 70 - 100
/||\ apotome (A) 101.96 – 125.41 100 - 130
/||| 2A-L 125.41 – 158.80 130 - 160
/|||\ 2A-D 158.80 – 192.18 160 - 180
X\ 1 tone (A+L) 192.18 – 215.64 180 - 220
or 2A-C

As you see, there is considerable agreement in these numbers.

I believe that you can have both the required symmetry and
flexibility of pitch without requiring the double apotome to be
equated with (i.e., the same number of degrees as) the whole tone.

> Yes, it is possible for me to learn Sagittal notation if I commit
myself to it starting from now on. But as a speaker of a particular
Chinese dialect spends all his childhood memorizing the glyphs and
their attributes, so will I have to face a similar task of `un-
learning` and `re-learning` notation from scratch. I was entertaining
the idea that `microtonality` should be viewed as a welcome addition
to the conventional musical knowledge, but I never imagined that it
would result in the obsolescence of the sharp and flat all together.

The change is not really as radical as you (and some others) might
think it to be. In the paper I have tried to emphasize that in the
pure Sagittal system it is only a few *symbols* that are changing.
Their *names* (sharp, flat) and *meanings* remain exactly the same as
before.

> Yes, you have preserved the sharp and flat in their original form
as a substitute to the understanding of Sagittal Notation, plus as a
means to quell any opposition to the contrary. However, these have
been reduced to a supplementary state rather than forming the basis
of the pragmatical comprehension of `intermediate tone`. I am not
sure if I can find it in my heart to abondon my `roots`. (Do forgive
my criticism if it sounds too harsh. I really appreciate and
congratulate your valuable efforts.)

Thank you for your honesty. Dave Keenan and I have tried to provide
options (such as the mixed symbol version) to make the notation as
broadly useful and acceptable as possible to a great diversity of
composers and musicians, and we realize that innovations may not be
accepted as quickly as we would like. For some the mixed-symbol
option may serve in a transitional capacity.

> The complete set of symbols which amounts to 53 per half-tone is a
bewildering number for someone like me who has a hard time with 72-
TET alone. Even the spartan symbol set with 13 symbols per half-tone
is too much given their similitude with each other. The distinctions
are simply not enough for a layman like me who wants to use
microtones for orchestral authentic maqam music.

I defer to your judgment that a 41-EDO symbol set will be sufficient
for your purposes.

> To the detriment of complexity, I venture to think that additional
symbols after 72-TET simply are a waste of notation space (I dare
risk being branded as a reactionary). Why would someone want to
persist in using 152-TET on the olde staff? If one wants to notate
music that no single professional musician could possibly play
correctly, I suggest that other methods for electronic notation be
developed aside from that which we wish to upgrade.

On the other hand, it would be very convenient, if at all possible,
to have the same sort of notation for both performance and electronic
music, for that would permit the latter to be converted to a printed
score that could be studied or (when technically feasible) to a
printed part that could be played on an acoustic instrument.
Conversely, it would also simplify the process of entering electronic
pitch data, since it would not be necessary to learn a completely
different notation for that purpose. Some of the more intricate
details regarding how this could best be accomplished are currently
in the process of being worked out.

> I am well pleased that we converge on the same number of symbols
for 41-edo. I also agree that your symbols take no less time than
mine to learn. However, I live in Turkey where thousands of musicians
are accustomed to the usage of the sharp as limma,

Then perhaps you should be one to educate them that this is incorrect
and that the sharp is actually an apotome. Ignorance about these
things also abounds in my part of the world, but I think that I
should then do something to remedy the problem.

> the slashed flat as the quarter-to-half tone and the reversed flat
as the comma. In order to preserve the application of these
microtones, I have to preserve the microtonal accidentals that go
along with them. The solution I found was not easy, and I expect to
deal with unsettling commotion from the instigators and
traditionalists who uphold the Yekta-Arel-Ezgi school with zeal. But
for the sake of clarity, transposability and flexibility, I can go no
further in number with the sharps and flats I proposed.

It sounds as if you're caught in a quagmire, with the symbols given
meanings at variance with what we have recognized for centuries in
the West. I cannot avoid the conclusion that the only way our
cultures might ever be able to have a notation in common would be to
start fresh with completely new symbols about which there would be no
dispute as to their meaning. As for how difficult it might be for
anyone to be persuaded to take that step, I will not lose any sleep
over that question; but I do what I can to make a difference.

> ...
> For those who still have no idea on what I'm drawling about, here
is a copy of my doctorate thesis report:
>
> http://www.musiki.org/yarman_tez_rap2.pdf

And I encourage them to look at the figure and tables on pages 13 and
16-19.

> I would love to read more about the keyboards you designed.

Please see these messages:
/tuning/topicId_39323.html#39399
/tuning/topicId_39323.html#39407
/tuning/topicId_36964.html#37151
as well as others in these subject threads.

In order to view the actual diagrams, you would have to join the
tuning-math group:
/tuning-math/
They are located here (inasmuch as there was no longer any room in
the files section for the alternate tunings group):
/tuning-math/files/secor/kbds/
Files: KbDec72.gif and KbScal31.gif

> ...
> As to your question:
>
> `...Would you teach the Turkish language using a simplified
> subset of your alhabet and then permit your students to
mispronounce
> words for the sake of independence? I think not.`
>
> I can tell you that the revolution which commenced about 80 years
ago did just that. ... Facing such a state of affairs, I have no
choice but to teach Turkish using as simplified an alphabet as
possible (29 in number since the Republic).

Well, I guess the joke is on me! I thought I was presenting a
hypothetical situation, but it was all too true! The point I was
trying to make is that there must enough symbols in the notation to
convey the various shades of meaning.

> As for our terminological dispute:
>
> I refer to the apotome based on the ditonic series starting from C
to its octave (series of white keys). Accordingly, the apotome is as
close a sharp to its neighbour as possible. Thus, D sharp is about
110 to 120 cents higher than D, and consequently about 90 to 80 cents
far from E. Maybe I'm using out-dated concepts, so please correct me
if I'm wrong. As far as I know, in western music the sharps raise and
flats lower an alternate tone in such a way that they by-pass each
other by 1/3 tone

The difference is generally much smaller than that, more like 1/9
tone -- but your 1/3-tone figure (which corresponds to a 17-tone
octave) is not out of the question. But this is only the case when
melodic expressiveness governs the nuances of pitch, which is only
half the story.

> although they are considered to be enharmonical. This is also a
necessity of the harmonic context as you describe.

The other half of the story is that when consonant harmony is of
primary concern, then sharps will be lower in pitch than
enharmonically related flats. In 31-EDO, for example, this
difference will amount to 1/5 tone:

0 +1 +2 +3 +4 +5
C C/|\ C# C#/|\ Cx
Dbb Db\|/ Db D\|/ D

> No, I'm not ignoring the subleties that enharmonical intervals
represent. On the contrary, by splitting the half-tone zone into two
sub-regions, I believe I am able to indicate the difference between
the sharp and the flat just the way it is meant to be.

Okay, I see that you have small triangles affixed to the sharp and
flat signs to give alternate spellings. These are equivalent to the
Sagittal ||\ sharp and !!/ flat without the left flag, such that:

D to F/||\ is a Pythagorean major 3rd (64:81), whereas
D to F||\ is a just major 3rd (4:5), and
C to E\!!/ is a Pythagorean minor 3rd (27:32), wherease
C to E!!/ is a just minor 3rd (5:6).

It appears that the only significant difference of opinion we have is:

1) Regarding the particular symbols to be used:
a) Present Turkish usage of the Tartini fractional symbols is at
variance with West European usage going back centuries (which puts us
at an impasse), but
b) The pure Sagittal symbols that would resolve that impasse are
too radical a change for you (and others) to accept.

2) You equate C-double-sharp with D in your 41-tone framework (if I
understand you correctly), whereas it is generally established (to
the extent that I'm completely confident that everyone knowledgeable
in this group would agree with me) that the double-sharp in that
division overshoots the major 2nd by one degree. I insist that a
double-apotome should *always* be twice as large as an apotome --
always!

> But I'm rather confused as to how a semi-tone could be a quarter-
tone in 22-edo, and an apotome a 3/4 tone. And how is it that the
syntonic comma (80:81) can be diverted to mean 55 cents when it is 20
cents? Please forgive my ignorance in these matters.

Since the fifths of 22-EDO are so wide, a chain of four fifths (e.g.,
C to E) will result in a wide major third of ~436 cents (8 degrees of
22, which is quite close to 7:9). Taking C as 1/1, we find that the
best approximation of 5/4 is 7 degrees (1 degree lower), so we
therefore notate this tone as E\! (E lowered by a 5-comma). The 5-
comma in 22-EDO therefore corresponds to a single degree (~54.5
cents). On the other hand, the 7-comma vanishes to zero.

Here is how we notate several degrees of 22:

0 +1 +2 +3 +4 +5 +6 +7
C C/| C||\ C/||\ C/||| CX\ C/X\ pure
D\!!! D\!!/ D!!/ D\! D D/| D||\ D/||\
C C/| C#\! C# C#/| Cx\! Cx mixed
Db\! Db Db/| D\! D D/| D#\! D#

Here, also, the double-apotome is twice as many degrees as the
apotome.

> I thank you for investing your valuable time to discipline my far-
fetched ideas. Our encounter has proved to be most worthy and
precious in my eyes.

Ozan, your ideas are not at all far-fetched, and I have learned some
valuable things from our discussion. There is much that I do not
understand concerning the way that you are seeking to apply your
notation to musical cultures with which I am for all practical
purposes unfamiliar, so I can only explain what we have done with
notation from a Western perspective and hope that it may be of
benefit to you.

Best wishes,

--George Secor

You can configure your subscription by sending an empty email to one
of these addresses (from the address at which you receive the list):
tuning-subscribe@yahoogroups.com - join the tuning group.
tuning-unsubscribe@yahoogroups.com - leave the group.
tuning-nomail@yahoogroups.com - turn off mail from the group.
tuning-digest@yahoogroups.com - set group to send daily digests.
tuning-normal@yahoogroups.com - set group to send individual emails.
tuning-help@yahoogroups.com - receive general help information.

Yahoo! Groups Sponsor
ADVERTISEMENT

------------------------------------------------------------------------------
Yahoo! Groups Links

a.. To visit your group on the web, go to:
/tuning/

b.. To unsubscribe from this group, send an email to:
tuning-unsubscribe@yahoogroups.com

c.. Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

---
Outgoing mail is certified Virus Free.
Checked by AVG anti-virus system (http://www.grisoft.com).
Version: 6.0.784 / Virus Database: 530 - Release Date: 27.10.2004

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George,
>
> I need some time to fully digest the science behind your concepts.
It also shows that I need to keep my spontenaity in check

Not at all -- we learn best and profit the most when we freely share
our ideas and opinions with others, with an attitude of mutual
respect.

> lest I ridicule myself further among my superiors.

Please regard everyone else here as peers who, having different
backgrounds and varying amounts of expertise in numerous areas of
knowledge and experience, can all learn from one other. While I have
the experience of many years with alternate tunings, another (such as
you) may bring a new viewpoint that has the effect of a breath of
fresh air.

> While I begin researching the subject and prepare a reply, I ask
you to take a look at my recent doodle which I named Spectral
Notation. I'm certain to benefit from your edifying directions and
look forward to continue our lessons.

If you are referring to using colors on a music page for various
shades of pitch, then I'm afraid I don't believe that would be very
practical, since color printing is much more expensive, less readily
available to many of us than monochrome, and can be somewhat
difficult to distinguish in poor light (such as some musicians
reading parts must endure, besides putting at a disadvantage those
who are partially color-blind). But I have used colors in diagrams
in association with prime and odd numbers: black or white (for
foreground or background, respectively) for 2, red for 3, yellow or
brown (for background or foreground, respectively) for 7, green for
9, cyan for 11, blue for 13, purple for 17, magenta or gray for
higher primes or complex ratios.

Best,

--George Secor

Dear George,

I must apologize for my belated reply. I could only just devour the immense Sagittal Notation text and comprehend the vicissitudinous contents therein. Having grasped the fundamentals of your notation system, allow me to propound my views:

I am glad we both concur that it is proper to regard `enharmonicities` as `concentrations of tones` restricted to certain `pitch classes` sanctioned by a pre-defined set of melodic and harmonic rules pertaining to the gestalt of `microtemperament`. Regarding the pitch-cluster boundaries, I am most surprised to find Mr. Keenan's intervallic values and mine to match so closely. You must understand that I accidentally chanced upon these findings while I floundered in my lack of proper theoretical information. It was rather my getting musically involved with maqam instruments such as the ney, bowed tanbur and qanun besides piano that showed me how equivocal `distinct` intervals could be performed.

Yet, with all due respect, I am not against the expansion of my system so as to allow the limma-apotome zone a wider berth. But in your system, the apotome, or any other interval for that matter, is restricted to a single pitch which is bound to the pre-selected temperament. Barring scalar transitions and miniscule flue-zones, the Sagittal Symbols correspond to fixed values rather than pitch-clusters along the register, as should be the case in a live performance, what with tone drift, comma pump, artistic expression and all that. My elder associate Can Akkoc's novel study on the stochastic scale-models for Traditional Turkish Music that has been published by the Journal of New Music Research elucidates this matter. Also, David Howard's study on "A Capella SATB Quartet In-Tune Singing: Evidence of Intonational Shift" cannot be summarily dismissed. Similarly, Hiroko Teresawa's `Pitch Drift` and Xaq Pitkow's "Why Do Octaves Sound The Same?" articles merit attention.

However, the Sagittal Notation's shortcoming in that respect have been realized in the article and a description of the intended temperament is advised for each composition beforehand. Tempering the syntonic comma differences on p. 19 using 7 analogous accidentals per measure is another story of course! If a simple definition is sufficient, why go to all the trouble of calculating commas, limmas and apotomes for each system when it is obvious that there shall arise a general inconsistency with the employment of the same symbols for diverse temperaments? And furthermore, why should anyone notate a regular C Major chord progression (no matter the temperament) with any accidentals at all? I, for one, find it objectionable that I cannot transcribe a diatonic, ditonic or enharmonic natural scale without the need for micro-accidentals when using Sagittal Notation.

I thank you for being so kind as to explain in depth how you arrive at different intervallic values each time you stack the best-fit fifths according to the chosen temperament. But as far as I understand, the terminology of the limma, apotome, comma and such are reserved strictly for the Pythagorean cycle of fifths. I beg to differ when it comes to the neology you introduce in order to ascribe concrete
pythagorean terms to the various degrees of any other system such as 22-edo, regardless of the fact that the respective formulae used to derive such intervals are isomorphic in character.

I am now very much familiar with how the Pythagorean diatonic scale is constructed. To me, (let me pronounce this at the expense of sounding ultra-conservative) it is the single basic diatonical series that makes any sense when teaching the difference between classical western music and other genres & periods. I regard the chain of pure fifths as the only system where one is entitled to extract the true values for limma, apotome, comma and such... In other systems where applying the same methodology leads to intervallic discrepancies, I am forced to resort to the tables presented by Mr. de Coul in his Scala program. Thus, my division of the whole-tone seems rather consistent with the amassed knowledge of scales and intervals thus far.

But I'll introduce a conversion scheme, where It will be possible to preserve your nomenclature:

5 positions forward in the cycle of `unperfect` fifths gives a "limma-substitute", 7 positions forward gives an "apotome-substitute", two "limma substitutes" added together gives a "minor whole tone substitute", a "limma substitute" plus "an apotome substitue" gives a "major whole tone substitute", 13 positions forward gives a "ditonic comma-substitute", 14 positions forward gives a double-apotome-substitute", the difference between the "diatoniaion-substitute" and "the sixth harmonic" gives the "syntonic comma-substitute", and so forth...

Therefore, my accidentals retain their `pythagorean identities`, and shall not be translated each time into a completely different system where their respective values otherwise require re-alignment according to the generator size of the quasi-fifth.

Perhaps our conception of `flexible` is not mutual at this point, as I have taken the 3-limit pythagorean diatonic scale as a nominal reference for the natural note-heads on the staff. This is acceptable, since the E-F and B-C are indicated as semi-tone intervals without accidentals on the staff, and hence are to be understood as natural tones in accordance with historical practice. No wonder Sagittal flexibility is irrevalent to my purposes! I shall invoke my whole-tone division to determine where the `substitute natural tones` and `intermediate tones` of any other temperament correspond in order to indicate their deviations using the same symbols regardless of the scale. Now that seems a lot more consistent compared to the employment of jumpy Sagittal arrows signifying the `apotome`, `limma`, `comma`, and whatnot as they jolt with every temperament due to the variation of the size of the fifth by only a few cents.

I do not favor the clumping of the apotome and the limma together actually, but it is possible to temper the limma out and adopt a 36-edo system. Seeing as this results in 12-edo elaborated exactly 3 times, or 72-edo simplified by 2, it appears to be the next step to take for the Maqam World which has adopted 24-edo since Allah knows when. Considering the lower boundaries Mr. Keenan gives in this table:

Square of lowerbound
.. 2-exponent
...... 3-exponent
................ Lowerbound (cents)
................................. Size range name
-------------------------------------------------
.. [ 0, 0 > ....... 0 ........... schismina
.. [-84, 53 > ..... 1.807522933 . schisma
.. [ 317,-200 > ... 4.499913461 . kleisma
.. [-19, 12 > .... 11.73000519 .. comma
.. [-57, 36 > .... 35.19001558 .. small-diesis
.. [ 8, -5 > ..... 45.11249784 .. (medium-)diesis
.. [-11, 7 > ..... 56.84250303 .. large-diesis
.. [-30, 19 > .... 68.57250822 .. small-semitone
.. [ 35, -22 > ... 78.49499048 .. limma
.. [-3, 2 > ..... 101.9550009 ... large-semitone
.. [ 62, -39 > .. 111.8774831 ... apotome
.. [-106, 67 > .. 115.492529

it is obvious that I have to calibrate the figures in this way:

~0-33 cents: Schisma-Comma accidental
~33-66 cents: Diesis accidental
~66-99 cents: Limma accidental

~100-133 cents: Apotome accidental
~133-166 cents: Sesquisharp/sesquiflat enharmonic
~166-199 cents: Minor whole-tone enharmonic.

As for the 41-edo table you were so kind to provide, the new boundaries should be like this:

Sagit Category DK boundaries OY boundaries
----- -------- --------------- -------------
/| comma (C) 11.73 – 35.19 0 - 33 2. degree
/|\ diesis (D) 35.19 - 68.57 33 - 66 3. degree
||\ limma (L) 68.57 – 101.96 66 - 99 4. degree (demitone)
/||\ apotome (A) 101.96 – 125.41 100 - 133 5. degree (semitone)
/||| 2A-L 125.41 – 158.80 133 - 166 6. degree
/|||\ 2A-D 158.80 – 192.18 166 - 199 7. degree
X\ 1 tone (A+L) 192.18 – 215.64 200 8. degree (wholetone)
or 2A-C
X Double A 200-233 9. degree

Indeed, I have corrected my mistake of considering the double apotome as an alternative method of expressing the wholetone, which should not have required any special accidental at all. So, the same number of common symbols suffice in my system, whereas you favor in your article the pure Sagittal symbols as a means to evade any misunderstandings that arise from the mixed version, and replace the sharps and flats altogether.

This brings us to the issue of 53-symbols per half-tone. You will notice that some of the criticisms you expressed for my Spectral approach are valid for the Sagittal Notation as well:

1. In poor lighting conditions, 7 colors are as difficult to discern as 53 symbols.
2. Just as there may be color-blind musicians, you will find the majority to be rather `symbol-blind`.
3. Printing the music sheets with dabs of color is only as expensive as printing scores double the size of what we mundane musicians are accustomed to.

Other than these, I am surprised that you did not consider applying colorific accidentals when you have used colors to distinguish prime and odd intervals already.

On the other hand, I concur that the sharp and flat are to be understood as pythagorean apotomes. According to a Turkish saying: `The hairs of my tongue died out` in the process of my repeating this fact over and over. The quagmire is boggier than one might think at first, but that is no reason for us to abondon the aesthetically beautiful and notorious sharp and flat symbols to the hands of ignorants, be they Westerners or Easterners. Forgive my saying so, but I see no way that the Sagittal Notation could be adopted right away as a means to unify our mutually opposed heritrices. Believe me that I am trying to be as objective as possible when I say this. Microtonality should be considered as a welcome addition to the available musical knowledge, not as a disendowment of traditional values.

Please accept my apology for any harsh comments that I may have uttered. I am most grateful to the attention you have shown and the valuable theoretical information you imparted to me.

Cordially,
Ozan Yarman

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> I do not favor the clumping of the apotome and the limma together
actually, but it is possible to temper the limma out and adopt a
36-edo system.

This is tempering out the Pythagorean comma, which means you are
looking at divisions which are multiples of 12.

Seeing as this results in 12-edo elaborated exactly 3 times, or 72-edo
simplified by 2, it appears to be the next step to take for the Maqam
World which has adopted 24-edo since Allah knows when. Considering the
lower boundaries Mr. Keenan gives in this table:

Why 36? Unless you are uninterested in the tuning of major thirds, 72
and 84 jump out at you. If you like {2,3,7} 36 is excellent.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George,
>
> I must apologize for my belated reply. I could only just devour the
immense Sagittal Notation text and comprehend the vicissitudinous
contents therein.

Dear Ozan, no apology is necessary. Otherwise I would be apologizing
to you for being so busy lately that I did not even look at the
messages for over a week -- and then you had already posted your
reply.

> Having grasped the fundamentals of your notation system, allow me
to propound my views:
>
> I am glad we both concur that it is proper to regard
`enharmonicities` as `concentrations of tones` restricted to certain
`pitch classes` sanctioned by a pre-defined set of melodic and
harmonic rules pertaining to the gestalt of `microtemperament`.
Regarding the pitch-cluster boundaries, I am most surprised to find
Mr. Keenan's intervallic values and mine to match so closely. You
must understand that I accidentally chanced upon these findings while
I floundered in my lack of proper theoretical information. It was
rather my getting musically involved with maqam instruments such as
the ney, bowed tanbur and qanun besides piano that showed me how
equivocal `distinct` intervals could be performed.

Theory must ultimately be based on what we discover in practice, but
in the process we find that much of what we find in practice is
explained by the mathematics underlying acoustical relationships.
While Dave Keenan's approach to size categories of intervals is
almost purely mathematical, it does correlate very nicely with things
that both you and I have observed.

> Yet, with all due respect, I am not against the expansion of my
system so as to allow the limma-apotome zone a wider berth. But in
your system, the apotome, or any other interval for that matter, is
restricted to a single pitch which is bound to the pre-selected
temperament. Barring scalar transitions and miniscule flue-zones, the
Sagittal Symbols correspond to fixed values rather than pitch-
clusters along the register, as should be the case in a live
performance, what with tone drift, comma pump, artistic expression
and all that.

I must apologize for a misunderstanding, because there is no such
restriction for the Sagittal symbols. I suspect that you may have
drawn your conclusion from this (or some similar) statement in our XH
paper:

<< Since the Sagittal notation symbolizes intervals that are allowed
to vary in size to accommodate many different tunings, it is
necessary that sufficient information be provided in a score to
specify the particular tuning that is intended, along with a pitch
reference. >>

We felt that it was necessary to emphasize that such indications be
provided in a composition precisely because the Sagittal system is,
without these indications, capable of a wide range of
interpretations, whereas the composer may have something very
specific in mind.

On the other hand, it is entirely possible that one might not wish to
specify a particular tuning or instrumentation for a composition, but
leave that entirely up to the performer(s). In fact, that's exactly
what I intend to do for a composition that I have planned, which I
will initially produce electronically in perhaps two or three
differently tuned versions. This approach may not work with
compositions written specifically to exploit the characteristics of
certain tunings (for which the intended tuning definitely should be
specified), but that limitation is a consequence of the composer's
intentions, not the notation itself. And even if a particular tuning
is specified for a composition notated with the Sagittal system, that
in no way prohibits a performer from taking artistic liberties with
pitch.

If you chose to use just pitches mapped to a 41-tone octave, then you
might use the 41-ET symbol set, which is only a very small part of
the entire Sagittal system. The 5-comma, 7-comma, 7:11-comma, and 55-
comma (which range in size from 21.5 to 33.1 cents) would
collectively be represented by a single symbol -- the one used for
the 5-comma, thus allowing considerable flexibility in interpreting
that symbol. The musical context would thereby determine which of
these commas is intended, and it would be necessary for you, the
composer, to list which ratios (or range of sizes) you are using for
each of the symbols, along with an indication that the fifths are to
be exact (2:3). I believe this is the sort of symbol flexibility you
had in mind, and I apologize if I have not previously made it clear
how this could be accomplished.

> However, the Sagittal Notation's shortcoming in that respect have
been realized in the article and a description of the intended
temperament is advised for each composition beforehand. Tempering the
syntonic comma differences on p. 19 using 7 analogous accidentals per
measure is another story of course! If a simple definition is
sufficient, why go to all the trouble of calculating commas, limmas
and apotomes for each system when it is obvious that there shall
arise a general inconsistency with the employment of the same symbols
for diverse temperaments? And furthermore, why should anyone notate a
regular C Major chord progression (no matter the temperament) with
any accidentals at all? I, for one, find it objectionable that I
cannot transcribe a diatonic, ditonic or enharmonic natural scale
without the need for micro-accidentals when using Sagittal Notation.

The figure to which you refer notates a comma pump in adaptive JI
using symbols defined as multiples of 1/4-comma. As you said, it's
not necessary to use any micro-accidentals at all, but suppose that
you're producing this music electronically or explaining adaptive JI
in a textbook? Would you not wish to have a notation capable of
expressing the departure of the desired pitches from a strict
pythagorean scale? The key concept here is flexibility: Sagittal
allows you to be as strict or liberal with pitch as you wish, and
whether or not you should use many, or a few, or no Sagittal
accidentals is entirely your choice.

Should a poet specify that a reading of his/her poem before an
audience be spoken by a person having a certain regional accent or
ethnic background, or should this be required for a character in a
stage play? Maybe, and maybe not -- it depends on the specifics. An
author may employ incorrect grammar and spell words in a peculiar
manner to express different mannerisms of speech. The same principle
would apply to musical tunings: perhaps all that would be necessary
is to specify the tuning (without using any special symbols), or
perhaps special micro-accidentals would be required to convey precise
meanings.

> I thank you for being so kind as to explain in depth how you arrive
at different intervallic values each time you stack the best-fit
fifths according to the chosen temperament. But as far as I
understand, the terminology of the limma, apotome, comma and such are
reserved strictly for the Pythagorean cycle of fifths. I beg to
differ when it comes to the neology you introduce in order to ascribe
concrete pythagorean terms to the various degrees of any other system
such as 22-edo, regardless of the fact that the respective formulae
used to derive such intervals are isomorphic in character.

We acknowledge the strict meaning of the term "apotome" in footnote 4
(p. 2) and thereby explain that we're using it in a more general
sense. The broader use of all of the terms you mentioned above to
designate the occurrence of analogous intervals in various
temperaments did not originate with us, having been in practice for
some years. Am I not entitled to use the term "perfect fifth" for a
tempered fifth? And if an "apotome" is defined as 7 perfect fifths
less 4 octaves, am I any less entitled to use that term for a
tempered apotome? The only requirement is that the harmonic identity
of the term be preserved.

> I am now very much familiar with how the Pythagorean diatonic scale
is constructed. To me, (let me pronounce this at the expense of
sounding ultra-conservative) it is the single basic diatonical series
that makes any sense when teaching the difference between classical
western music and other genres & periods. I regard the chain of pure
fifths as the only system where one is entitled to extract the true
values for limma, apotome, comma and such... In other systems where
applying the same methodology leads to intervallic discrepancies, I
am forced to resort to the tables presented by Mr. de Coul in his
Scala program. Thus, my division of the whole-tone seems rather
consistent with the amassed knowledge of scales and intervals thus
far.
>
> But I'll introduce a conversion scheme, where It will be possible
to preserve your nomenclature:
>
> 5 positions forward in the cycle of `unperfect` fifths gives
a "limma-substitute", 7 positions forward gives an "apotome-
substitute", two "limma substitutes" added together gives a "minor
whole tone substitute", a "limma substitute" plus "an apotome
substitue" gives a "major whole tone substitute", 13 positions
forward gives a "ditonic comma-substitute", 14 positions forward
gives a double-apotome-substitute", the difference between
the "diatoniaion-substitute" and "the sixth harmonic" gives
the "syntonic comma-substitute", and so forth...

This seems rather cumbersome. If you play the interval of an apotome
on an instrument while taking some liberty with the pitch, should I
still call that an "apotome" or should I insist on calling that
an "apotome substitute?" (Or should I always call it an "apotome
substitute", following the premise that it's not humanly possible for
anyone to sing or play an exact apotome on any instrument?)

The real issue is whether or not my liberal use of the term is liable
to lead a reader to confusion. I believe that in most cases the
context of a discussion should be sufficient to indicate whether this
term is being used in a strict or a broader sense. If you, on the
one hand, insist that Sagittal symbols should be allowed to be more
flexible in their interpretation than you presumed that I would allow
(and, of course, I agree), then why, on the other hand, would you
object to my using the same flexibility for the terms that identify
those symbols?

> Therefore, my accidentals retain their `pythagorean identities`,
and shall not be translated each time into a completely different
system where their respective values otherwise require re-alignment
according to the generator size of the quasi-fifth.

And in the context of your writing, that is exactly what I would
expect.

> Perhaps our conception of `flexible` is not mutual at this point,
as I have taken the 3-limit pythagorean diatonic scale as a nominal
reference for the natural note-heads on the staff. This is
acceptable, since the E-F and B-C are indicated as semi-tone
intervals without accidentals on the staff, and hence are to be
understood as natural tones in accordance with historical practice.
No wonder Sagittal flexibility is irrevalent to my purposes! I shall
invoke my whole-tone division to determine where the `substitute
natural tones` and `intermediate tones` of any other temperament
correspond in order to indicate their deviations using the same
symbols regardless of the scale. Now that seems a lot more consistent
compared to the employment of jumpy Sagittal arrows signifying the
`apotome`, `limma`, `comma`, and whatnot as they jolt with every
temperament due to the variation of the size of the fifth by only a
few cents.
>
> I do not favor the clumping of the apotome and the limma together
actually, but it is possible to temper the limma out and adopt a 36-
edo system.

As Gene Ward Smith observed, this is properly referred to
as "tempering out the Pythagorean comma."

> Seeing as this results in 12-edo elaborated exactly 3 times, or 72-
edo simplified by 2, it appears to be the next step to take for the
Maqam World which has adopted 24-edo since Allah knows when.

I believe that the 41 or 53 division is much better suited to your
purposes. Most of the ratios containing prime numbers 5 (5/4, 5/3,
6/5, 8/5, 9/5, 10/9, 7/5, 10/7, 15/8, 16/15) or 11 (11/8, 11/6, 11/7,
11/9, 12/11, 14/11, 16/11, 18/11) are very badly represented in 36-
edo, and I note that most of these ratios appear in your large
table. If you really must use a multiple of 12, then I would say
that a subset of 72-edo would be much better than 36-edo. The
difficulty with any multiple of 12 is that it doesn't allow you have
the following ratios fall naturally into an orderly sequence of
fifths (some approximate, yet close to 2:3):

40/27 10/9 5/3 5/4 15/8 7/5 21/20 128/81 32/27 16/9 4/3 1/1 3/2 9/8
27/16 81/64 243/128 10/7 16/15 8/5 6/5 9/5 27/20 81/80

(You don't have 9/5 in your large table, but you do have its octave-
complement, 10/9.)

Instead, the fifths would be in 3 separate circles of 12. In 36-edo,
the ratios 7/5 and 11/8 are both closest to 17deg36, but then you
would be moving 7/5 to 18deg36, along with 10/7.

Should the Maqam World consider taking a step beyond 24, I would
advise you to make a diligent effort to persuade them that they can
do better than 36.

> Considering the lower boundaries Mr. Keenan gives in this table:

Dave Keenan has very recently brought to my attention that the
boundary between the comma and diesis given in that table should
properly be at 33.38c rather than 35.19c and that the lower boundary
for the limma should be 80.30c rather than 78.49c. Up to this time
we were reluctant to have a category boundary occur between comma-
ratios that are represented by the same Sagittal symbol (|:

7:11 comma, 45056:45927, ~33.148c
comma/small diesis boundary, ~33.382c
29 small diesis, 256:261, ~33.487c
13:17 small diesis, 51:52, ~33.617c

For convenience we therefore agreed to use a comma-diesis boundary
that agreed with the upper boundary that we had established for the
(| symbol. That, however, has turned out to be somewhat arbitrary.
Dave's specific reason for making this change is not the least bit
arbitrary, since it involves the naming of small intervals based on
two things: 1) their prime factors, and 2) their sizes. I quote from
a recent message of his:

<< I recently noticed some examples that caused me to revise my
choice for the boundary between comma and small diesis (and therefore
also between small semitone and limma). That is, of the most popular
commas, there were fewer examples where there are two different n:m-
commas when the new boundary (below) is used.

The comma and small-diesis that caused this revision are:
5:49C 321489:327680 33.02 c
5:49S 98415:100352 33.74 c >>

We now agree that the present change was both inevitable and long
overdue. Here is the corrected table with this final change:

Square of lowerbound
.. 2-exponent
...... 3-exponent
................ Lowerbound (cents)
................................. Size range name
-------------------------------------------------
.. [ 0, 0 > ....... 0 ........... schismina
.. [-84, 53 > ..... 1.807522933 . schisma
.. [ 317,-200 > ... 4.499913461 . kleisma
.. [-19, 12 > .... 11.73000519 .. comma
.. [ 27, -17 > ... 33.38249264 .. small-diesis
.. [ 8, -5 > ..... 45.11249784 .. (medium-)diesis
.. [-11, 7 > ..... 56.84250303 .. large-diesis
.. [-30, 19 > .... 68.57250822 .. small-semitone
.. [-49, 31 > .... 80.30251341 .. limma
.. [-3, 2 > ..... 101.9550009 ... large-semitone
.. [ 62, -39 > .. 111.8774831 ... apotome
.. [-106, 67 > .. 115.492529

> it is obvious that I have to calibrate the figures in this way:
>
> ~0-33 cents: Schisma-Comma accidental
> ~33-66 cents: Diesis accidental
> ~66-99 cents: Limma accidental
>
> ~100-133 cents: Apotome accidental
> ~133-166 cents: Sesquisharp/sesquiflat enharmonic
> ~166-199 cents: Minor whole-tone enharmonic.
>
> As for the 41-edo table you were so kind to provide, the new
boundaries should be like this:

Dave's boundaries in that table must also be corrected, changing
35.19 to 33.38 and 192.18 to 193.99 (I have not attempted to decide
what upper boundary Dave might consider appropriate for the double-
apotome, so I have just given its exact pythagorean value in square
brackets):

Sagit Category DK boundaries OY boundaries
----- -------- --------------- -------------
/| comma (C) 11.73 – 33.38 0 - 33 2. degree
/|\ diesis (D) 33.38 - 68.57 33 - 66 3. degree
||\ limma (L) 68.57 – 101.96 66 - 99 4. degree (demitone)
/||\ apotome (A) 101.96 – 125.42 100 - 133 5. degree (semitone)
/||| 2A-L 125.42 – 158.80 133 - 166 6. degree
/|||\ 2A-D 158.80 – 193.99 166 - 199 7. degree
X\ 1 tone (A+L) 193.99 – 215.64 200 8. degree
(wholetone)
or 2A-C
/X\ Double A [227.37] 200-233 9. degree

Though these adjustments are slight, they are in fact closer to your
new 41-boundaries, so there is even better agreement than before.

The large table on page 17 in your thesis identifies both 12/11
(150.64c) and 11/10 (165.00c) with the name "mücenneb," which caused
me to question whether your previous boundary of 160 cents (given in
the table on page 16) was appropriate, but I see that you have now
changed that to 166 cents, which resolves that problem.

But now I have a question and an observation.

The question: The degree numbers you have at the far right are
actually one more than what they would be in the 41-division. What
do these numbers mean?

The observation: Regarding the placement of 12/11 and 11/10 into the
same category, I find that the 41 division does not agree with this
in that 12/11 is 5deg and 11/10 is 6deg of 41. This is perhaps just
an anomaly, since the categories that you establish must ultimately
depend on how these intervals are regarded or treated in actual
practice by you and other musicians knowledgeable and skilled in the
Maqam tradition.

> Indeed, I have corrected my mistake of considering the double
apotome as an alternative method of expressing the wholetone, which
should not have required any special accidental at all. So, the same
number of common symbols suffice in my system, whereas you favor in
your article the pure Sagittal symbols as a means to evade any
misunderstandings that arise from the mixed version, and replace the
sharps and flats altogether.
>
> This brings us to the issue of 53-symbols per half-tone.
>
> You will notice that some of the criticisms you expressed for my
Spectral approach are valid for the Sagittal Notation as well:
>
> 1. In poor lighting conditions, 7 colors are as difficult to
discern as 53 symbols.

How did you come up with that number of symbols? That number in
a "half-tone" would notate 624-edo, which is far more than what is
required for the majority of applications. The spartan symbol set
requires only 12 symbols for an apotome (pure version) or 7 in the
mixed version.

> 2. Just as there may be color-blind musicians, you will find the
majority to be rather `symbol-blind`.

I don't know how to interpret that statement. Do you mean that
musicians are only semi-literate?

> 3. Printing the music sheets with dabs of color is only as
expensive as printing scores double the size of what we mundane
musicians are accustomed to.

Preliminary testing has indicated that a 10 to 12-percent linear
enlargement of the staff size would be sufficient for reading
Sagittal symbols under adverse conditions.

> Other than these, I am surprised that you did not consider applying
colorific accidentals when you have used colors to distinguish prime
and odd intervals already.

I use colors only in diagrams. I don't think I would find it
convenient to keep a package of colored pencils with me wherever I
go, in case I get an idea for a melody and wish to write it down on a
piece of paper.

I apologize if I seem to be harsh in my reaction to your idea of
using color in notation, but I can't help thinking that you're
grasping at straws.

> On the other hand, I concur that the sharp and flat are to be
understood as pythagorean apotomes. According to a Turkish saying:
`The hairs of my tongue died out` in the process of my repeating this
fact over and over. The quagmire is boggier than one might think at
first, but that is no reason for us to abondon the aesthetically
beautiful and notorious sharp and flat symbols to the hands of
ignorants, be they Westerners or Easterners. Forgive my saying so,
but I see no way that the Sagittal Notation could be adopted right
away as a means to unify our mutually opposed heritrices. Believe me
that I am trying to be as objective as possible when I say this.

I could not expect it to be generally adopted right away, any more
than I could expect our cultures to become generally microtonal-
literate right away. With Sagittal we have made allowances for a
transition from the conventional symbols to mixed to pure (and also
from 72-edo Sims to Sagittal-Wilson to mixed Sagittal). Will it ever
happen? I don't know, but time will tell.

> Microtonality should be considered as a welcome addition to the
available musical knowledge, not as a disendowment of traditional
values.

We have tried to make the Sagittal symbols both logical and
aesthetically beautiful. If even a handful of people decide to use
them (as a few already have) and continue to do so, then in time they
will become a tradition, but I would say that our cherished values
are expressed in the music more than in the symbols.

> Please accept my apology for any harsh comments that I may have
uttered. I am most grateful to the attention you have shown and the
valuable theoretical information you imparted to me.

And I, in turn, am grateful for your interest and your comments,
which have been very helpful in showing me things that perhaps have
not been adequately explained in our documentation.

Best,

--George Secor

Dear Gene,

I am sorry that I have been unable to express myself using the correct phraseology. What I meant to say was that one can do away with the limma sharp and flat, slide the apotome sharp and flat to the tempered semitone and the microtonal/macrotonal accidentals will follow suit.

As for 36, it is the only number which is the simplest multiple of 12 with a resolution capable of expressing very crudely all the maqams one may find in the East. Yet, I am not talking about fixed points here, but rather pitch-clusters as delineated by my elder colleague Can Akkoc.

Cordially,
Ozan Yarman
----- Original Message -----
From: Gene Ward Smith
To: tuning@yahoogroups.com
Sent: 10 Kasım 2004 Çarşamba 2:40
Subject: [tuning] Re: Sagittal Notation

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> I do not favor the clumping of the apotome and the limma together
actually, but it is possible to temper the limma out and adopt a
36-edo system.

This is tempering out the Pythagorean comma, which means you are
looking at divisions which are multiples of 12.

Seeing as this results in 12-edo elaborated exactly 3 times, or 72-edo
simplified by 2, it appears to be the next step to take for the Maqam
World which has adopted 24-edo since Allah knows when. Considering the
lower boundaries Mr. Keenan gives in this table:

Why 36? Unless you are uninterested in the tuning of major thirds, 72
and 84 jump out at you. If you like {2,3,7} 36 is excellent.

You can configure your subscription by sending an empty email to one
of these addresses (from the address at which you receive the list):
tuning-subscribe@yahoogroups.com - join the tuning group.
tuning-unsubscribe@yahoogroups.com - leave the group.
tuning-nomail@yahoogroups.com - turn off mail from the group.
tuning-digest@yahoogroups.com - set group to send daily digests.
tuning-normal@yahoogroups.com - set group to send individual emails.
tuning-help@yahoogroups.com - receive general help information.

Yahoo! Groups Sponsor
ADVERTISEMENT

------------------------------------------------------------------------------
Yahoo! Groups Links

a.. To visit your group on the web, go to:
/tuning/

b.. To unsubscribe from this group, send an email to:
tuning-unsubscribe@yahoogroups.com

c.. Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

Dear George,

It is a real priviledge to converse with you on theory and practice. `I have been delighted to get enlightened` so to speak.

Now, we have reached a unanimous decision, I gather, that it is necessary to supply the temperament and reference pitch information preceding any microtonal work before one can correctly interpret the Sagittal symbols. However, the methodology you use to derive characteristic intervals allow variances which can be great and ubiquitous at the same time, making Sagittal Notation the ideal tool for those who wish to perform `trans-temperamental` music from only a single instance of transcription regardless of tuning specifications demanded by the composer. It is furthermore apparent that this delineates your actual expectations.

Nevertheless, Trans-temperamental notation is a bit confusing for the regular musicians in my part of the world, who feel more comfortable obeying the rules of a reference scale in order to comprehend which tone deviates how much from what pitch. Believe me, they would be greatly perplexed, even indignant, by the idea that an accidental is capable of being interpreted so liberally as you suggest. I myself think it is impermissable to contort a musical score outside its temperamental context unless it is part of a circus act.

To complicate matters even further, it is evident from your message, that I need to give minute directions to specify exactly ALL the ratios for the Sagittal Symbols you kindly provided for 41-Et if I expect to decipher the pitches indicated on the score right. But you see, in my blunderous attempt to define pitch-clusters, I was unable to make clear to you the fact that this is where the symbol flexibility is required! From the way you have partitioned the half-tone, it is not possible to indicate broad pitch-clusters which denote a particular tonal gestalt with only a single symbol without destroying the context of the temperament in question. If you say that I'm allowed to stretch the 41-Et set anyway I like, why then must I provide any intervallic specifications beforehand? If you say I do not need to, then what's stopping anyone else to interpret my music as 34-Et?

Thus, I have chosen the most inobjectionable historical system as the reference scale: The Pythagorean 12-tone. This leads to a system where microtonality is an emendment of the existing musical knowledge (instead of being a disendowment) by providing a feasible means for comparing temperaments. On the other hand (forgive my saying so, but) the comparison feature seems a little weak with the Sagittal System, although it most certainly seems like an ideal notation for other purposes.

I can therefore accept the Sagittal Notation as a means to complement my shortcomings in trans-temperamental music, but I hope you understand that I cannot throw away the classical terminology of the sharp and flat in favor of Sagittal symbols.

In order to justify the usage of Sagittal symbols for 12-tone just intonation, you ask:

"but suppose that you're producing this music electronically or explaining adaptive JI in a textbook? Would you not wish to have a notation capable of
expressing the departure of the desired pitches from a strict pythagorean scale?"

I do not think anyone would need to explain the departure of pitches from a strictly 12-et or 12-pt scale using any micro-accidentals when you have made clear the fact that tonal deviations must be implied by indicating the desired temperament and pitch information in advance. And what would happen if I were to take the liberty with the Sagittal symbols? Without any descriptions, the schismatic occurances could amount to as large an intervallic difference as dieses. As to the textbook argument, many will likely concur that the usage of so many symbols on the score is not visually pleasing for the expression of a 1/4 comma meantone temperament when there are more readily acceptable methods of doing so... such as listings of chord-structures, deviations expressed in cents, and/or a graphical network of zig-zag points underneath a regular staff notation.

As to the terminology of analogous intervals, I do not agree with you that a `perfect fifth` can be confounded to mean a `tempered fifth` or an `apotome substitute` could be confounded to mean `tempered apotome`, seeing as tempering indicates the partitioning and distribution of a supernumerary interval throughout a scale. (BTW, thanks for correcting my error regarding `tempering out the limma`). The margin-of-error allowed for these intervals in practice (as I have indicated in my division of the whole-tone continuum) has nothing to do with the methods used to derive their substitutes.

And if you find the substitution scheme cumbersome, what's stopping others from considering the Sagittal flexibility of the apotome, limma, comma, etc... in the same way? Is it then any the less cumbersome for a composer compelled to specify the tonal context each time, to use Sagittal Notation?

And if we are to hold on to the premise that

"it's not humanly possible for anyone to sing or play an exact apotome on any instrument?"

why then should we go to all the trouble of being so accurate with theory as you suggest? This is exactly the reason why I have been so concerned to preserve the concept of `pitch-cluster` as outlined by my elder Can Akkoc.

At this point, it is obvious that we differ on the concept of flexibility. Whereas you have insinuate flexibility to mean a `broad interpretation by way of substitution`, I have clinged to my definition of `pitch-cluster variations around fundamental tones outlined by a reference scale`.

As such, 36-tET is a preference of mine that should not be taken in its concrete form. I assure you, I much rather 72-edo, and it was not an easy decision to forsake so many details (come to think of it, my custom-tailored Qanun is based on a version of 72-edo). But if you consider each step of 36-tET a pitch-cluster in its own right, you will see that it can comprise all the important intervals that have been catalogued so far in a compact notational form. It is therefore possible to regard 72-edo as an elaborated version of 36. I will keep 36-tet as a stepping stone for higher resolutions for the while, seeing as it can approximate maqams quite well and it seems like the logical upgrade for Maqam Music. My personal choice, however, is 41-tET.

Regarding Mr. Keenan's intervallic values, the latest figures in my table of accidentals are as follows:

N
Sharps (+)
Appellation
Displacement Value
Flats (-)

Micro-tones
R
Comma
0 to 34 ¢ (~531441:524288)
r

v
Quarter-Tone (Diesis)
34 to 68 ¢ (~246:239)
V

Demitone
O
Pythagorean Limma
68 to 102 ¢ (minor ½ tone)
o

Semitone
5
Pythagorean Apotome
102 to 136 ¢ (major ½ tone)
6

Macro-tones
U / 3
Sesqui-Tone
136 to 170 ¢ (~12:11)
u / 1

C / 4
Minor Whole Tone
170 to 204 ¢ (~10:9)
c / 2

Double Sharp/Flat
9
Double Apotome [1]
>204 to 272 ¢ (~8:7)
8

--------------------------------------------------------------------------------

[1] As opposed to the 9:8 Major Whole Tone (Demitone+Semitone) which won’t require any accidental.

Here is how I notated 24-edo:

24-Tone Equal Divisions of the Octave

Step
Pitch
Cents
E24
INTERVAL
EXE24
Name

0
1/1
0.0000
C
with previous step
C
Dw
B5
unison, perfect prime

1
50.00000
50.0000
C| Db;
50.000 cents
CR
D1
BC

2
100.00000
100.0000
C# Db
50.000 cents
C5
D6
Bx

3
150.00000
150.0000
C#| D;
50.000 cents
CC
Dr
-

4
200.00000
200.0000
D
50.000 cents
D
Ew
Cx

5
250.00000
250.0000
D| Eb;
50.000 cents
DR
E1
-

6
300.00000
300.0000
D# Eb
50.000 cents
D5
E6
Fw

7
350.00000
350.0000
D#| E;
50.000 cents
DC
Er
F1

8
400.00000
400.0000
E
50.000 cents
E
F6
Dx

9
450.00000
450.0000
E| F;
50.000 cents
ER
Fr
-

10
500.00000
500.0000
F
50.000 cents
F
Gw
E5

11
550.00000
550.0000
F| Gb;
50.000 cents
FR
G1
EC

12
600.00000
600.0000
F# Gb
50.000 cents
F5
G6
Ex

13
650.00000
650.0000
F#| G;
50.000 cents
FC
Gr
-

14
700.00000
700.0000
G
50.000 cents
G
Aw
Fx

15
750.00000
750.0000
G| Ab;
50.000 cents
GR
A1
-

16
800.00000
800.0000
G# Ab
50.000 cents
G5
A6
-

17
850.00000
850.0000
G#| A;
50.000 cents
GC
Ar
-

18
900.00000
900.0000
A
50.000 cents
A
Bw
Gx

19
950.00000
950.0000
A| Bb;
50.000 cents
AR
B1
-

20
1000.00000
1000.0000
A# Bb
50.000 cents
A5
B6
Cw

21
1050.00000
1050.0000
A#| B;
50.000 cents
AC
Br
C1

22
1100.00000
1100.0000
B
50.000 cents
B
C6
Ax

23
1150.00000
1150.0000
B| C;
50.000 cents
BR
Cr
-

24
1200.0000
1200.0000
C
50.000 cents
C
Dw
B5
octave

and here is how I notated 36-edo:

36-Tone Equal Divisions of the Octave

Step
Pitch
Cents
E36
INTERVAL
EXE36
Name

0
1/1
0.0000
C
with previous step
C
Dw
BO
unison, perfect prime

1
33.33333
33.3333
C| Db;;
33.333 cents
CR
D2
B5

2
66.66667
66.6667
C|| Db;
33.333 cents
Cv
D1
BU

3
100.00000
100.0000
C# Db
33.333 cents
CO
D6
BC

4
133.33333
133.3333
C#| D;;
33.333 cents
C5
Do
Bx

5
166.66667
166.6667
C#|| D;
33.333 cents
CU
DV
-

6
200.00000
200.0000
D
33.333 cents
D
Ew
Cx

7
233.33333
233.3333
D| Eb;;
33.333 cents
DR
E2
-

8
266.66667
266.6667
D|| Eb;
33.333 cents
Dv
E1
-

9
300.00000
300.0000
D# Eb
33.333 cents
DO
E6
Fw

10
333.33333
333.3333
D#| E;;
33.333 cents
D5
Eo
F2

11
366.66667
366.6667
D#|| E;
33.333 cents
DU
EV
F1

12
400.00000
400.0000
E
33.333 cents
E
F6
Dx

13
433.33333
433.3333
E| F;;
33.333 cents
ER
Fo
-

14
466.66667
466.6667
E|| F;
33.333 cents
Ev
FV
-

15
500.00000
500.0000
F
33.333 cents
F
Gw
EO

16
533.33333
533.3333
F| Gb;;
33.333 cents
FR
G2
E5

17
566.66667
566.6667
F|| Gb;
33.333 cents
Fv
G1
EU

18
600.00000
600.0000
F# Gb
33.333 cents
FO
G6
EC

19
633.33333
633.3333
F#| G;;
33.333 cents
F5
Go
Ex

20
666.66667
666.6667
F#|| G;
33.333 cents
FU
GV
-

21
700.00000
700.0000
G
33.333 cents
G
Aw
Fx

22
733.33333
733.3333
G| Ab;;
33.333 cents
GR
A2
-

23
766.66667
766.6667
G|| Ab;
33.333 cents
Gv
A1
-

24
800.00000
800.0000
G# Ab
33.333 cents
GO
A6
-

25
833.33333
833.3333
G#| A;;
33.333 cents
G5
Ao
-

26
866.66667
866.6667
G#|| A;
33.333 cents
GU
AV
-

27
900.00000
900.0000
A
33.333 cents
A
Bw
Gx

28
933.33333
933.3333
A| Bb;;
33.333 cents
AR
B2
-

29
966.66667
966.6667
A|| Bb;
33.333 cents
Av
B1
-

30
1000.00000
1000.0000
A# Bb
33.333 cents
AO
B6
Cw

31
1033.33333
1033.3333
A#| B;;
33.333 cents
A5
Bo
C2

32
1066.66667
1066.6667
A#|| B;
33.333 cents
AU
BV
Cs

33
1100.00000
1100.0000
B
33.333 cents
B
C6
Ax

34
1133.33333
1133.3333
B| C;;
33.333 cents
BR
Co
-

35
1166.66667
1166.6667
B|| C;
33.333 cents
Bv
CV
-

36
2/1
1200.0000
C
33.333 cents
C
Dw
BO
octave

Distribution of notes were based on these criteria: Priority of the pythagorean whole tone and apotome, proximity of the best approximation for these intervals to the reference scale, tonal ranges conforming to each sharp and flat, and enharmonicities based on accidental pairs.

But I do not understand this question:

`The question: The degree numbers you have at the far right are
actually one more than what they would be in the 41-division. What do these numbers mean?`

As for the placement of 12/11 and 11/10 into the same category, please take a look at the diagram included with this message which shows how the 5th, 4th and 3rd degrees (from left to right) of a Zarlino temperament 7-tone major scale can vary in Maqam music. The pitch-clusters are interesting to behold in that the melodic mobility determines where a particular jump will land next.

Where did I come up with 53 Sagittal symbols? Am I mistaken when I count the complete Sagittal symbol superset from natural to apotome given on page 7, figure 3? I do not think it wrong to describe this superset as 53 symbols per half-tone. Also, if you don't mind my saying so, some of the Sagittal accidentals could be improved.

Coming back to the issue of adverse performance conditions, what I meant by `symbol-blind` is that the musicians do not pay much attention to novelty symbols nowadays. This is a result of my unfortunate
experience with a student symphony orchestra here in Istanbul that attempted to perform my maqam-infested piano concerto after a week's rehearsal.

As to the Spectral Notation idea of mine, I find the need to express my opposition:

1. Just as color-TV is not withdrawn from the market due to the fact that some people misperceive colors, the option of coloring accidentals should not be summarily dismissed.

2. It is no less convenient to carry your wallet with you than to carry around 8 colored pencils. Some pencils are even designed to contain many colors.

3. In the worst-case scenario, I allow the usage of numbers from 1 to 9 to signify which schismatic increment a particular pitch-cluster accidental refers to. You can therefore notate in black and white if you choose to. Heck, I'm very fond of black myself.

4. Colors help discern even finer nuances than additional symbols, and are easier to keep in memory when these are correlated to sub-pitch-clusters.

Say what dear George, I do not object to personally using Sagittal Notation straight away, would you object to giving the Spectral Notation idea a fair try? I rather think our respective cultures have a lot to gain by mutual accord and prospection.

I look forward to the next episode of our correspondence.

Cordially,
Ozan Yarma

---

File has not been scanned

Checked by AVG anti-virus system (http://www.grisoft.com).
Version: 6.0.797 / Virus Database: 541 - Release Date: 15.11.2004

Dear Ozan,

I have read with interest your exchanges with George Secor.

A few comments:

Before you understood much about Sagittal you complained that it was
too inflexible. Now you seem to be complaining that it is too
flexible. :-)

Just as a composer or transcriber to Sagittal can choose to specify
the pitches to extreme precision, or they can allow them to be
interpreted over a wide range of temperaments, so too are you free
to specify that they should be interpreted only over some small
pitch ranges, or indeed you might specify this in non-numerical
terms by simply saying that it is only to be interpreted as Maqam.

I am glad that we agree that conventional symbols (the nominals A to
G and sharps, flats and their doubles) should be understood as
notating a chain of fifths as
Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C#
G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx

I hope you now understand that there is no obligation to replace the
standard sharp and flat symbols or their doubles when using
Sagittal, in which case only single-shaft Sagittal symbols are used.

I am somewhat confused by your use of 36-ET for Maqam when you
otherwise seem to insist on Pythagorean fifths. 41-ET is of course
much closer to Pythagorean in the size of its fifths.

But in any case, you only need a single up/down pair of symbols in
addition to the standard ones, to notate 36-ET and only two such
pairs to notate 41-ET, and only 3 pairs to notate 72-ET. The 36-ET
notation and the 41-ET notation use different subsets of the symbols
used for 72-ET.

Quarter comma meantone (notated the same as 31-ET) also only
requires one pair of symbols in addition to the standard ones, and
then only if you wish to avoid double-sharps and flats. Double
sharps and flats are quite permissible in sagittal too, but the
other symbols are provided as a way of avoiding non-monotonic series
of nominals such as C Dbb C# Db Cx D. One can instead write
C C^ C# Db Dv D.

All this you can see in the Xenharmonikon article on the Sagittal
website, if you ignore the multi-shaft symbols and replace them by
the equivalent combinations of single-shaft Sagittals with
conventional sharps and flats as shown elsewhere in the article.

So I don't understand why you are concerned about the existence of
53 pairs of symbols. The other 50 pairs of symbols shown in the
article (only 31 of which are single-shaft) would not be needed for
your purposes and indeed may never be needed for _anyone's_
purposes. They would only be needed if someone needed to distinguish
more than about 500 pitches in the octave within a single work!

We are happy to hear any suggestions you may have for improving the
symbols.

Here I will use ASCII approximations of the 3 pairs of symbols that
do most of the work in Sagittal, to show representative Sagittal
notations for all the pitch clusters from A to C for some of the
systems mentioned. Of course alternative spellings are possible,
such as B^ instead of Cv in 72-ET, or A# instead od Bbv in 31-ET.

The pairs are
/ 5-comma up
\ 5-comma down
f 7-comma up
t 7-comma down
^ 11-medium-diesis up or semisharp
v 11-medium-diesis down or semiflat

72: A A/ Af A^ Bbt Bb\ Bb Bb/ Bbf Bv Bt B\ B B/ Bf Cv Ct C\ C
41: A A/ A^ Bb Bb/ Bv B\ B B/ C\ C
36: A Af Bbt Bb Bbf Bt B Bf Ct C
34: A A/ Bb Bb/ Bv B\ B C\ C
31: A A^ Bbv Bb Bv B B^ Cv C

By the way, silly though it may be, it is certainly permissible, in
the west, to refer to a tempered fifth as a "perfect fifth". The
term "perfect" here only means that it is not augmented or
diminished as regards the particular (possibly tempered) diatonic
scale it is in.

Regards,
-- Dave Keenan

Dear Dave,

It is a pleasure to correspond with you now that you have finally decided to join us. Please excuse my abundant ignorance when it comes to music theory. I have made a diligent effort to understand Sagittal Notation and I hope you will overlook my shortcomings if I somehow misconceived some facts. Also, I hope you will not infer anything as if my impressions were somehow influenced by `Sagittophobia`.

I am sad that you think I was complaining about Sagittal Notation. My `issues` with the Sagittal flexibility stems from the methodology and terminology of the authors more than the capacity of the notation itself. Let me summarize my views:

1. According to you, a `tempered fifth` (being the closest approximation to a `perfect fifth`) could be considered perfect. I would much rather that a tempered fifth is considered an `imperfect fifth`.

2. Thus, I do not agree that a tempered apotome is the same as an apotome-substitute. The latter would be the consequence of the `imperfect` generator size, while the prior is based on the best approximation of 2187:2048.

3. Following your line of reasoning, I'm confused as to why you disregard the closest approximations for apotome, limma, comma, so forth... in other temperaments while you pick the closest fifth as the acceptable generator size. For example, you pick the 13th step of 22-TET (710c) for the perfect fifth due to its proximity to 3:2 (an indisputably fundamental interval), but you do not deem the 2nd step of this temperament (110c) suitable for a tempered apotome (an interval no less fundamental than 3:2). Instead, the `cycle of imperfect fifths` result in a sesqui-tone interval (which I would rather call an apotome substitute) with a size of 164 cents which you express with the apotome sharp or flat.

4. In effect, you notate this apotome-substitute with the same double-shaft arrow symbol which would normally have to be an apotome (or a tempered apotome) with the nominal ratio of 2187:2048.

5. The lack of a fundamental reference system is what I found weak with the Sagittal Notation, which otherwise seems to be the perfect tool for trans-temperamental transcription.

6. I do not have any personal argument against adopting the Sagittal Notation straight away... heck, I accept it as a welcome addition to the existing musical knowledge and celebrate the contributors. But dear George seems to exercise constraint with my Spectral approach where other temperaments are expressed as deviations from the Pythagorean 12-tone.

7. If ever I or someone else decided to notate 500 different pitches in a single work, I do not see why we have to use 53 pairs of Sagittal symbols in order to do so when Spectral Notation allows the transcription of over 313-tones per octave using only 9 colored accidentals (minus the macrotonal enharmonics). Don't you think it is worth some consideration?

8. If I may say so, Sagittal is by no means the end to our notational problems. Furthermore, I have some grave issues as to the intelligibility of some symbols. Don't ask me how they can be improved, I hesitate to provide drawings and overstep my bounds when I am confronting the developers with my baubles. All I can say is, perhaps alternative sets could be developed to facilitate trans-cultural understanding, such as the ones I use.

As to 36-ET, I insist on 41 or even 82-edo with schismatic rational approximations personally for Maqam Music. But the Maqam World cannot possibly adopt such a high resolution over-night. 36-ET seems to be a transition temperament if you allow room for pitch-cluster calibrations the way I explained in my Spectral Notation proposal:

http://f2.pg.briefcase.yahoo.com/bc/ozanyarman/lst?.dir=/My+Documents

Forgive my criticism, but ASCII text seems to be a crude way of expressing notation. That is something that requires improvement too.

Cordially,
Ozan Yarman

Dear Paul,

I am glad that you have found the time to explain to me temperamental basics. I wonder, how would you define EDO as opposed to TET in one technical sentence?

With the equal semi-tone sharp/flat you'll only get 12-equal. I'm not clear on how you propose to get 36-equal.

I was assuming something in the order of what you said before about tempering out all commas to reach multiples of 12. If you can show me how to reach 24 or 36 equal tones per octave, I'm sure I can explain.

Yes, Maqam Music uses 12/11 most frequently. It is the sesqui-tone interval known as mujannab. I'm not so sure about 9/11 though. But you still think in terms of fixed points when I say 36 tones per octave. The tones should have a flexibility range for the correct (just) interpretation of maqams. That would make the system much more voluminous in actual practice of course. 36 is a number that is a crude best-fit in simplest terms.

If you have Scala, set the keyboard for 36-ET and try the following for the Saba Maqam:

D, E comma flat, F, F sesquisharp, A, Bb...

for Hijaz ascending:

D, D sesquisharp, F sharp, G, A, B comma flat, C, D

descending:

C, Bb, A, G, F sharp, D sesquisharp, D

for Huzzam:

E comma flat, F, G, G sesquisharp, B, and all the way to the tonic back.

The list can go on like this. The tones of course are very crude representations of what is actually desired of the maqam, but the notation is good enough to express them and give an idea of what they sound like. As I have stated in my reply to Dave, I much prefer 41, or even 82 equal tones per octave from a theoretical standpoint.

Looking forward to hearing more about your 20:19 and 19:18 'conduits',
Paul

I do not understand what you expect me to tell you more about these intervals.

>1. According to you, a `tempered fifth` (being the closest approximation to a `perfect >fifth`) could be considered perfect. I would much rather that a tempered fifth is >considered an `imperfect fifth`.

Ozan, your suggestion here runs counter to half a millenium of Western practice. The 'perfect fifth' is so called whether it is tempered or not. In the modern temperament of 12-equal, the temperament of the perfect fifth only amounts to 2 cents -- under most circumstances, not a noticeable difference. The other types of fifth recognized in Western practice (and all its tunings and temperaments) are the 'diminished fifth', the 'augmented fifth', 'doubly diminished fifth', etc. -- these are defined as 7 perfect fifths below, 7 perfect fifths above, 14 perfect fifths below, etc., the perfect fifth itself, regardless of how the perfect fifth is tuned. The term 'perfect' is needed to distinguish the common diatonic interval from these altered ones, regardless of which temperament is used.

To distinguish the 3:2 ratio from other nearby intervals, we say "just perfect fifth", "pure perfect fifth", etc. . . . and often the word "perfect" is omitted when confusion is unlikely to arise, so "just fifth" and "pure fifth" almost always refer to the exact 3:2 ratio.

I must voice my objection here, as there should be some allowable limit as to the `perfect fifth` in question. One or two cents may not affect the cycle much and I'm willing to consign the term `tempered fifth` to refer to all temperaments where the resultant apotome and limma fall within the ranges we are familiar with. But when the difference from a pure fifth amounts to as high as 5-8 cents, the apotome, limma and the rest exceed their classical boundaries beyond recognition.

>2. Thus, I do not agree that a tempered apotome is the same as an apotome-substitute. >The latter would be the consequence of the `imperfect` generator size, while the prior is >based on the best approximation of 2187:2048.

The apotome is defined with respect to a full chain of fifths, and the interval 2187:2048 is far too complex to tune without tuning most of the intervening fifths. So I can see no possible relevance for the "best approximation of 2187:2048" and certainly wouldn't use that as the definition of "tempered apotome". This goes against the entire practice and spirit of temperament.

Maybe I am far too confused with the concept of temperament here. Am I wrong to consider 700 cents a tempered fifth and 710 as a fifth substitute? Am I wrong when I say that the second degree (110c) of 22-TET is better suited to be called a tempered pythagorean apotome rather than the third degree (164c)? Or do you consider the third degree to be the tempered apotome instead of an apotome substitute, because the chain of imperfect fifths warrants it? As far as I remember, 2187:2048 is an interval called `apotome` by Pythagoras himself, which is the natural outcome of 7 pure fifths up minus 4 octaves, and I do not understand why you find it so complex to achieve. If the Pyhtagorean apotome, limma, comma (all tuned by the simplest arithmetic using the first three harmonics), are not fundamental intervals, then I do not know what is.

As to the definition of temperament, I was entertaining the idea that tempering involves contracting or expanding intervals of a scale away from their `natural` sizes by an acceptable margin. Meantone temperament involves tempering out the syntonic comma as you say, splitting its size to 3,4,5,6 or more parts and distributing the difference throughout, am I correct? Just intonation does so by the simplest approximation of whole number ratios, right? Equal Temperaments equalize the tones so that all consequent intervals are the same size, yes? But I'm assuming that the concept of apotome, limma, and the ditonic comma are reserved for 3-limit Pythagorean JI, where the basic ratios such as 2:1, 3:2 and 4:3 create them. So, I do not understand why you take 2:1 for granted as the octave and not 3:2 as the unalterable perfect fifth. According to your reasoning, it should also be possible to calibrate the octave away from its `natural` by a few cents and still call it an octave. But I do not think such an `octave tempering` is allowed anywhere in Western theory.

>3. Following your line of reasoning, I'm confused as to why you disregard the closest >approximations for apotome, limma, comma, so forth...

These small and complex intervals cannot be tuned precisely by ear without tuning a chain of intervening, simple-ratio intervals. When tuning a temperament, it is the simple-ratio intervals which are finely adjusted. If one is tuning two notes to an apotome, limma, or comma, there is no point in the tuning procedure where the tuner can hear the interval "lock in" to just intonation and say, "aha, this is just" or "this is tempered wide" or "this is tempered narrow". The whole point of temperament is to keep the tunable, simple-ratio intervals reasonably close to just intonation, while simplifying the field of resultant intervals. The tuning of the resultant intervals may indeed be very unfamiliar and/or different from JI in many cases, but there's nothing about the JI versions of these resultant intervals that is more "right" or "pure" in any audible way.

Agreed! We need the fundamental ratios to tune other intervals. As such, I do not think it possible to tune a cycle of imperfect fifths by ear, nor do I deem the perfect fifth as the more agreeable fifth in all cases.

>in other temperaments while you pick the closest fifth as the acceptable generator size. >For example, you pick the 13th step of 22-TET (710c) for the perfect fifth due to its >proximity to 3:2 (an indisputably fundamental interval), but you do not deem the >2nd step of this temperament (110c) suitable for a tempered apotome (an interval no >less fundamental than 3:2).

How can one consider the apotome, which is the interval between the impossibly high 2187th and 2048th harmonics, nearly as fundamental as 3:2? The only reason we're talking about 2187:2048 in the first place is because it is tunable by a long chain of 3:2s. Without the foundation of the 3:2, intervals like 2187:2048 would never enter the picture in the first place. 2187:2048 is the interval between two members of a chain of fifths, 7 fifths apart from one another, and if each of the fifths is altered by an equal and acceptable amount, the apotome will be altered 7 times as much. It makes no sense to consider a different interval to be the apotome, since the apotome's very function is so tied with the diatonical chain of 7 fifths, and attempting to use a different interval as the apotome would inevitably ruin at least one of the fifths in the chain. In particular, the apotome takes you from one end of the diatonic chain to a fifth beyond the other end, so it's the key to the simplest possible modulation from one fifth-based diatonic set to another. For example, from A aeolian (which has all naturals) to E aeolian (where the previous F becomes F#). In music which is based on such a diatonic chain of fifths (which is how the apotome became relevant in the first place), attempting to use a different interval as the apotome will result in the modulation being to a non-diatonic mode with at least one unacceptable rendition of a perfect fifth.

How can a pyhtagorean apotome not be a fundamental interval when as simple a procedure as a chain of three 3:2 up and four 4:3 down result in it? Even a singer can perform the apotome when written in such a rudimentary fashion. As I stated before, I do not consider an `out-of-range apotome` a tempered apotome, but an apotome-substitute just as I consider 710 cents a fifth-substitute instead of a perfect fifth. A tempered apotome should be somewhere about 90 to 130 cents. This conforms wıth historical practice, no?

Well, I'm wondering how one distinguishes 19:18 and 20:19 from othersimilar-sized intervals by ear without playing them as simultaneitiesin multi-part chords.>I think what you mean is, for example, how the SATB singers of a choir can intonate >such close-proximity intervals unless these are sounded within chords.

It's not the closeness of the proximity but the complexity of the ratios.

I do not believe these ratios to be very complex.

>I do not think >that it is practically possible, but I believe the ear can distinguish these >intervals upon >careful inspection.

The intervals are 5 cents different from one another, so under controlled circumstances, the ear can indeed distinguish them from one another. However, my skepticism concerns the ability to hear either of these intervals as "pure" and a tempered interval of, say, 91 cents as "impure" in a context outside large, multi-part chords. There's no qualititative difference between the just and tempered intervals in this range, so assigning ratios seems misleading or arbitrary. And what holds for 19:18 and 20:19 in this regard holds with even more certainty and strength for ratios as complex as 2187:2048. There are a myriad of ratios in this range of complexity -- what distinguishes a few from the rest is that they can be tuned by, and made musically meaningful in the context of, a chain of simple-ratio, ear-tunable intervals.
If the matter is one of `purity` of intervals, I consider all intervals, be they just, complex or tempered, acceptable by ear, as long as they conform to simple-ratio ear-tunable intervals as you so wonderfully put.

All the best,
Ozan

hi Ozan,

pardon me for butting in here after not following the tuning
list for several weeks (my internet connection was down) ...

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Dear Dave,
>
> <snip> I have made a diligent effort to understand
> Sagittal Notation and I hope you will overlook my shortcomings
> if I somehow misconceived some facts. Also, I hope you will
> not infer anything as if my impressions were somehow influenced
> by `Sagittophobia`.
>
> <snip>
>
> 1. According to you, a `tempered fifth` (being the closest
> approximation to a `perfect fifth`) could be considered perfect.
> I would much rather that a tempered fifth is considered an
> `imperfect fifth`.

it's not only according to Dave Keenan, it's according to
standard Western music-theory.

in standard terminology, there is no such thing as an
'imperfect fifth'. by definition, there are four 'perfect'
intervals (within one 'octave'): the unison/prime, 8ve, 4th,
and 5th. this is because in diatonic scales, with one exception
(the 'diminished-5th'), they only come in one basic size.

the other intervals (2nd, 3rd, 6th, and 7th) must be either
'major' or 'minor', which simply mean 'large' and 'small'
in Latin. therefore, these four are 'imperfect'.

you can see more in the relevant definitions in the
Tonalsoft Encyclopaedia:

http://tonalsoft.com/enc/index2.htm?perfect.htm
http://tonalsoft.com/enc/index2.htm?imperfect.htm

as i've emphasized to you before, there are *many* things
in standard musical notation which are not logical, but the
weight of tradition impels us to deal with them until
something better finally takes hold. at least Sagittal
notation does hold out some promise for that.

-monz

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George,
>
> It is a real priviledge to converse with you on theory and
practice. `I have been delighted to get enlightened` so to speak.

Ozan, I am still working on a reply to the many things you have
brought up in this message (and subsequent messages to me and
others). This has been between many other things that have been
making demands on my (unfortunately) very limited time. I hope to
have my reply finished some time next week, so we can continue our
most fruitful discussion.

I'm sorry that this could not be sooner, but these are issues that
cannot be treated adequately with a few words dashed off in haste.

Thank you for your patience.

Best,

--George

Dear Paul,

As I have remarked again and again, I'm a novice when it comes to the science of tuning. If I exceed my bounds at times due to my enthusiasm, I wish to be forgiven.

As to your questions: Yes, a square tonality diamond in SCALA based on primes 2,3 and 11 indeed gives quarter and sesquitones necessary for 24-ET, the rational values for which seems in perfect alignment with Maqam Music:

PITCHES

0: 1/1 C Dbb unison, perfect prime
1: 12/11 3/4-tone, undecimal neutral second
2: 4/3 F Gbb perfect fourth
3: 11/8 undecimal semi-augmented fourth
4: 16/11 undecimal semi-diminished fifth
5: 3/2 G Abb perfect fifth
6: 11/6 21/4-tone, undecimal neutral seventh
7: 2/1 C Dbb octave

CONSEQUENT INTERVALS

0: 12/11 150.637 3/4-tone, undecimal neutral second
1: 12/11 150.637 3/4-tone, undecimal neutral second
2: 11/9 347.408 undecimal neutral third
3: 33/32 53.273 undecimal comma, al-Farabi's 1/4-tone
4: 128/121 97.364 undecimal semitone
5: 33/32 53.273 undecimal comma, al-Farabi's 1/4-tone
6: 11/9 347.408 undecimal neutral third
7: 12/11 150.637 3/4-tone, undecimal neutral second

|
Interval class, Difference:
1: 4096/3993 44.091 cents
1: 33/32 53.273 cents undecimal comma, al-Farabi's 1/4-tone
1: 121/108 196.771 cents
2: 12/11 150.637 cents 3/4-tone, undecimal neutral second
2: 14641/13824 99.407 cents
2: 128/121 97.364 cents undecimal semitone
3: 32/27 294.135 cents Pythagorean minor third
3: 33/32 53.273 cents undecimal comma, al-Farabi's 1/4-tone
3: 128/121 97.364 cents undecimal semitone
4: 128/121 97.364 cents undecimal semitone
4: 33/32 53.273 cents undecimal comma, al-Farabi's 1/4-tone
4: 32/27 294.135 cents Pythagorean minor third
5: 128/121 97.364 cents undecimal semitone
5: 14641/13824 99.407 cents
5: 12/11 150.637 cents 3/4-tone, undecimal neutral second
6: 121/108 196.771 cents
6: 33/32 53.273 cents undecimal comma, al-Farabi's 1/4-tone
6: 4096/3993 44.091 cents

All 7 differences in sorted order:
4096/3993, 44.0912 cents /3.11.11.11
33/32, 53.2729 cents 3.11
128/121, 97.3641 cents /11.11
14641/13824, 99.4068 cents 11.11.11.11/3.3.3
12/11, 150.6371 cents 3/11
121/108, 196.7709 cents 11.11/3.3.3
32/27, 294.1350 cents /3.3.3

INTERVAL MATRIX:

1/1 : 12/11 4/3 11/8 16/11 3/2 11/6 2/1
12/11: 11/9 121/96 4/3 11/8 121/72 11/6 2/1
4/3 : 33/32 12/11 9/8 11/8 3/2 18/11 2/1
11/8 : 128/121 12/11 4/3 16/11 192/121 64/33 2/1
16/11: 33/32 121/96 11/8 3/2 11/6 121/64 2/1
3/2 : 11/9 4/3 16/11 16/9 11/6 64/33 2/1
11/6 : 12/11 144/121 16/11 3/2 192/121 18/11 2/1
2/1

As to a square tonality diamond based on primes 2, 3 and 7, we get this:

PITCHES

0: 1/1 0.000 unison, perfect prime
1: 8/7 231.174 septimal whole tone
2: 7/6 266.871 septimal minor third
3: 4/3 498.045 perfect fourth
4: 3/2 701.955 perfect fifth
5: 12/7 933.129 septimal major sixth
6: 7/4 968.826 harmonic seventh
7: 2/1 1200.000 octave
|

CONSEQUENT INTERVALS

0: 8/7 231.174 septimal whole tone
1: 8/7 231.174 septimal whole tone
2: 49/48 35.697 slendro diesis, 1/6-tone
3: 8/7 231.174 septimal whole tone
4: 9/8 203.910 major whole tone
5: 8/7 231.174 septimal whole tone
6: 49/48 35.697 slendro diesis, 1/6-tone
7: 8/7 231.174 septimal whole tone

Interval class, Difference:
1: 54/49 168.213 cents Zalzal's mujannab
1: 64/63 27.264 cents septimal comma, Archytas' comma
2: 54/49 168.213 cents Zalzal's mujannab
2: 64/63 27.264 cents septimal comma, Archytas' comma
3: 64/63 27.264 cents septimal comma, Archytas' comma
3: 54/49 168.213 cents Zalzal's mujannab
4: 54/49 168.213 cents Zalzal's mujannab
4: 64/63 27.264 cents septimal comma, Archytas' comma
5: 64/63 27.264 cents septimal comma, Archytas' comma
5: 54/49 168.213 cents Zalzal's mujannab
6: 64/63 27.264 cents septimal comma, Archytas' comma
6: 54/49 168.213 cents Zalzal's mujannab
All 2 differences in sorted order:
64/63, 27.2641 cents /3.3.7
54/49, 168.2132 cents 3.3.3/7.7

INTERVAL MATRIX:

1/1 : 8/7 7/6 4/3 3/2 12/7 7/4 2/1
8/7 : 49/48 7/6 21/16 3/2 49/32 7/4 2/1
7/6 : 8/7 9/7 72/49 3/2 12/7 96/49 2/1
4/3 : 9/8 9/7 21/16 3/2 12/7 7/4 2/1
3/2 : 8/7 7/6 4/3 32/21 14/9 16/9 2/1
12/7: 49/48 7/6 4/3 49/36 14/9 7/4 2/1
7/4 : 8/7 64/49 4/3 32/21 12/7 96/49 2/1
2/1

Again, we derived some very basic intervals that are required by Maqam Music. Truly, 24 TET is the crudest represantion of Maqam Music one can get. 36 tones per octave can comprise forsaken intervals better. The true values of Maqam Music need as voluminous a system as based on primes 2,3,5,7 and 11. I guess the simplest representation of such a system would be 82-TET. This is my recent preference which is doubly accurate as compared to 41 tones per octave, which I nevertheless notate with 4 sharps and 4 flats.

You ask how some of the maqams I have given translate to 36 equal. I provide the steps below:

>If you have Scala, set the keyboard for 36-ET and try the following for the Saba Maqam:

>D, E comma flat, F, F sesquisharp, A, Bb...

6, 11, 15, 19, 27, 30

>for Hijaz ascending:

>D, D sesquisharp, F sharp, G, A, B comma flat, C, D

6, 10, 18, 21, 27, 32, 36, 6!

>descending:

>C, Bb, A, G, F sharp, D sesquisharp, D

36, 30, 27, 21, 18, 10, 6

>for Huzzam:

>E comma flat, F, G, G sesquisharp, B, and all the way to the tonic back.

11, 15, 21, 26, 32 (25 on the way back)

I have decided to drop the argument in favor of your theoretical explanations regarding the semantic of the fifth, apotome, limma, so forth... Nevertheless, I will be unable to explain Maqam Musicians hereabouts how come Westerners attribute the apotome sharp and flat for the 70-90 cents interval named Classical Chroma in their meantone temperaments as opposed to the limma which is wrongly assigned the regular sharp and flat by the Yekta-Arel-Ezgi triumvirate. It would have been much easier to bridge the gap between our worlds if you would consign the sharp and flat to correlate mostly to the ~102-136 cent zone of the historically plausible ~68-136 cent half-tone region, which is meant to drive home the point that the sharp and flat correspond to the best rational approximations of the pythagorean apotome in vocal and instrumental practice. Afterall, the Classical Theory of Western Music is based entirely on the sharps being sharper and the flats being flatter as compared to the Baroque Period. Or not? We are, afterall, talking about the sharp and flat as the outcome of 12-tone temperaments of Western tradition.

So, I don't see why it is undesirable to assign the common sharp to the 2nd degree of 22TET when any modern Western musician reading the Csharp will automatically find the register closest to the upper zone of the half-tone region I outlined above. My approach has nothing to do with breaking the consonant chain of best-fit fifths. I take back what I said about the apotome-substitute being the 3rd degree of 22TET. For the sake of historical thrifiness, let us call the 3rd degree the apotome itself. Then we are able to reserve the second degree the position of the apotome-substitute as the closest deviation from the original 12P JI interval, since a musician accustomed to the classical terminology of sharp and flat cannot at a first glance interpret that pitch to be a sesquitone, and should not necessarily be made to!

This is where I draw the line for the sharp and flat: Whereas you have no issues with what these accidentals may correspond to given the chain of 7 `perfect fifths` or whatnot generator interval, I take the historical 12-tone temperaments of Western Music as the limit to how far above or below the middle half-tone a tempered apotome may fall. Also, I bear in mind that the apotome and limma are both attributed the same accidental throughout, while the Turkish tradition challenged this practice with Yekta a century ago.

So I don't understand what you imply by saying that I ignored your example of modulation. I am also anxious to learn where in historical Western Theory octave tempering is allowed.

As to the conformity, you are still thinking in terms of pitch-points. As I have mentioned before, I'm more inclined to think pitches and intervals in fuzzy terms since I have become acquainted with the `pitch-cluster` terminology of my senior colleague Can Akkoc. As such, The whole-tone continuum I have divided into limited portions can represented unlimited sets of interval classes and pitches.

Cordially,
Ozan
----- Original Message -----
From: wally paulrus
To: Ozan Yarman
Sent: 30 Kasım 2004 Salı 6:50
Subject: Re: Sagittal Notation

Hi Ozan,

>I am glad that you have found the time to explain to me temperamental basics. I >wonder, how would you define EDO as opposed to TET in one technical sentence?

ETs, like 12-tET (12-tone Equal Temperament), and other temperaments like meantone, result from taking just intonation and slightly altering the intervals so that fewer distinct pitches result . . . EDO indicates an equal division of the octave, without regard to whether or how the pitches may be derived from a just intonation foundation.

>With the equal semi-tone sharp/flat you'll only get 12-equal. I'm not clear on how >you propose to get 36-equal

>I was assuming something in the order of what you said before about tempering out all >commas to reach multiples of 12. If you can show me how to reach 24 or 36 equal >tones per octave, I'm sure I can explain.

24 and 36 equal arise most commonly as EDOs, not ETs. But 24-tET can be derived by tempering a just intonation based on primes 2, 3, and 11; 36-tET can be derived by tempering a just intonation based on primes 2, 3, and 7 . . . Are either of these a plausible foundation for Maqam music?

>If you have Scala, set the keyboard for 36-ET and try the following for the Saba Maqam:

>D, E comma flat, F, F sesquisharp, A, Bb...

How exactly does this translate into 36-equal?

>for Hijaz ascending:

>D, D sesquisharp, F sharp, G, A, B comma flat, C, D

Same question.

>descending:

>C, Bb, A, G, F sharp, D sesquisharp, D

same question.

>for Huzzam:

>E comma flat, F, G, G sesquisharp, B, and all the way to the tonic back.

same question
>>1. According to you, a `tempered fifth` (being the closest approximation to a >`perfect >fifth`) could be considered perfect. I would much rather that a tempered >fifth is >considered an `imperfect fifth`.

>Ozan, your suggestion here runs counter to half a millenium of Western practice. >The 'perfect fifth' is so called whether it is tempered or not. In the modern >temperament of 12-equal, the temperament of the perfect fifth only amounts to 2 >cents -- under most circumstances, not a noticeable difference. The other types >of fifth recognized in Western practice (and all its tunings and temperaments) are >the 'diminished fifth', the 'augmented fifth', 'doubly diminished fifth', etc. -- these >are defined as 7 perfect fifths below, 7 perfect fifths above, 14 perfect fifths below, >etc., the perfect fifth itself, regardless of how the perfect fifth is tuned. The term >'perfect' is needed to distinguish the common diatonic interval from these altered >ones, regardless of which temperament is used.

>To distinguish the 3:2 ratio from other nearby intervals, we say "just perfect fifth", >"pure perfect fifth", etc. . . . and often the word "perfect" is omitted when >confusion is unlikely to arise, so "just fifth" and "pure fifth" almost always refer to >the exact 3:2 ratio.

>I must voice my objection here, as there should be some allowable limit as to the >`perfect fifth` in question. One or two cents may not affect the cycle much and I'm >willing to consign the term `tempered fifth` to refer to all temperaments where the >resultant apotome and limma fall within the ranges we are familiar with. But when the >difference from a pure fifth amounts to as high as 5-8 cents, the apotome, limma >and the rest exceed their classical boundaries beyond recognition.

Nevertheless, the fifth remains a perfect fifth. For three hundred formative years, Western instruments were tuned with perfect fifths about 5 cents flat, and they continued to be called "perfect fifths" all along -- again, the only other kinds of fifths were "diminished fifths," "augmented fifths", etc. It is true that the terms "apotome" and "limma" might not be used in this context, but the apotome-substitute, as you'd call it, continued to be the object denoted by a sharp or flat in notation, respecting the demands of the Western tonal system. And it continued to be called an "augmented unison," and the limma continued to be called a "minor second," just as they were in the Pythagorean tuning. Thus, a perfect fifth continued to be called a "perfect fifth."

>>2. Thus, I do not agree that a tempered apotome is the same as an apotome->substitute. >The latter would be the consequence of the `imperfect` generator size, >while the prior is >based on the best approximation of 2187:2048.

>The apotome is defined with respect to a full chain of fifths, and the interval >2187:2048 is far too complex to tune without tuning most of the intervening fifths. >So I can see no possible relevance for the "best approximation of 2187:2048" >and certainly wouldn't use that as the definition of "tempered apotome". This >goes against the entire practice and spirit of temperament.

>Maybe I am far too confused with the concept of temperament here. Am I wrong to >consider 700 cents a tempered fifth and 710 as a fifth substitute?

Hmm . . . well, Western music has had virtually no use for perfect fifths wider than pure, due to its meantone affiliation. But certainly one could, from a modern perspective, desire a diatonic system where minor chords sound like 6:7:9s and one would use, say, 709-cent tempered fifths as one's perfect fifths.

>Am I wrong when I say that the second degree (110c) of 22-TET is better suited to be >called a tempered pythagorean apotome rather than the third degree (164c)?

Yes, because if 22-tET is indeed derived as a tempered Pythagorean system, then the apotome will indeed fall on its third degree.

>Or do you consider the third degree to be the tempered apotome instead of an apotome >substitute, because the chain of imperfect fifths warrants it?

Yes.

>As far as I remember, 2187:2048 is an interval called `apotome` by Pythagoras >himself, which is the natural outcome of 7 pure fifths up minus 4 octaves,

Correct -- this is precisely my point.

>and I do not understand why you find it so complex to achieve.

It's precisely as complex as you just said. Tuning up 7 fifths and 4 octaves is not too complex an operation, but it requires a chain of roughly 7 notes, each tuned to the previous one in the chain by tuning a consonant interval to it. In Western music, when an apotome or, if you prefer, an apotome-subsitute -- in other words, an augmented unison -- occurs, it's more likely than not that all the intervening notes in this chain will occur as well, and that they will occur in such combinations so that all these consonant intervals are heard in the music. For the augmented unison to take on a very different value than what results from the chain of 7 fifths and 4 octaves *as they are tuned in the music*, one would have to "break" one of the consonant intervals somewhere, and this would ruin the consonance which forms the whole reason for referencing just intonation in the first place. This is what I was pointing to with my example of modulating from A aeolian to E aeolian, which is reproduced below, but which you appear to have somewhat ignored.

>If the Pyhtagorean apotome, limma, comma (all tuned by the simplest arithmetic using >the first three harmonics), are not fundamental intervals, then I do not know what is.

The simple ratios are -- for example, the set of primes can be taken as fundamental, so 2:1, 3:1, 5:1, 7:1, 11:1, etc. All other just intonation intervals can be derived by tuning these in series.

>As to the definition of temperament, I was entertaining the idea that tempering involves >contracting or expanding intervals of a scale away from their `natural` sizes by an >acceptable margin. Meantone temperament involves tempering out the syntonic comma >as you say, splitting its size to 3,4,5,6 or more parts and distributing the difference >throughout, am I correct?

Essentially, yes.

>Just intonation does so by the simplest approximation of whole number ratios, right?

Just intonation is not temperament at all, and uses no approximations.

>Equal Temperaments equalize the tones so that all consequent intervals are the same >size, yes?

Yes.

>But I'm assuming that the concept of apotome, limma, and the ditonic comma are >reserved for 3-limit Pythagorean JI,

OK -- that is a valid opinion, and I will not quarrel with it any longer. In this message, both above and below, I've included your point of view by saying "or apotome-substitute" wherever I say apotome. The real issue for me is the augmented unison -- the interval that a sharp or flat indicates in notation -- and this is *always* derived from 7 perfect fifths up and 4 perfect octaves down *as they are tuned in the music*, regardless of how much these perfect fifths and perfect octaves are tempered in the system in question.

>where the basic ratios such as 2:1, 3:2 and 4:3 create them. So, I do not understand >why you take 2:1 for granted as the octave and not 3:2 as the unalterable perfect fifth.

I do not.

>According to your reasoning, it should also be possible to calibrate the octave away >from its `natural` by a few cents and still call it an octave. But I do not think such an >`octave tempering` is allowed anywhere in Western theory.

It is certainly allowed -- and expected -- on the piano and other instruments, and my recent theoretical work adapts it to improve many different temperament systems, old and new.
>>3. Following your line of reasoning, I'm confused as to why you disregard the >closest >approximations for apotome, limma, comma, so forth...

>These small and complex intervals cannot be tuned precisely by ear without >tuning a chain of intervening, simple-ratio intervals. When tuning a temperament, >it is the simple-ratio intervals which are finely adjusted. If one is tuning two notes >to an apotome, limma, or comma, there is no point in the tuning procedure >where the tuner can hear the interval "lock in" to just intonation and say, "aha, >this is just" or "this is tempered wide" or "this is tempered narrow". The whole >point of temperament is to keep the tunable, simple-ratio intervals reasonably >close to just intonation, while simplifying the field of resultant intervals. The >tuning of the resultant intervals may indeed be very unfamiliar and/or different >from JI in many cases, but there's nothing about the JI versions of these >resultant intervals that is more "right" or "pure" in any audible way.

>Agreed! We need the fundamental ratios to tune other intervals. As such, I do not think >it possible to tune a cycle of imperfect fifths by ear,

Only approximately, and with a lot of practice. Johnny Reinhard is convinced that 697-cent fifths were tuned with great regularity by the German organ tuners in Preatorius's day. The constant practice of acheiving (or failing to acheive) excellent approximations of 5:3 with three fifths minus an octave, and 5:2 with four fifths minus an octave, accumulated in a culture over generations, might indeed have resulted in such regularity. More recently, the practice of counting beats is cultivated in teaching piano tuners to approach regular 700-cent fifths.

You're correct that only the pure, just perfect fifth can be very accurately tuned by ear. But my point was that tempered fifths -- even when they're tempered by as much as 7 cents, as in 22-equal or 19-equal -- are still close enough to the true 3:2 ratio that they partake in some of the very special qualities -- like concordance -- that the simplest ratios possess. More complex ratios possess no such qualities and thus no amount of deviation can destroy them. It's true that melodic intervals are recognizable in their own right and can only stand so much alteration before they are heard as other melodic intervals. But this depends on the interval categories the listener has learned culturally (or otherwise), and not at all on ratios or just intonation.

>>in other temperaments while you pick the closest fifth as the acceptable generator >size. >For example, you pick the 13th step of 22-TET (710c) for the perfect fifth due to >its >proximity to 3:2 (an indisputably fundamental interval), but you do not deem the >>2nd step of this temperament (110c) suitable for a tempered apotome (an interval no >>less fundamental than 3:2).

>How can one consider the apotome, which is the interval between the impossibly >high 2187th and 2048th harmonics, nearly as fundamental as 3:2? The only >reason we're talking about 2187:2048 in the first place is because it is tunable by >a long chain of 3:2s. Without the foundation of the 3:2, intervals like >2187:2048 would never enter the picture in the first place. 2187:2048 is the >interval between two members of a chain of fifths, 7 fifths apart from one another, >and if each of the fifths is altered by an equal and acceptable amount, the >apotome will be altered 7 times as much. It makes no sense to consider a >different interval to be the apotome, since the apotome's very function is so tied >with the diatonical chain of 7 fifths, and attempting to use a different interval as >the apotome would inevitably ruin at least one of the fifths in the chain. In >particular, the apotome takes you from one end of the diatonic chain to a fifth >beyond the other end, so it's the key to the simplest possible modulation from >one fifth-based diatonic set to another. For example, from A aeolian (which has >all naturals) to E aeolian (where the previous F becomes F#). In music which is >based on such a diatonic chain of fifths (which is how the apotome became >relevant in the first place), attempting to use a different interval as the apotome >will result in the modulation being to a non-diatonic mode with at least one >unacceptable rendition of a perfect fifth.

>How can a pyhtagorean apotome not be a fundamental interval when as simple a >procedure as a chain of three 3:2 up and four 4:3 down result in it?

You seem to be missing my point above. And at this point, I hope it's clear I'm willing to replace "apotome" with "apotome-substitute" or "augmented unison" above, however you see fit.

>Even a singer can perform the apotome when written in such a rudimentary fashion.

Do you understand my scenario above, and how it relates to a modulating, chordal piece of music performed and notated in some temperament? We are discussing how to notate temperaments, I thought . . .

>As I stated before, I do not consider an `out-of-range apotome` a tempered apotome, >but an apotome-substitute just as I consider 710 cents a fifth-substitute instead of a >perfect fifth.

And I'll review my position -- apotome-substitute is fine, if you wish. But there are scenarios, where the perfect fifth may be tempered by as much as 7 cents, such as 19-equal and 22-equal, and still may be called a perfect fifth. The interval that results from 7 such perfect fifths will still be called an augmented unison, and will still be the interval that a sharp or flat alters a pitch by. If you are unhappy calling the augmented unison an apotome, and wish to call it an apotome-substitute, I'm perfectly happy to go along with that. You're not going against any Western historical practice that I know of. But the perfect fifths of Salinas and Costeley were still perfect fifths, though tuned in 19-equal, because the "perfect" signifier simply distinguishes the interval from the diminished fifth, augmented fifth, etc.

>A tempered apotome should be somewhere about 90 to 130 cents. This conforms wıth >historical practice, no?

Augmented unisons in the 300-year long meantone era were smaller than this. The sharp and flat in this period usually signified intervals around 70 to 90 cents -- the idealized "chroma" of ratio 25:24 being at the low end of this range. These intervals could be called "apotome substitutes". If you're unhappy calling them tempered apotomes, I will not argue with you, but I would argue with you if you took a *different* interval from the tempered system and called it a "tempered apotome" just because it happened to fall into your 90-130 cent range, you'd be misunderstanding the nature of temperament as applied to the Western musical system. Instead, I'd much prefer to claim that there is no apotome at all in meantone temperament, that it's only a part of Pythagorean tunings.

>Well, I'm wondering how one distinguishes 19:18 and >20:19 from other>similar-sized intervals by ear without playing them as simultaneities>in multi-part chords.>>I think what you mean is, for example, how the SATB singers of a choir can >intonate >such close-proximity intervals unless these are sounded within chords.

>It's not the closeness of the proximity but the complexity of the ratios.

>I do not believe these ratios to be very complex.

>>I do not think >that it is practically possible, but I believe the ear can >distinguish these >intervals upon >careful inspection.

>The intervals are 5 cents different from one another, so under controlled >circumstances, the ear can indeed distinguish them from one another. However, >my skepticism concerns the ability to hear either of these intervals as "pure" and >a tempered interval of, say, 91 cents as "impure" in a context outside large, >multi-part chords. There's no qualititative difference between the just and >tempered intervals in this range, so assigning ratios seems misleading or >arbitrary. And what holds for 19:18 and 20:19 in this regard holds with even more >certainty and strength for ratios as complex as 2187:2048. There are a myriad of >ratios in this range of complexity -- what distinguishes a few from the rest is that >they can be tuned by, and made musically meaningful in the context of, a chain >of simple-ratio, ear-tunable intervals.
>If the matter is one of `purity` of intervals, I consider all intervals, be they just, complex >or tempered, acceptable by ear, as long as they conform to simple-ratio ear-tunable >intervals as you so wonderfully put.

What do you mean by "conform to" in this context? How does an unlimited set "conform to" such a limited set? Please clarify . . .

Warm regards,
Paul

------------------------------------------------------------------------------
Do you Yahoo!?
Read only the mail you want - Yahoo! Mail SpamGuard.

Dear George, please take your time if my nonsensical arguments warrant a serious reply.

And dear Monz, I forwarded my reply to Paul Erlich to the list and I would like to hear your thoughts if you find the oppurtunity to oblige.

All the best,
Ozan
----- Original Message -----
From: George D. Secor
To: tuning@yahoogroups.com
Sent: 04 Aralık 2004 Cumartesi 0:13
Subject: [tuning] Re: Sagittal Notation

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George,
>
> It is a real priviledge to converse with you on theory and
practice. `I have been delighted to get enlightened` so to speak.

Ozan, I am still working on a reply to the many things you have
brought up in this message (and subsequent messages to me and
others). This has been between many other things that have been
making demands on my (unfortunately) very limited time. I hope to
have my reply finished some time next week, so we can continue our
most fruitful discussion.

I'm sorry that this could not be sooner, but these are issues that
cannot be treated adequately with a few words dashed off in haste.

Thank you for your patience.

Best,

--George

Dear Paul,

It's alright, I most certainly asked for it... I got just the smacking I needed and I thank you very much for your clarifications. I believe I now understand perfectly well the fundamentals of music theory as derived from the processes based on the concatenation of `perfect fifths`, `fourths`, `thirds` etc... I also comprehend how the apotome, or augmented unison as you call it, must always correspond to 7 `perfect fifths` up minus 4 octaves no matter the value, and how this affects the harmoniousness and modulation in music.

My effort was to provide regular musicians the means to sound the correct pitches in reference to the scales they were used to instead of the strict application of accidentals the way you explained. I concur with you that step numbers must repeat an exact pattern no matter the key in order to represent a particular mode. I somehow imagined that this obstacle could be overcome with some irregular practice though.

Forgive my ignorance if I made unfounded assertions for the Classical Theory of Western Music. It is only that I am used to performing sharps (apotomes) the way delineated by 12-tone pythagorean temperament and sharps and flats together as seen in 41-TET. Is not limma the difference between the `apotome` and the `major whole tone`? and is it not wrong to attribute the regular sharp to this interval? This is what is currently being done in the middle-east, a practice which I refer to as the `reverse diatonic progression`.

Deg Pitch Cents E12 INTERVAL E53 Name
0 1/1 0.0000 C Dbb C unison, perfect prime
1 2187/2048 113.6850 C# Db 2187/2048 C# Db/ apotome
2 9/8 203.9100 D Ebb 256/243 D major whole tone
3 19683/16384 317.5950 D# Eb 2187/2048 D# Eb/ Pythagorean augmented second
4 81/64 407.8200 E Fb 256/243 E Pythagorean major third
5 4/3 498.0450 F Gbb 256/243 F perfect fourth
6 729/512 611.7300 F# Gb 2187/2048 F# Gb/ Pythagorean tritone
7 3/2 701.9550 G Abb 256/243 G perfect fifth
8 6561/4096 815.6400 G# Ab 2187/2048 G# Ab/ Pythagorean augmented fifth
9 27/16 905.8650 A Bbb 256/243 A Pythagorean major sixth
10 16/9 996.0900 A# Bb 256/243 A#\ Bb Pythagorean minor seventh
11 243/128 1109.7750 B Cb 2187/2048 B Pythagorean major seventh
12 2/1 1200.0000 C Dbb 256/243 C octave

Yes, normally the sesquitone is not a pitch as you say, but the dyad resulting from this interval gives a pitch 150 cents away from natural, which could perhaps be referred to as a sesquitone. 12/11 is the pitch which is the simplest rational approximation of the sesquitone class of intervals. 22-TET apotome is close to this interval. This is what I wanted to imply by a sesquitone. A musician unaware of microtones or macrotones shall fail to sound this interval and the resulting pitch upon reading the 22-TET sharp. It was my naive idea of communicating this pitch to such musicians by filling the gaps of the major whole-tone regions in a rudimentary fashion that does not wreck their comprehension of the traditional tonal system.
The harmonic entropy concept intrigued me very much, and I would be very appreciative if you would be kind enough to provide me with at least one link where I could digest the concept of `octave tempering`.
Cordially,
Ozan

--------------------------------

Hi Ozan!

First, a correction -- 22-tET aeolian is 4-1-4-4-1-4-4, not 4-1-4-4-4-1-4.

Second, let me apologize for my rude and abrupt manner. I want to convey to you the spirit of warmest friendship as we embark on these interesting investigations together. Unfortunately, I'm much better at doing that in person than in typing :) And I want your dissertation and music career to be as successful and influential as possible, so it is a desire to help that lies behind my (overly :( ) sharp critiques.

Best,
Paul

----- Original Message -----
From: wally paulrus
To: Ozan Yarman
Sent: 05 Aralık 2004 Pazar 22:31
Subject: Re: Sagittal Notation

Hi Ozan,

>The true values of Maqam Music need as voluminous a system as based on primes 2,3,5,7 >and 11. I guess the simplest representation of such a system would be 82-TET.

You might want to say 82-EDO, and not 82-tET, because the frequency ratios in question could be used to derive 41-tET quite naturally, but the other notes of 82-equal would have to be derived from equal division as they do not arise from any of the 11-limit Just Intonation pitches in the process of temperament.

Meanwhile, 72-tET is a good deal more accurate than 41-tET (or 82-EDO) for the purpose of representing a system based on frequency ratios using all primes up to 11.

>You ask how some of the maqams I have given translate to 36 equal. I provide the steps below:

Thanks very much. I'll print these out and work with some musicians on them later.

>I have decided to drop the argument in favor of your theoretical explanations regarding the >semantic of the fifth, apotome, limma, so forth... Nevertheless, I will be unable to explain >Maqam Musicians hereabouts how come Westerners attribute the apotome sharp and flat for >the 70-90 cents interval named Classical Chroma in their meantone temperaments as >opposed to the limma which is wrongly assigned the regular sharp and flat by the Yekta-Arel->Ezgi triumvirate.

Actually, I think you will be able to explain this . . . if you bear with me and Dave and George long enough for us to explain this to you. You need to understand why Western music introduced sharps and flats in the first place, and you need to understand the Western system of key signatures. Also, you might want to take a peek at this paper of mine:

http://lumma.org/tuning/erlich/erlich-tFoT.pdf

> It would have been much easier to bridge the gap between our worlds if you would consign >the sharp and flat to correlate mostly to the ~102-136 cent zone of the historically plausible >~68-136 cent half-tone region, which is meant to drive home the point that the sharp and flat >correspond to the best rational approximations of the pythagorean apotome in vocal and >instrumental practice.

The purpose of the sharp and flat in Western music was not to represent a particular vocal or instrumental interval relative to the natural on the same nominal. Such intervals are quite rare anyhow.

The purpose of the sharp and flat in Western music was to provide consonant fifths and fourths where they would otherwise be impossible. The first instance in Western music was the distinction between B and B-flat. B-flat was introduced so that a note, roughly similar in scale position to B, could be sung against the note F so as to produce the concordant interval of a perfect fifth or perfect fourth against it. The exact distance between B and B-flat was never of primary concern; rather, the concern was that the interval between F and B-flat would be harmonically smooth and stable. E-flat was similarly introduced to be concordant with B-flat; F#, to be concordant with B; C#, to be concordant with F#; and so on.

Furthermore, the system of key signatures in Western music allows its basic diatonic scale to be transposed to any of several starting pitches. If you are not familiar with the system of key signatures, I suggest you study it, and carefully relate it to the considerations we've been discussing, as soon as you can. It will be very illuminating and is of crucial importance to understanding the Western notational system and its musical requirements.

>Afterall, the Classical Theory of Western Music is based entirely on the sharps being sharper >and the flats being flatter as compared to the Baroque Period. Or not?

I don't think so, Ozan. What is your reference for this so-called "Classical Theory of Western music"? What you're saying reminds me of the school of "expressive intonation" which has been visible in the West for about 200 years, but it doesn't even partially, let alone entirely, form the basis for the classical theory of Western music.

>We are, afterall, talking about the sharp and flat as the outcome of 12-tone temperaments of >Western tradition.

The sharp and flat date back to the Pythagorean and Meantone eras of Western tuning, and arise from the desire to apply consonant harmony more freely, and to transpose patterns of notes to other pitch levels, as I suggest above. The classical usage of the sharp and flat continue this same logic. 12-tone temperaments only came later and continued to make use of this same logic, with the result that the sharp and flat both came to represent an interval of 1 step out of the 12 in these temperaments. But that's only because moving 7 fifths up (and 4 octaves down) leaves you 1 step higher than you started, in a 12-tone temperament. It has virtually nothing to do with any desire for the melodic intervals B-flat to B, F to F-sharp, etc., to fall within a particular range of cents values. In fact, these melodic intervals are rare in common-practice Western music.

>So, I don't see why it is undesirable to assign the common sharp to the 2nd degree of 22TET >when any modern Western musician reading the Csharp will automatically find the register >closest to the upper zone of the half-tone region I outlined above.

22-tET is well outside the field of expertise of the vast majority of Western musicians. And many (including myself) have advocated wholly unconventional notation systems for it, for example dispensing with the conventional 7 nominals and using 10 instead. But let's proceed anyway. You are incorrect because C-sharp has to be consonant with F-sharp, which in turn has to be consonant with B. That's the raison d'etre of sharps (and flats) in the first place. If C-sharp is moved by one degree of 22-tET because of the very distant consideration that C to C-sharp must be in a certain range, one will likely destroy the harmoniousness of the music, since either C-sharp to F-sharp or F-sharp to B will now have to be a quite discordant interval. One also destroys the ability of the key signature system to perform its intended function and allow exact transposition of musical ideas from one pitch level to another.

>My approach has nothing to do with breaking the consonant chain of best-fit fifths.

But this would be the inevitable result in Western music.

>I take back what I said about the apotome-substitute being the 3rd degree of 22TET. For the >sake of historical thrifiness, let us call the 3rd degree the apotome itself. Then we are able to >reserve the second degree the position of the apotome-substitute as the closest deviation >from the original 12P JI interval, since a musician accustomed to the classical terminology of >sharp and flat cannot at a first glance interpret that pitch to be a sesquitone, and should not >necessarily be made to!

A sesquitone is not a pitch, it's an interval. But as I stated above, the relevant interval, the one the musician will be faced with performing in a pleasing way, is almost never the interval between F and F-sharp, etc. (for which different renditions differ only in their relative familiarity) but is far more likely to be a consonant fifth as between B and F-sharp (for which different renditions will sound vastly less concordant than the correct one). And this is one reason why following your suggested approach would be detrimental in Western music.

>So I don't understand what you imply by saying that I ignored your example of modulation.

Modulation/transposition is another key requirement of the Western notational system that would be destroyed by your proposal. Consider again a modulation from A aeolian to E aeolian. A aeolian is A-B-C-D-E-F-G-A, and E aeolian is E-F#-G-A-B-C-D-E. In 22-equal, we'd agree for the purposes of this discussion that A aeolian would have step sizes of 4-1-4-4-4-1-4 in degrees, right? Now, if the music modulates to E aeolian, or needs to be copied for an instrument, playing in unison with the original one, whose notation is a fifth away from concert pitch (a common scenario in the West), it's still in the same mode so has to have the same pattern of step sizes: 4-1-4-4-4-1-4. Starting from E, this means that F# is 4 degrees higher than E. But looking at A aeolian, we see that F is 1 degree higher than E. Hence the interval between F and F#, though it may never arise sequentially in the music, *must* be 3 degrees. If it is forced to be 2 degrees, the E aeolian mode will be entirely ill-proportioned melodically, won't produce the same result as A aeolian on the relevant transposing instrument, and of course, the interval B-F#, almost certain to occur as a consonance somewhere in the music, will sound highly discordant.

> I am also anxious to learn where in historical Western Theory octave tempering is allowed.

It is not only allowed, but practiced, in the tuning of virtually any piano. There is plenty of information on this on the Internet and the archives of the Tuning List. Let me know if your searches come up short.

Highest Regards,
Paul

------------------------------------------------------------------------------
Do you Yahoo!?
Read only the mail you want - Yahoo! Mail SpamGuard.

It was also my part of my training to get myself familiarized with 15 major and 15 minor keys. My fluency with Western accidentals have been hampered at first when I've delved into the microtonal world of maqam music. Also, my `absolute ear` lost its tonal stability due to the expanse of the pitch continuum. That is, I'm having difficulty at times pinpointing notes although I had no issues with them before. I recovered somewhat this past year with some practice but there is no telling what other musicians and singers might experience when I tell them that they have to forsake the tones and keys they have learned thus far to perform maqams correctly.

The point of the table is to show that the apotome is performed in the West more or less as seen in 3-limit 12P. In the East, it is performed more or less as the limma, the so-called `reverse diatonic progression`.

So let me rephrase my reservation with the 22-TET apotome: There is a 55 cent interval from G to Ab, a.k.a steps 13 to 14 in Ab Major, and the same interval from A to G#, a.k.a steps 17 to 16 in A Major. This is what I assume you infer by the simpler (sharped or flatted) diatonic interval as being part of a scale. I gather that you believe it is easy for a Western musician to adopt this practice of modified pitches upon reading the sharp and flat? I beg to differ on this.

So the answer to your question would be:

`Do you think my viewpoint entails wrecking the comprehension of the traditional tonal system?`

well, somewhat, yes... If I understand you correctly, you are inclined to interpret the sharp and flat as I explained above for 22-TET. Will it be possible for a choir singer or a violinist to sound the right pitch when you tell him that it is ok to perform the sharp and flat as part of such a xenharmonic scale? I do not think that they will be able to comprehend fully your intentions without a lenghty course on theory.

Thanks for the links, I will get to reading them when I can.

Cordially,
Ozan

----- Original Message -----
From: wally paulrus
To: Ozan Yarman
Sent: 07 Aralık 2004 Salı 22:43
Subject: Re: Sagittal Notation

>My effort was to provide regular musicians the means to sound the correct pitches in >reference to the scales they were used to instead of the strict application of accidentals the >way you explained.

Presumably, this is for Maqam music, where musicians are used to certain scales at certain fixed pitch levels. In Western music today, playing in all 12 keys is part of the training on almost any instrument, and is perhaps the primary means by which fluency with accidentals (sharps and flats) is acquired.

>Forgive my ignorance if I made unfounded assertions for the Classical Theory of Western >Music. It is only that I am used to performing sharps (apotomes) the way delineated by 12->tone pythagorean temperament and sharps and flats together as seen in 41-TET. Is not >limma the difference between the `apotome` and the `major whole tone`?

In Pythagorean (i.e., 3-limit) tuning or any temperament of it, the limma is the difference between the apotome and the whole tone, yes. In higher-limit systems, there are, in general, different kinds of whole tones, but also different kinds of limmas, etc.

>and is it not wrong to attribute the regular sharp to this interval?

In the West, it is wrong. The limma always represents an actual change in letter-name, whether B to C, E to F, F# to G, A to Bb, etc.

>This is what is currently being done in the middle-east, a practice which I refer to as the >`reverse diatonic progression`.

It seems there are some headings missing in this table? Or perhaps I'm missing the point of its inclusion . . .

>Yes, normally the sesquitone is not a pitch as you say, but the dyad resulting from this >interval gives a pitch 150 cents away from natural, which could perhaps be referred to as a >sesquitone. 12/11 is the pitch which is the simplest rational approximation of the sesquitone >class of intervals. 22-TET apotome is close to this interval. This is what I wanted to imply by >a sesquitone. A musician unaware of microtones or macrotones shall fail to sound this >interval and the resulting pitch upon reading the 22-TET sharp.

This is where I differed with your interpretation, from the perspective of a Western musician. You're tacitly assuming that "reading the sharp" implies that the relevant intervallic reference will be the interval between the sharp and the natural on the same nominal. But in Western music, this will seldom be the case, as augmented unisons (not just what I call them, but what they've been called for the better part of a millenium in the West) are quite rare melodically, let alone harmonically. Instead, the sharp in question will appear as part of a scale and/or chord in which simpler diatonic intervals, such as perfect fifths and major and minor seconds, will link the sharp to "known" pitches. So in many cases the intonation of the sharped note will be based on the musician's sense of *those* intervals, rather than the rare and distant augmented unison.

>It was my naive idea of communicating this pitch to such musicians by filling the gaps of the >major whole-tone regions in a rudimentary fashion that does not wreck their comprehension of >the traditional tonal system.

Do you think my viewpoint entails wrecking the comprehension of the traditional tonal system?

>The harmonic entropy concept intrigued me very much, and I would be very appreciative if you >would be kind enough to provide me with at least one link where I could digest the concept of >`octave tempering`.

Here is an assortment of links, relating to various possible interpretations of your request. I'm sure you'll find most of them irrelevant, but hopefully some will be relevant:

http://dkeenan.com/Music/DistributingCommas.htm

Bottom of
http://www.io.com/~hmiller/music/warped-canon.html

http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/octstretch.html

part about 'octave stretch' here:
http://www.pressenter.com/~trps/sbbjan04.html

http://www.hibberts.co.uk/nominals.htm

http://www.ptg.org/pipermail/caut/2001-February/003639.html

------------------------------------------------------------------------------
Do you Yahoo!?
Read only the mail you want - Yahoo! Mail SpamGuard.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> >Afterall, the Classical Theory of Western Music is based entirely
on the sharps being sharper >and the flats being flatter as compared
to the Baroque Period. Or not?
>
> I don't think so, Ozan. What is your reference for this so-called
"Classical Theory of Western music"? What you're saying reminds me of
the school of "expressive intonation" which has been visible in the
West for about 200 years, but it doesn't even partially, let alone
entirely, form the basis for the classical theory of Western music.

The sharps of 1/4-comma meantone are 5^(7/4)/16, which at 76 cents is
considerably flatter than the 100 cents of 12 equal. As you move from
1/4 comma to 12-et, the sharps on average would have to get sharper,
and the flats flatter.

> >So, I don't see why it is undesirable to assign the common sharp
to the 2nd degree of 22TET >when any modern Western musician reading
the Csharp will automatically find the register >closest to the upper
zone of the half-tone region I outlined above.
>
> 22-tET is well outside the field of expertise of the vast majority
of Western musicians.

If you simply treat it as a means of getting a diatonic scale, you end
up with a superpythagorean version, with supermajor thirds instead of
major thirds.

Thank you for the elucidation dear Gene. I guess that kind of vindicates my claim that the classical theory of Western music has sharper sharps and flatter flats as compared to the Baroque era.

BTW, how would you define a superpythagorean scale?

All the best,
Ozan Yarman
----- Original Message -----
From: Gene Ward Smith
To: tuning@yahoogroups.com
Sent: 09 Aralık 2004 Perşembe 2:36
Subject: [tuning] Re: Sagittal Notation

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> >Afterall, the Classical Theory of Western Music is based entirely
on the sharps being sharper >and the flats being flatter as compared
to the Baroque Period. Or not?
>
> I don't think so, Ozan. What is your reference for this so-called
"Classical Theory of Western music"? What you're saying reminds me of
the school of "expressive intonation" which has been visible in the
West for about 200 years, but it doesn't even partially, let alone
entirely, form the basis for the classical theory of Western music.

The sharps of 1/4-comma meantone are 5^(7/4)/16, which at 76 cents is
considerably flatter than the 100 cents of 12 equal. As you move from
1/4 comma to 12-et, the sharps on average would have to get sharper,
and the flats flatter.

> >So, I don't see why it is undesirable to assign the common sharp
to the 2nd degree of 22TET >when any modern Western musician reading
the Csharp will automatically find the register >closest to the upper
zone of the half-tone region I outlined above.
>
> 22-tET is well outside the field of expertise of the vast majority
of Western musicians.

If you simply treat it as a means of getting a diatonic scale, you end
up with a superpythagorean version, with supermajor thirds instead of
major thirds.

hi Ozan,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Thank you for the elucidation dear Gene. I guess that
> kind of vindicates my claim that the classical theory
> of Western music has sharper sharps and flatter flats
> as compared to the Baroque era.

i only have time to give a very brief summary here, but
basically, for the Western (i.e., Europe and its colonies)
musical reportoire, the "standard" tunings chronologically were:

ancient to c.1450: pythagorean - sharps higher than flats;

c.1450 to c.1700: meantone - flats higher than sharps;

c.1700 to c.1850 for keyboards: well/circulating, which
is limited to 12 chromatic notes where the "black keys"
function as either flats or sharps, thus depending on
the context/key either one could be higher;

c.1700 to c.1850 for orchestral instruments: "expressive"
intonation, which is basically the same as pythagorean;

c.1850 to the present: 12-ET for all instruments, except
that good orchestral performers are still taught to use
"expressive" for melodic purposes, and sometimes also elements
of JI for the harmony.

and of course, since about 1990 or so, microtonality has
definitely been on the increase! ... which means not only
that either sharps or flats can be higher (depending on the
specific tuning), but also that there can be lots of other
alterations to the basic diatonic set - "sesquisharps", etc.

;-)

-monz

Thanks for the great overview dear Monz! I think you should save this wonderfully articulate text for an introduction to Western Tonality. This information verifies that I'm very much partial to the expressiveness of the great romantics such as Liszt, Chopin, Strauss, Tchaikovsky, Mussorgsky, Bruckner, Mahler, Debussy and Rachmaninoff.

Cordially,
Ozan
----- Original Message -----
From: monz
To: tuning@yahoogroups.com
Sent: 11 Aralık 2004 Cumartesi 4:15
Subject: [tuning] sharps or flats higher? (was Re: Sagittal Notation)

hi Ozan,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Thank you for the elucidation dear Gene. I guess that
> kind of vindicates my claim that the classical theory
> of Western music has sharper sharps and flatter flats
> as compared to the Baroque era.

i only have time to give a very brief summary here, but
basically, for the Western (i.e., Europe and its colonies)
musical reportoire, the "standard" tunings chronologically were:

ancient to c.1450: pythagorean - sharps higher than flats;

c.1450 to c.1700: meantone - flats higher than sharps;

c.1700 to c.1850 for keyboards: well/circulating, which
is limited to 12 chromatic notes where the "black keys"
function as either flats or sharps, thus depending on
the context/key either one could be higher;

c.1700 to c.1850 for orchestral instruments: "expressive"
intonation, which is basically the same as pythagorean;

c.1850 to the present: 12-ET for all instruments, except
that good orchestral performers are still taught to use
"expressive" for melodic purposes, and sometimes also elements
of JI for the harmony.

and of course, since about 1990 or so, microtonality has
definitely been on the increase! ... which means not only
that either sharps or flats can be higher (depending on the
specific tuning), but also that there can be lots of other
alterations to the basic diatonic set - "sesquisharps", etc.

;-)

-monz

hi Ozan,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Thanks for the great overview dear Monz! I think you should
> save this wonderfully articulate text for an introduction
> to Western Tonality. This information verifies that I'm
> very much partial to the expressiveness of the great romantics
> such as Liszt, Chopin, Strauss, Tchaikovsky, Mussorgsky,
> Bruckner, Mahler, Debussy and Rachmaninoff.
>
> Cordially,
> Ozan

well, we certainly have musical tastes in common!
you just named 8 of my all-time favorite composers, along
with the person i consider the Greatest Genius Of All Time.

(hint ... his name begins with "M" ...)

:)

however, please note that that particular composer lamented
to his friend Schoenberg that "European music, in giving up
Meantone tuning, had suffered a great loss".

i've made a MIDI-file of the 1st movement of Mahler's 7th
Symphony, which i hope someday to painstakingly retune to
something which makes good sense of the harmonies used in it.

-monz

By the majesty of Allah, we do have a common taste in the great masters it seems my dear Monz! I too consider Mahler the pinnacle of all the Romantic symphonists. Pity he did not have the proper appreciation for expressive intonation, which is a dilemma considering that his epic works sound best when directed by such free-stylists as Bernstein and performed by the most prominent philarmonic orchestras such as New York or Berlin. Aside from some temperaments of Werckmeister and Kirnberger, meantone temperament is too rough for my taste by half(tone). Now that you mentioned it, `correct intonation` is a fundamental problem with the orchestras and ensembles in Turkey, particularly because the musicians receive virtually no education on the theory of proper intonation. Haphazard intervals may thus occur frequently!

All the best,
Ozan
----- Original Message -----
From: monz
To: tuning@yahoogroups.com
Sent: 11 Aralık 2004 Cumartesi 11:14
Subject: [tuning] sharps or flats higher? (was Re: Sagittal Notation)

hi Ozan,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Thanks for the great overview dear Monz! I think you should
> save this wonderfully articulate text for an introduction
> to Western Tonality. This information verifies that I'm
> very much partial to the expressiveness of the great romantics
> such as Liszt, Chopin, Strauss, Tchaikovsky, Mussorgsky,
> Bruckner, Mahler, Debussy and Rachmaninoff.
>
> Cordially,
> Ozan

well, we certainly have musical tastes in common!
you just named 8 of my all-time favorite composers, along
with the person i consider the Greatest Genius Of All Time.

(hint ... his name begins with "M" ...)

:)

however, please note that that particular composer lamented
to his friend Schoenberg that "European music, in giving up
Meantone tuning, had suffered a great loss".

i've made a MIDI-file of the 1st movement of Mahler's 7th
Symphony, which i hope someday to painstakingly retune to
something which makes good sense of the harmonies used in it.

-monz

You can configure your subscription by sending an empty email to one
of these addresses (from the address at which you receive the list):
tuning-subscribe@yahoogroups.com - join the tuning group.
tuning-unsubscribe@yahoogroups.com - leave the group.
tuning-nomail@yahoogroups.com - turn off mail from the group.
tuning-digest@yahoogroups.com - set group to send daily digests.
tuning-normal@yahoogroups.com - set group to send individual emails.
tuning-help@yahoogroups.com - receive general help information.

Yahoo! Groups Sponsor
ADVERTISEMENT

------------------------------------------------------------------------------
Yahoo! Groups Links

a.. To visit your group on the web, go to:
/tuning/

b.. To unsubscribe from this group, send an email to:
tuning-unsubscribe@yahoogroups.com

c.. Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

On Saturday 11 December 2004 03:38 am, Ozan Yarman wrote:
> By the majesty of Allah, we do have a common taste in the great masters it
> seems my dear Monz!

Don't you mean Zeus? ;) <grin> Forgive me, I'm one of those agnostic
'infidels' so reviled by all the worlds religions...I'm fed up with all the
hate and murder that the idea of a 'God' of any sort seems to inspire.

> Aside from
> some temperaments of Werckmeister and Kirnberger, meantone temperament is
> too rough for my taste by half(tone).

You ought consider trying out meantone for the type of music it fits best
with: Renaissance masters such as Byrd, Josquin, etc. It really is quite a
beautiful temperament for what it does.

> Now that you mentioned it, `correct
> intonation` is a fundamental problem with the orchestras and ensembles in
> Turkey, particularly because the musicians receive virtually no education
> on the theory of proper intonation. Haphazard intervals may thus occur
> frequently!

Yikes! Sounds like it might be kind of fun to listen to, huh?

BTW, Ozan, I like your website very music. I listened to 'Balkan Carnival'.
Great work!

Warmly,
Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

Dear Paul, please take your time. But I would like you to explain to me
later just what you infer by `expressive intonation` just in case we are not
on the same wavelenght.

By the way, have you visited my new website www.ozanyarman.com?

Cordially,
Ozan

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: "Ozan Yarman" <ozanyarman@superonline.com>
Sent: 12 Aral�k 2004 Pazar 0:50
Subject: sharps or flats higher? (was Re: Sagittal Notation)

Hi Ozan, I'm sorry I'm behind in responding to several of your e-
mails. I just wanted to point out to you that I believe Monz is
making a quite false assertion below, and I e-mailed him, writing:

"> c.1700 to c.1850 for orchestral instruments: "expressive"
> intonation, which is basically the same as pythagorean;

According to all the evidence I've seen, this is false,
and "expressive" intonation did not come about until the very late
18th century. On what basis are you extending its domination, let
alone its appearance, back almost a full century?"

Additionally, Monz himself has been quite vocal that Mahler preferred
meantone tuning, and Monz has been working on a meantone realization
of a Mahler symphony for some time now.

Cheers,
Paul

In a message dated 12/11/2004 6:38:42 PM Eastern Standard Time,
ozanyarman@superonline.com writes:
Additionally, Monz himself has been quite vocal that Mahler preferred
meantone tuning, and Monz has been working on a meantone realization
of a Mahler symphony for some time now.

Cheers,
Paul
You know, history is on the side of extended sixth comma meantone as the
meantone intended by the non-piano works of the Romantic period. Expressivity,
seen in light of this, would be in the direction toward quarter comma meantone.

best, Johnny Reinhard

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

>
> c.1700 to c.1850 for orchestral instruments: "expressive"
> intonation, which is basically the same as pythagorean;
>

oops ... my bad. Paul Erlich pointed out to me that i made
an error there ... it was just a simple typo. the dates
should be c.1800 to c.1850 (and as i wrote immediately after,
continuing into the present day with good musicians).

"expressive intonation" was first advocated right around
the time of Mozart's death, 1791.

-monz

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> By the majesty of Allah, we do have a common taste in
> the great masters it seems my dear Monz! I too consider
> Mahler the pinnacle of all the Romantic symphonists.

he's much more than that to me ... but what you wrote is
good for a start. ;-)

(remember, i said "Greatest Genius Of All Time"? ...)

> Pity he did not have the proper appreciation for expressive
> intonation,

well, be cautious about saying that -- there's absolutely
no evidence either way regarding his opinions about
"expressive intonation". in fact, the only evidence i
know of in connection with Mahler's preference for intonation
is the remark that Schoenberg made to Peter Yates which
i quoted, where Mahler lamented the loss of meantone.

i love to speculate, and when i found out that Josef Petzval
was teaching about 31-ET and demonstrating it on instruments
he invented, at the University of Vienna at exactly the
time that Mahler attended, i extrapolated that that's how
Mahler had such knowledge of meantone. it wasn't common
for musicians of his time (1860-1911) to know much about it.

however, it's not a simple matter to just say that Mahler's
symphonies should be performed in meantone tuning. his
harmonic style ranges from simple diatonicity to near-atonality,
and he certainly had 12-ET in mind at least some of the time,
particularly after he struck up a friendship with Schoenberg.

which is why what you say here is correct:

> which is a dilemma considering that his epic works
> sound best when directed by such free-stylists as Bernstein
> and performed by the most prominent philarmonic orchestras
> such as New York or Berlin.

i agree. Bernstein had an insight into Mahler that
surpassed nearly every other conductor i've ever heard
do Mahler. it was Bernstein who was largely responsible
for making Mahler into the superstar composer he is today.

i believe that Bernstein's "free-style" approach is very
close indeed to Mahler's own temperament (pun intended).

-monz

hi Aaron and Ozan,

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:

> On Saturday 11 December 2004 03:38 am, Ozan Yarman wrote:
> >
> > By the majesty of Allah, we do have a common taste in
> > the great masters it seems my dear Monz!
>
> Don't you mean Zeus? ;) <grin> Forgive me, I'm one of
> those agnostic 'infidels' so reviled by all the worlds
> religions...I'm fed up with all the hate and murder that
> the idea of a 'God' of any sort seems to inspire.

me too ... i circumvent that whole problem by worshiping Mahler.

:)

Ozan quoted a private email from Paul Erlich:

> Additionally, Monz himself has been quite vocal that
> Mahler preferred meantone tuning, and Monz has been working
> on a meantone realization of a Mahler symphony for some time now.

don't anyone get their hopes up too soon ... i've been
*planning* a meantone realization of Mahler's 7th, 1st movement
... but still a long way off from actually working on it.

too much work to do on the Musica software yet ...

-monz

On Saturday 11 December 2004 09:54 am, Aaron K. Johnson wrote:

> BTW, Ozan, I like your website very music. I listened to 'Balkan Carnival'.
> Great work!

Kurt pointed out to me that I made this slip...wow! That's pretty fuunny that
I didn't notice it!

Best,
Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

A heavenly intervention may be at work here with Mahler's genius, justifying to an extent your desire to deify him dear Monz, but I'm sure it is just your own way of expressing great admiration for a great man, who is, no doubt, an incarnation of other-worldly spiritual forces, a gift from Allah Almighty to mankind. I view Mahler as the last inheritor of Beethoven's legacy and the capitulation of the Classical Era. Thus I hope you do not infer that I refrain from conferring upon him the honor he very much deserves. He was, most certainly, an artistic intellect of sublime nature, a musical messenger of God so to speak, or if you prefer the metaphors of mythological phenomena, an avatar of Hermes or Apollo.

I regard Mahler's excursion into meantone as an attempt to break the shackles of 12-TET, or better said, 12 arbitrary Western tones. So you see, I am no less a speculator when it comes to Mahler. Neverthless, I am glad we share the same taste concerning the `Mahlerian flavor`, namely: Bernstein, a virtuoso no less important if you ask my opinion. Pity that exceptional men die young and worn out. The toll of life in a world as stupid as this may be the culprit.

But what is this Musica software you mentioned?

Cordially,
Ozan
----- Original Message -----
From: monz
To: tuning@yahoogroups.com
Sent: 12 Aralık 2004 Pazar 6:38
Subject: [tuning] Mahler tuning (was: sharps or flats higher?)

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> By the majesty of Allah, we do have a common taste in
> the great masters it seems my dear Monz! I too consider
> Mahler the pinnacle of all the Romantic symphonists.

he's much more than that to me ... but what you wrote is
good for a start. ;-)

(remember, i said "Greatest Genius Of All Time"? ...)

> Pity he did not have the proper appreciation for expressive
> intonation,

well, be cautious about saying that -- there's absolutely
no evidence either way regarding his opinions about
"expressive intonation". in fact, the only evidence i
know of in connection with Mahler's preference for intonation
is the remark that Schoenberg made to Peter Yates which
i quoted, where Mahler lamented the loss of meantone.

i love to speculate, and when i found out that Josef Petzval
was teaching about 31-ET and demonstrating it on instruments
he invented, at the University of Vienna at exactly the
time that Mahler attended, i extrapolated that that's how
Mahler had such knowledge of meantone. it wasn't common
for musicians of his time (1860-1911) to know much about it.

however, it's not a simple matter to just say that Mahler's
symphonies should be performed in meantone tuning. his
harmonic style ranges from simple diatonicity to near-atonality,
and he certainly had 12-ET in mind at least some of the time,
particularly after he struck up a friendship with Schoenberg.

which is why what you say here is correct:

> which is a dilemma considering that his epic works
> sound best when directed by such free-stylists as Bernstein
> and performed by the most prominent philarmonic orchestras
> such as New York or Berlin.

i agree. Bernstein had an insight into Mahler that
surpassed nearly every other conductor i've ever heard
do Mahler. it was Bernstein who was largely responsible
for making Mahler into the superstar composer he is today.

i believe that Bernstein's "free-style" approach is very
close indeed to Mahler's own temperament (pun intended).

-monz

You can configure your subscription by sending an empty email to one
of these addresses (from the address at which you receive the list):
tuning-subscribe@yahoogroups.com - join the tuning group.
tuning-unsubscribe@yahoogroups.com - leave the group.
tuning-nomail@yahoogroups.com - turn off mail from the group.
tuning-digest@yahoogroups.com - set group to send daily digests.
tuning-normal@yahoogroups.com - set group to send individual emails.
tuning-help@yahoogroups.com - receive general help information.

Yahoo! Groups Sponsor
ADVERTISEMENT

------------------------------------------------------------------------------
Yahoo! Groups Links

a.. To visit your group on the web, go to:
/tuning/

b.. To unsubscribe from this group, send an email to:
tuning-unsubscribe@yahoogroups.com

c.. Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

I didn't notice it till you mentioned. Duh! I'm pretty blind nowadays, paying no heed or attention whatsoever to the details.

Ozan
----- Original Message -----
From: Aaron K. Johnson
To: tuning@yahoogroups.com ; ozanyarman@superonline.com
Cc: kkb@breathsense.com
Sent: 12 Aralık 2004 Pazar 17:29
Subject: Re: [tuning] sharps or flats higher? (was Re: Sagittal Notation)

On Saturday 11 December 2004 09:54 am, Aaron K. Johnson wrote:

> BTW, Ozan, I like your website very music. I listened to 'Balkan Carnival'.
> Great work!

Kurt pointed out to me that I made this slip...wow! That's pretty fuunny that
I didn't notice it!

Best,
Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

hi Ozan,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> A heavenly intervention may be at work here with Mahler's
> genius, justifying to an extent your desire to deify him dear
> Monz, but I'm sure it is just your own way of expressing
> great admiration for a great man, who is, no doubt, an
> incarnation of other-worldly spiritual forces, a gift from
> Allah Almighty to mankind. I view Mahler as the last inheritor
> of Beethoven's legacy and the capitulation of the Classical
> Era. Thus I hope you do not infer that I refrain from
> conferring upon him the honor he very much deserves. He was,
> most certainly, an artistic intellect of sublime nature, a
> musical messenger of God so to speak, or if you prefer the
> metaphors of mythological phenomena, an avatar of Hermes or
> Apollo.

i think the two of us are pretty much in agreement about
Mahler. OK, no further need to belabor the readers of the
tuning list about it -- there's a Yahoo Mahler List for that.

> But what is this Musica software you mentioned?

it's pretty hard to visit my website without finding out
about it.

http://tonalsoft.com

it's now scheduled for commercial release around July 2005.

-monz

Oh dear, I overdid it again... my apologies. But do tell me dear Monz, where do I make the payment for this excellent program of yours upon its release?

All the best,
Ozan
----- Original Message -----
From: monz
To: tuning@yahoogroups.com
Sent: 13 Aralık 2004 Pazartesi 0:17
Subject: [tuning] Re: Mahler tuning (was: sharps or flats higher?)

hi Ozan,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> A heavenly intervention may be at work here with Mahler's
> genius, justifying to an extent your desire to deify him dear
> Monz, but I'm sure it is just your own way of expressing
> great admiration for a great man, who is, no doubt, an
> incarnation of other-worldly spiritual forces, a gift from
> Allah Almighty to mankind. I view Mahler as the last inheritor
> of Beethoven's legacy and the capitulation of the Classical
> Era. Thus I hope you do not infer that I refrain from
> conferring upon him the honor he very much deserves. He was,
> most certainly, an artistic intellect of sublime nature, a
> musical messenger of God so to speak, or if you prefer the
> metaphors of mythological phenomena, an avatar of Hermes or
> Apollo.

i think the two of us are pretty much in agreement about
Mahler. OK, no further need to belabor the readers of the
tuning list about it -- there's a Yahoo Mahler List for that.

> But what is this Musica software you mentioned?

it's pretty hard to visit my website without finding out
about it.

http://tonalsoft.com

it's now scheduled for commercial release around July 2005.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
>
> i only have time to give a very brief summary here, but
> basically, for the Western (i.e., Europe and its colonies)
> musical reportoire, the "standard" tunings chronologically were:
>
> ancient to c.1450: pythagorean - sharps higher than flats;
>
> c.1450 to c.1700: meantone - flats higher than sharps;
>
> c.1700 to c.1850 for keyboards: well/circulating, which
> is limited to 12 chromatic notes where the "black keys"
> function as either flats or sharps, thus depending on
> the context/key either one could be higher;
>
> c.1700 to c.1850 for orchestral instruments: "expressive"
> intonation, which is basically the same as pythagorean;
>
> c.1850 to the present: 12-ET for all instruments, except
> that good orchestral performers are still taught to use
> "expressive" for melodic purposes, and sometimes also elements
> of JI for the harmony.
>
> and of course, since about 1990 or so, microtonality has
> definitely been on the increase! ... which means not only
> that either sharps or flats can be higher (depending on the
> specific tuning), but also that there can be lots of other
> alterations to the basic diatonic set - "sesquisharps", etc.
>
> ;-)
>

Ah yes, this question what is higher... I had ground my teeth on that
one, too. Last time I tried to answer it (including asking here), the
answer seemed to be more complicated than is indicated above - if I
understand

/tuning/topicId_52504.html#52506

correctly...
Can someone give more enlightenmemt?!

Hans Straub

Hi Hans,

In our website

www.hermode.com

you will find in the chapter "historical" in English
or "historisch" in German

an introdution in the subject of tuning and temperament
and its historical evolution.
Including easy-to-understand diagrams
and including sound examples.
I hope, this will be helpful, too.

Werner
> -----Urspr�ngliche Nachricht-----Von: hstraub64
[mailto:hstraub64@telesonique.net]
> Gesendet: Montag, 13. Dezember 2004 13:14
> An: tuning@yahoogroups.com
> Betreff: [tuning] sharps or flats higher? (was Re: Sagittal Notation)

>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>>
>> i only have time to give a very brief summary here, but
>> basically, for the Western (i.e., Europe and its colonies)
>> musical reportoire, the "standard" tunings chronologically were:
>>
>> ancient to c.1450: pythagorean - sharps higher than flats;
>>
>> c.1450 to c.1700: meantone - flats higher than sharps;
>>
>> c.1700 to c.1850 for keyboards: well/circulating, which
>> is limited to 12 chromatic notes where the "black keys"
>> function as either flats or sharps, thus depending on
>> the context/key either one could be higher;
>>
>> c.1700 to c.1850 for orchestral instruments: "expressive"
>> intonation, which is basically the same as pythagorean;
>>
>> c.1850 to the present: 12-ET for all instruments, except
>> that good orchestral performers are still taught to use
>> "expressive" for melodic purposes, and sometimes also elements
>> of JI for the harmony.
>>
>> and of course, since about 1990 or so, microtonality has
>> definitely been on the increase! ... which means not only
>> that either sharps or flats can be higher (depending on the
>> specific tuning), but also that there can be lots of other
>> alterations to the basic diatonic set - "sesquisharps", etc.
>>
>> ;-)
>>

> Ah yes, this question what is higher... I had ground my teeth on that
> one, too. Last time I tried to answer it (including asking here), the
> answer seemed to be more complicated than is indicated above - if I
> understand
>
/tuning/topicId_52504.html#52506
>
> correctly...
> Can someone give more enlightenmemt?!
> Hans Straub

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:
> On Saturday 11 December 2004 03:38 am, Ozan Yarman wrote:
> > By the majesty of Allah, we do have a common taste in the great
masters it
> > seems my dear Monz!
>
> Don't you mean Zeus? ;) <grin> Forgive me,

Oh, dear! For a brief moment I saw a glimmer of hope that our
efforts have not, thus far, been in vain:

http://dkeenan.com/sagittal/gift/GiftOfTheGods.htm

May Zeus forgive you! It would be a terrible thing for a promising
young composer such as yourself to be shunned by the Muses! :-(

> I'm one of those agnostic
> 'infidels' so reviled by all the worlds religions...I'm fed up with
all the
> hate and murder that the idea of a 'God' of any sort seems to
inspire.

Perhaps, once the story behind the notation is fully revealed, you
may have a change of heart. ;-)

--GS

hi Ozan,

when release 1.0 comes out (planned for July/August 2005),
you'll be able to pay for Tonalsoft Musica by credit card
by simply clicking on the "purchase" button.

our plan is to have a free demo available for download,
which will do everything the regular program does, except
save and print. paying for it will give you an unlock code
which unlocks the save and print features of the demo
program you've already downloaded.

we plan on an initial sale price of around $99.99 (US).
(there will be a discount off of the $169.99 listed right now
... and please don't anyone try to pay for it yet -- rest
assured that i'll let you all know when it's available.)

future releases will also include a "Professional" version
at a rather higher price, which will print engraver-quality
scores and parts.

please note that release 1.0 will be for Windows platforms only.
we hope to eventually port it to other platforms (Mac and Linux),
but there are no plans for that yet.

-monz

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Oh dear, I overdid it again... my apologies. But do tell me
> dear Monz, where do I make the payment for this excellent
> program of yours upon its release?
>
> All the best,
> Ozan

Gosh, the price is a bit too steep Monz, I wonder if you would consider half that much for me to be able to pay for it?

All the best,
Ozan
----- Original Message -----
From: monz
To: tuning@yahoogroups.com
Sent: 14 Aralık 2004 Salı 1:07
Subject: [tuning] Tonalsoft Musica software (was Re: Mahler tuning)

hi Ozan,

when release 1.0 comes out (planned for July/August 2005),
you'll be able to pay for Tonalsoft Musica by credit card
by simply clicking on the "purchase" button.

our plan is to have a free demo available for download,
which will do everything the regular program does, except
save and print. paying for it will give you an unlock code
which unlocks the save and print features of the demo
program you've already downloaded.

we plan on an initial sale price of around $99.99 (US).
(there will be a discount off of the $169.99 listed right now
... and please don't anyone try to pay for it yet -- rest
assured that i'll let you all know when it's available.)

future releases will also include a "Professional" version
at a rather higher price, which will print engraver-quality
scores and parts.

please note that release 1.0 will be for Windows platforms only.
we hope to eventually port it to other platforms (Mac and Linux),
but there are no plans for that yet.

-monz

hi Ozan,

there's still plenty of time to save ... it won't be out
until this coming summer.

-monz

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Gosh, the price is a bit too steep Monz, I wonder if you
> would consider half that much for me to be able to pay for it?
>
> All the best,
> Ozan

--- In tuning@yahoogroups.com, "Werner Mohrlok" <wmohrlok@h...>
wrote:
> Hi Hans,
>
> In our website
>
> www.hermode.com
>
> you will find in the chapter "historical" in English
> or "historisch" in German
>
> an introdution in the subject of tuning and temperament
> and its historical evolution.
> Including easy-to-understand diagrams
> and including sound examples.
> I hope, this will be helpful, too.
>
> Werner

Thanks for the l! Yes, the pythagorean case is quite clear there -
sharps higher than flats, no doubt. The 5-limit case is less clear -
I originally had derived that flats are higher, but there is this
complication with the two sizes of major seconds and minor thirds -
I still have to find a concise formulation...

Hans Straub

I think I will wait for fluctuations as well as competition in the market for a favourable discount dear Monz.

Cordially,
Ozan
----- Original Message -----
From: monz
To: tuning@yahoogroups.com
Sent: 14 Aralık 2004 Salı 17:56
Subject: [tuning] Tonalsoft Musica software (was Re: Mahler tuning)

hi Ozan,

there's still plenty of time to save ... it won't be out
until this coming summer.

-monz

Dear Ozan,

My apologies for not replying sooner, but I have at least 3 other
microtonal projects going on at the same time and still precious
little time to devote to them. But the delay has been profitable for
me to think carefully, once again, before replying.

I see that Paul Erlich has, in the meantime, addressed some of the
things that I am covering here, but since I am taking these topics in
a logical order, you may then consider some of this as a review.

I. "Perfect" vs. "imperfect"
----------------------------

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> ...
> As to the terminology of analogous intervals, I do not agree with
you that a `perfect fifth` can be confounded to mean a `tempered
fifth` or an `apotome substitute` could be confounded to mean
`tempered apotome`, seeing as tempering indicates the partitioning
and distribution of a supernumerary interval throughout a scale.

The term "perfect," as applied to intervals, is one that originated
many centuries ago in the early stages of polyphony. In the
Pythagorean tuning the intervals of a unison, octave, fifth, and
fourth (1:1, 1:2, 2:3, 3:4, with the pitches sounding simultaneously)
were observed to be consonant, and the other intervals in the scale
were judged dissonant. With the passage of time thirds and sixths
began to be judged semi-consonant, particularly when some liberty was
taken with intonation ("temperament" on instruments of fixed pitch)
so as make these intervals closer to exact 5-prime ratios, and these
intervals were called "imperfect" consonances in order to distinguish
them from the "perfect" (3-limit) consonances.

Since consonant 3rds and 6ths each come in two sizes, they were
distinguished by the prefixes "major" and "minor." There were
intervals of a fourth and a fifth in the diatonic scale that were not
consonant, and those were prefixed with "augmented" and "diminished,"
respectively. In situations where it was necessary to emphasize that
one was referring specifically to consonant fifths or fourths, the
prefix "perfect" was used for these (and also for unisons and
octaves) to distinguish them from augmented and/or diminished
intervals. These terms have been used this way in theory books for
many centuries to refer to classes of intervals, and the
term "perfect" has been used for the 3-limit consonances, the unison,
octave, fifth, and fourth, regardless of whether or not they are
tempered.

So if you object to my calling a tempered fifth a perfect fifth, your
argument is with a centuries-old Western European tradition, and not
with Paul or me in particular. In our East-meets-West exchange of
ideas, I share your opinion that tempered intervals are in a certain
sense "imperfect," but I must caution you that if you aren't willing
to accept established musical terminology in the English-speaking
world, then you will run the risk of being misunderstood or, worse,
of not being taken seriously.

II. Notating consonances vs. dissonances
----------------------------------------

Consonant intervals are distinguished from dissonant intervals in
that the former are defined by frequency ratios consisting of small
numbers. (While the boundary between small and large is often a
matter of personal choice, that's not at issue in our present
discussion.) In order to identify approximations to consonances in a
particular EDO, we find the number of degrees in that division that
is closest in size to the exact ratio. Should we then decide to
substitute a subminor 3rd (6:7) for a minor third (5:6) in some
particular division in which both of these consonances are
represented (e.g., 22, 31, or 41-EDO), we would always be
substituting a smaller interval for a slightly larger one, because
consonances are defined (and perceived) independently of any scale
context.

The limma and apotome are defined with such large numbers that no one
(to my knowledge) defines these intervals as consonances. They are
invariably considered dissonant intervals that owe their existence to
the prior establishment of a diatonic scale formed by a chain of
fifths that, when reduced to a single octave, has adjacent tones
separated by major or minor 2nds. The limma is defined as the
interval of a minor 2nd (or diatonic semitone) and the apotome (or
chromatic semitone) as the difference between a major 2nd (or whole
tone) and a minor 2nd. These relationships hold regardless of
whether the fifths in the scale are just or tempered (either narrow
or wide). If they are wider than those in 12-ET, then the apotome
will be larger than the limma, but if they are narrower, then the
reverse will be true. The apotome has also been called an augmented
unison or augmented prime, and alteration of pitch by this amount is
indicated by a sharp or flat symbol.

Our point of disagreement is that you wish the limma and apotome to
be independently existing intervals that are capable of having
separate ranges of size narrow enough that they do not overlap. More
specifically, you wish that the allowable sizes for the limma and
apotome each be clustered around their exact (or Pythagorean) sizes.
The difficulty that we in the Western tradition have with this is
that this would require narrow-fifth tunings to be notated in a way
that makes little sense. If I may use 31-ET as an example, you would
require that a whole tone be subdivided as follows (using ^ and v to
symbolize a diesis up and down, respectively):

C C^ C#v C# C#^ Cx
Dbb Dbv Db Db^ Dv D

whereas we do it this way:

C C^ C# C#^ Cx
Dbb Dbv Db Dv D

Since a chain of fifths of 31-ET is very nearly the same thing as the
historical meantone temperament that was commonly in use in the West
until the end of the 18th century, what I have to say about meantone
will also generally apply to 31. The purpose of the meantone
temperament was to distribute the error of Pythagorean 3rds and 6ths
from 5-consonances such that no 5-limit consonance would exceed 1/4-
comma. By tempering out the 5-comma (80:81) we have, in essence,
made the *functional* Pythagorean major 3rd (i.e., the major 3rd
arrived at by a chain of four perfect 5ths) the same interval as the
*just* major 3rd (i.e., the interval closest in size to 4:5). Since
these two intervals are not distinguished in meantone temperament (or
31-ET, or most other narrow-fifth tunings), a 5-comma symbol would be
interpreted as representing a unison (i.e., zero cents) and would
have no use, since the comma vanishes.

Now a Pythagorean chain of fifths is notated thus:

... Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B
F# C# G# D# A# E# B# Fx Cx ...

but if we were to adopt your notation for subdividing the whole tone
(as I showed it above), then I expect that this chain of fifths would
then be notated this way:

... Bbv Fb^ Cb^ F# C# G# D# A# F C G D A E B
Gb Db Ab Eb Bb E#v B#v Fx^ Cx^ ...

This is completely contrary to the way sharps and flats have ever
been used (to the best of my knowledge) by anyone, anywhere! Major
thirds in the meantone temperament have been notated D-F#, A-C#, Bb-
D, Eb-G, etc. for centuries, and likewise perfect fifths as B-F# and
Bb-F. To propose that we change the notation for 5-limit (harmonic)
consonances in narrow-fifth tunings, just so that the size range of
the limma and apotome (i.e., dissonances) may conform to a
Pythagorean (melodic) scheme leads to some very bizarre results.

However, upon more closely examining the 31-ET table of pitches in
your Spectral Notation paper, I see that they would be arranged in a
chain of fifths this way:

... Bbv E B F# C# G# D# A# F C G D A E\ B\
Gb Db Ab Eb Bb E^ B^ F#^ C#^ ...

which is even more chaotic than what I expected. In your 31-ET table
the whole tone from C to D, F to G, and G to A is 5 degrees, but the
whole tone from D to E and from A to B is 6 degrees, while the
interval from A to E is not a perfect fifth! You are also
distinguishing pairs of pitches E and E\ and likewise B and B\ as if
they were separated by a 5-comma, whereas this is a division in which
the 5-comma vanishes. Is this what you really intended, or is this a
mistake?

I also have more questions about some of the divisions in your
Spectral Notation tables.

In the 24-ET table on pages 8-9 you are using comma-symbols for
alterations of 50 cents, whereas I understood that this amount would
require diesis symbols. Please explain.

In the 36-ET table on pages 9-10 you have the sharp and flat each as
4 degrees instead of 3, so that a 12-ET subset of 36-ET would be
notated differently than 12-ET itself. Please explain.

There are too many inconsistencies in your proposal, such that I
cannot seriously consider its adoption.

Ozan, you are a talented composer -- I downloaded four compositions
of yours and have listened to each of them several times, and the
most important thing I will say in this message is that I thoroughly
enjoyed them! Please keep on with your work -- I believe you have a
promising future ahead of you.

But I think that you need to get the theory behind your notation on a
more solid footing before you make any more proposals about a
notation that we might all consider adopting.

III. Substitution vs. functionality
-----------------------------------

This is actually a continuation of the previous topic, in which I
will attempt to explain how larger variations in the sizes of
dissonances (specifically the limma and apotome) in tempered-fifth
tunings might become acceptable to you. I think that you may be a
bit more prejudiced than you have been willing to admit, so I am
hoping to show you a more "international" viewpoint.

Your assertion that an apotome *must* be larger than a limma strikes
me as having a parallel in language -- as if an Italian had come to
me insisting that the English-speaking world is not only
mispronouncing the vowels -- long "e" as if it were "i", long "a" as
if it were "e", short "o" as if it were "a", etc. -- but also doing
it inconsistently (to say nothing of regional variations and foreign
accents). If speakers of English use the *Roman* alphabet, should
they not abide by the "proper" (i.e., Latin or Italian) pronunciation
of its vowels?

Should we all now speak English with an Italian accent? No, but
should we decide to live in Rome, it would be advisable to do as the
Romans do and learn Italian as best we can. And we would hope that
an emigrant from Rome would do likewise! But in any case, once we
have taken into account the background of the speaker, we can make
mental allowances for differences in pronunciation that enable us to
understand what is being said, given the fact that each sound is
heard in a context of words and sentences, not in isolation.

Ozan, in the course of your composing, should you decide to journey
to any of the cities of meantone-land, such as 19-ET or 31-ET (where
one hears somewhat different dialects of meantone uttered), I am
hoping that you will recognize the limma and apotome, although
different in size to what you are accustomed, also occur in a
diatonic scale context where they *function* with *exactly* the same
relationships as in Pythagorean-land (where one may visit 41-ET and
53-ET). You may find it peculiar that in meantone-land the
Pythagorean major third (arrived at through a chain of 4 fifths) has
the same sound as a just major third (best 4:5), but this is no
different than a native of China speaking English without making a
distinction between "l" and "r": while we find it a bit peculiar, we
still understand what is being said.

A similar thing may be said for super-Pythagorean-land, where in the
city of 22-ET we find that there is no distinction between a
Pythagorean major 3rd and a supermajor 3rd (7:9). And although the
apotome is three times as large as the limma, we are not confused or
disoriented when we hear these intervals in a 22-ET diatonic scale,
once we learn to allow for a 22-ET "accent".

And we should not insist that in these foreign lands the inhabitants
are using *substitute* limmas or apotomes in place of the real ones,
for they are simply using the *functional* equivalents of those
intervals as they occur in a diatonic scale. While it is true they
are melodically different from the older Pythagorean intervals, that
is simply how Pythagorean intervals sound if one journeys to a
foreign land.

For those who have not had occasion to travel outside Pythagorean-
land (of which 12-ET is a border city), I can understand this
reaction that you describe (which I would call a form of culture-
shock):

> Nevertheless, Trans-temperamental notation is a bit confusing for
the regular musicians in my part of the world, who feel more
comfortable obeying the rules of a reference scale in order to
comprehend which tone deviates how much from what pitch. Believe me,
they would be greatly perplexed, even indignant, by the idea that an
accidental is capable of being interpreted so liberally as you
suggest. I myself think it is impermissable to contort a musical
score outside its temperamental context unless it is part of a circus
act.

Ozan, whether or not you accept the "international" viewpoint that I
have presented, you can take consolation in the fact that, should you
decide to confine your travels to those places where the fifth varies
no more than 700 cents (12-ET) to 703.45 cents (29-ET), then your
limma will range only from 100 to 82.7 cents, while your apotome will
be 100 to 124.1 cents. This should be the land in which the Maqam
will most often dwell.

Is this something that you might be able to accept?

IV. Functional vs. interpretive pitch flexibility
-------------------------------------------------

You seem to have the impression that I don't accept your idea of
flexible pitch through pitch-clusters, when nothing could be further
from the truth. My concept of pitch flexibility for Sagittal
accidentals is twofold:

1) The *tuning* (or more specifically, the size of the fifth)
determines the "functional" size of the interval represented by a
particular symbol, e.g., the interval that functions as an apotome in
31-ET is 2deg, or ~77.42c, while the one that functions as a 5-comma
vanishes to zero. I would therefore encourage you to use the
term "functional" apotome instead of "substitute" apotome as a label
for it in tunings outside the Pythagorean family, where it departs
significantly in size from 2048:2187.

What is important to remember is that *functional* flexibility of
Sagittal accidentals operates only *across* tunings, not *within*
them. But there is yet another type of pitch flexibility of the type
you are advocating.

2) Once the functional sizes of the accidentals is established, the
*number* of accidentals in the notation then determines the
resolution of pitch, which in turn determines the maximum amount of
*interpretive* (or discretionary) pitch flexibility (or pitch
clustering) that will be meaningful for the tuning. In 12-ET the
maximum acceptable interpretive flexibility is usually (but not
always!) much less than +-50 cents (at which extremes one is
generally judged to be "out of tune"), whereas in athenian-level
(medium-precision) JI the maximum flexibility would encompass the
entire range (of ~5.4 cents) between symbol boundaries.

For Maqam music you are advocating a flexibility on the order of +-15
cents. This is in line with the pitch resolution of the 41 division,
and any set of symbols for 41-ET (Sagittal or otherwise) should serve
your purpose. There is nothing in the Sagittal philosophy that is
contrary to what you wish to accomplish, and the reason that I said
that it would be necessary to specify the meanings of the symbols
would be mostly for the benefit of someone outside the Maqam
tradition who wished to study Maqam music notated with Sagittal
symbols. If minute directions are more confusing than helpful for
Maqam musicians, then they should be eliminated.

> I can therefore accept the Sagittal Notation as a means to
complement my shortcomings in trans-temperamental music, but I hope
you understand that I cannot throw away the classical terminology of
the sharp and flat in favor of Sagittal symbols.

What is your opinion of the mixed-symbol version of Sagittal, where
we retain the existing sharp and flat symbols and combine them with
single-shaft Sagittal accidentals? A 41-tone octave would require
only two (up/down) pairs of Sagittal accidentals.

> Where did I come up with 53 Sagittal symbols? Am I mistaken when I
count the complete Sagittal symbol superset from natural to apotome
given on page 7, figure 3? I do not think it wrong to describe this
superset as 53 symbols per half-tone.

Ah, so! Perhaps it is not "wrong" to call this 53 symbols per half-
tone, but it would be misleading, since the *complete* symbol
superset as it appears in that figure is, as best as I can remember,
never actually used to notate any particular tuning. At present only
subsets of this superset are used for actual tunings, sometimes with
accent marks (for more complicated tunings). Some symbols are more
appropriate for certain tunings than for others, and they have been
selected on that basis. Your posed the following question:

> And what would happen if I were to take the liberty with the
Sagittal symbols? Without any descriptions, the schismatic occurances
could amount to as large an intervallic difference as dieses.

As a matter of fact, a 5-schisma accent mark would equate to a single
degree of 22-ET (55-cents) if used in that division, so your
hypothetical case is not all that far-fetched! But the point that I
make is that we have selected symbols for each division that are
*meaningful* and *useful* for that division.

In the case of 22-ET we use the 5-comma symbol for a single degree.
You will object that 55 cents is more like a diesis than a comma, to
which I reply that, while a single degree may function either as a
diesis or a comma in 22, the comma function is the one that will be
more frequently desired and used in this division, and so the best
approximation of a just major triad (4:5:6) will be notated as C E\ G
(with a 5-comma symbol). Had I used a diesis symbol instead, C Ev G,
you might get the impression that the lower interval of the chord
(7deg22) would be close to 9:11 (instead of 4:5), but that is not the
case. (Actually 6deg22 is closest to 9:11, while 22:27 is almost
exactly 6.5 degrees.)

Sagittal accidentals for various divisions are thus selected so that
they will be most meaningful in notating the closest approximations
to consonant intervals (which have simple ratios), as opposed to
notating the closest approximations to dissonances such as limmas,
apotomes, dieses, and commas (which result from more complex, or
functional, tonal relationships).

V. Symbol cosmetics
--------------------

> Also, if you don't mind my saying so, some of the Sagittal
accidentals could be improved.

Leaving aside everything I have said previously (and also the issue
of color), let's take a look at the symbols on page 2 of your Spectal
Notation paper. I immediately find several problems with these:

1) I previously said that the comma symbols would not be accepted in
the West, since these have been understood for centuries in the West
(and currently by Arabic musicians) to represent dieses or half-
apotomes.

2) The fractional flat symbols for diesis, limma, sesqui-tone, and
minor whole tone are basically embellished flat symbols of the sort
that Ivan Wyschnegradsky proposed for 72-ET (which we rejected as too
cumbersome). To be more specific, the two problems I have with these
are that:

a) They are too much alike in appearance, and the embellishments
are rather small compared to the overall size of the symbol, making
them relatively difficult to distinguish from one another;

b) The eye must be focused (and the attention centered) on the
embellishment in order to read it. Inasmuch as each embellishment is
inappropriately centered approximately 3 staff positions above the
notehead that is being altered, this requires one to glance first at
the main part of the symbol (to identify the staff position), then
away from the desired staff position to read the embellishment (to
identify the amount of pitch alteration). It would be much better if
all of the information to be conveyed by the symbol could be taken in
at a single glance, with the most important detail of the symbol
being centered on the same line or space as the notehead being
modified (something we have scrupulously observed in designing the
Sagittal symbols).

Ozan, you have caused me to think through many of these issues from
more angles and far more rigorously than I had ever expected. This
reply has now gone about 7 pages, and if I did not stop soon, you
might never see it. If there is anything I have left unanswered or
have not answered to your satisfaction, please let me know.

My very best wishes,

--George

Dear George,

Let me express my profound admiration for the patience you have shown with my ignorance. Please know that I have benefitted immensely from our mutual discourse. I realize that I have unknowingly practiced sciolism on the matter of music theory. I shall not contradict you further without any foundations and I hope you will forgive my blunders that arise from my unstoppable curiosity.

So allow me try to correct my position with renewed vigor.

1. The terminology of the perfect and imperfect is pretty much clear to me now. I propound no more arguments in this direction.

2. The issue of consonance and dissonance is elucidated in minute detail. I acknowledge the practice of the Western world.

3. Our East-meets-West-meets-East dialogue shows that the established precepts of Western music theory are still elusive to musicians like myself at a first glance just the same way Eastern music theory is elusive to musicians educated solely by Western standards.

4. Your explanations on notating consonances and dissonances clears up the matter of the limma, apotome and comma. I concur entirely, save for the fact that I have come to perceive Western tonal practice to execute the apotome as the diatonic semitone and the limma as the chromatic semitone.

5. I understand that the sharp or flat is expressed by the augmented prime 7 fifths up minus 4 octaves no matter how wide or narrow the generator interval is. Thus, I abondon my previous insistance on this matter. Mine was an effort to help choir singers and orchestra members in order that they could play the right pitches in reference to the tonal scale they were educated with. For the sake of mathematical correctness, I retreat from this erroneous approach.

6. I accept every scenario where the limma may be larger than the apotome depending on the fifth's narrowness starting from 700 cents. I accept the fact that they may overlap.

7. The pitch-cluster phenomenon led me to conceive the whole-tone division that I have given previously. No longer do I imagine the apotome and limma to cluster around their pythagorean sizes.

8. I understand that meantone eliminates the distinction between 5/4 and 81/64 by tempering out the syntonic comma. Thus, I follow you that a 5-comma symbol has no place in such a temperament. But, does this mean that the Zarlino diatonic scale composed of 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 can be notated as a natural C Major scale on the staff?

9. In my division of the whole tone, I don't understand how you notated the chain of fifths in this way:

... Bbv Fb^ Cb^ F# C# G# D# A# F C G D A E B
Gb Db Ab Eb Bb E#v B#v Fx^ Cx^ ...

According to the displacement values, F# should come after 7 fifths up, followed by the rest of the sharps. Perhaps you are referring to the narrower fifth in 31-TET? Please disregard the distasteful notation that arises from my blunder in that respect.

10. As I explained previously, I had in mind the habits of the regular musicians in the Spectral Notation paper. Please disregard any and all bizzarre/chaotic consequences of my solecism.

11. For 24-TET I had in mind the standard of applying the diesis symbols recognized in the West and the Arab world. I will have to discard many pages and start over with the directions I received and hopefully will continue to receive from you.

12. With 36-TET I entertained the idea of punctuating the apotome difference as outlined in the whole-tone division. This will go down the chute as well.

13. All of the inconsistencies you were so kind to point out will have to be dealt with severely. However, it should be clear that I never assumed that this portion of the proposal could be taken seriously. The core of the proposal still remains, and that is to identify finer divisions of the octave by way of colors which you have neglected to address in your reply.

14. I am grateful that you had a chance to listen to my compositions. They are my humble works which I can feel proud of if not anything else. There are some more, but I dare not upload any until they are made more presentable.

15. My lack of theoretical knowledge, on the other hand, should and does not preclude the possibility that a `stupid idea` such as indicating subtle nuances of pitch by way of color is worthy of some attention. I understand that you did not have much to say about it, but perhaps there is some chance that a truly professional theory can be built on this idea? Would you consider my position seriously then?

16. Yes, I am willing to admit that I have been more prejudiced than I presumed, provided that you also are ready to admit that you have been overlooking the possibility that Sagittal Notation might not be the best microtonal alternative so far. I understand how much work and wisdom went into it, congratulate your efforts and celebrate your accumulations. Nevertheless I still feel uncomfortable with the semiology although I now completely agree with the theory behind it. If the matter boils down to which symbols should be used and where, then I believe democratical contributions from the entire microtonal world ought to be received.

17. I, for one, have the following issue: The entire world of microtonality should not be made to require notational subsets filled with symbols out of a superset whose certain symbols are hardly distinguishable from the others. This is very confusing to me, and I wish to believe that further simplification is possible as well as plausible. This is where you will perhaps, hopefully, begin to consider the spectral idea if you deem that it is not totally wrecked.

18. Yes, I agree with the international viewpoint and `tonal city-scapes` illustrations. I now broaden my scope to include all temperaments regardless of how the apotome and limma overlap or how the comma differences vanish. I now imagine situations where the Maqam may dwell in 22-TET or 19-TET lands. I am grateful to you for enlarging my perspective so artfully.

19. Ok, I agree that all pitches should have functional and interpretive flexibility. The functional flexibility is trans-temperamental, while the latter is dependent on pitch-cluster variations of tone. Wonderful. There is nothing else we disagree with on the theory of tuning by fifths. I adopt your jargon of `functional`, instead of the biased `substitute`.

20. I wish you explain to me though, what makes an `acceptable interpretive flexibility`? Why should I not have the liberty of playing the 12-ET sharps and flats within a zone 50 cents wide whose center is 100 cents? Perhaps this matter could be elucidated further in the Sagittal paper?

21. You should know that I am seriously considering personally adopting the Sagittal system. How I will be able to explain what I write is another story of course. I may have reached an impasse even with the mixed version. I still advocate some other solution that can be embodied by the `cycle of fifths and co` theory seen in Sagittal Notation. Alternative symbols based on quarter-tone accidentals would be a very welcome addition for those of us in the Maqam world.

22. I will be frank with you considering your question with the mixed version: I think that the sharps and flats accompanied by the apotome complement Sagittal symbols look pretty makeshift. I regress from employing them on the grounds that they are more confusing than intelligible. Hereabouts we employ arabic numerals to signify the step number of the sharp and flat based on 53-TET. I would rather revert to numbers if you will forgive my bluntness.

23. If this matter could be resolved without causing a mayhem, I am eager to imagine a hypothetical Cairo Music Congress on its the 75th anniversary. It was last organized in 1932 and Rauf Yekta was the Turkish participant back then. It would be a festive occasion, a grand revolution in microtonal music, if we could somehow arrive at a consensus and conceive of uniting our `microtonal forces`.

24. I am prepared to ditch the accidentals I have been using so far in favor of others. Thank you for pointing out their weaknesses. I am more inclined now then ever of adopting the quarter-tone notation of the Arab world. The matter remains as to their elaboration with colors perhaps, or other means mayhap resembling the Reinhard Notation.

25. Let me ask a direct question at this point. Are you completely comfortable with all the symbols contained in Sagittal Notation?

Dear George, it has been a most beneficial discussion and I hope we can prolong our fulfillling interaction throughout our remaining lives.

Cordially,
Ozan

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George,
>
> Let me express my profound admiration for the patience you have
shown with my ignorance. Please know that I have benefitted immensely
from our mutual discourse. ...

I am likewise ignorant of many things about the music outside (and in
some cases inside!) my own culture, so your observations are also of
great benefit to me (and doubtless to others following this thread).

>
> So allow me try to correct my position with renewed vigor.
> ...
> 4. Your explanations on notating consonances and dissonances clears
up the matter of the limma, apotome and comma. I concur entirely,
save for the fact that I have come to perceive Western tonal practice
to execute the apotome as the diatonic semitone and the limma as the
chromatic semitone.

Perhaps it will take some time in meantone-land for you to hear these
as we do, so get ready for some culture shock. Have you heard Aaron
Johnson's realization of John Bull's Ut Re Mi in 19-ET? The ogg
version is preferred (I remember AKJ mentioning that the mp3 version
is incomplete):

http://www.akjmusic.com/audio/bull_ut_re.ogg
http://www.akjmusic.com/audio/bull_ut_re.mp3

This may sound out of tune to you if you've never heard 19-ET before,
but when I first heard this I thought nothing of the sort -- I found
it absolutely delightful! Yes, the limmas are quite large, but this
tends, along with the relatively small error of the 3rds and 6ths
(compared to 12-ET), to give the piece a very relaxed mood, as if to
say that there's no hurry to resolve a leading tone to the tonic
chord.

> 7. The pitch-cluster phenomenon led me to conceive the whole-tone
division that I have given previously. No longer do I imagine the
apotome and limma to cluster around their pythagorean sizes.

But I see no reason why you cannot continue to do so for Maqam music,
or, for that matter, for anything in just intonation. It's only when
you travel to distant tuning-lands that you must try to perceive
things differently.

> 8. I understand that meantone eliminates the distinction between
5/4 and 81/64 by tempering out the syntonic comma. Thus, I follow you
that a 5-comma symbol has no place in such a temperament. But, does
this mean that the Zarlino diatonic scale composed of 1/1 9/8 5/4 4/3
3/2 5/3 15/8 2/1 can be notated as a natural C Major scale on the
staff?

Yes, but only on the condition that there would also be an
explanation about this (and also that there are no additional tones
in the example that would differ from any of the previous ones by a 5-
comma). Otherwise, we in the West we would assume that a C Major
scale would be played, by default, in equal temperament.

But in any tuning in which there are pairs of tones separated by a 5-
comma, I would hope to see 5-comma symbols used for the ratios of 5
and that all of the natural tones (without comma-accidentals) be in a
single Pythagorean chain of fifths. To do otherwise would invite
confusion, in my opinion.

> 9. In my division of the whole tone, I don't understand how you
notated the chain of fifths in this way:
>
> ... Bbv Fb^ Cb^ F# C# G# D# A# F C G D A E B
> Gb Db Ab Eb Bb E#v B#v Fx^ Cx^ ...

I had looked only at your accidentals between C and D and had assumed
that all of the naturals would be 5 degrees apart, with the
intervening accidentals appearing in the same order, which results in
the above sequence. In the course of working this out I discovered
that my assumption was not correct.

> According to the displacement values, F# should come after 7 fifths
up, followed by the rest of the sharps. Perhaps you are referring to
the narrower fifth in 31-TET?

Yes, I was referring to your table for 31-ET.

> 13. All of the inconsistencies you were so kind to point out will
have to be dealt with severely. However, it should be clear that I
never assumed that this portion of the proposal could be taken
seriously. The core of the proposal still remains, and that is to
identify finer divisions of the octave by way of colors which you
have neglected to address in your reply.

Yes, I said I had to stop somewhere. There were other issues to
address before I could take up the question of color.

> 14. I am grateful that you had a chance to listen to my
compositions. They are my humble works which I can feel proud of if
not anything else. There are some more, but I dare not upload any
until they are made more presentable.

What I have heard so far is very impressive, and I will look forward
to hearing more! We can talk theory day in and day out, but what is
the point if it does not enable us (or at least help others) to write
any music?

> 15. My lack of theoretical knowledge, on the other hand, should and
does not preclude the possibility that a `stupid idea` such as
indicating subtle nuances of pitch by way of color is worthy of some
attention. I understand that you did not have much to say about it,
but perhaps there is some chance that a truly professional theory can
be built on this idea? Would you consider my position seriously then?

There are several separate issues you are facing when you consider
color, but I think they fall into two categories:

1) Is it practical and convenient to implement?

Here you must consider the requirements of producing printed music
with full-color reproduction. It is more expensive than monochrome
(black) printing, and a color photocopier is required to copy parts.
I, for one, do not have quick access to a color photocopier, and if I
wished to make a copy of one of my compositions, I would be out in
the cold!

And if a musician is color-blind, what then? I don't quite recall
whether you mentioned some other sort of markings that could be used
instead -- but if that works, then why not use that alternative for
everyone?

But you don't have to go by my opinion alone. IS THERE ANYONE ELSE
OUT THERE WHO HAS ANYTHING TO SAY ABOUT THIS, ONE WAY OR ANOTHER?

2) Is it sufficiently meaningful and capable of being processed
easily when read?

I'm not color-blind, but when I look at the tables (Spectral Set of
Accidental Pairs) beginning on page 5 of your Spectral Notation
paper, I have trouble *quickly* distinguishing some of the darker
colors on my computer monitor (brown from violet, indigo from black,
and brown from black): I have to stare at one of these symbols for
too long to be sure of its color (and then I begin to wonder if color
fatigue has begun to set in). I don't think that this would be
improved with colors printed on paper; if anything, it would probably
be worse, because the lighting might not be as good. Perhaps this
could be remedied by modifying your choice of colors (magenta and
cyan might be good choices).

You have a color-size correlation, but I'm not sure if there is much
else to help me remember which color stands for what (except black),
since I don't see any useful color-to-prime correlation. Nor do I
find any help in the observation that black (5-comma) plus indigo (7-
comma) equals yellow-green (35-diesis) -- the equivalent of /| +|)
= /|) in Sagittal, where flag arithmetic is meaningful.

If you have not already done so, I would encourage you to read George
A. Miller's "The Magical Number Seven," which discusses some of the
dynamics of human perception and information processing:

http://www.well.com/user/smalin/miller.html

A portion of this deals with color recognition, and I think that
having 9 different colors (all of which must be dark enough to show
up well on a white background) is taking it pretty close to the
limit. Some of this will also not be relevant, since the experiments
involving "absolute judgments of stimuli" allowed the subjects an
unlimited amount of time to make their judgments, whereas in reading
musical notation for performance, time is critical.

The main problem I have with your Spectral Notation proposal is that
the way you are using them, the specific colors do not, in
themselves, have specific meanings, nor is it meaningful to add
colors together (such as red + blue = magenta, green + blue = cyan,
or green + black = dark green).

I hope that you have not failed to notice the advantages of Sagittal
flag arithmetic and that the symbols often give an indication of
prime number content, e.g., that /| + /| = //| (two 5-commas) and /|
+ |) = /|) (35-diesis as 5 times 7). Even the higher primes 17, 19,
and 23 have distinctive flags, ~|, )|, and |~, respectively, that are
meaningful even when they appear in combinations such as ~|(, )/|, )
|), /|~, and |~).

While there are a lot of symbols, there are very few types of symbol
components to remember:

4 types of flags: barb, arc, scroll, boathook
2 possible positions for each flag: left, right
2 flags maximum per symbol
4 possible numbers of shafts in pure version: |, ||, |||, X
5 possible mixed-version associations: bb, b, (nothing), #, x
2 possible directions of alteration: up, down

With these parameters it would be possible to have 72 different
single-shaft symbols, but we have limited our superset to less than
half of this (31), which is not very much more than there are letters
in the alphabet. We expect that less than half that number (12) will
be generally used for just intonation and even fewer than that (8 or
less) for the most popular temperaments. Please don't get hung up on
the multitude of symbols in the superset; we have them available to
satisfy the very few who think that they might want to do things that
most of us would never seriously consider.

The physical size of a Sagittal symbol also gives you a general idea
of how large the alteration is.

By comparison, with Spectral Notation you have these parameters:

2 symbol categories: sharps, flats
7 symbol variations per category
9 possible colors for each symbol

There are fewer parameters than with Sagittal, but there are more
values for each parameter. Taking all of this into account, I'm not
convinced that Spectral would be any simpler to comprehend or quicker
to read than Sagittal.

> 16. Yes, I am willing to admit that I have been more prejudiced
than I presumed, provided that you also are ready to admit that you
have been overlooking the possibility that Sagittal Notation might
not be the best microtonal alternative so far. I understand how much
work and wisdom went into it, congratulate your efforts and celebrate
your accumulations. Nevertheless I still feel uncomfortable with the
semiology although I now completely agree with the theory behind it.
If the matter boils down to which symbols should be used and where,
then I believe democratical contributions from the entire microtonal
world ought to be received.

Is anyone holding any elections soon? ;-)

Notation is a free product in a free market, and composers will vote
with their pens (whether ink or virtual). We present our ideas and
proposals, and we then see how that market will react. We have
already made cosmetic improvements to some of the Sagittal symbols in
response to criticism. Likewise, we are open to suggestions as to
the particular selection of symbols that should be included in the
set for various tunings; for some tunings the selection is fairly
obvious, but for others it helps to have some hands-on (and
especially ears-on) experience to make the best choice.

> 17. I, for one, have the following issue: The entire world of
microtonality should not be made to require notational subsets filled
with symbols out of a superset whose certain symbols are hardly
distinguishable from the others. This is very confusing to me, and I
wish to believe that further simplification is possible as well as
plausible. This is where you will perhaps, hopefully, begin to
consider the spectral idea if you deem that it is not totally wrecked.

As I pointed out above,.there are a couple of very important features
that Sagittal offers (flag-prime correlation and meaningful flag
arithmetic) that you don't have with Spectral. The Sagittal symbols
seem cryptic only until you learn the meanings of the components, and
you need only a small fraction of the superset for most applications
(in which you can gladly ignore the other symbols).

> 18. Yes, I agree with the international viewpoint and `tonal city-
scapes` illustrations. I now broaden my scope to include all
temperaments regardless of how the apotome and limma overlap or how
the comma differences vanish. I now imagine situations where the
Maqam may dwell in 22-TET or 19-TET lands.

But I, for one, don't imagine that this would be very likely. As
long as you're able to express all of your intervals as rational
ratios, you shouldn't be bothered by the melodic quirks of these
tempered tunings.

> 19. Ok, I agree that all pitches should have functional and
interpretive flexibility. The functional flexibility is trans-
temperamental, while the latter is dependent on pitch-cluster
variations of tone. Wonderful. There is nothing else we disagree with
on the theory of tuning by fifths. I adopt your jargon of
`functional`, instead of the biased `substitute`.

Thank you.

> 20. I wish you explain to me though, what makes an `acceptable
interpretive flexibility`? Why should I not have the liberty of
playing the 12-ET sharps and flats within a zone 50 cents wide whose
center is 100 cents? Perhaps this matter could be elucidated further
in the Sagittal paper?

It has nothing to do with notation, and everything to do with what is
acceptable in the context of a particular style or tradition.
Symphony musicians are generally not allowed pitch flexibility of
more than a comma or so, +-21 cents, if that much (excluding
_portamento_), while jazz and "blues" musicians routinely bend
pitches as much as they like.

> 21. You should know that I am seriously considering personally
adopting the Sagittal system. How I will be able to explain what I
write is another story of course. I may have reached an impasse even
with the mixed version. I still advocate some other solution that can
be embodied by the `cycle of fifths and co` theory seen in Sagittal
Notation. Alternative symbols based on quarter-tone accidentals would
be a very welcome addition for those of us in the Maqam world.

Yes, you will need to explain the nature of that impasse with the
mixed version. Is it that you object to using two accidentals to
modify a single note? In your next point you said:

> 22. I will be frank with you considering your question with the
mixed version: I think that the sharps and flats accompanied by the
apotome complement Sagittal symbols look pretty makeshift. I regress
from employing them on the grounds that they are more confusing than
intelligible. Hereabouts we employ arabic numerals to signify the
step number of the sharp and flat based on 53-TET. I would rather
revert to numbers if you will forgive my bluntness.

From the very start Dave Keenan and I have had a polite (yet firm and
ongoing) disagreement about whether the pure or mixed-symbol of
Sagittal would be more readily accepted. He believes that discarding
conventional sharp and flat symbols is too radical a change, whereas
I believe that having two symbols to modify a single note is
inefficient. In agreeing to disagree, the only compromise we could
arrive at was to offer the notation in two forms, neither of which
satisfies both of our preferences.

Does this describe the problem you have with Sagittal (as it would be
used for the 41-division), or is there something else?

> 23. If this matter could be resolved without causing a mayhem, I am
eager to imagine a hypothetical Cairo Music Congress on its the 75th
anniversary. It was last organized in 1932 and Rauf Yekta was the
Turkish participant back then. It would be a festive occasion, a
grand revolution in microtonal music, if we could somehow arrive at a
consensus and conceive of uniting our `microtonal forces`.

That sounds like a worthy goal. I'm not entertaining the idea of
modifying or abandoning Sagittal as it already exists, but I do
realize that (at least in its pure form) it may be too radical a
change to be made in a single step. We came up with the Sagittal-
Wilson symbol set (in a mixed-symbol implementation) to have
something that would not conflict with either the Sagittal or Sims
notation, but could serve as a transitional step to mixed-symbol
Sagittal.

It it possible that you're looking for some sort of 41-tone hybrid
notation in which the symbols possess characteristics of both the
symbols in your paper (a couple of your fractional sharp symbols are
rather nice, by the way!) and of pure Sagittal symbols (quartertone
arrows would not conflict with Maqam usage)? It would be confusing
for a player to be familiar with two different notations if the same
symbol is used for two different things, but I think a hybrid
notation having no conflicts with either Maqam, Arabic, or Sagittal
might be possible, if you're interested in pursuing that. While I
would view something like this as a possible transition to pure
Sagittal, you would likely view it as an end in itself.

> 24. I am prepared to ditch the accidentals I have been using so far
in favor of others. Thank you for pointing out their weaknesses. I am
more inclined now then ever of adopting the quarter-tone notation of
the Arab world.

You have said previously that other Turkish musicians would not
accept that, but if you can demonstrate that they are at odds with
the rest of the world (both Arabic and Tartini) and offer a good
theoretical foundation to support your view, then perhaps you have a
chance of convincing them. Perhaps a small chance?

> 25. Let me ask a direct question at this point. Are you completely
comfortable with all the symbols contained in Sagittal Notation?

No, only about half of them at present, because I haven't had much
occasion to use the others. This is not to say that I'm not
*satisfied* with the appearance of the others (I do know all of them
at sight), only that I'm not *completely* comfortable with actually
using so many symbols. In my opinion, it would take an
extraordinarily complicated application to warrant their use.

Best,

--George

Dear George,

Meantone is by no means new to me. I have been listening for some years to Baroque music performed by those historically conscientous musicians who are expressive in the flavor of contracted apotomes the size of pythagorean limmas. Neither am I baffled or shocked by the almost sweet sounds of 19-ET that you gave the link for, except for one instance of awkward modulation at some point in the beginning of the piece.

I forsook all arguments in favor of adopting the practice of the cycle of `perfect fifths`, remember? I have every intention of applying this concept to Maqam Music now that we have resolved that issue. Trans-temperamental expression of the sharps and flats is no longer incompatible with my current view of tones and leading-tones.

But I do not understand how you came to the conclusion that the colors of Spectral Notation do not have specific meanings. The energy level of the color itself gives an idea as to the size of the chosen interval. For example, It is not impossible to attribute a diesis sharp the color red to signify that the deviation ought to be understood as a small-diesis with the nominal (Sagittal) ratio of 45:44 as opposed to the widely acknowledged 128:125 expressed with black (which is not comprised by Sagittal Notation by the way).

Furthermore the addition of two colors out of the whole sequence cannot possibly result in the sum total of two intervals as you expect (whose ratios are actually multiplied, not added). It is very well known that the addition of two colors gives the mean, not the product. Thus, red + green = yellow. It is not possible to directly correlate interval arithmetic with the spectral distribution of colors. However, it is feasible to calculate mentally that a nominal black syntonic comma sharp + an indigo septimal comma sharp would equal a septimal diesis 36:35 which falls to the category of a medium-diesis according to Dave Keenan (look up comma in `Monzopedia`), and hence to the category of the middle sequence of colors in the spectrum.

After our most valuable discussion, I have decided to attribute four sharps and flats to the following categories:

Square of lowerbound
.. 2-exponent
...... 3-exponent
............... Lowerbound (cents)
................................ Size range name
-------------------------------------------------

COMMA

.. [ 0 0 > ....... 0 ........... schismina
.. [-84 53 > ..... 1.807522933 . schisma
.. [ 317 -200 > .. 4.499913461 . kleisma
.. [-19 12 > .... 11.73000519 .. comma
.. [ 27 -17 > ... 33.38249264 ..

DIESIS

.. [ 27 -17 > ... 33.38249264 .. small-diesis
.. [ 8 -5 > ..... 45.11249784 .. (medium-)diesis
.. [-11 7 > ..... 56.84250303 .. large-diesis
.. [-30 19 > .... 68.57250822 ..

LIMMA

.. [-30 19 > .... 68.57250822 .. small-semitone
.. [-49 31 > .... 80.30251341 .. limma
.. [-3 2 > ..... 101.9550009 ...

APOTOME

.. [-3 2 > ..... 101.9550009 ... large-semitone
.. [ 62 -39 > .. 111.8774831 ... apotome
.. [-106 67 > .. 115.492529

I also revised the sharps and flats so that the Tartini-Couper quarter-tone symbols are untouched while I translated the Arabic quartertone (fraction) flat to the limma flat as used in Turkish Maqam Music and re-interpreted my previous selection of the quarter-tone sharp as the limma sharp. In black, these accidentals would correspond to these ratios respectively:

Comma: 81:80
Diesis: 128:125
Limma: 256:243
Apotome: 2187:2048

This is a compromise for both the Arabs and the Turks in as fair a deal as I can make.

When colors are applied, the wavelenght would give an idea as to which size is to be inferred. I will consider again which colors are best to utilize, and what other method is most agreeable when color-impaired musicians are concerned. Otherwise, I do not think that the Spectral Notation would be too much of a burden to the printing industry.

But it is not fair that you compare the flag arithmetic of Sagittal with the sharps and flats I consigned to use for Spectral Notation. It is not I who designed the sharp and flat symbology in the first place! For all I know, you could still preserve the Sagittal flag arithmetic and employ colors as I attempted above to arrive at fewer categories of paremeters, which I assume would be much less confusing to the general reader.

My objections concerning the Sagittal system can be summarized in this fashion:

1. The superset on page 7 needs some cosmetic-aesthetical touches.
2. Employing colors for voluminous systems could be helpful. The superset would then be reduced to a handful of symbols.
3. A mixed version that employs the pure Sagittal symbols next to the revisions I suggested above would be very convenient for everyone. If the whole thing could be so designed that Maqam Music could be expressed with the sharps and flats we all are accustomed to next to their Sagittal Alternatives, such a hybrid system would be a relief.

It is possible that I might make a ferocious argument here in favor of the revision I mentioned in Maqam Music notation, provided that it is equally possible for you to consider some of the objections I outlined above.

Happy New Year!

Cordially,
Ozan

----- Original Message -----
From: George D. Secor
To: tuning@yahoogroups.com
Sent: 22 Aralık 2004 Çarşamba 20:51
Subject: [tuning] Re: Sagittal Notation

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George,
>
> Let me express my profound admiration for the patience you have
shown with my ignorance. Please know that I have benefitted immensely
from our mutual discourse. ...

I am likewise ignorant of many things about the music outside (and in
some cases inside!) my own culture, so your observations are also of
great benefit to me (and doubtless to others following this thread).

>
> So allow me try to correct my position with renewed vigor.
> ...
> 4. Your explanations on notating consonances and dissonances clears
up the matter of the limma, apotome and comma. I concur entirely,
save for the fact that I have come to perceive Western tonal practice
to execute the apotome as the diatonic semitone and the limma as the
chromatic semitone.

Perhaps it will take some time in meantone-land for you to hear these
as we do, so get ready for some culture shock. Have you heard Aaron
Johnson's realization of John Bull's Ut Re Mi in 19-ET? The ogg
version is preferred (I remember AKJ mentioning that the mp3 version
is incomplete):

http://www.akjmusic.com/audio/bull_ut_re.ogg
http://www.akjmusic.com/audio/bull_ut_re.mp3

This may sound out of tune to you if you've never heard 19-ET before,
but when I first heard this I thought nothing of the sort -- I found
it absolutely delightful! Yes, the limmas are quite large, but this
tends, along with the relatively small error of the 3rds and 6ths
(compared to 12-ET), to give the piece a very relaxed mood, as if to
say that there's no hurry to resolve a leading tone to the tonic
chord.

> 7. The pitch-cluster phenomenon led me to conceive the whole-tone
division that I have given previously. No longer do I imagine the
apotome and limma to cluster around their pythagorean sizes.

But I see no reason why you cannot continue to do so for Maqam music,
or, for that matter, for anything in just intonation. It's only when
you travel to distant tuning-lands that you must try to perceive
things differently.

> 8. I understand that meantone eliminates the distinction between
5/4 and 81/64 by tempering out the syntonic comma. Thus, I follow you
that a 5-comma symbol has no place in such a temperament. But, does
this mean that the Zarlino diatonic scale composed of 1/1 9/8 5/4 4/3
3/2 5/3 15/8 2/1 can be notated as a natural C Major scale on the
staff?

Yes, but only on the condition that there would also be an
explanation about this (and also that there are no additional tones
in the example that would differ from any of the previous ones by a 5-
comma). Otherwise, we in the West we would assume that a C Major
scale would be played, by default, in equal temperament.

But in any tuning in which there are pairs of tones separated by a 5-
comma, I would hope to see 5-comma symbols used for the ratios of 5
and that all of the natural tones (without comma-accidentals) be in a
single Pythagorean chain of fifths. To do otherwise would invite
confusion, in my opinion.

> 9. In my division of the whole tone, I don't understand how you
notated the chain of fifths in this way:
>
> ... Bbv Fb^ Cb^ F# C# G# D# A# F C G D A E B
> Gb Db Ab Eb Bb E#v B#v Fx^ Cx^ ...

I had looked only at your accidentals between C and D and had assumed
that all of the naturals would be 5 degrees apart, with the
intervening accidentals appearing in the same order, which results in
the above sequence. In the course of working this out I discovered
that my assumption was not correct.

> According to the displacement values, F# should come after 7 fifths
up, followed by the rest of the sharps. Perhaps you are referring to
the narrower fifth in 31-TET?

Yes, I was referring to your table for 31-ET.

> 13. All of the inconsistencies you were so kind to point out will
have to be dealt with severely. However, it should be clear that I
never assumed that this portion of the proposal could be taken
seriously. The core of the proposal still remains, and that is to
identify finer divisions of the octave by way of colors which you
have neglected to address in your reply.

Yes, I said I had to stop somewhere. There were other issues to
address before I could take up the question of color.

> 14. I am grateful that you had a chance to listen to my
compositions. They are my humble works which I can feel proud of if
not anything else. There are some more, but I dare not upload any
until they are made more presentable.

What I have heard so far is very impressive, and I will look forward
to hearing more! We can talk theory day in and day out, but what is
the point if it does not enable us (or at least help others) to write
any music?

> 15. My lack of theoretical knowledge, on the other hand, should and
does not preclude the possibility that a `stupid idea` such as
indicating subtle nuances of pitch by way of color is worthy of some
attention. I understand that you did not have much to say about it,
but perhaps there is some chance that a truly professional theory can
be built on this idea? Would you consider my position seriously then?

There are several separate issues you are facing when you consider
color, but I think they fall into two categories:

1) Is it practical and convenient to implement?

Here you must consider the requirements of producing printed music
with full-color reproduction. It is more expensive than monochrome
(black) printing, and a color photocopier is required to copy parts.
I, for one, do not have quick access to a color photocopier, and if I
wished to make a copy of one of my compositions, I would be out in
the cold!

And if a musician is color-blind, what then? I don't quite recall
whether you mentioned some other sort of markings that could be used
instead -- but if that works, then why not use that alternative for
everyone?

But you don't have to go by my opinion alone. IS THERE ANYONE ELSE
OUT THERE WHO HAS ANYTHING TO SAY ABOUT THIS, ONE WAY OR ANOTHER?

2) Is it sufficiently meaningful and capable of being processed
easily when read?

I'm not color-blind, but when I look at the tables (Spectral Set of
Accidental Pairs) beginning on page 5 of your Spectral Notation
paper, I have trouble *quickly* distinguishing some of the darker
colors on my computer monitor (brown from violet, indigo from black,
and brown from black): I have to stare at one of these symbols for
too long to be sure of its color (and then I begin to wonder if color
fatigue has begun to set in). I don't think that this would be
improved with colors printed on paper; if anything, it would probably
be worse, because the lighting might not be as good. Perhaps this
could be remedied by modifying your choice of colors (magenta and
cyan might be good choices).

You have a color-size correlation, but I'm not sure if there is much
else to help me remember which color stands for what (except black),
since I don't see any useful color-to-prime correlation. Nor do I
find any help in the observation that black (5-comma) plus indigo (7-
comma) equals yellow-green (35-diesis) -- the equivalent of /| +|)
= /|) in Sagittal, where flag arithmetic is meaningful.

If you have not already done so, I would encourage you to read George
A. Miller's "The Magical Number Seven," which discusses some of the
dynamics of human perception and information processing:

http://www.well.com/user/smalin/miller.html

A portion of this deals with color recognition, and I think that
having 9 different colors (all of which must be dark enough to show
up well on a white background) is taking it pretty close to the
limit. Some of this will also not be relevant, since the experiments
involving "absolute judgments of stimuli" allowed the subjects an
unlimited amount of time to make their judgments, whereas in reading
musical notation for performance, time is critical.

The main problem I have with your Spectral Notation proposal is that
the way you are using them, the specific colors do not, in
themselves, have specific meanings, nor is it meaningful to add
colors together (such as red + blue = magenta, green + blue = cyan,
or green + black = dark green).

I hope that you have not failed to notice the advantages of Sagittal
flag arithmetic and that the symbols often give an indication of
prime number content, e.g., that /| + /| = //| (two 5-commas) and /|
+ |) = /|) (35-diesis as 5 times 7). Even the higher primes 17, 19,
and 23 have distinctive flags, ~|, )|, and |~, respectively, that are
meaningful even when they appear in combinations such as ~|(, )/|, )
|), /|~, and |~).

While there are a lot of symbols, there are very few types of symbol
components to remember:

4 types of flags: barb, arc, scroll, boathook
2 possible positions for each flag: left, right
2 flags maximum per symbol
4 possible numbers of shafts in pure version: |, ||, |||, X
5 possible mixed-version associations: bb, b, (nothing), #, x
2 possible directions of alteration: up, down

With these parameters it would be possible to have 72 different
single-shaft symbols, but we have limited our superset to less than
half of this (31), which is not very much more than there are letters
in the alphabet. We expect that less than half that number (12) will
be generally used for just intonation and even fewer than that (8 or
less) for the most popular temperaments. Please don't get hung up on
the multitude of symbols in the superset; we have them available to
satisfy the very few who think that they might want to do things that
most of us would never seriously consider.

The physical size of a Sagittal symbol also gives you a general idea
of how large the alteration is.

By comparison, with Spectral Notation you have these parameters:

2 symbol categories: sharps, flats
7 symbol variations per category
9 possible colors for each symbol

There are fewer parameters than with Sagittal, but there are more
values for each parameter. Taking all of this into account, I'm not
convinced that Spectral would be any simpler to comprehend or quicker
to read than Sagittal.

> 16. Yes, I am willing to admit that I have been more prejudiced
than I presumed, provided that you also are ready to admit that you
have been overlooking the possibility that Sagittal Notation might
not be the best microtonal alternative so far. I understand how much
work and wisdom went into it, congratulate your efforts and celebrate
your accumulations. Nevertheless I still feel uncomfortable with the
semiology although I now completely agree with the theory behind it.
If the matter boils down to which symbols should be used and where,
then I believe democratical contributions from the entire microtonal
world ought to be received.

Is anyone holding any elections soon? ;-)

Notation is a free product in a free market, and composers will vote
with their pens (whether ink or virtual). We present our ideas and
proposals, and we then see how that market will react. We have
already made cosmetic improvements to some of the Sagittal symbols in
response to criticism. Likewise, we are open to suggestions as to
the particular selection of symbols that should be included in the
set for various tunings; for some tunings the selection is fairly
obvious, but for others it helps to have some hands-on (and
especially ears-on) experience to make the best choice.

> 17. I, for one, have the following issue: The entire world of
microtonality should not be made to require notational subsets filled
with symbols out of a superset whose certain symbols are hardly
distinguishable from the others. This is very confusing to me, and I
wish to believe that further simplification is possible as well as
plausible. This is where you will perhaps, hopefully, begin to
consider the spectral idea if you deem that it is not totally wrecked.

As I pointed out above,.there are a couple of very important features
that Sagittal offers (flag-prime correlation and meaningful flag
arithmetic) that you don't have with Spectral. The Sagittal symbols
seem cryptic only until you learn the meanings of the components, and
you need only a small fraction of the superset for most applications
(in which you can gladly ignore the other symbols).

> 18. Yes, I agree with the international viewpoint and `tonal city-
scapes` illustrations. I now broaden my scope to include all
temperaments regardless of how the apotome and limma overlap or how
the comma differences vanish. I now imagine situations where the
Maqam may dwell in 22-TET or 19-TET lands.

But I, for one, don't imagine that this would be very likely. As
long as you're able to express all of your intervals as rational
ratios, you shouldn't be bothered by the melodic quirks of these
tempered tunings.

> 19. Ok, I agree that all pitches should have functional and
interpretive flexibility. The functional flexibility is trans-
temperamental, while the latter is dependent on pitch-cluster
variations of tone. Wonderful. There is nothing else we disagree with
on the theory of tuning by fifths. I adopt your jargon of
`functional`, instead of the biased `substitute`.

Thank you.

> 20. I wish you explain to me though, what makes an `acceptable
interpretive flexibility`? Why should I not have the liberty of
playing the 12-ET sharps and flats within a zone 50 cents wide whose
center is 100 cents? Perhaps this matter could be elucidated further
in the Sagittal paper?

It has nothing to do with notation, and everything to do with what is
acceptable in the context of a particular style or tradition.
Symphony musicians are generally not allowed pitch flexibility of
more than a comma or so, +-21 cents, if that much (excluding
_portamento_), while jazz and "blues" musicians routinely bend
pitches as much as they like.

> 21. You should know that I am seriously considering personally
adopting the Sagittal system. How I will be able to explain what I
write is another story of course. I may have reached an impasse even
with the mixed version. I still advocate some other solution that can
be embodied by the `cycle of fifths and co` theory seen in Sagittal
Notation. Alternative symbols based on quarter-tone accidentals would
be a very welcome addition for those of us in the Maqam world.

Yes, you will need to explain the nature of that impasse with the
mixed version. Is it that you object to using two accidentals to
modify a single note? In your next point you said:

> 22. I will be frank with you considering your question with the
mixed version: I think that the sharps and flats accompanied by the
apotome complement Sagittal symbols look pretty makeshift. I regress
from employing them on the grounds that they are more confusing than
intelligible. Hereabouts we employ arabic numerals to signify the
step number of the sharp and flat based on 53-TET. I would rather
revert to numbers if you will forgive my bluntness.

From the very start Dave Keenan and I have had a polite (yet firm and
ongoing) disagreement about whether the pure or mixed-symbol of
Sagittal would be more readily accepted. He believes that discarding
conventional sharp and flat symbols is too radical a change, whereas
I believe that having two symbols to modify a single note is
inefficient. In agreeing to disagree, the only compromise we could
arrive at was to offer the notation in two forms, neither of which
satisfies both of our preferences.

Does this describe the problem you have with Sagittal (as it would be
used for the 41-division), or is there something else?

> 23. If this matter could be resolved without causing a mayhem, I am
eager to imagine a hypothetical Cairo Music Congress on its the 75th
anniversary. It was last organized in 1932 and Rauf Yekta was the
Turkish participant back then. It would be a festive occasion, a
grand revolution in microtonal music, if we could somehow arrive at a
consensus and conceive of uniting our `microtonal forces`.

That sounds like a worthy goal. I'm not entertaining the idea of
modifying or abandoning Sagittal as it already exists, but I do
realize that (at least in its pure form) it may be too radical a
change to be made in a single step. We came up with the Sagittal-
Wilson symbol set (in a mixed-symbol implementation) to have
something that would not conflict with either the Sagittal or Sims
notation, but could serve as a transitional step to mixed-symbol
Sagittal.

It it possible that you're looking for some sort of 41-tone hybrid
notation in which the symbols possess characteristics of both the
symbols in your paper (a couple of your fractional sharp symbols are
rather nice, by the way!) and of pure Sagittal symbols (quartertone
arrows would not conflict with Maqam usage)? It would be confusing
for a player to be familiar with two different notations if the same
symbol is used for two different things, but I think a hybrid
notation having no conflicts with either Maqam, Arabic, or Sagittal
might be possible, if you're interested in pursuing that. While I
would view something like this as a possible transition to pure
Sagittal, you would likely view it as an end in itself.

> 24. I am prepared to ditch the accidentals I have been using so far
in favor of others. Thank you for pointing out their weaknesses. I am
more inclined now then ever of adopting the quarter-tone notation of
the Arab world.

You have said previously that other Turkish musicians would not
accept that, but if you can demonstrate that they are at odds with
the rest of the world (both Arabic and Tartini) and offer a good
theoretical foundation to support your view, then perhaps you have a
chance of convincing them. Perhaps a small chance?

> 25. Let me ask a direct question at this point. Are you completely
comfortable with all the symbols contained in Sagittal Notation?

No, only about half of them at present, because I haven't had much
occasion to use the others. This is not to say that I'm not
*satisfied* with the appearance of the others (I do know all of them
at sight), only that I'm not *completely* comfortable with actually
using so many symbols. In my opinion, it would take an
extraordinarily complicated application to warrant their use.

Best,

--George

You can configure your subscription by sending an empty email to one
of these addresses (from the address at which you receive the list):
tuning-subscribe@yahoogroups.com - join the tuning group.
tuning-unsubscribe@yahoogroups.com - leave the group.
tuning-nomail@yahoogroups.com - turn off mail from the group.
tuning-digest@yahoogroups.com - set group to send daily digests.
tuning-normal@yahoogroups.com - set group to send individual emails.
tuning-help@yahoogroups.com - receive general help information.

Yahoo! Groups Sponsor

Get unlimited calls to

U.S./Canada

------------------------------------------------------------------------------
Yahoo! Groups Links

a.. To visit your group on the web, go to:
/tuning/

b.. To unsubscribe from this group, send an email to:
tuning-unsubscribe@yahoogroups.com

c.. Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

Dear Ozan,

I am now ready to continue our "ferocious" discussion. ;-)

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George,
> ... I do not understand how you came to the conclusion that the
colors of Spectral Notation do not have specific meanings. The energy
level of the color itself gives an idea as to the size of the chosen
interval. For example, It is not impossible to attribute a diesis
sharp the color red to signify that the deviation ought to be
understood as a small-diesis with the nominal (Sagittal) ratio of
45:44 as opposed to the widely acknowledged 128:125 expressed with
black (which is not comprised by Sagittal Notation by the way).

I'm sorry. I should have explained my statement more clearly.

The color red does not have a particular *harmonic* meaning
everywhere that it is used. As 351:352 it signifies a 7-prime-limit
ratio; as 44:45 an 11-limit ratio, as 23:24 a 23-limit ratio, as
31:33 a 31-limit ratio, etc. The essence of the Sagittal philosophy
is that the accidentals provide, wherever possible, an indication of
*harmonic* function.

In Spectral notation you are using color to indicate *melodic*
refinement of pitch. One difficulty I have with this is that, should
you choose to carry over the *harmonic* meanings of the colors, then
such an ordering of colors will not necessarily be constant in all
contexts. For an example, I refer you once again to our
Xenharmonikon article:

http://dkeenan.com/sagittal/Sagittal.pdf

At the bottom of figure 5 (on page 10) there appears a sequence of
symbols for athenian-level JI in which the symbols (which modify
nominal pitches in a Pythagorean sequence) are in order of their
rational sizes. If you compare that with the trojan (12-relative)
symbol sequence in figure 10 (on page 18, in which the symbols modify
the pitches of 12-ET), you will observe a few qualitative
differences, i.e., (| is now smaller than both |) and |~. Similar
size reversals also occur in our symbol sequences for some of the
more unusual or obscure divisions of the octave (e.g., in figure 8,
observe the 51, 56, and 63 divisions, in which |), the functional 7-
comma, is smaller than /|, the functional 5-comma. While Dave Keenan
and I have considered it desirable to minimize such occurrences of
size-reversal in these symbol sequences, we have not done that in
instances where it would have introduced other, more serious
complications, such as the substitution of more obscure symbols
(which would not indicate the simplest, most desired harmonic
functions) in place of the most familiar ones. We recognize that
certain families of temperaments possess somewhat contorted tonal
relationships that are reflected in their symbol sequences, so we are
under no obligation to apologize for this.

On the other hand, in dealing with these situations in Spectral
notation, should you choose to preserve the sequence of the colors by
(melodic) size (which I think would make more sense), then you must
abandon the *harmonic* meanings of (and correlation with) some of
those colors. Inasmuch as our main objective in designing the
Sagittal notation was to have a common (or generalized) notation that
could express simple harmonic relationships *independent of any
particular tuning*, I cannot escape the conclusion that the Spectral
approach to pitch refinement would, by its very nature, fall short of
accomplishing that objective. For a multi-tuning notation, you must
choose whether your symbols will be defined harmonically or
melodically, and you must then accept that there will be variances in
the element that you have not chosen for that definition.

Your observation about 125:128 (the meantone diesis) being "not
comprised by Sagittal Notation" is not entirely correct. First I
must point out that the upper boundary for the Sagittal symbol (|(
representing 44:45 is ~40.602c, whereas your upper boundary for the
corresponding colored symbol is ~41.536c. This means that 125:128 is
represented in athenian-level (medium-resolution) JI by the
symbol //| (which has a primary value of 6400:6561). 125:128 does in
fact have an exact symbol in high-resolution JI: .//| (observe the
period to the left, which indicates reduction in size by a 5-schisma,
32768:32805). Since athenian-level JI (with approximately 5.4-cent
resolution) does not make the distinction of the 5-schisma, the same
symbol is used for both 6400:6561 and 125:128 (and also for 512:525,
which you show in your table of Spectral accidentals).

FYI, here is our current table of athenian-level boundaries, as given
in the file Sag_ji1.par (which is supplied with Scala):

! Athenian Sagittal JI notation. Dave Keenan & George Secor. 2003-Mar-
2
! This file is still under construction
! The average step is ln(3^7/2^11)/ln(2)*1200/21 = 5.413571717 cents
!
max_abs_slope 8.28
c_exp_range -6 10
nc_exp_range -1 5
!
!Lower bound Symbol Value Short 21-EDA
-67.66964646 (!/ 8192/8505 w -12
-62.044 (!) 704/729 o -11
-56.84250303 \!/ 32/33 v -10
-51.641 \!) 35/36 u -9
-46.01535959 \\! 6400/6561 _ -8
-40.60178788 (!( 44/45 d -7
-35.270 (! 45056/45927 j -6
-29.820 !) 63/64 t -5
-24.36107273 \! 80/81 \ -4
-18.94750101 ~!( 4096/4131 h -3
-13.480 )!( 891/896 i -2
-8.120357576 !( 5103/5120 c -1
-2.706785859
2.706785859 |( 5120/5103 r 1
8.120357576 )|( 896/891 * 2
13.480 ~|( 4131/4096 p 3
18.94750101 /| 81/80 / 4
24.36107273 |) 64/63 f 5
29.820 (| 45927/45056 ? 6
35.270 (|( 45/44 q 7
40.60178788 //| 6561/6400 = 8
46.01535959 /|) 36/35 n 9
51.641 /|\ 33/32 ^ 10
56.84250303 (|) 729/704 @ 11
62.044 (|\ 8505/8192 m 12
67.66964646

> Furthermore the addition of two colors out of the whole sequence
cannot possibly result in the sum total of two intervals as you
expect (whose ratios are actually multiplied, not added). It is very
well known that the addition of two colors gives the mean, not the
product. Thus, red + green = yellow. It is not possible to directly
correlate interval arithmetic with the spectral distribution of
colors.

My point exactly. I'm not saying that this additive property is
essential in a notation, only that it's something helpful that we
already have in Sagittal (as flag arithmetic) that would be lost if
colors were used instead.

> However, it is feasible to calculate mentally that a nominal black
syntonic comma sharp + an indigo septimal comma sharp would equal a
septimal diesis 36:35 which falls to the category of a medium-diesis
according to Dave Keenan (look up comma in `Monzopedia`), and hence
to the category of the middle sequence of colors in the spectrum.
>
> After our most valuable discussion, I have decided to attribute
four sharps and flats to the following categories:
>
>
> Square of lowerbound
> .. 2-exponent
> ...... 3-exponent
> ............... Lowerbound (cents)
> ................................ Size range name
> -------------------------------------------------
>
> COMMA
>
> .. [ 0 0 > ....... 0 ........... schismina
> .. [-84 53 > ..... 1.807522933 . schisma
> .. [ 317 -200 > .. 4.499913461 . kleisma
> .. [-19 12 > .... 11.73000519 .. comma
> .. [ 27 -17 > ... 33.38249264 ..
>
> DIESIS
>
> .. [ 27 -17 > ... 33.38249264 .. small-diesis
> .. [ 8 -5 > ..... 45.11249784 .. (medium-)diesis
> .. [-11 7 > ..... 56.84250303 .. large-diesis
> .. [-30 19 > .... 68.57250822 ..
>
> LIMMA
>
> .. [-30 19 > .... 68.57250822 .. small-semitone
> .. [-49 31 > .... 80.30251341 .. limma
> .. [-3 2 > ..... 101.9550009 ...
>
>
> APOTOME
>
> .. [-3 2 > ..... 101.9550009 ... large-semitone
> .. [ 62 -39 > .. 111.8774831 ... apotome
> .. [-106 67 > .. 115.492529
>
>
> I also revised the sharps and flats so that the Tartini-Couper
quarter-tone symbols are untouched while I translated the Arabic
quartertone (fraction) flat to the limma flat as used in Turkish
Maqam Music and re-interpreted my previous selection of the quarter-
tone sharp as the limma sharp. In black, these accidentals would
correspond to these ratios respectively:
>
> Comma: 81:80
> Diesis: 128:125
> Limma: 256:243
> Apotome: 2187:2048
>
> This is a compromise for both the Arabs and the Turks in as fair a
deal as I can make.

I don't think I am following all of this. A diagram would help.

> When colors are applied, the wavelenght would give an idea as to
which size is to be inferred. I will consider again which colors are
best to utilize, and what other method is most agreeable when color-
impaired musicians are concerned. Otherwise, I do not think that the
Spectral Notation would be too much of a burden to the printing
industry.

Ozan, I can see how Spectral notation could be useful in Maqam Music
with symbols for the 41 division (broken down into 9 colors, as you
now have it). I could also envision it in other applications where
it's desirable to indicate expressive intonation and/or adaptive JI --
e.g., for 31-ET (using the Tartini-Couper symbol set with 7 colors)
with alterations in increments of ~5.5 cents up to +-3 steps to
achieve 217-ET, or for 19-ET (using only conventional sharps and
flats with 9 colors) in increments of ~7.9 cents up to +-4 steps to
achieve 152-ET (Paul Erlich might be interested). So I'm not writing
it off entirely. I just don't see Spectral notation accomplishing
the specific (harmonic-based) purposes that we intended for
Sagittal. Nor have I seen anyone else in this group offering any
positive input in its behalf. However, we must keep in mind the fact
that the viewpoint and requirements of most of those here have a
Western harmonic-based point of view, whereas Turkish Maqam Music is
more melodic-oriented and thus more naturally suited to the Spectral
approach. We develop what we can and then hope that others will use
it.

> But it is not fair that you compare the flag arithmetic of Sagittal
with the sharps and flats I consigned to use for Spectral Notation.
It is not I who designed the sharp and flat symbology in the first
place! For all I know, you could still preserve the Sagittal flag
arithmetic and employ colors as I attempted above to arrive at fewer
categories of paremeters, which I assume would be much less confusing
to the general reader.

I have tried to keep an open mind in bringing up the above-mentioned
possibilities, but I must say that I find the introduction of color
more confusing than helpful when taking a harmonic approach to
microtonality.

> My objections concerning the Sagittal system can be summarized in
this fashion:
>
> 1. The superset on page 7 needs some cosmetic-aesthetical touches.

If you don't give any specific suggestions, then I have no idea what
you have in mind. We've already made cosmetic improvements several
times, to the point where we're now completely satisfied with them.

> 2. Employing colors for voluminous systems could be helpful. The
superset would then be reduced to a handful of symbols.

As I already explained, I don't believe that color works well to
convey certain harmonic information, but I could conceive of it to
indicate expressive intonation (as small melodic alterations). If
you're thinking of that "handful of symbols" as a set capable of
notating the 41-division, then I think that it would be necessary for
us to agree on what those symbols would look like.

> 3. A mixed version that employs the pure Sagittal symbols next to
the revisions I suggested above would be very convenient for
everyone. If the whole thing could be so designed that Maqam Music
could be expressed with the sharps and flats we all are accustomed to
next to their Sagittal Alternatives, such a hybrid system would be a
relief.

I don't understand exactly what you mean. Are you talking about a
redundant notation that uses two different symbols for each note,
with each symbol representing the same thing -- so you decide which
symbol you want to read and then disregard the other one? That
sounds like a lot of clutter, and I think we can do better than that.

I would prefer to see some sort of intermediate or transitional
symbol set for the 41 division, such as I suggested in my previous
message (with subsets also usable for some lesser divisions, e.g.,
17, 22, 29, and 31), which I will repeat here (in context):

<< > > 23. If this matter could be resolved without causing a
mayhem, I am
> eager to imagine a hypothetical Cairo Music Congress on its the
75th
> anniversary. It was last organized in 1932 and Rauf Yekta was the
> Turkish participant back then. It would be a festive occasion, a
> grand revolution in microtonal music, if we could somehow arrive
at a
> consensus and conceive of uniting our `microtonal forces`.
>
> That sounds like a worthy goal. I'm not entertaining the idea of
> modifying or abandoning Sagittal as it already exists, but I do
> realize that (at least in its pure form) it may be too radical a
> change to be made in a single step. We came up with the Sagittal-
> Wilson symbol set (in a mixed-symbol implementation) to have
> something that would not conflict with either the Sagittal or
Sims
> notation, but could serve as a transitional step to mixed-symbol
> Sagittal.
>
> It it possible that you're looking for some sort of 41-tone
hybrid
> notation in which the symbols possess characteristics of both the
> symbols in your paper (a couple of your fractional sharp symbols
are
> rather nice, by the way!) and of pure Sagittal symbols
(quartertone
> arrows would not conflict with Maqam usage)? It would be
confusing
> for a player to be familiar with two different notations if the
same
> symbol is used for two different things, but I think a hybrid
> notation having no conflicts with either Maqam, Arabic, or
Sagittal
> might be possible, if you're interested in pursuing that. While
I
> would view something like this as a possible transition to pure
> Sagittal, you would likely view it as an end in itself. >>

If such a notation cannot be free of conflicts, at least we could
minimize the conflicts. If you're open to this, I can offer some
suggestions with a diagram that I have already prepared.

> ...
> Happy New Year!

And a happy and very productive New Year to you also.

Best regards,

--George

Back to Sagittal...

Dear George,

I understand that the flag arithmetic of some Sagittal symbols provides a means to understand which intervals are added together to produce a third one, whereas colors are not utilizable in the same manner. But, as I understand it, this method applies ONLY to the intervals that are derived by `natural generators`, whose sizes are unaltered (untempered) such as those with ratios 3:2 or 5:4.

Yet, I do not understand why you think that Spectral Notation does not provide an harmonic context when I choose to attribute colors to the interval types such as apotome, limma, ditonic comma, so forth, instead of my previous approach where I employed colors to express cent sizes of these intervals in pythagorean context. It all comes down to engraving unto memory the pattern of colors that are correlated to the interval types in question. If the correspondence is consistent in itself, then the harmonic function will be stable throughout for all systems of tuning, no?

I am unable to notice much qualitative differences between the Athenian and Trojan sets I'm afraid. If it boils down to refinement of pitch, I fail to see where colors fall short of doing the same thing as Sagittal Symbols. If you will re-visit my page at the academical studies section, you will see that I'm revising the Spectral Notation following the guidance I recieved from you and other microtonalist colleagues. Though incomplete, the latest touches convince me that I'm somewhat on the right track this time. So, it will be possible to represent harmonic relationships `independent of any particular tuning` with Spectral Notation too. Thus, I do not think I need to choose between `melodic` or `harmonic` contexts when it is possible to preserve both.

I find it undesirable to express as common a vanishing interval as 128:125 by . //| when it is possible to assign a single color to a quarter-tone sharp or flat to represent it. Besides, how was I supposed to know if this interval was comprised (however obscurely) by Sagittal Notation when I don't see it mentioned on page 9, table 1?

Please disregard the colored table I prepared aforetime. The concept of the Spectral Notation should be based on no different a music theory as used by Sagittal Notation. We are agreed, once and for all, that symbols (and colors) should represent interval types and not interval sizes. Please look at the Spectral Notation paper again to see what I mean. Maybe you can suggest which 40-50 vanishing interval types should have priority within the half-tone?

Thank you for acknowledging the fact that my Spectral idea is not a waste of time entirely. But please realize that I do not anymore insist on a 41-tone division anymore than I insist on 36, 53 or 81. These are all approximations to the actual practice on maqams with varying degrees of certainty. I much prefer if I could prepare large table of just intervals like that produced by Partch. Also, I am not content anymore with a monophonic approach. A staff notation should accomodate polyphony wherever desired.

But it is not fair that you bring up the issue of the lack of any positive input on behalf of Spectral Notation. If it comes down to it, I don't see any negative input against the core philosophy of using colors with accidentals either, aside from one particular remark made by yourself some weeks ago where you likened my desire to employ colors to `grasping at straws`. As you said, I develop what I can, just as you develop what you can, and I'm very much open to endless improvement.

Still, I regress from making awry comments concerning symbol cosmetics. If you are completely satisfied with the superset, then I have nothing else to say further.

About redundancy of symbols, I take back what I said aforetime.

Yes, I would like to see the diagram you prepared.

Cordially,
Ozan

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Back to Sagittal...

Ozan, I am just making a brief reply now -- more later.

> ... If you will re-visit my page at the academical studies section,
you will see that I'm revising the Spectral Notation following the
guidance I recieved from you and other microtonalist colleagues.
Though incomplete, the latest touches convince me that I'm somewhat
on the right track this time.

Please give me a link to this. The last thing I have is dated 23
Nov, and you are undoubtedly referring to something more recent.

> Yes, I would like to see the diagram you prepared.

Here it is, with a brief explanation. See folder:
/tuning-math/files/secor/notation/
The filename is Symset41.gif

The top version has the Tartini/Couper accidentals for the even-
numbered degrees and Erv Wilson's version of Bosanquet's symbols for
1 degree. The remaining odd-numbered symbols feature alternately
sloping lines to indicate commatic differences (according to the
direction of the slope) from natural, sharp/flat, and double
sharp/flat accidentals. You will observe that the +3 and +5 symbols
are modifications of ones you have already used. (A couple of
these, -3 and +7, are pure Sagittal symbols, because I couldn't think
of anything else.)

The second version is a modification of the top version with Sagittal
arrows replacing the Tartini/Couper fractional symbols (thus avoiding
conflict with current Turkish meanings) and the Sagittal comma-down
symbol replacing the Bosanquet symbol. I believe that the
progression of physical size of the symbols is better than in the top
version and that the fractional sharps are also easier to distinguish
from one another due to their more varied appearance.

The objective for these first two sets was to incorporate good
features of existing notations, including retention of the
conventional sharp and flat symbols. Subsets of these symbols may
also be used for the 17, 19, 22, 24, 29, 31, and 34 divisions.

The third version is simply the pure Sagittal symbols as they
presently exist, shown for comparison. Observe that it would be
relatively easy for a musician to learn all three of these sets
without becoming confused about the meanings of any of the symbols.

Best regards,

--George

The link to the address I mentioned is http://www.ozanyarman.com/files/SPECTRAL%20NOTATION.pdf

I know it isn't quite finished, but something tells me that it's going to work this time once I am able to correspond colors with accidentals.

I like the symbols you prepared. Nicely done! This is the first attempt, as far as I can see it, that aims to bridge the gap of music theories between East and West. I opt for the Tartini-Couper-Wilson set if you can present alternatives for 7. -7. and -3. degrees.

Dear George, I think this is important that deserves a chapter in the Sagittal Notation paper.

Cordially,
Ozan

Dear Ozan,

On first reading, I was truly delighted by this.

Best of luck to you,

Pete

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> The link to the address I mentioned is
http://www.ozanyarman.com/files/SPECTRAL%20NOTATION.pdf
>
> I know it isn't quite finished, but something tells me that it's...

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> The link to the address I mentioned is
http://www.ozanyarman.com/files/SPECTRAL%20NOTATION.pdf
>
> I know it isn't quite finished, but something tells me that it's
going to work this time once I am able to correspond colors with
accidentals.

Thanks, Ozan. I'll let you know what I think once I get time to read
it.

> I like the symbols you prepared. Nicely done!

Yikes! I'm having mixed feelings about these, because now you've
gotten me hooked into working on something that has the potential of
competing with Sagittal. I find it difficult to resist this sort of
challenge, especially since Dave and I already stated in our paper
that "various attempts to extend these [Tartini] symbols in some
logical fashion to even finer divisions [than 24 and 31] are, in our
opinion, too cumbersome." With my further modifications (see below),
this is the symbol set I wished I had 30 years ago, but,
unfortunately, back then it didn't occur to me to define the symbols
in terms of comma-ratios (of JI) rather than as fractions of a tone
(temperament).

> This is the first attempt, as far as I can see it, that aims to
bridge the gap of music theories between East and West. I opt for the
Tartini-Couper-Wilson set if you can present alternatives for 7. -7.
and -3. degrees.

Challenge accepted! See folder:
/tuning-math/files/secor/notation/
As before, the filename is Symset41.gif -- if it doesn't look any
different than before, then you need to click on your browser's
refresh button.

The -3 symbol is a flat symbol with an upward-sloping slash through
it to signify a down-apotome reduced by a comma-up. Normally I don't
like to add something to a symbol to give it less alteration, nor to
have parts of the same symbol indicating alterations in opposite
directions, but in this case the meaning is so clear that I think it
would be justified. The -7 symbol is a combination of a flat and -3,
pushed together so that the lowest part of the slash blends into the
flat.

The +7 symbol is a sort of asterisk * with the downward-sloping line
longer and bolder than the other two in order to emphasize that this
is a double-sharp less a comma, which is equivalent (if the schisma
is disregarded) as an apotome plus a limma (which, you will recall,
equals a whole tone), so that a unison raised by this symbol would be
a respelling of a major 2nd.

I've also brought all three of these symbols down into the second
(Conventional-Sagittal Hybrid) symbol set, so the only way the first
two sets differ now is for -6, -2, -1, +2, and +6 degrees. There is
a subtle difference for +1, and you might want this symbol in the top
set changed to match that in the 2nd, which I prefer. I would also
like to see the -1 symbol from the 2nd set brought into the 1st
(since all the other down-symbols have upward-pointing stems), but
that's up to you.

Only 4 more symbols would be required for the 53 division (which
would also take care of 39 and 46), and I already have ideas for
those -- if you're interested.

> Dear George, I think this is important that deserves a chapter in
the Sagittal Notation paper.

Yes, both of these symbol sets will now have to be included in the
Sagittal documentation, along with the Sagittal-Wilson (mixed-symbol)
notation, as options for those who don't wish to make a complete jump
into Sagittal.

I've whimsically tacked both of our names onto the label for this top
set of symbols, so now it's "Tartini/Couper/Wilson/Yarman/Secor", not
to mention that I probably should also have had "Bosanquet" in there,
or perhaps "et al" at the end. There are so many individuals to whom
we're indebted for the ideas that went into this that it would be
impractical to give them all credit. For the time being maybe we
should just call this the "Extended Tartini/Bosanquet" notation until
we can come up with a better name. (The word "Sagittal" shouldn't be
used at all, because none of the symbols have arrows.)

Best,

--George

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> The link to the address I mentioned is
http://www.ozanyarman.com/files/SPECTRAL%20NOTATION.pdf
>
> I know it isn't quite finished, but something tells me that it's
going to work this time once I am able to correspond colors with
accidentals.

Hi Ozan,

Thanks for linking to some of my web pages. You should however replace
any ocurrences of
http://dkeenan.com or
http://www.uq.net.au/~zzdkeena with http://dkeenan.com

I am no longer paying for the uq.net.au site and it may be deleted at
any time.

While your 16 colours are probably a good attempt at maximal
distinctness for people with normal colour vision viewing a computer
screen (although you should try an LCD screen when looking from above
or below), I forsee a number of problems.

(1) They can look different (and so some may be less distinct) on Mac
versus PC.

(2) They can look different (and so some may be less distinct) when
printed by various different printing processes.

(3) About 1/12 of males and about 1/250 of females have some kind of
colour vision deficiency. About 1/50 of males are dichromats,
completely missing one kind of receptor, either red or green (very
rarely blue). Either the red inputs to the brain are being driven by
more green receptors or the green inputs are being driven by more red
receptors. You could get some idea of how distinct your colours might
look to these people by setting both R and G components to the same
value, calculated as 0.23*R + 0.77*G (to keep the luminance the same).

(4) Sheet music often has to be read under low light conditions where
the colour vision of even normal-sighted people is of little use. You
can get an idea of how they will look in that case, by setting R, G
and B components to the same value, the luminance, calculated as
0.21*R + 0.72G + 0.07*B.

(5) Even for normally sighted people in good light, there would be no
consensus about the natural linear ordering of these 16 colours,
except for possibly those saturated colours which are truly
"spectral", i.e. appearing in a sunlight spectrum or natural rainbow.

(6) The ability to recognise (and therefore distinguish) colours is
dependent on the area (or visual angle subtended). Musical accidentals
represent very small areas.

What all this means is that a practical upper limit for this purpose
is five colours (in addition to black and white).

It's interesting to note that a dichromat recognises only two hues,
not a continuum of hues like a trichromat. To a dichromat a solar
spectrum appears to have a saturated colour at each end but both
colours fade towards white near the middle. So to them, the sequence
dark-blue, blue, light-blue, white, light-red, red, dark red might
well seem a naturally ordered sequence. Presumably this could also be
wrapped around, and broken at white (the background colour) thereby
giving light-red, red, dark red, black, dark-blue, blue, light-blue as
a natural sequence. Or enhancing this for trichromats it could be
light-green, yellow, dark-red, black, violet, blue, light-blue.

But that puts black in the middle. If you want black at one end of the
sequence then the best is probably Cynthia Brewer's sequence black,
indigo, blue, turquoise, light green, light yellow, white. This gives
a hue change recognisable by a dichromat (and enhanced for a
trichromat), at the same time as a luminance sequence (barely)
recognisable by anyone in low light.

Cynthia Brewer has done some interesting work on this, in regard to maps.
http://www.personal.psu.edu/faculty/c/a/cab38/ColorSch/Schemes.html

-- Dave

Dear Pete, I'm glad you enjoyed it. Thanks for the positive input.

Cordially,
Ozan
----- Original Message -----
From: ambassadorbob
To: tuning@yahoogroups.com
Sent: 27 Ocak 2005 Perşembe 11:48
Subject: [tuning] Re: Sagittal Notation

Dear Ozan,

On first reading, I was truly delighted by this.

Best of luck to you,

Pete

Oops... It seems that Sagittal has a new contender now. :) I am very pleased to see that the East and West do actually converge.

I'm greatly honored that you include my name in this new notation project. But you know, even a little acknowledgement in an obscure corner would suffice. :)

Your splendid efforts in this direction are not in vain dear George. Here are some further suggestions:

The modifications for -7 and -3 are almost there. But I urge you to move the fraction sign up where the stem of the flat is, just as is used in Maqam Music today. Also, please consider limiting the lenght of the fraction to one per flat symbol. So, -7 should be bb with the second b slashed.

I agree that the 1. degree of the `Extended Tartini-Bosanquet Set` naturally deserves to be replaced with that of the hybrid set.

I recommend that you treat the flat of Ext. Set degree -1 as that of degree -2, using a d with a backslash moved up to the stem. This way, All the minus degrees will resemble the original flat.

Words like `New` and `Improved` come to mind as we go along. Let us complete now the superset!

Cordially,
Ozan

Dear Dave,

The RGB color code is, I expect, a standard for all RGB-based color systems. Of course that does not mean that there is one-to-one correlation with other color systems such as CMYK. Here is an extensive list of available color conversions:

http://www.psc.edu/~burkardt/src/colors/colors.html

Yet, I do not personally see a drastic change when modifying colors between these. The differences seem to be very slight.... as slight as a minute change in the lighting of an environment.

Likewise, the printing issue you brought up can be easily evaded by quality inkjet, color-laser or off-set printing.

Since I am no expert with colors, I can only assume that people with normal (or even those with slightly abnormal) color vision will `imagine` the obvious difference between the different color names I put forth when they correlate these names with the colors they have experienced beforehand in the real-world.

Thus, I don't think any person with a normal vision can fail to distinguish gold from lime, or orange from brown. For quick reference, I will attempt to reproduce the Spectral Notation color palette here to see how the internet conveys the color code:

MAGENTA
VIOLET
INDIGO
BLUE (very common, perhaps for # / b 256:243 and t / d 128:125?)
LIGHT BLUE
TURQUOISE
GREEN (quite common, perhaps for # / b 25:24 and t /d 64:63)
LIGHT GREEN
LIME
YELLOW
GOLD
ORANGE
RED (also very common, perhaps for # / b 648:625 and t / d 80/81?)
DARK RED
BROWN
BLACK (most common, definitely for # / b Apotome and t / d 33:32)

Notice that I made some correlations with well-known intervals. In this context, (t) and (d) stand for the Tartini-Couper-Secor sharp and flat.

These addresses give a good idea of what color-impaired people see: http://www.toledo-bend.com/colorblind/Ishihara.html
http://colorvisiontesting.com/
http://www.iamcal.com/toys/colors/

Here are the statistics for the color-impaired:

a.. 1 in 12 people have some sort of color deficiency. About 8% of men and 0.4% of women in the US.

a.. 0.38% of women are deuteranomalous (around 95% of all color deficient women).

a.. 0.005% of the population are totally colour blind.

a.. 0.003% of the population have tritanopia.

a.. Protanomaly occurs in about 1% of males.

a.. Deuteranomaly occurs in about 5% of males. It's the most common color deficiency.

a.. Protanopia occurs in about 1% of males.

a.. Deuteranopia occurs in about 1% of males.

Now, just as a language is not changed because of people of little or no hearing, but a seperate sign-language is developed for their needs by specialists, I cannot be expected to supplicate the deficiency of color-blind individuals who may also be musicians. Do you consider it imperative to communicate interval arithmetic to the visually-impaired dear Dave? Do not forget that, I already suggested the usage of letters when colors cannot be distinguished.

I like Brewer's colors but I don't understand how you reached the number 7 for the optimal number of colors. I can very well differentiate 16 colors I bought from the grocery on a music sheet.

As for the consensus, I need to hear more comments.

At this juncture, I urge the venerable participants of the tuning list to point out to me if there are colors that demand modification or need to be discarded entirely.

Cordially,
Ozan Yarman

Hi Ozan,

By "venerable participants" do you mean the long-term participants or only
the ancient ones? :-)
If the latter, then maybe I qualify! My comments, for what they're worth,
follow.

1. I have no trouble distinguishing any of these colours as printed in
your message and displayed on my PC.

2. The colour yellow on a white background has very little contrast. When
printed onto a tinted paper, the contrast is almost zero.

3. Many of these differences may be harder to spot when we have, not a
whole word, but only only a tiny note or accidental to read. The problem
will be exacerbated at smaller font sizes. As a writer, and a teacher of
writers, I take pains to ensure that wherever possible, text is at least 11
points in size, to enhance the likelihood of it being readable by the
majority of our audience. Venerable persons, such as I, have difficulty
with smaller text.

4. I personally have great difficulty in distinguishing your Indigo from
Black when given only a small sample, such as would appear in a note or
accidental. My wife tells me that my response to blue is subnormal, as she
classifies many blue-greens as Blue that I classify as Green. Do other
members have difficulty distinguishing these colours in small samples?

5. Let me ask whether it is not possible in every case to use TWO colours
rather than one? To enhance contrast on all backgrounds, consider using
whatever colours you like, but only as a fill within a thin outline. On
light backgrounds, the outline should be black; on dark backgrounds, the
outline should be white. Colours of a mid-range tone (greyscale value) are
almost always unsuitable as backgrounds.

6. No, you cannot be expected to supply a way to overcome a deficiency that
some individuals suffer. However, you may wish to choose to create a
system usable by as many as possible.

7. To test the effective contrast of any colour combination for people with
normal colour vision, take a screen print of your sample text or music in
it, then convert it to greyscale using any of the thousands of graphics or
photo manipulation programs available. Viewing the greyscale result will
give you a clear picture of the effective contrast. Alternatively, you
could use a formula for luminance based on the RGB components of a colour,
such as the one that Dave Keenan supplied recently (some authorities use
slightly different factors, but the results would be similar).

Regards,
Yahya
------------------------------------------------
Yahya Abdal-Aziz
Yahya@MelbPC.Org.Au
Melbourne PC User Group
Member 1075
Convener, Graphics Interest Group
Convener, Music Interest Group
------------------------------------------------

-----Original Message-----
From: Ozan Yarman
Sent: Monday 31 January 2005 19:51 pm
To: tuning
Subject: Re: [tuning] Colour in music notation (was: Sagittal Notation)

Dear Dave,

The RGB color code is, I expect, a standard for all RGB-based color
systems. Of course that does not mean that there is one-to-one correlation
with other color systems such as CMYK. Here is an extensive list of
available color conversions:

http://www.psc.edu/~burkardt/src/colors/colors.html

Yet, I do not personally see a drastic change when modifying colors
between these. The differences seem to be very slight.... as slight as a
minute change in the lighting of an environment.

Likewise, the printing issue you brought up can be easily evaded by
quality inkjet, color-laser or off-set printing.

Since I am no expert with colors, I can only assume that people with
normal (or even those with slightly abnormal) color vision will `imagine`
the obvious difference between the different color names I put forth when
they correlate these names with the colors they have experienced beforehand
in the real-world.

Thus, I don't think any person with a normal vision can fail to
distinguish gold from lime, or orange from brown. For quick reference, I
will attempt to reproduce the Spectral Notation color palette here to see
how the internet conveys the color code:

MAGENTA
VIOLET
INDIGO
BLUE (very common, perhaps for # / b 256:243 and t / d 128:125?)
LIGHT BLUE
TURQUOISE
GREEN (quite common, perhaps for # / b 25:24 and t /d 64:63)
LIGHT GREEN
LIME
YELLOW
GOLD
ORANGE
RED (also very common, perhaps for # / b 648:625 and t / d 80/81?)
DARK RED
BROWN
BLACK (most common, definitely for # / b Apotome and t / d 33:32)

Notice that I made some correlations with well-known intervals. In this
context, (t) and (d) stand for the Tartini-Couper-Secor sharp and flat.

These addresses give a good idea of what color-impaired people see:
http://www.toledo-bend.com/colorblind/Ishihara.html
http://colorvisiontesting.com/
http://www.iamcal.com/toys/colors/

Here are the statistics for the color-impaired:

a.. 1 in 12 people have some sort of color deficiency. About 8% of men and
0.4% of women in the US.

a.. 0.38% of women are deuteranomalous (around 95% of all color deficient
women).

a.. 0.005% of the population are totally colour blind.

a.. 0.003% of the population have tritanopia.

a.. Protanomaly occurs in about 1% of males.

a.. Deuteranomaly occurs in about 5% of males. It's the most common color
deficiency.

a.. Protanopia occurs in about 1% of males.

a.. Deuteranopia occurs in about 1% of males.

Now, just as a language is not changed because of people of little or no
hearing, but a seperate sign-language is developed for their needs by
specialists, I cannot be expected to supplicate the deficiency of
color-blind individuals who may also be musicians. Do you consider it
imperative to communicate interval arithmetic to the visually-impaired dear
Dave? Do not forget that, I already suggested the usage of letters when
colors cannot be distinguished.

I like Brewer's colors but I don't understand how you reached the number 7
for the optimal number of colors. I can very well differentiate 16 colors I
bought from the grocery on a music sheet.

As for the consensus, I need to hear more comments.

At this juncture, I urge the venerable participants of the tuning list to
point out to me if there are colors that demand modification or need to be
discarded entirely.

Cordially,
Ozan Yarman

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.300 / Virus Database: 265.8.3 - Release Date: 31/1/05

> Here are the statistics for the color-impaired:
>
> a.. 1 in 12 people have some sort of color deficiency. About 8% of
men and 0.4% of women in the US.

1 in 12 seems like a large enough number to me, that any system that
uses colour to convey information ought to consider them. Cynthia
Brewer and major map makers clearly think so too.

> Now, just as a language is not changed because of people of little
or no hearing, but a seperate sign-language is developed for their
needs by specialists,

A quick web search suggests that the numbers of such people in the US
are about 1 in 200, and most of these are over the age of 64. It's
more like 1 in 500 for working age people. That's very different from
1 in 12.

> I cannot be expected to supplicate the deficiency of color-blind
individuals who may also be musicians.
>

Of course that's entirely up to you. I'm just trying to be helpful by
pointing out some things that perfect trichromats often forget.
Including the fact that low light conditions make monochromats of us all.

Looking at RGB converted to CMYK (or whatever) and then back again, on
your computer's CRT is not comparable to actually printing it on paper
by various different processes, e.g. inkjet, laser, offset press, and
looking at it on an LCD screen from various angles and under less than
perfect lighting.

If you run Cynthia Brewer's "ColourBrewer" applet you will see that
for every colour scheme she gives ticks, question marks and crosses
for whether they are colour-blind friendly, print friendly, LCD
display friendly, photocopier friendly etc. She and her associates
have actually tested them on these devices.

Also you need to include among your tests, actual size accidentals on
paper (not large overlapping swatches) under dim two-phosphor
flourescent lighting.

> I like Brewer's colors but I don't understand how you reached the
number 7 for the optimal number of colors.
>

I didn't say it was optimal, but rather a practical upper limit. If
you look at the sequential schemes in ColorBrewer that are CRT, LCD
and print-friendly I think you will find there are none with more than
7 colours. Fortunately, one can also insist on colour-blind
friendliness without reducing this number.

-- Dave

Dear Yahya,

Thanks for the comments! Below is my reply in articles:

1. I tried to darken yellow as much as possible without confusing it with orange. Do you think we would be better off without it?

2. I encourage you to try some colors on actual paper with a colored-pencil set that can be obtained in a market. I did this, and the results quite satisfied me.

3. There is some vital problem with the color indigo with RGB monitors. The actual indigo in real life is much more vibrant, but I fail to achieve it digitally. Perhaps we should discard it?

4. I like your idea of encasing a colored accidental in a thin black outline on a white background, but I do not know if that would just cause more confusion. Besides, I hardly know if it would be possible to integrate such a thing in a notation program.

5. I am inclined to design a system that is optimal for a large number of musicians. The question at this point is, are there more color-blind musicians than are deaf-musicians? If the numbers are even, than I should have no more concern than a record-company publishing music for people with normal hearing.

6. As far as I gather, you can easily distinguish 14 colors out of 16. If I reduced the number to 12 and made minute adjustments would that be acceptable?

Cordially,
Ozan

Dear Dave,

I admit that I have failed to generate sufficient empathy for the color-deficient in my non-dysfunctional trichromatic condition due to the fact I cannot begin to understand how they see the world. But could I assume that the right question is `how many musicians are afflicted with visual disorders affecting their perception of colors`?

If my memory does not betray me, I believe you mentioned Mrs. Brewer to be your acquaintance. Do you think it would be appropriate for me to contact her on this issue and ask for advice? It is without doubt that I am willing to adopt a system that best suits the needs of everyone.

Nevertheless, you should not forget that the Spectral Notation is not yet complete, so I have not yet attempted to reproduce colored accidentals on the staff before agreeing on the optimal swatches.

Perhaps the practical upper limit for colors - for both color-deficient and color-functional viewers, could be extended to a higher number so that I can easily include some fundamental invervals within the compartments I delineated?

Regards,
Ozan

In a message dated 2/3/2005 2:26:10 PM Eastern Standard Time,
ozanyarman@superonline.com writes:
But could I assume that the right question is `how many musicians are
afflicted with visual disorders affecting their perception of colors`?
Hello Ozan,

I'm glad you framed the question this way. Some of the players I works with
in the AFMM Ensemble are indeed color blind. I have wondered, annecdotally,
if nature provides extreme gifts in one area (e.g., pitch) and much like
rubato, takes it away from another (e.g., color).
I would rather not mention names here as they may be sensitive to their color
disadvantages.

In my own case, my mother was a professional color mixer. For reasons only a
psychologist could understand, I do not have confidence with naming colors at
all. For this reason I have not strayed too deeply into your discussions.
In principle, using colors to inform, much like traffic lights, makes sense.
But I feel more secure about discerning between only a few contrasting colors,
rather than between as many as you have suggested.

all best, Johnny Reinhard

Hello Johhny,

Thank you for sharing this important piece of information. This calls for a poll in my opinion. What do you and the musicians of AFMM think are the maximal number of colors that people with color-deficiency prefer to see in a musical score?

Cordially,
Ozan
----- Original Message -----
From: Afmmjr@aol.com
To: tuning@yahoogroups.com
Sent: 03 Şubat 2005 Perşembe 22:06
Subject: Re: [tuning] Re: Colour in music notation

Hello Ozan,

I'm glad you framed the question this way. Some of the players I works with in the AFMM Ensemble are indeed color blind. I have wondered, annecdotally, if nature provides extreme gifts in one area (e.g., pitch) and much like rubato, takes it away from another (e.g., color).
I would rather not mention names here as they may be sensitive to their color disadvantages.

In my own case, my mother was a professional color mixer. For reasons only a psychologist could understand, I do not have confidence with naming colors at all. For this reason I have not strayed too deeply into your discussions. In principle, using colors to inform, much like traffic lights, makes sense. But I feel more secure about discerning between only a few contrasting colors, rather than between as many as you have suggested.

all best, Johnny Reinhard

I'll ask some of them if you like. But the color blind folk would likely
simply reject the idea. I'm not sure what I would say for that matter. Johnny

Perhaps it is best if you asked them anyhow dear Johnny. As for you, I had been entertaining the idea that you were not so dismissive of the idea at first.

All the best,
Ozan
----- Original Message -----
From: Afmmjr@aol.com
To: tuning@yahoogroups.com
Sent: 04 Şubat 2005 Cuma 4:29
Subject: Re: [tuning] Re: Colour in music notation

I'll ask some of them if you like. But the color blind folk would likely simply reject the idea. I'm not sure what I would say for that matter. Johnny

human beings can see more variations of yellow and green than any of the others

From: "Ozan Yarman" <ozanyarman@superonline.com>
Subject: Re: Re: Colour in music notation

Hello Johhny, Thank you for sharing this important piece of information. This calls for a poll in my opinion. What do you and the musicians of AFMM think are the maximal number of colors that people with color-deficiency prefer to see in a musical score?

Cordially,

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear Dave,
>
> I admit that I have failed to generate sufficient empathy for the
color-deficient in my non-dysfunctional trichromatic condition due to
the fact I cannot begin to understand how they see the world.
>

Well you can get a pretty good approximation, or so I'm told, by
applying that function I gave earlier in this thread, to the R and G
components of your proposed colours.

> But could I assume that the right question is `how many musicians
are afflicted with visual disorders affecting their perception of
colors`?
>

There's little reason to think that their statistics are much
different from the rest of the population. Except that, since music
does not (as yet) require much in the way of colour vision, or for the
reason Johnny gave, one might expect their numbers to be slightly
higher than 1 in 12.

> If my memory does not betray me, I believe you mentioned Mrs. Brewer
to be your acquaintance.

Not at all. She doesn't know me from a bar of soap. I just found her
website by searching and following links.

> Do you think it would be appropriate for me to contact her on this
issue and ask for advice? It is without doubt that I am willing to
adopt a system that best suits the needs of everyone.
>

I think she already gives enough information on her site, particularly
in the "Color Brewer" applet. But if there's something else you want
to know, I guess it can't hurt to ask.

Sorry to have been the bearer of bad news, but I thought it best to
tell you sooner, rather than have you go to a lot of trouble only to
find out later, the hard way.

-- Dave

I actually work with artist that are color blind and one who is so sensitive to value can match color way better than you could imagine. Another artist i worked with years ago stuck to charcoal drawing and did some amazing work in this regard. the naming of colors is not standardized except for a few, past that point the names can refer to a quite broad range.

BTW i remember scriabin placing C as orange and not green, i thought. From: Afmmjr@aol.com
Subject: Re: Re: Colour in music notation

I'm glad you framed the question this way. Some of the players I works with in the AFMM Ensemble are indeed color blind. I have wondered, annecdotally, if nature provides extreme gifts in one area (e.g., pitch) and much like rubato, takes it away from another (e.g., color).
I would rather not mention names here as they may be sensitive to their color disadvantages.

In my own case, my mother was a professional color mixer. For reasons only a psychologist could understand, I do not have confidence with naming colors at all. For this reason I have not strayed too deeply into your discussions. In principle, using colors to inform, much like traffic lights, makes sense. But I feel more secure about discerning between only a few contrasting colors, rather than between as many as you have suggested.

all best, Johnny Reinhard

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Dear Dave, please do not think that you are discouraging in any way, on the contrary, I benefitted immensely from our discussion.

I contacted Cynthia Brewer about the optimal palette to use and color-conversion tables for the color-impaired people. Let's see how things will turn out.

Cordially,
Ozan
----- Original Message -----
From: Dave Keenan
To: tuning@yahoogroups.com
Sent: 05 Şubat 2005 Cumartesi 7:47
Subject: [tuning] Re: Colour in music notation

Sorry to have been the bearer of bad news, but I thought it best to
tell you sooner, rather than have you go to a lot of trouble only to
find out later, the hard way.

-- Dave

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Oops... It seems that Sagittal has a new contender now. :) I am
very pleased to see that the East and West do actually converge.

And I am pleased to report that Dave Keenan has also decided to
converge on this latest notational effort. I thought that I would be
replying sooner, but since Dave has begun providing his valuable
input (in which he often takes a somewhat different viewpoint than I
do), there have been a few things between us that have had to
be "ferociously" debated (mercifully, off-list, in order to spare you
the excruciating details).

As it turned out, there were a few things on which we were not able
to reach agreement, so Dave and I have agreed to disagree and are
therefore submitting two sets of symbols for the 41 division of the
octave (with subsets usable for 17, 19, 22, 24, 29, 31, and a few
others). We have carried our spirit of co-operative disagreement so
far as to critique each other's symbols so that they would be more
acceptable to the other. I am happy to report that both sets have
been improved in the process.

One thing we did agree on was that I would set aside my proposal in
which Sagittal arrows replace the Tartini-Couper symbols.

> I'm greatly honored that you include my name in this new notation
project. But you know, even a little acknowledgement in an obscure
corner would suffice. :)

That long string of names (I would now have to add Keenan also) was
supposed to be a joke. In the course of our debating, Dave and I
also discussed names, and we settled on one that I suggested:

<< Better yet, "Tartini-plus" would have a double meaning,
indicating 1) an expansion of the Tartini symbols, and 2) the comma-
up symbol looks like a plus sign. >>

As it turns out, only one of these symbol sets has the Wilson-Wolf-
style plus sign for comma-up, but the name can still be used for
either one.

The diagram you last saw has been updated, with the two new symbol
sets on the top staff and the particular version of Sagittal that
more closely resembling each one directly below that. See folder:
/tuning-math/files/secor/notation/
where the filename is still Symset41.gif.

I thought that it would be best to keep any explanation of the
symbols to a minimum until you have had a chance to look at them and
offer your comments.

A few of the symbols have changed only slightly since you last saw
them, and these are shown on the bottom staff. I will refer to some
of these below.

Dave and I thought that it would be appropriate to make the
conventional double-sharp symbol somewhat larger. You may be aware
that there is an older symbol for the double-sharp that is indeed
larger in size (but which lacked the blobs on the ends of the "X";
see "8 older" on the bottom staff). The new versions combine the
appearance of the more modern symbol with the larger size of the
older one.

The rest of this message will be a reply to the suggestions you
made. I have discussed these with Dave, and he is in general
agreement with my comments that follow.

> Your splendid efforts in this direction are not in vain dear
George. Here are some further suggestions:
>
> The modifications for -7 and -3 are almost there. But I urge you to
move the fraction sign up where the stem of the flat is, just as is
used in Maqam Music today. Also, please consider limiting the lenght
of the fraction to one per flat symbol.

We all agree that this symbol should consist of a flat, embellished
by an upward slash of some sort. I show the Maqam version of this on
the bottom staff (at the right), for which I will now offer a
critique.

One thing clearly in its favor is that it's easy to distinguish from
everything else (but less so if we adopted your suggestion for -1,
which would resemble its lateral mirror-image; but more about that
below).

Another point is that it's already familiar to Maqam musicians, but
I'm not sure whether that's good or not. If the size of alteration
that this currently represents is not a limma, then could there be a
problem in getting them to accept a new meaning for this symbol?

Whatever the case, I previously expressed another difficulty that I
had with this type of symbol:

<< The eye must be focused (and the attention centered) on the
embellishment in order to read it. Inasmuch as each embellishment is
inappropriately centered approximately 3 staff positions above the
notehead that is being altered, this requires one to glance first at
the main part of the symbol (to identify the staff position), then
away from the desired staff position to read the embellishment (to
identify the amount of pitch alteration). It would be much better if
all of the information to be conveyed by the symbol could be taken in
at a single glance, with the most important detail of the symbol
being centered on the same line or space as the notehead being
modified (something we have scrupulously observed in designing the
Sagittal symbols). >>

By itself this might not be too serious an objection, particularly
since we had eliminated (in my last diagram) the differing
embellishments you previously had in some of the other symbols, with
which this one no longer has to compete for recognition.

But I have still another difficulty with this symbol in that
opposites should resemble one another. The slash in -3 "Maqam"
resembles the cross-bars of the +2 symbol much more than the comma-
slash in the +3 symbol (all versions).

In summary, the slash in the -3a Maqam symbol leaves something to be
desired in that it differs from those in all of the other odd-
numbered symbols in that it 1) is much shorter than the others, 2)
does not slope nearly as much, and 3) is not located at the same
height as the notehead that is being modified.

We all seemed to agree that what I last proposed for -3 (shown
as "old") needed improvement. You will observe that the two new
versions being proposed are very similar to one another.

> So, -7 should be bb with the second b slashed.

In both new sets the -3 symbols have been placed them to the *left*
of a conventional flat for -7. This is to follow the pattern
established in -5 and -6 (and also in mixed Sagittal), where the
smaller component is combined with a conventional flat (or sharp) by
placing it to the left.

> I agree that the 1. degree of the `Extended Tartini-Bosanquet Set`
naturally deserves to be replaced with that of the hybrid set.

That has been done in the first version.

> I recommend that you treat the flat of Ext. Set degree -1 as that
of degree -2, using a d with a backslash moved up to the stem. This
way, All the minus degrees will resemble the original flat.

I found the same difficulties with this suggestion that I had with -
3 "Maqam", above (where I also mentioned the problem of lateral
mirroring), but I find an even more serious problem here involving
semantics. The "d" (semiflat symbol) combined with a backslash
implies that this is an 11-diesis (half-flat) down combined with a 5-
comma down, which is equivalent to -3 degrees, not -1.
Alternatively, a "d" with a slash (11-diesis down combined with a 5-
comma up) would be -1 degree; although this would now be the correct
number of degrees of 41, it is not really what is desired. The -1
symbol is supposed to indicate a 5-comma down (~21.5 cents), which is
not the same as an 11-diesis minus a 5-comma (~31.8 cents), which is
to say that the 5-comma (represented by -1) is farther away from the
11-diesis (represented by -2) than it is from a unison (represented
by the natural sign). Not only would the implied semantics be
incorrect, but the size of the interval represented by the slash
would also be larger here than in all of the other occurrences. In
addition, the upward slope of the slash, while technically correct as
to direction, would not maintain the pattern of alternating the
direction of the slope for consecutive odd-numbered symbols.

This discrepancy becomes even more serious if we attempt to use this
symbol for other divisions of the octave. In the 22-division the 11-
diesis and 5-comma are each 1 degree. For the single degree of 22 we
need a 5-comma symbol, but the d with backslash would be 2 degrees,
while the d with a slash would be zero. So neither of those
combinations would be correct. We would also have the same problem
if we tried to employ this symbol for the 72 division (e.g., in
Spectral notation), in which the 5-comma is 1 degree and the 11-
diesis is 3 degrees (but neither the sum nor difference of these
comes out to a single degree of 72). We would therefore have a
symbol in which the components are not valid for multiple tunings.

So once again I recommend changing the -1 symbol to that shown in
either proposed set, so that all of the down-symbols would have
upward-pointing stems. This would, in fact, make it almost an exact
mirror image of Fokker's symbol for -1deg of 31, which I cite as
evidence that he did not think its appearance so different from that
of a flat symbol as to consider it aesthetically out of place..
Moreover, it would be a suitable counterpart to the +1 symbol (in
either proposal). So it happens to look like a Sagittal half-arrow!
Is there anything wrong with that? Please do not fail to remember
that Didymus the Musician thought that this was one of our best
symbols, indicating a downward direction of alteration by 1) its
appearance as a downward-pointing half-arrow and 2) by the downward
slope of the diagonal slash, which symbolizes a comma. (Yes, I will
concede that Didymus was a little biased, this being *his* comma ;-).

> Words like `New` and `Improved` come to mind as we go along. Let us
complete now the superset!

We have taken the next step by submitting these proposals, and we
look forward to your comments, questions, and suggestions.

Best,

--George