A Sagittal notation for myna[31]

With Dave Keenan's new Sagittal spreadsheet, I've started taking another look at notating regular temperaments. One of the temperaments that's always been a problem to notate is myna. With 4 nominals and accidentals for �4, �8, and �12 generator steps, you can only notate 28 of the 31-note myna scale. Ideally you'd want at least 27 nominals to notate myna, but that doesn't fit with the 24-nominal system. On the other hand, a notation based on the standard chain of fifths used to run into trouble because there weren't enough accidentals to fill the gaps. But now it seems that there are. This system can even be extended to myna[58] or myna[89], but the basic 31-note MOS should be enough to start with.

myna [<1, -1, 0, 1], <0, 10, 9, 7]> TOP 1198.828458, 309.8926610

E)\!/ 49/45 (+4, -15) (+0.360601c)
G~!) 64/49 (+4, -14) (+5.085190c)
A.(|\ 14/9 (+4, -13) (-2.542592c)
C)||( 50/27 (+4, -12) (-10.891572c)
E\! 10/9 (+3, -11) (-5.445786c)
G 4/3 (+3, -10) (+0.000001c)
B!!/ 8/5 (+3, -9) (+5.445788c)
D)!!( 48/25 (+3, -8) (+10.891575c)
E|) 8/7 (+2, -7) (+2.542595c)
G~|) 49/36 (+2, -6) (-5.085188c)
B)\!/ 49/30 (+2, -5) (+0.360599c)
D~!) 96/49 (+2, -4) (+5.085189c)
F!) 7/6 (+1, -3) (-2.542594c)
G)||( 25/18 (+1, -2) (-10.891574c)
B\! 5/3 (+1, -1) (-5.445787c)
D 1/1 (+0, +0) (+0.000000c)
F/| 6/5 (+0, +1) (+5.445787c)
A)!!( 36/25 (+0, +2) (+10.891574c)
B|) 12/7 (+0, +3) (+2.542594c)
D~|) 49/48 (-1, +4) (-5.085189c)
F)/|\ 60/49 (-1, +5) (-0.360599c)
A~!) 72/49 (-1, +6) (+5.085188c)
C!) 7/4 (-1, +7) (-2.542595c)
D)||( 25/24 (-2, +8) (-10.891575c)
F||\ 5/4 (-2, +9) (-5.445788c)
A 3/2 (-2, +10) (-0.000001c)
C/| 9/5 (-2, +11) (+5.445786c)
E)!!( 27/25 (-3, +12) (+10.891572c)
G'(!/ 9/7 (-3, +13) (+2.542592c)
A~|) 49/32 (-3, +14) (-5.085190c)
C)/|\ 90/49 (-3, +15) (-0.360601c)

Not bad, only one accent required (for 9/7, 14/9). Even that can be eliminated using the [<1, -1, 0, 1, 5], <0, 10, 9, 7, -6]> 11-limit mapping of myna:

A)||~ 11/7 (+4, -13) (+15.033539c)
G)!!~ 14/11 (-3, +13) (-15.033539c)

Now the high-precision 5-limit temperaments (vishnu and luna) seem to be the most problematic ones remaining. Vishnu could be notated with a 16-nominal system and a symbol for an approx. 30.8-cent interval; )|) is in the right size range. Luna, with a 6-nominal system, needs more accidentals. But since these are 5-limit systems, there shouldn't be any confusion using the higher-limit accidentals as approximations, I'd guess.

Aside from vishnu and luna, most temperaments should be able to be notated using Sagittal notation directly, which has some advantages (chords near your "home key" are easier to read, and the notation of intervals in general is more familiar), but has the drawback that most notes and intervals have alternative notations that differ by one or more of the commas being tempered out. The half-octave symmetry of temperaments having two periods to the octave is also a problem. So I think the mixed Sagittal notation with accented nominals still has a place.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> With Dave Keenan's new Sagittal spreadsheet, I've started taking
another
> look at notating regular temperaments.
> One of the temperaments that's
> always been a problem to notate is myna. With 4 nominals and
accidentals
> for ±4, ±8, and ±12 generator steps, you can only notate 28 of the
> 31-note myna scale. Ideally you'd want at least 27 nominals to notate
> myna, but that doesn't fit with the 24-nominal system. On the other
> hand, a notation based on the standard chain of fifths used to run into
> trouble because there weren't enough accidentals to fill the gaps. But
> now it seems that there are. This system can even be extended to
> myna[58] or myna[89], but the basic 31-note MOS should be enough to
> start with.
>
> myna [<1, -1, 0, 1], <0, 10, 9, 7]> TOP 1198.828458, 309.8926610
>
> E)\!/ 49/45 (+4, -15) (+0.360601c)
> G~!) 64/49 (+4, -14) (+5.085190c)
> A.(|\ 14/9 (+4, -13) (-2.542592c)
> C)||( 50/27 (+4, -12) (-10.891572c)
> E\! 10/9 (+3, -11) (-5.445786c)
> G 4/3 (+3, -10) (+0.000001c)
> B!!/ 8/5 (+3, -9) (+5.445788c)
> D)!!( 48/25 (+3, -8) (+10.891575c)
> E|) 8/7 (+2, -7) (+2.542595c)
> G~|) 49/36 (+2, -6) (-5.085188c)
> B)\!/ 49/30 (+2, -5) (+0.360599c)
> D~!) 96/49 (+2, -4) (+5.085189c)
> F!) 7/6 (+1, -3) (-2.542594c)
> G)||( 25/18 (+1, -2) (-10.891574c)
> B\! 5/3 (+1, -1) (-5.445787c)
> D 1/1 (+0, +0) (+0.000000c)
> F/| 6/5 (+0, +1) (+5.445787c)
> A)!!( 36/25 (+0, +2) (+10.891574c)
> B|) 12/7 (+0, +3) (+2.542594c)
> D~|) 49/48 (-1, +4) (-5.085189c)
> F)/|\ 60/49 (-1, +5) (-0.360599c)
> A~!) 72/49 (-1, +6) (+5.085188c)
> C!) 7/4 (-1, +7) (-2.542595c)
> D)||( 25/24 (-2, +8) (-10.891575c)
> F||\ 5/4 (-2, +9) (-5.445788c)
> A 3/2 (-2, +10) (-0.000001c)
> C/| 9/5 (-2, +11) (+5.445786c)
> E)!!( 27/25 (-3, +12) (+10.891572c)
> G'(!/ 9/7 (-3, +13) (+2.542592c)
> A~|) 49/32 (-3, +14) (-5.085190c)
> C)/|\ 90/49 (-3, +15) (-0.360601c)

Hi Herman,

That's beautiful. Wonderful to see the information being put to use so
quickly. Sorry I took so long to finish it.

I notice you use the pure sagittal. I'm afraid I still have to think
in mixed and then translate. I find it difficult to remember the
double-shaft apotome complements of any but the most common single shafts.
/| ||\ 5-comma
|) ||) 7-comma
/|\ (|) 11-M-diesis

So I'm thinking there may be others who would benefit from seeing it
developed in the mixed notation too.

To notate a linear temperament having N generators to the prime 3,
using chain-of-fifth nominals, you only need Floor(N/2) pairs of
single-shaft accidentals in addition to conventional sharps and flats.
(However when N is small or the number of notes is large we may use
more pairs, to avoid double-sharps and double-flats.)

So for Myna we only need 5 pairs. This should be easy to understand if
we visualise the single chain of minor third generators as being made
by braiding 10 chains of fifths, where each chain of fifths uses a
different accidental. Chains of fifths are shown vertically below.

I went for /|) 35M rather than ~|) 49S for 4 generators as it notates
ratios with lower product complexity and results in fewer flags and
therefore easier-to-follow flag arithmetic.

E)\!/
G\!)
Bb!)
C#\\!
E\!
G
Bb/|
Db//|
E|)
G/|)
B)/|\
D\!)
F!)
G#\\!
B\!
D
F/|
Ab//|
B|)
D/|)
F#)\!/
A\!)
C!)
D#\\!
F#\!
A
C/|
Eb//|
F#|)
A/|)
C#)/|\

Now if we translate that into pure Sagittal, we'll find it is exactly
what Herman has above, except for the substitution of /|) for ~|) and
some perfectly valid alternate spellings using the 7L symbol .(|\ or
'(!/ instead of the 7C symbol |) or !)

We see that to do 31 notes of Myna, one of the 10 chains of fifths
must have 4 notes in it. It might make sense to make that the chain
with no accidentals. In that case we would not be centered on D, but
on the D-A or G-D fifth.

> Not bad, only one accent required (for 9/7, 14/9). Even that can be
> eliminated using the [<1, -1, 0, 1, 5], <0, 10, 9, 7, -6]> 11-limit
> mapping of myna:
>
> A)||~ 11/7 (+4, -13) (+15.033539c)
> G)!!~ 14/11 (-3, +13) (-15.033539c)

You don't even have to do that, as you can see above.

-- Dave K

Of course you have to use the "Show message option/Use fixed width
font" thing to see what I'm talking about in my previous message in
this thread.

-- Dave K

Hi Dave,

>To notate a linear temperament having N generators to the prime 3,
>using chain-of-fifth nominals, you only need Floor(N/2) pairs of
>single-shaft accidentals in addition to conventional sharps and flats.

Does this include cases where the period is a fraction of the
octave?

-Carl

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> With Dave Keenan's new Sagittal spreadsheet, I've started taking
another
> look at notating regular temperaments. One of the temperaments
that's
> always been a problem to notate is myna. With 4 nominals and
accidentals
> for ±4, ±8, and ±12 generator steps, you can only notate 28 of the
> 31-note myna scale. Ideally you'd want at least 27 nominals to
notate
> myna, but that doesn't fit with the 24-nominal system. On the other
> hand, a notation based on the standard chain of fifths used to run
into
> trouble because there weren't enough accidentals to fill the gaps.
But
> now it seems that there are. This system can even be extended to
> myna[58] or myna[89], but the basic 31-note MOS should be enough to
> start with.
>
> myna [<1, -1, 0, 1], <0, 10, 9, 7]> TOP 1198.828458, 309.8926610
>
> E)\!/ 49/45 (+4, -15) (+0.360601c)

Could you put in a key? It would be nice if this was translated into
something where a single ascii symbol corresponded to single sagittal
symbol, and then there was a key explaining what the symbols were.

I'm already lost; )\!/ appears to be a 392/405 symbol, which is 35
myna generators down. "&" is suggested as a shorthand, which is
ascii, though in general I guess you need unicode. What the 49/45,
(+4, -15) then mean I don't know.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> Now the high-precision 5-limit temperaments (vishnu and luna) seem to
be
> the most problematic ones remaining.

What you are calling "luna" I was still calling semithirds just now
over on tuning. Should I consider names here

/tuning-math/message/14640

to be officially adopted and put them on a web page, I wonder?

Dave Keenan wrote:

> I notice you use the pure sagittal. I'm afraid I still have to think
> in mixed and then translate. I find it difficult to remember the
> double-shaft apotome complements of any but the most common single shafts.
> /| ||\ 5-comma
> |) ||) 7-comma
> /|\ (|) 11-M-diesis
> > So I'm thinking there may be others who would benefit from seeing it
> developed in the mixed notation too.

Understandable. I like the pure notation because the size of the symbol gives you a rough idea of how big the interval is, and I can never remember the exact intervals of any of the multiple-flag symbols without looking them up anyway. Also, that lets me reserve the mixed notation for compound nominals if I need to use them.

> We see that to do 31 notes of Myna, one of the 10 chains of fifths
> must have 4 notes in it. It might make sense to make that the chain
> with no accidentals. In that case we would not be centered on D, but
> on the D-A or G-D fifth.

Certainly one advantage of the fifth-based notation is that you can more easily set the reference pitch to any of the traditional note names (A, G, and C are some choices likely to be popular). When I originally started playing with regular temperaments, I used A as a starting pitch and went up from there; now I have scales centered on D.

>> Not bad, only one accent required (for 9/7, 14/9). Even that can be >> eliminated using the [<1, -1, 0, 1, 5], <0, 10, 9, 7, -6]> 11-limit >> mapping of myna:
>>
>> A)||~ 11/7 (+4, -13) (+15.033539c)
>> G)!!~ 14/11 (-3, +13) (-15.033539c)
> > You don't even have to do that, as you can see above.

I see what you're doing there. I don't like to use accidentals larger than an apotome unless the step sizes of the scale are fairly large to begin with, but they do come in handy for ratios like 9/7 and 7/5.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> >> Now the high-precision 5-limit temperaments (vishnu and luna) seem to > be >> the most problematic ones remaining. > > What you are calling "luna" I was still calling semithirds just now > over on tuning. Should I consider names here
> > /tuning-math/message/14640
> > to be officially adopted and put them on a web page, I wonder?
> I think pretty much all of these are the names from Paul's paper without the -pent and -sept suffixes (if any). I don't recall if I've seen "luna" in any other contexts... I haven't been paying as much attention to micro-/nano- temperaments as I probably should have....

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>> myna [<1, -1, 0, 1], <0, 10, 9, 7]> TOP 1198.828458, 309.8926610
>>
>> E)\!/ 49/45 (+4, -15) (+0.360601c)
> > Could you put in a key? It would be nice if this was translated into > something where a single ascii symbol corresponded to single sagittal > symbol, and then there was a key explaining what the symbols were.

I can never remember the single ascii symbols; they're pretty much arbitrary and you have to memorize each one. Besides, without using the mythological names, which are also not always easy to remember, how else is it possible to represent the symbols in a recognizable way? I'm using the standard ASCII representation of the flags, which I believe is described in the Sagittal paper.

> I'm already lost; )\!/ appears to be a 392/405 symbol, which is 35 > myna generators down. "&" is suggested as a shorthand, which is > ascii, though in general I guess you need unicode. What the 49/45, > (+4, -15) then mean I don't know. 49/45 is the interval from D (which I'm identifying with 1/1 in this notation; any note can be used). E without any symbol is 9/8, so E)\!/ represents (9/8)(392/405), which is 49/45.

+4, -15 is the number of myna generators to reach this note: 4 periods up, 15 generators down. In TOP tuning this is 4(1198.83) - 15(309.89), or 146.92 cents. (Here I'm stretching the octave to 1200.0 cents and keeping the relative size of the generator, which is why the cents error doesn't seem to add up. I forgot I was doing that or I would have mentioned it....)

The number in parentheses is the difference between the JI value represented by the Sagittal notation and the tuning of the note in TOP tuning stretched to have exact 1200.0 cent octaves. So that's 147.4281 cents for 49/45 compared with 4(1200.0) - 15(310.1955), or 147.0675 cents, a difference of about 0.36 cents.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> Hi Dave,
>
> >
>
> Does this include cases where the period is a fraction of the
> octave?

Hi Carl,

Good point! I guess I can retrospectively claim to have been using the
term linear temperament in the strict Erlich/Secor manner. ;-)

For multi-linears (general rank 2) it seems you would just have to
multiply by the number of periods per octave. So I think I should have
written:

To notate any rank-2 temperament using chain-of-fifth nominals, the
minimum required number of pairs of sagittal accidentals (in addition
to conventional sharps and flats) is

Floor(M*N/2)

where M is the number of periods per prime 2 and
N is the number of generators per prime 3

For example, Srutal/Pajara/Diaschismic has 2 periods per octave and
one generator per fifth so you only need a single accidental (the
5-comma accidental) to switch between chains.

Herman mentioned Vishnu. That's pretty scary for a chian-of-fifths
nominal notation since it has 2 periods per octave and 7 generators
per fifth, hence needing 7 pairs of single-shaft sagittals. A good
argument for a MOS/DE nominals approach.

-- Dave K

At 04:08 PM 3/4/2007, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>> Hi Dave,
>>
>> >
>>
>> Does this include cases where the period is a fraction of the
>> octave?
>
>Hi Carl,
>
>Good point! I guess I can retrospectively claim to have been using the
>term linear temperament in the strict Erlich/Secor manner. ;-)

Drat, you got me there. I've been meaning to read that term the
strict way ever since Paul beat me up over it. I'm glad to see
I'm not alone in still tending towards the older usage.

>For multi-linears (general rank 2) it seems you would just have to
>multiply by the number of periods per octave. So I think I should have
>written:
>
>To notate any rank-2 temperament using chain-of-fifth nominals, the
>minimum required number of pairs of sagittal accidentals (in addition
>to conventional sharps and flats) is
>
>Floor(M*N/2)
>
>where M is the number of periods per prime 2 and
> N is the number of generators per prime 3
>
>For example, Srutal/Pajara/Diaschismic has 2 periods per octave and
>one generator per fifth so you only need a single accidental (the
>5-comma accidental) to switch between chains.

Gotcha.

-Carl