Notating chains of minor thirds

One problem with using a chain of fourths or fifths as the basis for a notation is that some tunings either have fewer than 7 fifths in a row (multiples of 5-ET), or no good fifths at all (e.g. 11-ET). However, some of these tunings do have reasonably good minor thirds. A chain of seven notes separated by minor thirds can be notated using each of the 7 standard note names, even though more accidentals are required.

E ? 125/108
G)||( 25/18
B\! 5/3
D 1/1
F/| 6/5
A)!!( 36/25
C ? 216/125

One difficulty with this approach is that any other accidentals needed for reaching the other notes need to be combined with these. Another is that there doesn't seem to be a way to represent 250/243. So, as a special case of the "compound nominal" idea, it might be useful to notate chains of minor thirds as #E #G B D F bA bC (using the actual sharp ♯ and flat ♭ symbols where available). This particular pattern of sharps and flats should be enough to signify a chain of minor thirds.

Then, for 15-ET and 11-ET you could have something like:

0 D 0 D
1 D )||( 1 D )//||
2 #E )!!( 2 #E
3 #E 3 F
4 F 4 F )//||
5 F )||( 5 #G
6 #G )!!( 6 bA
7 #G 7 B )\\!!
8 bA 8 B
9 bA )||( 9 bC
10 B )!!( 10 D )\\!!
11 B 11 D
12 bC
13 bC )||(
14 D )!!(
15 D

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> One problem with using a chain of fourths or fifths as the basis
for a
> notation is that some tunings either have fewer than 7 fifths in a
row
> (multiples of 5-ET), or no good fifths at all (e.g. 11-ET).
However,
> some of these tunings do have reasonably good minor thirds. A chain
of
> seven notes separated by minor thirds can be notated using each of
the 7
> standard note names, even though more accidentals are required.
>
> E ? 125/108
> G)||( 25/18
> B\! 5/3
> D 1/1
> F/| 6/5
> A)!!( 36/25
> C ? 216/125
>
> One difficulty with this approach is that any other accidentals
needed
> for reaching the other notes need to be combined with these.
Another is
> that there doesn't seem to be a way to represent 250/243.

Hi Herman,

I'm sorry I haven't been acknowledging any of the good theoretical
work you've doing here. I've been spending a lot of time lately
expanding a highly detailed Sagittal diagram that makes a lot of
information available at a glance (which will help to address some of
the issues you've been raising), so I haven't had much chance to
delve into the specifics of the notations you're proposing.

So for the time being I'm going to have to restrict my comments
mainly to how ratios are notated in Sagittal.

The symbol /|)' is defined as 125M (243:250), and its apotome-
complement (|\. is defined as 125L (512000:531441, which equals three
5-commas) Dropping the right accents (for a lower resolution of
pitch) gives /|) and (|\, which are defined as 35M (35:36) and 35L
(8192:8505) respectively. The difference represented by the right
accent amounts to 4374:4375, or ~0.396 cents. Thus, in less-than-
olympian resolution you'd be using unaccented symbols in secondary
roles for:

E /|) 125/108
C \!) 216/125

> So, as a
> special case of the "compound nominal" idea, it might be useful to
> notate chains of minor thirds as #E #G B D F bA bC (using the
actual
> sharp ♯ and flat ♭ symbols where available). This particular
pattern of
> sharps and flats should be enough to signify a chain of minor
thirds.
>
> Then, for 15-ET and 11-ET you could have something like:
>
> 0 D 0 D
> 1 D )||( 1 D )//||
> 2 #E )!!( 2 #E
> 3 #E 3 F
> 4 F 4 F )//||
> 5 F )||( 5 #G
> 6 #G )!!( 6 bA
> 7 #G 7 B )\\!!
> 8 bA 8 B
> 9 bA )||( 9 bC
> 10 B )!!( 10 D )\\!!
> 11 B 11 D
> 12 bC
> 13 bC )||(
> 14 D )!!(
> 15 D

Since I'm a bit wary of using symbols representing large alterations
in opposite directions, I'd tend to prefer using a greater number of
accidentals. (That's also because I'm already familiar with the
meaning of all of those accidentals.) But you're much more familiar
with the 15- and 11-ET territory (and thus the meaning of those
compound nominals), so I don't think my opinion should count for much
at this point in time. :-)

However, I was thinking that perhaps the # and b symbols in those
compound nominals could go into a kind of key signature, rather than
appearing alongside each note.

No doubt you'll be using whatever works best for you, which is fine
with me, as long as the Sagittal accidentals are being used for
ratios for which they're valid.

--George

George D. Secor wrote:

> The symbol /|)' is defined as 125M (243:250), and its apotome-
> complement (|\. is defined as 125L (512000:531441, which equals three > 5-commas)

I'd forgotten about that one; it's even in the Sagittal paper. I've avoided using it because I've been focusing on 7-limit temperaments (in which 36/35 comes up frequently).

I was thinking the right accent was 4096/4095 (or does it depend on the context?)

> Since I'm a bit wary of using symbols representing large alterations > in opposite directions, I'd tend to prefer using a greater number of > accidentals. (That's also because I'm already familiar with the > meaning of all of those accidentals.) But you're much more familiar > with the 15- and 11-ET territory (and thus the meaning of those > compound nominals), so I don't think my opinion should count for much > at this point in time. :-) > > However, I was thinking that perhaps the # and b symbols in those > compound nominals could go into a kind of key signature, rather than > appearing alongside each note.

The apparent contradiction of directions is one of the big problems with compound nominals (which is one advantage of the differently shaped note heads). Ideally you'd learn to think of #E as a basic note on its own, but #E )!!( does look like you're starting with E, sharpening it, and then flattening it -- is the result higher or lower than E?

I was thinking that a minor third based notation might be appropriate for tunings without good fifths, but actually the standard notation for 11-ET already is a good notation for some of the minor thirds.

G\! G/| A\!
A/| B\! C
C||\ D E!!/
E F/| G\!
G/| A\! A/|

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
>
> > The symbol /|)' is defined as 125M (243:250), and its apotome-
> > complement (|\. is defined as 125L (512000:531441, which equals
three
> > 5-commas)
>
> I'd forgotten about that one; it's even in the Sagittal paper. I've
> avoided using it because I've been focusing on 7-limit temperaments
(in
> which 36/35 comes up frequently).
>
> I was thinking the right accent was 4096/4095 (or does it depend on
the
> context?)

Yes, and yes. A single right accent is defined as 4095:4096, but it
can cover anything between ~0.244c and ~0.732c. A double right
accent is defined as 2079:2080, but it can cover anything between
~0.732c and ~1.220c. Exact "mina" boundaries are odd-number
multiples of 1/466 apotome. Each symbol (both unaccented and
accented) is usually defined as the most popular and/or least complex
ratio falling between two consecutive mina boundaries, so the exact
value of the right accent in a given right-accented symbol depends on
the definition of both that particular symbol and its unaccented
definition, and the value of a right accent in a given symbol may
change according to the role (or ratio) that the (accented) symbol is
filling.

The primary role (or definition) of a symbol is usually clear-cut,
but there are instances such as (|' where it becomes a judgment call:
51:52 (with lower complexity) vs. 256:261 (with a higher popularity);
51:52 was chosen as its definition on account of its lower prime
limit and its near-equidistance from mina boundaries.

> > Since I'm a bit wary of using symbols representing large
alterations
> > in opposite directions, I'd tend to prefer using a greater number
of
> > accidentals. (That's also because I'm already familiar with the
> > meaning of all of those accidentals.) But you're much more
familiar
> > with the 15- and 11-ET territory (and thus the meaning of those
> > compound nominals), so I don't think my opinion should count for
much
> > at this point in time. :-)
> >
> > However, I was thinking that perhaps the # and b symbols in those
> > compound nominals could go into a kind of key signature, rather
than
> > appearing alongside each note.
>
> The apparent contradiction of directions is one of the big problems
with
> compound nominals (which is one advantage of the differently shaped
note
> heads). Ideally you'd learn to think of #E as a basic note on its
own,

Yes, that's precisely why I suggested putting the # into a key
signature, since it's not really functioning as an accidental.

> but #E )!!( does look like you're starting with E, sharpening it,
and
> then flattening it -- is the result higher or lower than E?

It helps to remember that )!!( converts to b//|, giving #Eb//|, which
then becomes E//|, since the # and b cancel one another.

I'm noticing that using fewer symbol cores means that there would be
fewer apotome-complements to remember.

> I was thinking that a minor third based notation might be
appropriate
> for tunings without good fifths, but actually the standard notation
for
> 11-ET already is a good notation for some of the minor thirds.
>
> G\! G/| A\!
> A/| B\! C
> C||\ D E!!/
> E F/| G\!
> G/| A\! A/|

That's the subset-of-22-ET notation, if I remember correctly.

--George