explanation of Monzo's preferred notations

Joe [Pehrson],

Here is a collective response (it got rather long...) to a number
of your posts, in which I give a comprehensive explanation of my
notational systems. (I do make use of several.)

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_20929.html#21383

> It seems that Wolf's notation gets a little bizarre in terms of
> accidental usage, now?? It looks a little as though somebody fell
> the wrong way on the typewriter....

Well, that's the main reason I prefer plain old prime-factors
instead.

But as I said before, giving credit where's it's due, Wolf's symbols
make his notation more compact and more immediate than mine.

> AND, the "Monzotone" notation is still beyond me. I have read that
> page 20 times and I still don't understand it.
>
> Is it possible, Monz, to describe your Just system in some kind
> of terms so that somebody like me can understand it?? It seems
> a little obfuscatory...

OK - but *you* are therefore the one responsible for the length of
*this* post!

:-P

Hopefully, I've become a better writer in the 6 years that
have passed since I originally wrote that paper:
http://www.ixpres.com/interval/monzo/article/article.htm

Actually, my own personal system is extremely simple if the
prime-factor accidentals are simply used to accompany regular
12-EDO (= 12-tET) notation. (But these accidentals can also be
used in conjunction with more sophisticated or more complex
staff-notations... more on that at the end of this post.)

One simply adds the list of prime-factors and their appropriate
exponents before the note-head of a regular 12-EDO pitch according
to either of two ways:

1. The 12-EDO pitch is determined according to a system based on
the pitches I define as representing each succesive prime-
limit. For example, if "middle-C" is our reference n^0
(= 1/1 ratio), I call 5^1 "E", 7^1 "Bb", 11^1 "F#", 13^1 "Ab",
17^1 "Db", 19^1 "D#", and so on (... my JI music often stays
within 19-limit).

Letter-names of pitches arrived at thru expansion in any of
those dimensions can be easly determined by applying the
same mathematics consistently: so 5^1 7^1 goes with "D",
7^1 11^1 goes with "E", etc.

I like this system because the letter-names correspond
to those that are familiar from any serious study of
traditional triadic harmony.

Note, however, that because both 7^1 and 11^1 are so far from
their 12-EDO representations (~31 and ~49 cents respectively),
this last example is actually much closer to the 12-EDO Eb in
pitch (in fact at ~320 cents it's nearly a 6/5). This is an
inconsistency, not in the sense of Erlich's definition of
consistency, but definitely still some kind of inconsistency
in terms of what those letter-names are supposed to mean.
(Perhaps Paul will elaborate on this.) Therefore...

2. It might be preferable to simply use the 12-EDO pitch that is
nearest to the pitch defined by the ratio. This will
definitely result in Erlichian inconsistencies in a few
places, but generally should work well; seems to me it would
be better than the incongruous example I describe above
(320-cent "E").

This would result in a situation somewhat like Schoenberg's
use of the most easily-readable notation for highly chromatic
music: i.e., he might spell a chord G#:C:Eb instead of the
traditional way of recognizing triadic harmony by using
G#:B#:D#.

I never really liked this method because of my disagreement
with Schoenberg on this, but might shift to it after all to
avoid the incongruities discussed above.

(Hmmm... if that type of "congruity/incongruity" is different
enough from Erlich's "consistency/inconsistency" concept,
maybe I have another new tuning term to define... Paul, help!)

For a rather extreme example of how my accidentals work: if
"middle-C" is our reference n^0 (= 1/1), then a note whose pitch
is a 35:33 above it would have as prime-factors 3^-1 5^1 7^1
11^-1, and would be attached to the letter-name "Db", because a
3:2 below C (= 3^-1) is "F", a 5:4 above that is "A", a 7:4 above
that is "G", and an 11:8 below that is "Db". The cent value is
~102 cents, so in this particular case either method (of the two
listed above) works equally well.

(I deliberately chose this ratio because it's a sort of perverse
distortion of why I decided on prime-factor notation in the first
place; that is, the prime-factor accidental is longer/bigger than
the ratio. Here's a converse example that shows precisely *why*
I like this notation so much: if "C" is 1/1, then 1148175/720896
would be 3^8 5^2 7^1 "Cx". Which one is easier for *you* to
understand? ... I thought so!)

In my scores I don't have to follow the ASCII conventions that
are necessary here on the Tuning List, so the accidental itself
is a little more compact than what I write here. The exponents
are small superscript numbers attached to the prime-factors (as
demonstrated in my article and book), and I don't need so much
space between them to keep them separated. Normally when we give
the prime-factorization here on the list we also add the
multiplication sign "*" between each set of factors to show the
relevant mathematical operation.

The way I write the accidentals in the score (i.e., with
superscripts for the exponents), I can follow the regular
algebraic convention of simply butting the terms up against
each other to show multiplication, and eliminating any extra
multiplication symbol. This makes the whole accidental symbol
more compact.

Perhaps some of the difficulty you are having understanding my
system, Joe, is that in my article and book I also mention another
way of using the same concept, where one may dispense with the
prime-factors themselves in the accidental, and indicate only
the values of the exponents; so that, for example, the 35:33
illustrated above would be rendered as -1 1 1-1. This should
be very familiar to anyone who's made use of the vector notation
of the prime-factors, because that's exactly what it is, without
the enclosing parentheses or brackets that are usually included.

Using this method, it is understood that each successive number
represents the exponent of a prime in the prime series, starting
with 3 on the left and continuing thru 5, 7, 11 (as in this
example) and beyond to 13, 17, 23, 29, 31, 37, 41, 43, 47, etc.,
if more places are used.

It is necessary to include the use of zero as a place-holder when
using this method, whereas when the prime-factors are included as
in the earlier examples using the other method, those factors with
a value of zero can be omitted entirely.

I find that the version which includes the prime-factor and omits
those factors with a zero exponent is usually more compact and
therefore more useful for actual staff-notation, because in any
given section of a composition, most ratios have only a few (1, 2,
or 3) factors, thus there is no need to have long strings of zeros
in the accidental. All of the actual musical examples in my book,
and all of my scores written in prime-factor notation, use this
method.

The vector method is more useful when creating tables of pitches
in an entire scale (as when I enter exponents into cells on a
spreadsheet to create a lattice diagram), or for use in computer
algorithms.

> I will try to find the Perspectives article on Ben Johnston's
> notation... although I am beginning to doubt that it is as
> "direct" as these "improved" systems.

I can guarantee that you'll only become comfortable with
Johnston's notation if you are a hard-core 5-limit JI enthusiast
with a healthy interest in memorizing lattice diagrams. :)

If you're not, then read those _Perpectives_ articles to get the
superficial understanding of Johnston's system that you feel you
need as a minimum, put them in your library for future reference
if that need arises, and move on.

If you're having trouble understanding how Johnston's notation
works in 5-limit, multiply that difficulty by each new dimension
you add to the lattice with each new prime-factor. :( Even tho
his accidentals for all prime-factors higher than 5 follow more
logical rules, that problemmatic 2-dimensional basic scale pattern
is replicated in every other added dimension. (I'll go into more
detail on this below.)

If you're not interested in analyzing Johnston's (or his
students's) music, and not interested in composing music like
theirs, forget about that notational system, other than for
having knowledge of it for general purposes. If your primary
purpose is to learn a useful new tool for use in composing your
own extended JI music, work with the HEWM Helmholtz/Ellis/Wolf/
Monzo] notation instead. (This newly-coined acronym has just
been added to my Tuning Dictionary.)

(With all this negativity, I should probably remind everyone here
that I have the utmost respect for just about everything else
Johnston has written, musical or theoretical, and that I value
his friendship highly. I just don't like his notation, even tho
it *was* the catalyst that finally gave me the insight into how
to develop *my* system to adapt staff-notation for JI, and so I
still appreciate it for that reason too.)

> The reason, though, that I am so interested in it is, obviously,
> Johnston's stature as a composer and xenharmonic thinker and the
> usage of such composers as Doty and Gann...
>
> It can't just be "overlooked..."

Yes, Joe, that's exactly why I address this issue so strongly.
Johnston was an influential teacher (he's retired now, composing
full-time) who had a lot of students. I feel that his notation
is being accepted unquestioningly by a large (and ever-increasing)
number of JI composers and theorists, and that if they put a
little more thought into the problem, they too would probably
prefer the HEWM system instead. Paul Erlich advocates HEWM over
Johnston ... and you can be sure that *his* is not an uncritical
opinion! :)

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_20929.html#21387

> perhaps it would be best to go for "Monzotone" rather than
> Wolf... since I am in more frequent communication with you...

Well, that sure makes sense. I've put a lot of thought into the
microtonal notations that suit my particular uses, and it would
really make me happy to see my proposals adopted by others here
who share similar concerns. And I'm around here a lot, always
available to provide help when it's requested.

In brief, some of my preferences are:

- Helmholtz/Ellis/Wolf/Monzo (HEWM) prime-factor notation for
any-prime-limit extended JI. Extremely useful as an analytical
tool, as it can represent clearly *any* rational system than can
be devised. Just about the only sensible way to notate very-
high-prime music (i.e., La Monte Young's) on a staff. (Example
near the end of my book.)

- Sims 72-EDO notation for up to 17-limit JI, with subsets that
work fine for 48-, 36-, 24-, 18-, 12-, 9-, and 8-EDO notations
and/or tunings. (My ASCII adaptation of 72-EDO is great for
Tuning List posts.)

- Stearns/Sims 144-EDO for the type of poly-EDOs division of the
pitch-continuum favored by Dan Stearns, or for cases of JI
music where the context demands smaller-than-72-EDO accuracy
(as in my piece _A Noiseless Patient Spider_).

All three of these are intended to function as representations of
the infinite pitch continuum, whose subdivisions make up the scales
and harmonies that we analyze in various musical pieces or
performances.

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_21267.html#21394

> Quote from Johnny Reinhard:
>
> "Cents makes the most sense:
>
> 1200 divisions of the octave being the very threshold of human
> hearing for pitch differentiation. Additionally, cents allows
> for the total intellectualization of all pitch "points" on the
> line of frequency to be immediately apprehended. Most
> importantly, players always reference new pitches to constants
> -- like open strings and harmonics -- in order to find exotics.
> Microtonal notations that represent moving relationships can
> lose their constants in pitch drift, making it difficult to come
> up with the necessary hand positions or fingerings to produce
> the appropriate sounds. The solution is for all pitches to
> relate in a unified field in order to be fully prescriptive for
> players. Cents also anchors us to present music tradition --
> no small feat. Of course, by using enharmonic quartertones,
> one need not use numbers larger than 25 to indicate cents
> distinctions (recently pointed out by Paul Erlich, a subscriber
> to the internet's busy Tuning List)."

Johnny makes an important point here: that his 12-EDO system
is a closed one, whereas the HEWM system is not.

If I were as heavily involved in the practical side of
polymicrotonal music-making as Johnny Reinhard, I'd probably like
his 1200-EDO notation a lot too. This system is also intended by
Johnny and his followers to function as a representation of the
infinite pitch continuum.

But since I generally prefer to think of cents with the decimal
point made manifest (as in my use of Semitones... see my
Dictionary entry), 1200-EDO is more accurate than necessary for
my *practical* purposes. (Of course, I use 1200-EDO, as cents,
all the time in my theoretical work.)

I believe that it's unlikely that live performance on non-fixed-
pitch instruments is more accurate than +/- 5 cents anyway, so
that would make 240-EDO good enough. Since I already use 144-EDO,
with its great properties of multiple subdivision (into 72-, 48-,
36-, 24-, 18-, 16-, 12-, 9-, 8-, 6-, 4-, 3-, and 2-EDO) and an
accuracy of 8-&-1/3 cents, 144-EDO is good enough for me. It
also gives pretty good representations of 11-, 13-, 29-, 47-, and
49-EDO, and a few on either side of 72-EDO, so it's handy for
exploring Stearns-like uses of the pitch-continuum.

The big advantage Johnny's 1200-EDO has over all other systems is
that it harnesses the power and biological inherency of the decimal
number system, and this is not something that can or should be
easily dismissed.

Albeit limited, our hands and fingers are a built-in abacus.
1200-EDO, thought of as (100*12)-EDO, is thus so easy to
conceptualize and apply for musicians who have already become
comfortable with 12-EDO that it's certainly the most quickly-
learned representation of the infinite pitch continuum for a
trained musician. (And as I contend in my book, even without
familiarity with 12-EDO, a musician dealing with a Pythagorean
chromatic system would naturally tend to feel the "octave" as
divided into 12 parts, because the Pythagorean Comma is 3^12.)

There is thus a fundamental difference between the way one
understands 1200-EDO and, for example, 72- or 144-EDO.

72-/144-EDO is based on continuous subdivision of 12-EDO,
similar to the way in the "English" system of linear measurement
1 foot is divided into 12 inches, because 12 is easily divided
into 2, 3, 4, or 6, so that people could think in terms of 1/2,
1/3, 1/4, or 1/6 of a foot, or the way an hour is divided into
60 minutes and a minute into 60 seconds, because 60 is even more
readily divisible (believe it or not, this is actually a
surviving relic of Sumerian mathematics, 5000 years ago).

1200-EDO, on the other hand, while not as easy to divide into
fractional subsets, is a sort of "metric system" for dividing
12-EDO. So 72-/144-EDO is thus admittedly more difficult.

I would say that all other EDO divisions are even more difficult
to comprehend intutively than 144-/72-/48-/36-/24-/18-/16-/12-
etc.-EDO, because they use more unwieldy divisions of the "octave"
than 12-EDO. (Of course, any notation can become easy and
familiar with repeated exposure and practice.)

Because of its relatively small size, 19-EDO is a good notational
alternative to 12 that gives a good representation of meantone
harmonic theory, the basis of the so-called "common practice".

(Generally, "standard" theory/harmony texts already use a notation
that is consistent with 19-EDO-as-1/4-comma-meantone, so all one
needs to do to *hear* meantone in action as the foundation of
European "classical" harmony is play the musical illustrations
in these books in 19-EDO tuning!

Just don't try this with Schoenberg's _Harmonielehre_... that
one *requires* 12-EDO for realization of the music examples if
one intends to follow Schoenberg's line of argument properly to
the end of the book. The ability to use 19-EDO effectively for
the European "classical" repertoire begins to break down with
Beethoven's music, and with Wagner's music from _Tristan_ [1865]
and after it won't work.)

The HEWM notation is more difficult still, because the prime-factor
representation is even harder to conceptualize, but it is in many
ways the most useful, especially for analyzing harmony if one
accepts either (or both) the small-integer-ratio and/or the
overtone-series-as-paradigm harmonic theories.

(Note that the latter, albeit oddly distorted because of his
adherence to the errors of 12-EDO, was indeed Schoenberg's basis
too.)

And I think Paul Erlich's 22-EDO decatonic notations are really
ingenious and useful for those who work in that tuning.

Fokker's 31-EDO notation is good for exploring that EDO - and
there's info about it on the web at
http://bikexprt.com/music/notation.htm

And Erv Wilson has devised a number of interesting microtonal
notations for various-sized "moduli", as he calls them (see his
_Xenharmonikon_ articles, most of them available at
http://www.anaphoria.com

The only other EDO of real historical importance that I haven't
covered is 53-EDO. I'd very much like to know more about
notations for this, especially (if I'm correct that he had one)
Tanaka's.

... back to
/tuning/topicId_20929.html#21387

> > [me, monz]
> > In Johnston's notation, both the #/b *and* +/- accidentals
> > *always* indicate an adjustment which includes *both*
> > prime-factors 3 and 5, because his basic scale includes both.
> > This is my primary difficulty with his system.
> >
>
>
> You're saying, then, that it's not entirely clear what's implied
> when using either the #/b or +/- ??
>

NO!!!!!!

What's implied by the #/b or +/- in Johnston's notation is always
*crystal clear*!

It's the 2-dimensional "lattice shape" of the *basic scale* that's
faulty!

In both the Johnston and HEWM systems, every accidental in the
system refers to an alteration in a specific set of dimensions,
in lattice terms.

Differing fundamentally from the Johnston system in only one
aspect, the HEWM notation uses a 1-dimensional scale (the 3-limit
Pythagorean diatonic) as its basis.

But - this is another important difference - in the HEWM system,
each unique prime-factor accidental above 3 moves the tuning only
*one* exponent further along *that* axis! This is not the case
with prime-factor 5 in Johnston's system.

The Johnston notation uses a 2-dimensional scale as its basis.
So The #/b requires a shift of *two* degrees (exponents) along
the 5-axis, because 25/24 == 3^-1 * 5^2. This is a further
complication.

Because the HEWM basic scale is 1-dimensional, and because all
*required* accidentals (i.e., prime-factors above 3) follow the
same consistent operation, the lattice movements of the
alterations caused by every *required* accidental are
*intuitively* easy to visualize once they are learned. A
huge advantage is that the basic scale doesn't have to be
learned, because *it* follows the rules too.

This property of logical consistency also makes it easy to adapt
HEWM notation to computer algorithms. I wrote a bunch of small
programs in Microsoft QBasic several years ago which worked great,
once I figured out how to make QBasic understand the "accidental
rules". The programs gave me printouts of lattices complete with
the letter-name notation, so that I didn't have to figure that
out for hundreds of notes.

Having made all that fuss about how the Johnston and HEWM systems
differ in representing prime-factor 5, note that in both systems,
accidentals representing exponents of 3 do not follow the rule of
increasing the exponent by 1. This is due to the use of different
letter-names to represent different powers of 3.

Exponents of 3 never have to be designated explicitly in either
system, (altho it is frequently useful to include them anyway
in my version of HEWM), because the #/b and their extensions
x/bb (and x#/bbb, xx/bbbb, and beyond, which are actually found
in Johnston's music) carry with them implicit 3-limit lattice
movements, in both systems.

However, in Johnston's system there is no specific way to
identify *only* 3-limit movement by itself, because of the
2-dimensional nature of the basic scale; the "lower accidentals"
#/b and +/- *both* incorporate both prime-factors 3 *and* 5.

As Erlich has repeatedly pointed out, if one creates a 3-limit
Pythagorean chain of 5ths in Johnston notation, plusses and
minuses will have to be added to the accidentals with every few
steps along the 3-axis as one expands away from the reference
tone and towards either end of the linear system.

In the HEWM notation, strictly 3-limit movement is designated
strictly by the "cycle-of-5ths" progression familiar to any
musician who understands standard music notation. Exponents
of 3 are increased or decreased by 1 as one encounters each new
letter-name in the "cycle-of-5ths", and sharps and flats
consistently mean an alteration of +/- 3^7 [= ~113.685 cents,
the "apotome"] from the basic A B D C E F G Pythagorean diatonic
pitches. (It's advisable to use "cycle" here instead of "circle",
because in JI it's really a spiral instead of a circle.) +/- are
used *only* in connection with prime-factor 5.

Thus, another advantage in the HEWM system is that both types of
Pythagorean semitones (the limma and apotome... look them up my
Dictionary if you're not familiar) are easily notated in a
systematic way, which is not the case with Johnston's system.
This makes it much more useful for those who work a lot in
Pythagorean tuning, as the historical Pythgorean origin of our
notational system of letter-names, sharps, and flats is retained.

(Hmmm... Thinking about this right now makes me particularly
interested in how this notation might be good for exploring
microtonal aspects of Ives's music, if one accepts Johnny
Reinhard's postulate that Ives preferred Pythgorean, which I'm
willing to entertain. The basic notation gives Ives's basic
Pythagorean scale, and the prime-factor accidentals using 5 and
above would provide the basis for analysis of rational
implications of Ives's occassionally-used 1/4- and 1/8-tones.)

The only way HEWM could have been made *entirely* consistent
would have been to let the successive letter-names represent
increasing/decreasing exponents of 3. For example, this time
let's let "A" = n^0 = 1/1, which was the situation when our
European notation was first invented.

"B" would be 3^1 = 3/2,
"C" would be 3^2 = 9/8,
"D" would be 3^3 = 27/16,
"E" would be 3^4 = 81/64,
"F" would be 3^5 = 243/128,
"G" would be 3^6 = 729/512.

From this point the system could probably just keep using the
alphabet, not needing to invoke a sharp or flat until "Z" is
reached. Or, more probably, it would make sense to invoke the
accidental to represent the Pythagorean comma [== 3^12 =
531441/524288 = ~23.46 cents]. But both of these seem too
bizarre to be useful to anyone who already knows regular notation.

(Note that some medieval theorists, such as Boethius, did use
diagrams in their treatises which were labelled similarly to this,
to explain monochord division.)

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_20929.html#21390

> Well, the point is that I should KNOW the Johnston AND MonzoWolf
> system thoroughly as part of my xenharmonic studies... As my
> music evolves and develops more in these directions, I will,
> thereby, know the right kind of notation to use... So some of
> this is a bit in the "R&D" area...

I applaud this healthy catholic attitude, Joe.

Keep in mind what I wrote above: you'll probably only find
Johnston's notation really useful yourself if you intend to work
a lot in 5-limit JI, and you'd better enjoy looking at lattices,
because you'll have to use them constantly until you memorize them.

But by all means, if you're intrigued enough by Johnston's system,
don't allow me to discourage you from learning it well and using it
when you feel the need. (But I always end up cursing it when I'm
working on Johnston's music...)

> > [Paul Erlich:]
> > Or you might even want to use 72-tET notation (if you're not
> > moving across vast stretches of the lattice)
>
>
> I never had any trouble conceptializing 72-tET... and there is
> quite a bit of PRACTICAL literature out about it. That's a LOT
> different, from a practical standpoint, than these "comma based"
> systems, in MY opinion...

On the contrary, Joe, once you understand how the HEWM notation
works, you can apply it to the large literature of rational-tuning
harmonic theory.

For just one example, Partch's entire book could be rewritten
translating all the musical examples and pages filled with ratios
into HEWM notation (and, of course, the associated lattice
diagrams, all of which I would present as subsets implanted within
the big 43-tone lattice). That, to me, would make it *much*
easier to grasp Partch's theories. (This is a project that I
actually hope to accomplish some day.)

So while it's true that there's not a lot published which actually
already *uses* HEWM notation, it's very easy to adapt the notation
to any rational system that *has* already been published, and
there are tons of those, going back at least to the ancient Greeks
if not further.

As I've made clear here before, I totally agree with Paul Erlich
about the utility of 72-EDO notation for a wide variety of
microtonal applications, grounded in either JIs or ETs. If I
were doing a lot more composing than theorizing, I'd probably be
using it all the time.

I mentioned near the beginning that the HEWM system of accidentals
can be easily combined with pitch-height notations more
sophisticated than 12-EDO.

I like to combine the 72-EDO accidentals with my prime-factor
accidentals when I use a regular 5-line staff, supplementing the
sharps and flats of 12-EDO with +/-, >/<, and ^/v to indicate
1/12-, 1/6-, and 1/4-tones respectively. It helps give a clearer
picture of the pitch-height of a given pitch.

In some cases, where the musical context demands recognition of
pitch differences smaller than the 16-&-2/3-cent basic step-size
of 72-EDO (for example, as in my _A Noiseless Patient Spider_),
I use my adaptation of Dan Stearn's 144-EDO notation (which itself
is a simple adaptation of the Sims 72-EDO notation).

(In the copy of the score you have, Joe ...which is still the
only one that exists :) ...I didn't bother to specify the
prime-factors, because the music was already realized on computer.
Some day, I hope to get around to writing an accurate Monzo-
notation score that could be used for live performance.)

Actually, given the ~5-cent acceptance of variability in intended
intonation that I mentioned above, the 144-EDO notation, and
usually even the 72-EDO notation, is accurate enough to ensure an
intonationally-reasonable live performance of my music anyway.
(Ezra Sims has written that when he listens to computer mock-ups
of his 37-limit-JI pieces, tuned in both JI and 72-EDO, he can't
hear a difference between the two.)

So the prime-factor accidentals in my scores are only really
valuable for analytical purposes, to study my JI harmonic
techniques, exactly as they're irreplaceable when analyzing
someone else's harmony.

A more radical alternative I like is the notation I invented in
connection with my work on my MIDI-file of Haba's _2nd Quartet_.
I showed an example of this at my Microfest presentation. Each
semitone is represented by a staff-line, and the quartertones go
in the spaces. This gives the advantage of presenting the pitch-
height graphical element of the notation quantized to the quarter-
tone level.

Other accidentals can then also be added. For example:

- 48-EDO could be easily represented by one accidental symbol, to
represent either the raising or lowering of any pitch by a
1/8-tone.

- 72-EDO could be represented very simply by using two more
symbols, to represent the raising and lowering of any pitch
by 1/12-tones. See a very crude example of the 72-EDO scale
on the quarter-tone-based staff, and some 72-EDO quarter-
tone-based notations of segments of the harmonic series, here:
http://www.ixpres.com/interval/monzo/72edo/72edo-on-qt.jpg

- Prime-factor notation can be easily incorporated by simply
adding the prime-factor accidentals.

The quarter-tone-based notation simply gives a more accurate
quantization of the pitch-height aspect of staff-notation.

There's one more important advantage to it, the disadvantage of
which in our regualr staff-notation is often overlooked: it
equalizes the meaning of the vertical spacing of the lines in
the staff.

To modern sensibilities used to thinking in terms of
logarithmically equal divisions of the pitch-continuum, regular
staff notation has the defect that moving between a line and a
space on the staff can mean either of two intervals: a semitone
(between B:C and E:F) or a whole-tone (between A:B, C:D, D:E,
F:G, and G:A). Thus the correlation between the visual aspect
and the musical meaning is not consistent.

This happened because staff-notation developed during the early
part of the last millenium to represent diatonic music, which had
a combination of semitones and whole-tones built into it, and the
inventors of the notation simply didn't consider this inconsistency
to be important. The staff clearly made it much easier to remember
and communicate music, and so it was quickly adopted and then used
as the basis for most of the notational development which followed.

In my quarter-tone-based notation, the staff has a line missing
at regular intervals on the page to represent each "octave" of "C"
(Hmmm... maybe I should make it "A" instead, for even more logical
consistency.). This acts a separator for the staves, so that each
staff covers the span of an "octave".

Then there are two different thicknesses of lines: the thin lines
represent the 12-EDO pitches of the white keys on the piano
keyboard, and the thicker lines represent 12-EDO pitches of the
black keys. The spaces are for the quarter-tones, which thus
never need accidental symbols.

Stretching the actual vertical space which represents an "octave"
on the paper has another nice result: I can stack an 8-"octave"
pitch-height space onto one page (which equals one "system", in
music-score language), and have plenty of room to write several
different parts legibly on the same system/page. So the original
score is simultaneously also a kind of "reduction" that shows the
interrelationship of the different parts. For an example, see my
renotation of the first system of Haba's _2nd Quartet_ (with all
four parts written on the same staff) at:

http://www.ixpres.com/interval/monzo/haba/monzo-qt-notation.jpg

(My quarter-tone-based staff is adapted from the new 12-EDO staff
notation invented by Max Meyer in _The Musician's Arithmetic_
[1929].

Schoenberg also invented a very interesting new 12-EDO notation
which used a very compressed staff and so cannot be further
adapted to microtonality. But it makes a lot of sense for his
12-EDO music. The article, with illustrative examples, is in
_Style and Idea_.)

Last note: one drawback of *all* of these systems is that they all
assume "octave"-equivalence. I haven't even begun to wrestle with
the problem of notating non-"octave" scales.

I sure hope that helps, Joe (and anyone else who needed it).
Took me all day to write it. I basically rewrote a lot of
what's in my old article. The fact that I went on about it
for such a long time shows you how passionately I care about
the subject.

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

--- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> Note, however, that because both 7^1 and 11^1 are so far from
> their 12-EDO representations (~31 and ~49 cents respectively),
> this last example is actually much closer to the 12-EDO Eb in
> pitch (in fact at ~320 cents it's nearly a 6/5). This is an
> inconsistency, not in the sense of Erlich's definition of
> consistency, but definitely still some kind of inconsistency
> in terms of what those letter-names are supposed to mean.
> (Perhaps Paul will elaborate on this.)

Here's an example that better illustrate the connection with my
definition of consistency.

You notate 11 as F#, and 13 as Ab, but 13/11 is actually much closer
to Eb than to D. This is because 12-tET is grossly inconsistent in
the 13-limit.

Therefore...
>
>
> 2. It might be preferable to simply use the 12-EDO pitch that is
> nearest to the pitch defined by the ratio. This will
> definitely result in Erlichian inconsistencies in a few
> places,

OK . . . this is the _other_ kind of inconsistency, that came up
before when we disagreed on how to notate Partch's 43 tones in 144-
tET notation . . . it's not the kind of inconsistency with which my
name is usually associated (better illustrated above).

> (Hmmm... if that type of "congruity/incongruity" is different
> enough from Erlich's "consistency/inconsistency" concept,
> maybe I have another new tuning term to define... Paul, help!)

Are you referring to the above? It's actually related to Paul Hahn's
higher-level consistency definition. Feel free to elaborate and I'll
clarify what I can.
>
> Perhaps some of the difficulty you are having understanding my
> system, Joe, is that in my article and book I also mention another
> way of using the same concept, where one may dispense with the
> prime-factors themselves in the accidental, and indicate only
> the values of the exponents; so that, for example, the 35:33
> illustrated above would be rendered as -1 1 1-1. This should
> be very familiar to anyone who's made use of the vector notation
> of the prime-factors, because that's exactly what it is, without
> the enclosing parentheses or brackets that are usually included.

Vector notation is also covered in my _Gentle Introduction to
Periodicity Blocks_.

> Since I already use 144-EDO,
> with its great properties of multiple subdivision (into 72-, 48-,
> 36-, 24-, 18-, 16-, 12-, 9-, 8-, 6-, 4-, 3-, and 2-EDO) and an
> accuracy of 8-&-1/3 cents, 144-EDO is good enough for me. It
> also gives pretty good representations of 11-, 13-, 29-, 47-, and
> 49-EDO, and a few on either side of 72-EDO, so it's handy for
> exploring Stearns-like uses of the pitch-continuum.

I'm puzzled by why you would focus so much attention on JI and ETs,
which are the two "extremes" as it were, while
ignoring "intermediate" systems. For example, meantone can't be
adequately notated in 144-tET notation. But since Joseph is going to
use JI, assuming he's not going beyond the 17-limit, 72-tET notation
may be ideal for him.
>
>
> The HEWM notation is more difficult still,

This was a strange leap, Monz . . . from 19-tET as meantone to HEWM.

You can't even notate meantone in HEWM.

> because the prime-factor
> representation is even harder to conceptualize, but it is in many
> ways the most useful, especially for analyzing harmony if one
> accepts either (or both) the small-integer-ratio and/or the
> overtone-series-as-paradigm harmonic theories.

I accept both but I reject HEWM for analysing music in the meantone
era (roughly 1480-1720) because it would be forced to make notational
distinctions with no musical meaning.
>
> And I think Paul Erlich's 22-EDO decatonic notations are really
> ingenious and useful for those who work in that tuning.

Note that the correct key signatures for these are only available in
my new paper, which, as soon as I get them back from Bill Alves, I'll
be sending you (Monz) and Joseph copies of.
>
> Fokker's 31-EDO notation is good for exploring that EDO - and
> there's info about it on the web at
> http://bikexprt.com/music/notation.htm

And it's also consistent with how you would play music from the
meantone era in 31-tET, since 31-tET is essentially 1/4-comma
meantone.
>
> And Erv Wilson has devised a number of interesting microtonal
> notations for various-sized "moduli", as he calls them (see his
> _Xenharmonikon_ articles, most of them available at
> http://www.anaphoria.com

These are fascinating and should be studied as well, for the
intellectual beauty if nothing else. Interestingly, Wilson proposes
notation for moduli 7, 8, 9, 11, 12, and 13 . . . but skipped 10!
Curious how my work happened to fit into Erv's work so neatly (and
totally accidentally!)

> The only other EDO of real historical importance that I haven't
> covered is 53-EDO. I'd very much like to know more about
> notations for this

Well, 53-tET _is_ for all intents and purposes 5-limit JI, and Ellis
often wrote with 53 in mind . . . hence 5-limit HEWM is
the "standard" 53-tET notation.

Actually, if I recall correctly, I'm going to have to add Larry
Hanson to the list of notational "renegades" (including Johnston and
Fokker) . . . Larry Hanson's 34-tET notation at least, and I would
assume his 53-tET notation too, departed from the 5-limit HEWM-like
notation normally proposed for these tunings.

>
>
> The only way HEWM could have been made *entirely* consistent
> would have been to let the successive letter-names represent
> increasing/decreasing exponents of 3. For example, this time
> let's let "A" = n^0 = 1/1, which was the situation when our
> European notation was first invented.
>
> "B" would be 3^1 = 3/2,
> "C" would be 3^2 = 9/8,
> "D" would be 3^3 = 27/16,
> "E" would be 3^4 = 81/64,
> "F" would be 3^5 = 243/128,
> "G" would be 3^6 = 729/512.

Hmm . . . I don't know if I'd call that "consistent" -- actually I'd
call it "screwy" from a melodic point of view -- which is probably
the most important point of view to keep in mind when designing a
notation (if nothing else, the musician has got to know whether she's
going up or down!)

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_21424.html#21449

Hi Paul. Thanks for responding to this. I'm going to have to
think more (when I have more time) about your consistency
explanations, before I reply to them. But here are some other
responses:

> > [me, monz]
> > The HEWM notation is more difficult still,
>
> This was a strange leap, Monz . . . from 19-tET as meantone to
> HEWM.
>
> You can't even notate meantone in HEWM.

Well... In *my* version of HEWM you can - remember a while back
when I asked you to show me how to factor meantone calculations?
Well, those would be my accidentals. But it's probably not
worth it, as they would be full of fractions as well as exponents,
IOW, ridiculously complicated. But anyway, if you *have* to
notate a meantone in HEWM, you can do it with my version.

>
> > because the prime-factor
> > representation is even harder to conceptualize, but it is in many
> > ways the most useful, especially for analyzing harmony if one
> > accepts either (or both) the small-integer-ratio and/or the
> > overtone-series-as-paradigm harmonic theories.
>
> I accept both but I reject HEWM for analysing music in the
> meantone era (roughly 1480-1720) because it would be forced to
> make notational distinctions with no musical meaning.

Hmmm... I might disagree with you there, Paul. HEWM would
obviously not be entirely accurate without some kind of
modification to take meantone into consideration, but the
whole point of meantone was to give good approximations of
prime-factor 5 while retaining some resemblance to the 3-limit
scale.

Seems to me that HEWM could be harnessed for use in analyzing
this music, if for no other reason than its broad flexibility
(I'm thinking particularly of my version here). For example,
a high-prime factor could be invoked to give a close approximation
to certain meantone intervals. I'll have to think about it more,
maybe come up with a few examples for you.

> > Fokker's 31-EDO notation is good for exploring that EDO - and
> > there's info about it on the web at
> > http://bikexprt.com/music/notation.htm
>
> And it's also consistent with how you would play music from the
> meantone era in 31-tET, since 31-tET is essentially 1/4-comma
> meantone.

Of course. Thanks for that... I should have mentioned it.

> >
> > And Erv Wilson has devised a number of interesting microtonal
> > notations for various-sized "moduli", as he calls them (see his
> > _Xenharmonikon_ articles, most of them available at
> > http://www.anaphoria.com
>
> These are fascinating and should be studied as well, for the
> intellectual beauty if nothing else. Interestingly, Wilson
> proposes notation for moduli 7, 8, 9, 11, 12, and 13 . . .
> but skipped 10! Curious how my work happened to fit into
> Erv's work so neatly (and totally accidentally!)

Since I'm the one who brought up Wilson's notations, I might
as well admit that I haven't really studied them all that much.
(You proabably know them much better than I, Paul.)

So many tunings... so little time...

>
> > The only other EDO of real historical importance that I haven't
> > covered is 53-EDO. I'd very much like to know more about
> > notations for this
>
> Well, 53-tET _is_ for all intents and purposes 5-limit JI, and
> Ellis often wrote with 53 in mind . . . hence 5-limit HEWM is
> the "standard" 53-tET notation.

OK, sure, that makes total sense. Should have thought of that
myself too. (... really late at night, lack of sleep, etc.)

My brain's too foggy at the moment... could you say a little
on how to determine the closure of the 53-tone system?, because
in a JI system that big, you *do* end up with small "leftover"
intervals at the edges of the periodicity-block.

I should also mention that Johnston's first JI system was
a 53-tone 5-limit system. That was the starting-point for
*his* notation.

>
> Actually, if I recall correctly, I'm going to have to add
> Larry Hanson to the list of notational "renegades" (including
> Johnston and Fokker) . . . Larry Hanson's 34-tET notation at
> least, and I would assume his 53-tET notation too, departed
> from the 5-limit HEWM-like notation normally proposed for
> these tunings.

Hmmm... I'd like to know more about those. Any references?

>
> >
> >
> > The only way HEWM could have been made *entirely* consistent
> > would have been to let the successive letter-names represent
> > increasing/decreasing exponents of 3. For example, this time
> > let's let "A" = n^0 = 1/1, which was the situation when our
> > European notation was first invented.
> >
> > "B" would be 3^1 = 3/2,
> > "C" would be 3^2 = 9/8,
> > "D" would be 3^3 = 27/16,
> > "E" would be 3^4 = 81/64,
> > "F" would be 3^5 = 243/128,
> > "G" would be 3^6 = 729/512.
>
> Hmm . . . I don't know if I'd call that "consistent" -- actually
> I'd call it "screwy" from a melodic point of view -- which is
> probably the most important point of view to keep in mind when
> designing a notation (if nothing else, the musician has got to
> know whether she's going up or down!)

Yeah, I would never actually advocate a "screwy" system like this.

Just wanted to present it for the sake of completeness, because
it struck me as very interesting how HEWM's treatment of 3
differs from its treatment of all other prime-factors, which
I suppose is simply confirmation that it is indeed "Pythagorean
based".

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

--- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> --- In tuning@y..., PERLICH@A... wrote:
>
> /tuning/topicId_21424.html#21449
>
>
> Hi Paul. Thanks for responding to this. I'm going to have to
> think more (when I have more time) about your consistency
> explanations, before I reply to them. But here are some other
> responses:
>
> > > [me, monz]
> > > The HEWM notation is more difficult still,
> >
> > This was a strange leap, Monz . . . from 19-tET as meantone to
> > HEWM.
> >
> > You can't even notate meantone in HEWM.
>
>
> Well... In *my* version of HEWM you can - remember a while back
> when I asked you to show me how to factor meantone calculations?
> Well, those would be my accidentals. But it's probably not
> worth it, as they would be full of fractions as well as exponents,
> IOW, ridiculously complicated. But anyway, if you *have* to
> notate a meantone in HEWM, you can do it with my version.

I can't agree about the "HEW" part . . . but of course for the "M"
part, M is correct . . . and yes, it would look ridiculously
complicated, and not even be uniquely defined (remember, the
Fundamental Theorem of Arithmetic only applies to _integers_).

> > > because the prime-factor
> > > representation is even harder to conceptualize, but it is in
many
> > > ways the most useful, especially for analyzing harmony if one
> > > accepts either (or both) the small-integer-ratio and/or the
> > > overtone-series-as-paradigm harmonic theories.
> >
> > I accept both but I reject HEWM for analysing music in the
> > meantone era (roughly 1480-1720) because it would be forced to
> > make notational distinctions with no musical meaning.
>
>
> Hmmm... I might disagree with you there, Paul. HEWM would
> obviously not be entirely accurate without some kind of
> modification to take meantone into consideration, but the
> whole point of meantone was to give good approximations of
> prime-factor 5 while retaining some resemblance to the 3-limit
> scale.
>
> Seems to me that HEWM could be harnessed for use in analyzing
> this music, if for no other reason than its broad flexibility
> (I'm thinking particularly of my version here). For example,
> a high-prime factor could be invoked to give a close approximation
> to certain meantone intervals. I'll have to think about it more,
> maybe come up with a few examples for you.

Please do. My example is the comma-pump sequence -- in HEW you'd be
forced to either have the progression drift or have a comma shift
between the two occurences of D -- neither of which occur in the
meantone rendition, and neither of which would have anything to do
with anything going on in the mind of any meantone-era composer.
>
> My brain's too foggy at the moment... could you say a little
> on how to determine the closure of the 53-tone system?, because
> in a JI system that big, you *do* end up with small "leftover"
> intervals at the edges of the periodicity-block.

Well, as you might expect, there are several different ways to
construct a 53-tone periodicity block . . . you need two 5-limit
unison vectors . . . one pair of unison vectors that might have
relevance for Johnston's first JI system is the schisma and the
Mercator's comma. The former is 2 cents and the latter is less than 4
cents . . . so the fifths and thirds straddling the edges of the
block would be off by this amount . .. so this 53-tone just system
might be said to be indistinguishable from 53-tET in a performance
setting.
>
> >
> > Actually, if I recall correctly, I'm going to have to add
> > Larry Hanson to the list of notational "renegades" (including
> > Johnston and Fokker) . . . Larry Hanson's 34-tET notation at
> > least, and I would assume his 53-tET notation too, departed
> > from the 5-limit HEWM-like notation normally proposed for
> > these tunings.
>
> Hmmm... I'd like to know more about those. Any references?

I saw the Hanson notation in Xenharmonikon 17, the same issue in
which my paper was published . . . it's been well over a year since I
had a copy of that, though . . .

Paul Erlich wrote,

<<meantone can't be adequately notated in 144-tET notation.>>

While there's no good reason why you would want to just because you
can, nevertheless you can. (Notate meantones in 144 that is... I've
used it for 19 many times, 31 too.)

My whole rationale behind the 144 notation was that any tuning could
be pressed through it and work. (For the degree of tuning accuracy
that my music would generally need to make itself manifest anyway.)

The convenience of a one-size-fits-all notation that was (relatively)
easy to internalize overruled most other concerns at that point.

However, once I didn't much think about performances anymore (a dash
of self-preservation is a good thing sometimes), I also didn't much
care about that convenience anymore! So I started using different
notations to fit the different tunings and pieces. And now a piece
that uses several different tunings mixes notations as well.

Actually not having to notate anything seems to work best... but I
still like to chip away at it occasionally, so I do.

It's fun (unless it ain't).

--Dan Stearns

--- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> Joe [Pehrson],
>
> Here is a collective response (it got rather long...) to a number
> of your posts, in which I give a comprehensive explanation of my
> notational systems. (I do make use of several.)
>

Thank you so much, Monz for your thorough ideas concerning notation.
It IS a comprehensive post... in fact it is SO comprehensive that
I've had to move over to Internet Explorer to answer it... Netscape
won't let me quote from such a large post (!!)
>
> Perhaps some of the difficulty you are having understanding my
> system, Joe, is that in my article and book I also mention another
> way of using the same concept, where one may dispense with the
> prime-factors themselves in the accidental, and indicate only
> the values of the exponents; so that, for example, the 35:33
> illustrated above would be rendered as -1 1 1-1. This should
> be very familiar to anyone who's made use of the vector notation
> of the prime-factors, because that's exactly what it is, without
> the enclosing parentheses or brackets that are usually included.
>

This is exactly right... the use of the exponents thoroughly confused
me... In fact, THAT is the usage that I MOST remember about the
system...

>
> I find that the version which includes the prime-factor and omits
> those factors with a zero exponent is usually more compact and
> therefore more useful for actual staff-notation, because in any
> given section of a composition, most ratios have only a few (1, 2,
> or 3) factors, thus there is no need to have long strings of zeros
> in the accidental. All of the actual musical examples in my book,
> and all of my scores written in prime-factor notation, use this
> method.
>

Yes... like I mentioned, I remember this vector approach best from
your book.

>
>
> > I will try to find the Perspectives article on Ben Johnston's
> > notation... although I am beginning to doubt that it is as
> > "direct" as these "improved" systems.
>
>
> I can guarantee that you'll only become comfortable with
> Johnston's notation if you are a hard-core 5-limit JI enthusiast
> with a healthy interest in memorizing lattice diagrams. :)
>
> If you're not, then read those _Perpectives_ articles to get the
> superficial understanding of Johnston's system that you feel you
> need as a minimum, put them in your library for future reference
> if that need arises, and move on.
>

I just ordered the Fonville article from the 1991 Perspectives, and I
feel it is a good addition to my library. Even if the Johnston
notation is somewhat "flawed," it is certainly a noble attempt at
notation.

Frankly, I am *very* interested in notation. I know that David Doty
says he ISN'T... but then perhaps he isn't so interested in working
with live instrumentalists... (??)

>
> If you're not interested in analyzing Johnston's (or his
> students's) music, and not interested in composing music like
> theirs, forget about that notational system, other than for
> having knowledge of it for general purposes. If your primary
> purpose is to learn a useful new tool for use in composing your
> own extended JI music, work with the HEWM Helmholtz/Ellis/Wolf/
> Monzo] notation instead. (This newly-coined acronym has just
> been added to my Tuning Dictionary.)
>

This sounds reasonable... HEWM works well. I also affectionately
call it "Monzowolfellholz" notation...

>
> > The reason, though, that I am so interested in it is, obviously,
> > Johnston's stature as a composer and xenharmonic thinker and the
> > usage of such composers as Doty and Gann...
> >
> > It can't just be "overlooked..."
>
>
> Yes, Joe, that's exactly why I address this issue so strongly.
> Johnston was an influential teacher (he's retired now, composing
> full-time) who had a lot of students. I feel that his notation
> is being accepted unquestioningly by a large (and ever-increasing)
> number of JI composers and theorists, and that if they put a
> little more thought into the problem, they too would probably
> prefer the HEWM system instead.

Paul made a good point that somebody's "stature" is a pretty bs
reason for advocating what is intellectually specious. Good call,
Paul... However, I felt like you did.

>
> Well, that sure makes sense. I've put a lot of thought into the
> microtonal notations that suit my particular uses, and it would
> really make me happy to see my proposals adopted by others here
> who share similar concerns. And I'm around here a lot, always
> available to provide help when it's requested.
>

Believe me, your commentary and assistance is VERY much appreciated.
You are an amazing asset to the entire xenharmonic community!
>
>
>
> The big advantage Johnny's 1200-EDO has over all other systems is
> that it harnesses the power and biological inherency of the decimal
> number system, and this is not something that can or should be
> easily dismissed.

That makes cents, literally...

>
> Albeit limited, our hands and fingers are a built-in abacus.
> 1200-EDO, thought of as (100*12)-EDO, is thus so easy to
> conceptualize and apply for musicians who have already become
> comfortable with 12-EDO that it's certainly the most quickly-
> learned representation of the infinite pitch continuum for a
> trained musician.

I believe this too, and it has been proven to me. They young lady in
Moscow who performed my piece had NO rehearsals with me! She just
got the music, saw the cents notation and said to my friend Anton
Rovner in Moscow, "Oh, no problem... I know how to do this... I've
played this kind of music before..."

>
> 72-/144-EDO is based on continuous subdivision of 12-EDO,
> similar to the way in the "English" system of linear measurement
> 1 foot is divided into 12 inches, because 12 is easily divided
> into 2, 3, 4, or 6, so that people could think in terms of 1/2,
> 1/3, 1/4, or 1/6 of a foot, or the way an hour is divided into
> 60 minutes and a minute into 60 seconds, because 60 is even more
> readily divisible (believe it or not, this is actually a
> surviving relic of Sumerian mathematics, 5000 years ago).
>
> 1200-EDO, on the other hand, while not as easy to divide into
> fractional subsets, is a sort of "metric system" for dividing
> 12-EDO. So 72-/144-EDO is thus admittedly more difficult.
>

This is an interesting analogy... I don't know if it holds up
completely, but I "liked" it...

>
>> In both the Johnston and HEWM systems, every accidental in the
> system refers to an alteration in a specific set of dimensions,
> in lattice terms.
>
> Differing fundamentally from the Johnston system in only one
> aspect, the HEWM notation uses a 1-dimensional scale (the 3-limit
> Pythagorean diatonic) as its basis.
>
>
> But - this is another important difference - in the HEWM system,
> each unique prime-factor accidental above 3 moves the tuning only
> *one* exponent further along *that* axis! This is not the case
> with prime-factor 5 in Johnston's system.
>
> The Johnston notation uses a 2-dimensional scale as its basis.
> So The #/b requires a shift of *two* degrees (exponents) along
> the 5-axis, because 25/24 == 3^-1 * 5^2. This is a further
> complication.
>

Thanks, Monz... This really is a good explanation of the "problem..."

> Because the HEWM basic scale is 1-dimensional, and because all
> *required* accidentals (i.e., prime-factors above 3) follow the
> same consistent operation, the lattice movements of the
> alterations caused by every *required* accidental are
> *intuitively* easy to visualize once they are learned. A
> huge advantage is that the basic scale doesn't have to be
> learned, because *it* follows the rules too.
>

Well, Paul Erlich really showed this with just a few triads. It's
amazing what a few choice examples can do.

>
>
> However, in Johnston's system there is no specific way to
> identify *only* 3-limit movement by itself, because of the
> 2-dimensional nature of the basic scale; the "lower accidentals"
> #/b and +/- *both* incorporate both prime-factors 3 *and* 5.
>

Got it!

> As Erlich has repeatedly pointed out, if one creates a 3-limit
> Pythagorean chain of 5ths in Johnston notation, plusses and
> minuses will have to be added to the accidentals with every few
> steps along the 3-axis as one expands away from the reference
> tone and towards either end of the linear system.
>

I can see the problem almost "visually..."

>
> >
> --- In tuning@y..., jpehrson@r... wrote:
>
> /tuning/topicId_20929.html#21390
>
>
> > Well, the point is that I should KNOW the Johnston AND MonzoWolf
> > system thoroughly as part of my xenharmonic studies... As my
> > music evolves and develops more in these directions, I will,
> > thereby, know the right kind of notation to use... So some of
> > this is a bit in the "R&D" area...
>
>
> I applaud this healthy catholic attitude, Joe.
>
> Keep in mind what I wrote above: you'll probably only find
> Johnston's notation really useful yourself if you intend to work
> a lot in 5-limit JI, and you'd better enjoy looking at lattices,
> because you'll have to use them constantly until you memorize them.
>
> But by all means, if you're intrigued enough by Johnston's system,
> don't allow me to discourage you from learning it well and using it
> when you feel the need. (But I always end up cursing it when I'm
> working on Johnston's music...)
>

It seems like the references will be important additions to my
library... to sit under the bust of Haydn and, hopefully, only
visited at "odd" hours when I'm feeling a bit "ideosyncratic..."

> >
> A more radical alternative I like is the notation I invented in
> connection with my work on my MIDI-file of Haba's _2nd Quartet_.
> I showed an example of this at my Microfest presentation. Each
> semitone is represented by a staff-line, and the quartertones go
> in the spaces. This gives the advantage of presenting the pitch-
> height graphical element of the notation quantized to the quarter-
> tone level.
>
> Other accidentals can then also be added. For example:
>
> - 48-EDO could be easily represented by one accidental symbol, to
> represent either the raising or lowering of any pitch by a
> 1/8-tone.
>
> - 72-EDO could be represented very simply by using two more
> symbols, to represent the raising and lowering of any pitch
> by 1/12-tones. See a very crude example of the 72-EDO scale
> on the quarter-tone-based staff, and some 72-EDO quarter-
> tone-based notations of segments of the harmonic series, here:
> http://www.ixpres.com/interval/monzo/72edo/72edo-on-qt.jpg
>
> - Prime-factor notation can be easily incorporated by simply
> adding the prime-factor accidentals.
>
> The quarter-tone-based notation simply gives a more accurate
> quantization of the pitch-height aspect of staff-notation.
>
>
> There's one more important advantage to it, the disadvantage of
> which in our regualr staff-notation is often overlooked: it
> equalizes the meaning of the vertical spacing of the lines in
> the staff.
>
>
> In my quarter-tone-based notation, the staff has a line missing
> at regular intervals on the page to represent each "octave" of "C"
> (Hmmm... maybe I should make it "A" instead, for even more logical
> consistency.). This acts a separator for the staves, so that each
> staff covers the span of an "octave".
>
> Then there are two different thicknesses of lines: the thin lines
> represent the 12-EDO pitches of the white keys on the piano
> keyboard, and the thicker lines represent 12-EDO pitches of the
> black keys. The spaces are for the quarter-tones, which thus
> never need accidental symbols.
>
>
> Stretching the actual vertical space which represents an "octave"
> on the paper has another nice result: I can stack an 8-"octave"
> pitch-height space onto one page (which equals one "system", in
> music-score language), and have plenty of room to write several
> different parts legibly on the same system/page. So the original
> score is simultaneously also a kind of "reduction" that shows the
> interrelationship of the different parts. For an example, see my
> renotation of the first system of Haba's _2nd Quartet_ (with all
> four parts written on the same staff) at:
>
> http://www.ixpres.com/interval/monzo/haba/monzo-qt-notation.jpg
>

This is very cool and, actually, very "electronically" looking. I
loved looking at this. Now whether anybody can play it is another
matter, but it sure is fun to look at...

> I sure hope that helps, Joe (and anyone else who needed it).
>

Thanks you for this important post, which I have saved, and which I
will place ON TOP of the Johnston references!

I'm all for Monzowolfellholz!

________ _____ ______ ______
Joseph Pehrson

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul Erlich wrote,
>
> <<meantone can't be adequately notated in 144-tET notation.>>
>
> While there's no good reason why you would want to just because you
> can, nevertheless you can. (Notate meantones in 144 that is... I've
> used it for 19 many times, 31 too.)

As I see it, half the meantone intervals would fall "in the cracks"
of 144, requiring 288. The meantone fifth, generator of meantone
temperament, would be 83 1/2 steps of 144-tET.

If you use it to notate 19 and 31, that's fine, given that you
probably had instruments that were really "in" 19 and 31, or else
your use of said tunings was not one that included sweet-sounding
diatonic triadic progressions. But if you trained string players to
play in 144-tET, and then "translated" a score from meantone, I
wouldn't assume (or even expect) it would come out right.

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_21424.html#21481

> --- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Paul Erlich wrote,
> >
> > <<meantone can't be adequately notated in 144-tET notation.>>
> >
> > While there's no good reason why you would want to just because
you
> > can, nevertheless you can. (Notate meantones in 144 that is...
I've
> > used it for 19 many times, 31 too.)
>
> As I see it, half the meantone intervals would fall "in the cracks"
> of 144, requiring 288. The meantone fifth, generator of meantone
> temperament, would be 83 1/2 steps of 144-tET.
>

This is actually pretty funny... since when I joined this list
probably a year and a half ago... Paul Erlich wrote to me in an off-
list post that Joe Monzo was trying to figure out meantone in his
JustMusic system and it was incredibly complex.

And here everything is cycling all over again!

_______ ______ ______
Joseph Pehrson

Paul Erlich wrote,

<<But if you trained string players to play in 144-tET, and then
"translated" a score from meantone, I wouldn't assume (or even expect)
it would come out right.>>

I really doubt that it would be anywhere near as not right as you
might think.

Listen it's not that I'm advocating 144 notation as a meantone
notation, I'm not. But meantone is a good example when comparing 144
to 72 of a system that shoehorns in with a lot more agreeable results.

And once again my aim was in the direction of the most tolerably
pliable, one-size-fits-all tuning with the least difficult learning
curve when compared to standard notation.

--Dan Stearns

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

/tuning/topicId_21424.html#21424

>
> Joe [Pehrson],
>
> Here is a collective response (it got rather long...) to a number
> of your posts, in which I give a comprehensive explanation of my
> notational systems. (I do make use of several.)
>

I would like to thank Monz again for this wonderful post, #21424,
which is one of the most interesting posts for me that I have ever
yet seen on the Tuning List! Maybe it's partially because I am so
interested in notation AS IT RELATES to xenharmonic systems. As we
have seen in the Johnston, there is an implication that NOTATION CAN
DETERMINE the nature of the writing! In a way, it's a little like
the criticism of MIDI composition that "quantizes" so much to a
steady beat... the "convenience" is there, and the tendency is
to "go with the flow..."

So emphasis on notation, in it's systematic manifestations cannot AT
ALL be minimized by anybody seriously trying to write this stuff...

> Hopefully, I've become a better writer in the 6 years that
> have passed since I originally wrote that paper:
> http://www.ixpres.com/interval/monzo/article/article.htm
>

I wanted to tell you, Monz, that I have now read your post over a few
times and it is ENTIRELY UNDERSTANDABLE, even the "vector" part.
It's TOTALLY CLEAR.

May I PLEASE encourage you to have this introductory material up on
the site demonstrating your Prime Factor notation (which I
affectionately call "Monzowolfellholtz.")

>
> All three of these are intended to function as representations of
> the infinite pitch continuum, whose subdivisions make up the scales
> and harmonies that we analyze in various musical pieces or
> performances.
>

I love, Monz, when you continually bring this up, since that is
the "excitement" of Xenharmonics for me... and your 1/4 tone
notation has that kind of "electronic" look... which just means that
that kind of notation has been generally associated with scores and
pieces that are concerned with the entire sound spectrum. (Think
Stockhausen's Studie II... but that had its own tuning in, I believe,
a non-octave scale...)

> The big advantage Johnny's 1200-EDO has over all other systems is
> that it harnesses the power and biological inherency of the decimal
> number system, and this is not something that can or should be
> easily dismissed.
>
> Albeit limited, our hands and fingers are a built-in abacus.

You know, I even *forgot* about us having 10 fingers.... of course,
perhaps we could have developed with a base 20 system, as well, if we
had used our toes?? ...

> 1200-EDO, thought of as (100*12)-EDO, is thus so easy to
> conceptualize and apply for musicians who have already become
> comfortable with 12-EDO that it's certainly the most quickly-
> learned representation of the infinite pitch continuum for a
> trained musician. (And as I contend in my book, even without
> familiarity with 12-EDO, a musician dealing with a Pythagorean
> chromatic system would naturally tend to feel the "octave" as
> divided into 12 parts, because the Pythagorean Comma is 3^12.)
>

This ie exceptionally clear and it illustrates, as Johnny says,
why "cents make sense..."

> Because of its relatively small size, 19-EDO is a good notational
> alternative to 12 that gives a good representation of meantone
> harmonic theory, the basis of the so-called "common practice".
>
> (Generally, "standard" theory/harmony texts already use a notation
> that is consistent with 19-EDO-as-1/4-comma-meantone, so all one
> needs to do to *hear* meantone in action as the foundation of
> European "classical" harmony is play the musical illustrations
> in these books in 19-EDO tuning!
>

I thought that 19-EDO was 1/3rd rather than 1/4 comma meantone... Am
I confused about something??

>
> What's implied by the #/b or +/- in Johnston's notation is always
> *crystal clear*!
>
> It's the 2-dimensional "lattice shape" of the *basic scale* that's
> faulty!
>
>
> In both the Johnston and HEWM systems, every accidental in the
> system refers to an alteration in a specific set of dimensions,
> in lattice terms.
>
> Differing fundamentally from the Johnston system in only one
> aspect, the HEWM notation uses a 1-dimensional scale (the 3-limit
> Pythagorean diatonic) as its basis.
>
>
> But - this is another important difference - in the HEWM system,
> each unique prime-factor accidental above 3 moves the tuning only
> *one* exponent further along *that* axis! This is not the case
> with prime-factor 5 in Johnston's system.
>
> The Johnston notation uses a 2-dimensional scale as its basis.
> So The #/b requires a shift of *two* degrees (exponents) along
> the 5-axis, because 25/24 == 3^-1 * 5^2. This is a further
> complication.
>

This is a really great explanation, Monz, of this Johnston problem!
Thanks!

Your post is so fascinating, and so "jam-packed" that it required TWO
responses from me as I've been "digesting" it!

___________ ______ _______ ____
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_21424.html#21511\

> > [me, monz]
> > All three of these are intended to function as representations
> > of the infinite pitch continuum, whose subdivisions make up
> > the scales and harmonies that we analyze in various musical
> > pieces or performances.
> >
>
> I love, Monz, when you continually bring this up, since that
> is the "excitement" of Xenharmonics for me... and your 1/4 tone
> notation has that kind of "electronic" look... which just means
> that that kind of notation has been generally associated with
> scores and pieces that are concerned with the entire sound
> spectrum. (Think Stockhausen's Studie II... but that had its
> own tuning in, I believe, a non-octave scale...)

The tuning Stockhausen used in that piece is 5^(1/25), called
"25th root of 5", which is a logarithmically equal division
of a 5:1 ratio into 25 parts. This makes it ~10.767-EDO,
or just over 10-&-3/4 equal steps per "octave". The basic
step size is ~111 cents.

I've been thinking that I should point out that, after
Schoenberg's microtonal experiments of 1908-9 and his decision
to stick with 12-EDO, he thought of 12-EDO as a representation
of the infinite pitch continuum, in the same way that those
of us who use the microtonal notations I've been describing
think of *them*. He simply quantized the continuum to 12 equal
steps instead of using smaller (i.e., 72-EDO) or variable
(i.e., HEWM or "Monzowolfellholtz") divisions.

Reflecting on this, I'd say that it's imperative to label this
tuning as 12-EDO rather than 12-tET in connection with Schoenberg
and his legions of followers.

By the time Schoenberg started writing "atonal" music in 1908,
he was no longer really thinking of 12-EDO as a tempering of
*clearly-implied* just-intonation intervals (the "closer
overtones", in Schoenberg's lingo), which was how it had
always been used before.

It was still a tempering of just intervals according to
Schoenberg's harmonic thinking, but he emphasized that during
this stage of his writing he was thinking exclusively in terms
of the "more remote overtones, which are not as easily
comprehensible as the closer ones" (paraphrased from
_Harmonielehre_ [1911]).

So it was really a way of using the 12-EDO tuning to vaguely
represent *any* subset of the virtual pitch continuum that the
composer wished to use. That's precisely how Schoenberg
thought of it.

> You know, I even *forgot* about us having 10 fingers....
> of course, perhaps we could have developed with a base 20
> system, as well, if we had used our toes?? ...

In fact, there are cultures around the world which *do* use
base-20 number systems.

There are even remnants of it in European languages, as in
the English use of "score" to mean "20" (cf. Lincoln's
_Gettysburg Address_: "Four score and seven years ago..."
to mean 87 years, which is a direct reflection of French,
where 80 is "quatre vignts", or "four 20s"). These usages
are the result of employing the toes as well as the fingers.

Conversely, some counting systems are base-5, using each hand
and foot as a separate system. In fact, that's probably where
the idea of the abacus originated.

(I've figured out a base-6 system where the fingers of one
hand each represent 6^0 [= 1, as in regular counting], the
fingers of the other hand 6^1, the toes of one foot 6^2, and
the toes of the other foot 6^3, which, if one could learn how
to use it, would make the fingers/toes system an accurate
digital computer able to calculate any number up to 1295!

Perhaps this is the origin of the substantial historical
use of 12 in counting: 12 units in a dozen, 12 inches in a
foot, etc.

I realize it seems odd to count in such a way, but people get
used to this kind of thing thru constant use. The Sumerian
mathematicians were able to easily navigate base-60 math
4000 years ago.)

> > (Generally, "standard" theory/harmony texts already use a notation
> > that is consistent with 19-EDO-as-1/4-comma-meantone, so all one
> > needs to do to *hear* meantone in action as the foundation of
> > European "classical" harmony is play the musical illustrations
> > in these books in 19-EDO tuning!
> >
>
> I thought that 19-EDO was 1/3rd rather than 1/4 comma meantone...
> Am I confused about something??

OOPS! My bad! (And even Paul didn't catch that one!)

You're correct, Joe.
19-EDO is nearly identical to 1/3-comma meantone.
31-EDO is the one that's nearly identical to 1/4-comma.

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

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:

> Listen it's not that I'm advocating 144 notation as a meantone
> notation, I'm not. But meantone is a good example when comparing 144
> to 72 of a system that shoehorns in with a lot more agreeable >
results.
>
Agreed. And 288 would be much better than 144 in this regard. In
fact, 288 would be completely flawless for not only meantone, but
also diatonic adaptive 5-limit JI.

--- In tuning@y..., jpehrson@r... wrote:

> >
> > (Generally, "standard" theory/harmony texts already use a notation
> > that is consistent with 19-EDO-as-1/4-comma-meantone, so all one
> > needs to do to *hear* meantone in action as the foundation of
> > European "classical" harmony is play the musical illustrations
> > in these books in 19-EDO tuning!
> >
>
> I thought that 19-EDO was 1/3rd rather than 1/4 comma meantone...
Am
> I confused about something??

You're right, as Monz knows . . . I suspect Monz really
meant "meantone" in general rather than something as precise as 1/4-
comma meantone.

However, the diatonic semitones of 19-tET or 1/3-comma meantone are
_very_ wide . . . so it might be a bit of an extreme choice to
illustrate the foundation of European "classical" harmony. 1/6-comma
meantone enjoyed a "standard" status in some places in the Baroque,
perhaps partially due to the more "incisive" diatonic semitones, the
melodic intervals which had come to play such an important role in
the then-new tonal music.

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_21424.html#21538

> --- In tuning@y..., jpehrson@r... wrote:
>
> > > [me, monz]
> > > (Generally, "standard" theory/harmony texts already use
> > > a notation that is consistent with 19-EDO-as-1/4-comma-
> > > meantone, so all one needs to do to *hear* meantone in
> > > action as the foundation of European "classical" harmony
> > > is play the musical illustrations in these books in 19-EDO
> > > tuning!
> > >
> >
> > I thought that 19-EDO was 1/3rd rather than 1/4 comma
> > meantone... Am I confused about something??
>
> You're right, as Monz knows . . . I suspect Monz really
> meant "meantone" in general rather than something as precise
> as 1/4-comma meantone.

Yup. I've already responded to this, admitting my error.
And now I'm thinking that the whole idea here was wrong
(see below).

>
> However, the diatonic semitones of 19-tET or 1/3-comma meantone
> are _very_ wide . . . so it might be a bit of an extreme choice
> to illustrate the foundation of European "classical" harmony.
> 1/6-comma meantone enjoyed a "standard" status in some places
> in the Baroque, perhaps partially due to the more "incisive"
> diatonic semitones, the melodic intervals which had come to
> play such an important role in the then-new tonal music.

You're absolutely right, Paul. I suppose I was thinking of
meantone in a general sense, and if I *was* thinking of any
specific meantone it certainly would have been 1/4-comma,
which AFAIK was by far the most widely used.

So, my idea about listening to traditional harmony examples
as they represented 19-EDO is probably not a good one.
Any more extensive thoughts on that?

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

>So, my idea about listening to traditional harmony examples as they
represented 19-EDO is probably not a good one. Any more extensive thoughts
on that?

Well, 1/3-comma meantone has been used for some performances (I heard a
Froberger piece in 1/3-comma -- thanks Manuel). But it's definitely at the
extreme end of meantone temperaments that have been used through the ages --
even though more extreme ones like 26-tET are great in more of a xenharmonic
sense. Typically, I'd place the small end of "reasonable" meantone fifths
around Zarlino's 2/7-comma, Smith's 5/18-comma, and Woolhouse's 7/26-comma
meantone temperaments. The large end would have to be at 1/6-comma. Anything
between those two extremes would serve wonderfully for illustrating
traditional harmony, as long as enharmonic equivalencies were not used in
the example. Also, some adaptive tuning (a la John deLaubenfels) or adaptive
JI (a la Vicentino) would be acceptable -- especially desirable if the music
has long, sustained chords.

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_21424.html#21546

> > So, my idea about listening to traditional harmony examples
> > as [if] they represented 19-EDO is probably not a good one.
> > Any more extensive thoughts on that?
>
> Well, 1/3-comma meantone has been used for some performances
> (I heard a Froberger piece in 1/3-comma -- thanks Manuel).
> But it's definitely at the extreme end of meantone temperaments
> that have been used through the ages -- even though more extreme
> ones like 26-tET are great in more of a xenharmonic sense.
> Typically, I'd place the small end of "reasonable" meantone
> fifths around Zarlino's 2/7-comma, Smith's 5/18-comma, and
> Woolhouse's 7/26-comma meantone temperaments. The large end
> would have to be at 1/6-comma. Anything between those two
> extremes would serve wonderfully for illustrating traditional
> harmony, as long as enharmonic equivalencies were not used in
> the example. Also, some adaptive tuning (a la John deLaubenfels)
> or adaptive JI (a la Vicentino) would be acceptable -- especially
> desirable if the music has long, sustained chords.

Thanks, Paul. This is a very good summary in (corrective)
response to my thoughts on the subject.

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

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

/tuning/topicId_21424.html#21528

> Reflecting on this, I'd say that it's imperative to label this
> tuning as 12-EDO rather than 12-tET in connection with Schoenberg
> and his legions of followers.
>
> By the time Schoenberg started writing "atonal" music in 1908,
> he was no longer really thinking of 12-EDO as a tempering of
> *clearly-implied* just-intonation intervals (the "closer
> overtones", in Schoenberg's lingo), which was how it had
> always been used before.
>
> It was still a tempering of just intervals according to
> Schoenberg's harmonic thinking, but he emphasized that during
> this stage of his writing he was thinking exclusively in terms
> of the "more remote overtones, which are not as easily
> comprehensible as the closer ones" (paraphrased from
> _Harmonielehre_ [1911]).
>
> So it was really a way of using the 12-EDO tuning to vaguely
> represent *any* subset of the virtual pitch continuum that the
> composer wished to use. That's precisely how Schoenberg
> thought of it.
>

Thanks so much, Monz, for this interesting historical observation,
and explanation of the way Schoenberg viewed the pitch continuum with
12-EDO!
>
> (I've figured out a base-6 system where the fingers of one
> hand each represent 6^0 [= 1, as in regular counting], the
> fingers of the other hand 6^1, the toes of one foot 6^2, and
> the toes of the other foot 6^3, which, if one could learn how
> to use it, would make the fingers/toes system an accurate
> digital computer able to calculate any number up to 1295!
>

This is quite excellent, Monz... but how do you sit when you do
this... I surely hope not in "lotus position..." (??)

________ _____ _______
Joseph Pehrson