Help: description of Ben Johnston's notation

I have taken the job of engraving a score that contains
accidentals that the composer says are in the just-intonation
notation of Ben Johnston. I would like to understand what
the symbols mean.

The symbols '+', '-', and '7' seem to be prefixed to notes.
They may be combined: I see a combination of flat, '7', and '-'
in one case.

Can anyone point me to a description of what the symbols
mean? I have found descriptions of just intonation, and
I understand the idea; I just can't see what ratios these
symbols are trying to express.

Henry Rich

--- In tuning@egroups.com, glasss@m... wrote:
> I have taken the job of engraving a score that contains
> accidentals that the composer says are in the just-intonation
> notation of Ben Johnston. I would like to understand what
> the symbols mean.
>
> The symbols '+', '-', and '7' seem to be prefixed to notes.
> They may be combined: I see a combination of flat, '7', and '-'
> in one case.
>
> Can anyone point me to a description of what the symbols
> mean? I have found descriptions of just intonation, and
> I understand the idea; I just can't see what ratios these
> symbols are trying to express.

+ and - mean raise or lower the ratio by 81:80; 7 means raise it by
64:63, and an upside-down 7 means lower it by 64:63. Or . . . I may
have the 7 and upside-down 7 reversed . . .

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

> http://www.egroups.com/message/tuning/15258
>
> --- In tuning@egroups.com, Henry Rich <glasss@m...> wrote:
>
> > http://www.egroups.com/message/tuning/15256
> >
> > I have taken the job of engraving a score that contains
> > accidentals that the composer says are in the just-intonation
> > notation of Ben Johnston. I would like to understand what
> > the symbols mean.
> >
> > The symbols '+', '-', and '7' seem to be prefixed to notes.
> > They may be combined: I see a combination of flat, '7', and '-'
> > in one case.
> >
> > Can anyone point me to a description of what the symbols
> > mean? I have found descriptions of just intonation, and
> > I understand the idea; I just can't see what ratios these
> > symbols are trying to express.
>
> + and - mean raise or lower the ratio by 81:80; 7 means raise
> it by 64:63, and an upside-down 7 means lower it by 64:63.
> Or . . . I may have the 7 and upside-down 7 reversed . . .

Paul, your explanation of Johnston's + and - is correct,
but yes, you do have the operation of the '7' backwards, and
also the ratio is not correct. The adjustment indicated
by Johnston's numerical accidentals is always measured in
relation to the 5-limit diatonic major scale.

I use ASCII symbols to represent Johnston's symbols up to
11-limit as follows:

raise lower amount ~cents

+ - 81:80 21.5

L 7 36:35 49

^ v 33:32 53

The capital 'L' is supposed to represent the upside-down '7',
and the caret and lower-case 'v' represent the respective arrows
that Johnston uses for ratios containing 11 as a factor.
'Amount' means the adjustment from the basic 5-limit scale.

For primes higher than 11 (Johnston's later music is 31-limit),
the example of 7 is followed: the prime-number itself is used
for the otonal (harmonic) adjustment, and the number is written
upside-down for the utonal (subharmonic) adjustment.

Here, I'll use 'RU' to mean 'right side up' and 'UD' to
mean 'upside down':

raise lower amount ~cents

13 RU UD 65:64 27

17 RU UD 51:50 34

19 UD RU 96:95 18

23 RU UD 46:45 38

29 RU UD 145:144 12

31 RU UD 31:30 57

A good explanation of Johnston's notation is given in:

Fonville, John. 1991.
'Ben Johnston's Extended Just Intonation: A Guide for Interpreters'.
_Perspectives of New Music_ 29.2 [Summer], p 106-137.

There's an error (a simple typo) on p. 114: Fonville says:

> To raise the 4/3 (cent value 498) by 11/8, or 53 cents,
> one arrives at 551, the cent value of 11/8 notated F^.

The 4/3 is in fact raised by 33/32 (~53 cents), not 11/8,
so that 4/3 *becomes* 11/8 (~551 cents).

Even tho I love Johnston's music and must acknowledge that
his theories are very similar to my own, I want to emphasize,
in agreement with Daniel Wolf, that Johnston's notation,
while admirably compact, is needlessly complicated. It forces
the reader to understand the 5-limit lattice _a priori_ and
to calculate all other prime-factor adjustments thru a 2-step
process in relation to the 5-limit lattice.

Wolf and I both propose modifications of Johnston's ideas
which utilize the linear 3-limit (Pythagorean) tuning as
the _a priori_ basis, since 'standard' notation developed
on that basis anyway: each distinctive letter-name and all
of the associated (and multiple) sharps and flats by themselves
clearly indicate an extended Pythagorean system. Use of prime-
factor 5 is indicated by further accidentals (which are also
required by Johnston anyway), and the other primes follow in
a similar manner.

These systems have the big advantage of requiring _a priori_
systemic knowledge only of the linear Pythagorean system: any
pitch can be calculated by its relation to the Pythagorean tuning,
which in turn can also be calculated from 1/1 by knowledge of the
simple 'rules' of the generating cycle of 4ths/5ths and the
letter-names and sharps/flats.

For a detailed description of my version of this notation see:
http://www.ixpres.com/interval/monzo/article/article.htm

Wolf's system is identical in concept and differs only in
the symbols he uses to indicate the prime-factors: I use
the primes themselves (like Johnston), accompanied by the
prime's exponent, with negative or positive exponents indicating
sharpening or flattening; Wolf uses various typographic symbols
to indicate the primes and the operations. His system is
outlined in:

Wolf, Daniel. 1996.
Letter in _1/1, the journal of the Just-Intonation Network_ 9:3
[Summer], p 15.

All high-prime (5 and above) accidentals beyond the usual
sharps and flats indicate clearly the adjustment for that pitch
from the *Pythagorean* tuning, which is certainly a more 'natural'
mental calculation than Johnston's system, and in fact is the
cause of Paul Erlich's mistake in thinking that Johnston's '7'
accidental indicated a 64:63 adjustment.

Henry, you might want to try to talk this composer into letting
you copy his score using one of these notation systems instead.
In my opinion, it would ultimately encourage more performances,
even tho at the present time I must admit that certainly more
performers are familiar with Johnston's notation than with mine
or Wolf's.

Perhaps the only way to make a microtonal notation more 'natural'
is to show its pitches's adjustments in relation to 12-tET,
because that's the scale with which everyone is familiar,
and because the accursed hardware and software designers insist
on building it into every instrument and program they create.

But it's very difficult to indicate JI tunings *systematically*
in relation to 12-tET, because of the inherent differences
between ETs (which are closed systems) and JI (which are open,
i.e., potentially infinite).

(I'm wondering here if you're using notation software. I use
Finale, which is incredibly flexible, and it's still a bitch to
notate JI music, which must be written as deviations from 12-tET.)

-monz
http://www.ixpres.com/interval/monzo/homepage.html

Monz wrote,

>Even tho I love Johnston's music and must acknowledge that
>his theories are very similar to my own, I want to emphasize,
>in agreement with Daniel Wolf, that Johnston's notation,
>while admirably compact, is needlessly complicated. It forces
>the reader to understand the 5-limit lattice _a priori_ and
>to calculate all other prime-factor adjustments thru a 2-step
>process in relation to the 5-limit lattice.

>Wolf and I both propose modifications of Johnston's ideas
>which utilize the linear 3-limit (Pythagorean) tuning as
>the _a priori_ basis, since 'standard' notation developed
>on that basis anyway: each distinctive letter-name and all
>of the associated (and multiple) sharps and flats by themselves
>clearly indicate an extended Pythagorean system. Use of prime-
>factor 5 is indicated by further accidentals (which are also
>required by Johnston anyway), and the other primes follow in
>a similar manner.

>These systems have the big advantage of requiring _a priori_
>systemic knowledge only of the linear Pythagorean system: any
>pitch can be calculated by its relation to the Pythagorean tuning,
>which in turn can also be calculated from 1/1 by knowledge of the
>simple 'rules' of the generating cycle of 4ths/5ths and the
>letter-names and sharps/flats.

I agree completely.