612-ET notation

Other than sagittal, how else can you notate 612-EDO and other high-order ET's?

(I use numeric notation for my own work, like [4 -1 1] for the 5/4 major third and [10 -1 -2] for 7/4.)

hi Danny,

--- In tuning@yahoogroups.com, "Daniel A. Wier" <dawiertx@s...> wrote:

> Other than sagittal, how else can you notate 612-EDO and
> other high-order ET's?

i'd be surprised if there has ever been any other proposal
for notating 612-edo or others of such high cardinality.

> (I use numeric notation for my own work, like [4 -1 1] for
> the 5/4 major third and [10 -1 -2] for 7/4.)

exactly how does this numeric notation work? it looks like
the first number in each set is the generator "5th" number ...
what are the other two?

-monz

From: "monz": (after I wrote)

>> (I use numeric notation for my own work, like [4 -1 1] for
>> the 5/4 major third and [10 -1 -2] for 7/4.)
>
> exactly how does this numeric notation work? it looks like
> the first number in each set is the generator "5th" number ...
> what are the other two?

The first number is the number of Pythagorean semitones from 1/1 (C/Do/Rast):

1/1 [0 0 0]
256/243 [1 0 0]
9/8 [2 0 0]
32/27 [3 0 0]
81/64 [4 0 0]
4/3 [5 0 0]
1024/729 [6 0 0]
3/2 [7 0 0]
128/81 [8 0 0]
27/16 [9 0 0]
16/9 [10 0 0]
243/128 [11 0 0]
2/1 [12 0 0]
and so on.

Negative numbers are used for pitches below 1/1; simply subtract 12 from the first number for every octave lower as you would add 12 for an octave higher:
243/256 [-1 0 0]
8/9 [-2 0 0]
27/32 [-3 0 0]
etc.

The second number is the number of Pythagorean commas higher or lower:
(assuming 1/1 = C)
E = [4 0 0]
E\ = [4 -1 0]
Gb or F#\ = [6 0 0]
F# or G#/ = [6 1 0]

The third number is the number of twelfths of a Pythagorean comma (or schismas) higher or lower.

If the third number is a zero, it can be omitted; if the second and third number are zero, both can be left out. So 4/3 can be [5 0 0], [5 0] or [5].

The first number could be replaced by a note name like C/Do for 0, D/Re for 2 or Bb/Sib for -2 or 10. but I prefer to use numbers. When indicating comma and fraction shifts in staff notation, I'd put the number of commas as a superscript and the number of schismas as a subscript, for example to the left of a 4 1/12-comma flat (this looks better using a monospaced font):

1 |
|_
-1|/

Or maybe the numbers should be written to the right of the sharp/flat/natural, in between the accidental and the note....

--- In tuning@yahoogroups.com, "Daniel A. Wier" <dawiertx@s...> wrote:

> Other than sagittal, how else can you notate 612-EDO and other
high-order
> ET's?

I've suggested an ennealimmal based notation which would be well
adapted to 612-et. This has nine nominals, each of which corresponds
exactly to a step of 9-et, and to that you can add a symbol for 21/20,
and then various combinations of steps and 21/20 symbol as
abbreviations--(21/20)^2/2^(1/9), etc. Then a 2^(1/18) symbol if you
want to notate 11-limit. The idea is that you can express anything in
the 7 (extendible to 11) limit to within ennealimmal's accuracy, which
for most purposes is much better than really needed.

In 612-et, the 1-5/4-3/2-7/4 chord is 0-197-358-494. A nominal step is
612/8 = 68, and we have 197 = 3*68 - 7, and a 7 is two nominals up and
three 21/20 down, for which we would abbreviate. Similarly 358 is
5*68+18, and an 18 is two 21/20 up and a nominal down, for which we
would also abbreviate. Finally, 494 is 7*68+18.

--- In tuning@yahoogroups.com, "Daniel A. Wier" <dawiertx@s...> wrote:
> From: "monz": (after I wrote)
>
> >> (I use numeric notation for my own work, like [4 -1 1] for
> >> the 5/4 major third and [10 -1 -2] for 7/4.)
> >
> > exactly how does this numeric notation work? it looks like
> > the first number in each set is the generator "5th" number ...
> > what are the other two?
>
> The first number is the number of Pythagorean semitones from 1/1
> (C/Do/Rast):
>
> 1/1 [0 0 0]
> 256/243 [1 0 0]
> 9/8 [2 0 0]

This seems like a problematic choice for a numeric notation, because
you're equating two different "Pythagorean semitones" with a single
vector, [1 0 0]. Wouldn't you instead want a notation where a chain
of two superimposed [1 0 0]s yields a [2 0 0]? Otherwise it seems
you'd tie yourself in knots.

From: "wallyesterpaulrus":

>> The first number is the number of Pythagorean semitones from 1/1
>> (C/Do/Rast):
>>
>> 1/1 [0 0 0]
>> 256/243 [1 0 0]
>> 9/8 [2 0 0]
>
> This seems like a problematic choice for a numeric notation, because
> you're equating two different "Pythagorean semitones" with a single
> vector, [1 0 0]. Wouldn't you instead want a notation where a chain
> of two superimposed [1 0 0]s yields a [2 0 0]? Otherwise it seems
> you'd tie yourself in knots.

I probably need to use a different symbol than brackets then, since I mean to simply draw a "Pythagorean roadmap" rather than indicate a vector.

There are alternative systems, one of them having the first number indicate 12-TET steps. I just thought of another that has the first number actually indicate 9/8 major seconds, then a flat or sharp could be placed next to it if you want to shift the note up or down by 2187/2048. But that would make a perfect fifth [b3 0 0] and a perfect octave [6 -1 0], which looks odd to me.