As a first attempt, I'm starting with the assumption that the notation
system I'm looking for will have names for 4/3 and 3/2 -- for simplicity
I'll call the 1/1 "D" so that I can use "G" for 4/3, "A" for 3/2.
Pitches near these notes will use accidentals. 7/5 can be notated G +
21/20 or A - 15/14, so it won't be necessary to include it in the basic
scale unless the scale has enough notes that the gap between 4/3 and 3/2
is too large.

So, with a list of intervals likely to be used in 7-limit scales, I can
identify 81/80, 64/63, 36/35, 28/27, 25/24, 21/20, 135/128, 16/15, and
15/14 as some of the more important accidentals that will be needed. The
importance of superparticular ratios is one of the ideas that I've
established for Zireen music, so I'll want to take another look at that
135/128.

21/20 * 225/224 = 135/128

Since the difference is relatively small, the 225/224 could be
represented as an extra mark (e.g. a hook or a slash) added to the main
21/20 accidental. Note that the pairs (28/27 25/24) and (16/15 15/14)
are also separated by 225/224.

Note that 50/49 and 49/48 are not included in the list of accidentals.
Since these are so close together (2401/2400 apart), it would be
convenient not to need both of them. The easiest way to do this would be
to use a temperament that tempers out 2401/2400.

49/48 is the difference between 8/7 and 7/6. 50/49 seems to be mainly
needed for ratios containing 49 (e.g. 49/40 * 50/49 = 5/4).

Not counting 49/48, you've got 5 basic size classes of accidentals.

81/80, 64/63
36/35
28/27, 25/24
21/20, 135/128
16/15, 15/14

This suggests something around 53-ET as a first approximation. 53-ET has
some useful properties, but tempering out 2401/2400 isn't one of them.
Another approach is back to 171-ET, which does temper out 2401/2400 and
has a step size near 225/224.

Graham Breed wrote:
> Herman Miller wrote:
>> As a simplification, you really only need a notation for the pitches
>> from 1/1 to 3/2. The rest of the pitches can be reached by inverting the
>> notation down from 2/1, or transposing up by 3/2.
>
> I don't follow this.  Decimal notation, for example, doesn't
> have a 4/3 from the root.

Well, that's an example of symmetrical notation. You can go 4 steps up
and 5 steps down (or vice versa).

>> 5/4 and 6/5 are also pretty basic intervals, but you don't need both of
>> them: the difference between them is 25/24, which is a useful accidental
>> to have. E.g., 25/18 = 4/3 * 25/24. You could represent these in other
>> ways, e.g. 5/4 as 4/3 * 15/16, 6/5 as 7/6 * 36/35, but at least one of
>> these would be useful to have a basic notation for. (There's always the
>> option of the chain of fifths notation using 81/80 as an accidental, but
>> I'm looking for alternative notations.)
>
> You do need both of them to get a major triad, which is nice
> to have.

You don't need both of them to be notated from the same root without an
accidental.

>> An essential thing for a 7-limit notation is being able to represent 7/6
>> and 8/7 reasonably well. 7/6 could be 6/5 * 35/36, 8/7 could be 9/8 *
>> 64/63, but it would be nice to have one of these as a basic pitch of the
>> notation system, with 49/48 as an accidental to notate the other one.
>
> 8:7 is useful because it breaks down into 15:14 and 16:15 --
> two secors or toes.

Okay, that's a point in its favor. Either 15:14 could be another basic
step of the notation, or it could be an accidental.

>> Ideally, the kind of scale I'm looking for should have all relatively
>> simple ratios, largest/smallest step size ratio less than 2, and
>> strictly proper.
>>
>> This is the sort of scale that might be appropriate, but there doesn't
>> seem to be anything obvious or special about it. I'm hoping to find
>> something better.
>>
>> 1/1 15/14 8/7 5/4 4/3 7/5 3/2 8/5 12/7 15/8 2/1
>
> Move 5/4 and 4/3 down a quomma and you have a decimal scale.

Hmm, well I suppose I could give that a look.

Another thing to try is finding notations for scales that I've already
used in Zireen music and see what fits. Here's one possibility for
lemba[26]:

(+3, -6)  5/4 / 225/224   56/45
(+4, -6)  7/4 *  64/63    16/9
(+2, -5) 10/9 /  16/15    25/24
(+3, -5)  3/2 /  36/35    35/24
(+2, -4) 10/9 *  21/20     7/6
(+3, -4)  7/4 /  21/20     5/3
(+2, -3)  4/3              4/3
(+3, -3)  7/4 *  16/15    28/15
(+1, -2) 10/9 /  25/24    16/15
(+2, -2)  3/2 *  25/24    25/16
(+1, -1)  5/4              5/4
(+2, -1)  7/4              7/4
(+0, +0)  1/1              1/1
(+1, +0)  3/2 /  15/14     7/5
(+0, +1) 10/9 *  36/35     8/7
(+1, +1)  3/2 *  16/15     8/5
(+0, +2)  4/3 /  64/63    21/16
(+1, +2)  7/4 *  15/14    15/8
(-1, +3) 10/9 /  28/27    15/14
(+0, +3)  3/2              3/2
(-1, +4)  5/4 /  25/24     6/5
(+0, +4)  7/4 /  49/48    12/7
(-2, +5)  1/1 * 225/224  225/224
(-1, +5)  3/2 /  16/15    45/32
(-2, +6) 10/9 *  81/80     9/8
(-1, +6)  3/2 *  15/14    45/28

Probably not the notation I'll end up using, but this should give me an
idea of which nominals and accidentals are most productive.

Herman Miller wrote:
> One of the things I've had some interest in, but not much success, is
> coming up with a notation system that might have developed in a musical
> culture where intervals like 7/4 and 7/6 are used prominently. One of
> the issues is actually coming up with a basic 7-limit scale that makes
> sense and has more than 5 notes. Something like 1/1 7/6 4/3 3/2 7/4 2/1
> would work, but has so few notes that many different accidentals would
> be required.

That approximates as every other step of Negri10, or as the
5 note MOS of one of these no-fives "bug" type temperaments.
   It's not a very accurate temperament but it puts every
interval within the 9-limit.  Which is a start.

There's also the "Pygmie scale" which approximates to every
other note of decimal (miracle10) or 5 notes of wonder
temperament:

1/1 8/7 21/16 3/2 7/6 2/1

And you can forget the sevens and use a classic pentatonic.

> Maybe, in a manner similar to Graham Breed's tripod notation, a 5-note
> scale can be a basis for notating 10 or 15 different pitches. Or maybe I
> can find some reasonable scale with around 7-9 pitches that makes sense
> for notating 7-limit music.

Yes, the scales above will work as negri or miracle by
dividing the steps into two almost equal parts.

> As a simplification, you really only need a notation for the pitches
> from 1/1 to 3/2. The rest of the pitches can be reached by inverting the
> notation down from 2/1, or transposing up by 3/2.

I don't follow this.  Decimal notation, for example, doesn't
have a 4/3 from the root.

> 4/3 and 3/2 are such basic intervals that they'd probably be represented
> in any notation system. With accidentals, you can use those to represent
> nearby pitches such as:
>
> 7/5 = 4/3 * 21/20
> 9/7 = 4/3 * 27/28
>
> etc.
>
> 5/4 and 6/5 are also pretty basic intervals, but you don't need both of
> them: the difference between them is 25/24, which is a useful accidental
> to have. E.g., 25/18 = 4/3 * 25/24. You could represent these in other
> ways, e.g. 5/4 as 4/3 * 15/16, 6/5 as 7/6 * 36/35, but at least one of
> these would be useful to have a basic notation for. (There's always the
> option of the chain of fifths notation using 81/80 as an accidental, but
> I'm looking for alternative notations.)

You do need both of them to get a major triad, which is nice
to have.

> An essential thing for a 7-limit notation is being able to represent 7/6
> and 8/7 reasonably well. 7/6 could be 6/5 * 35/36, 8/7 could be 9/8 *
> 64/63, but it would be nice to have one of these as a basic pitch of the
> notation system, with 49/48 as an accidental to notate the other one.

8:7 is useful because it breaks down into 15:14 and 16:15 --
two secors or toes.

> Ideally, the kind of scale I'm looking for should have all relatively
> simple ratios, largest/smallest step size ratio less than 2, and
> strictly proper.
>
> This is the sort of scale that might be appropriate, but there doesn't
> seem to be anything obvious or special about it. I'm hoping to find
> something better.
>
> 1/1 15/14 8/7 5/4 4/3 7/5 3/2 8/5 12/7 15/8 2/1

Move 5/4 and 4/3 down a quomma and you have a decimal scale.


                      Graham

One of the things I've had some interest in, but not much success, is
coming up with a notation system that might have developed in a musical
culture where intervals like 7/4 and 7/6 are used prominently. One of
the issues is actually coming up with a basic 7-limit scale that makes
sense and has more than 5 notes. Something like 1/1 7/6 4/3 3/2 7/4 2/1
would work, but has so few notes that many different accidentals would
be required.

Maybe, in a manner similar to Graham Breed's tripod notation, a 5-note
scale can be a basis for notating 10 or 15 different pitches. Or maybe I
can find some reasonable scale with around 7-9 pitches that makes sense
for notating 7-limit music.

As a simplification, you really only need a notation for the pitches
from 1/1 to 3/2. The rest of the pitches can be reached by inverting the
notation down from 2/1, or transposing up by 3/2.

4/3 and 3/2 are such basic intervals that they'd probably be represented
in any notation system. With accidentals, you can use those to represent
nearby pitches such as:

7/5 = 4/3 * 21/20
9/7 = 4/3 * 27/28

etc.

5/4 and 6/5 are also pretty basic intervals, but you don't need both of
them: the difference between them is 25/24, which is a useful accidental
to have. E.g., 25/18 = 4/3 * 25/24. You could represent these in other
ways, e.g. 5/4 as 4/3 * 15/16, 6/5 as 7/6 * 36/35, but at least one of
these would be useful to have a basic notation for. (There's always the
option of the chain of fifths notation using 81/80 as an accidental, but
I'm looking for alternative notations.)

An essential thing for a 7-limit notation is being able to represent 7/6
and 8/7 reasonably well. 7/6 could be 6/5 * 35/36, 8/7 could be 9/8 *
64/63, but it would be nice to have one of these as a basic pitch of the
notation system, with 49/48 as an accidental to notate the other one.

Ideally, the kind of scale I'm looking for should have all relatively
simple ratios, largest/smallest step size ratio less than 2, and
strictly proper.

This is the sort of scale that might be appropriate, but there doesn't
seem to be anything obvious or special about it. I'm hoping to find
something better.

1/1 15/14 8/7 5/4 4/3 7/5 3/2 8/5 12/7 15/8 2/1

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Paul H wrote:
>
> > Does anyone know alot about Myhills' Property? Mixing the black
> > pentachord and its subsets with the white diaton and its subsets
in a
> > controlled way could have mathematical significance....
>
> Myhill's property is another kind of MOS, but definitely
> assuming period=octave IIRC.  There's an obvious link
> between MOS scales and propriety which I gave proofs for here:
>
> http://x31eq.com/proof.html
>
>
>                    Graham

Thanks, your proof is a real cracker, Graham

PGH

George D. Secor wrote:
> As you point out, it would nice to use the same accidentals for both
> 27-edo & beatles[27].  How about if we we to notate 27-edo this way?
>
> 27:  /|  )/|\  ||\  /||\
>
> Since )/|\ is defined as 5:49M (392:405), this would use 7-limit
> symbols.  I've come around to the idea that it's good to use )/|\ for
> half of /||\ in places where /|\ isn't valid for that purpose.

Yes, all of those are consistent with beatles. The full list of symbols
my program came up with for the (-2, +7) interval is:

)| '|) )/|\ .||)

of which )| is clearly unsuitable due to its size, and the other two are
accented (which is inconvenient if unaccented symbols are available).

The other 2-step interval in beatles[27], (+6, -20), has only the )||(
symbol as an option.

It might be worth checking other temperaments with 27-note MOS for
consistency: e.g. augene, superpyth, sensisept, myna, ennealimmal.

I've already got a list for myna.

(-8, +31)   /|
(-9, +35)   )/|\
(-10, +39)  ||\
(-18, +70)  /||\

None of these are really relevant in a 27-note subset of myna, so
there's no real conflict. But if for some reason we want to notate 27-ET
as myna, with a 7/27 generator, the math works out.

(31 * 7) - (8 * 27)  = 1
(35 * 7) - (9 * 27)  = 2
(39 * 7) - (10 * 27) = 3
(70 * 7) - (18 * 27) = 4

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> I've started to take an interest in the beatles 7-limit temperament
>
> [<1, 1, 5, 4], <0, 2, -9, -4]>
> TOP-MAX P = 1197.104145, G = 354.720338
> TOP-RMS P = 1196.641985, G = 354.908182
>
> Since this is practically 27-ET, it would be nice to use 27-ET
Sagittal
> notation for it. Unfortunately, Sagittal notation uses /| for 1
step of
> 27-ET and /|) for 2 steps, but these both represent the same
interval in
> beatles temperament since |) is tempered out.

Hi Herman,

As I recall, Dave & I chose symbols for the 27-equal notation that
corresponded to the lowest primes that 1) are most accurately
represented, and 2) don't vanish, which would be 5 and 13, giving the
following sequence:

27:  /|  /|)  ||\  /||\

Since |), which vanishes, doesn't appear in the symbol set to upset
the symbol arithmetic, we decided that it would be okay to use /|)
for the medium-size 13-diesis (1024:1053), although it's technically
a 35-diesis (35:36).  Now it appears that our decision has come back
to bite us. :-(

As you point out, it would nice to use the same accidentals for both
27-edo & beatles[27].  How about if we we to notate 27-edo this way?

27:  /|  )/|\  ||\  /||\

Since )/|\ is defined as 5:49M (392:405), this would use 7-limit
symbols.  I've come around to the idea that it's good to use )/|\ for
half of /||\ in places where /|\ isn't valid for that purpose.

--George

Graham Breed wrote:
> Herman Miller wrote:
>> I've started to take an interest in the beatles 7-limit temperament
>>
>> [<1, 1, 5, 4], <0, 2, -9, -4]>
>> TOP-MAX P = 1197.104145, G = 354.720338
>> TOP-RMS P = 1196.641985, G = 354.908182
>>
>> Since this is practically 27-ET, it would be nice to use 27-ET Sagittal
>> notation for it. Unfortunately, Sagittal notation uses /| for 1 step of
>> 27-ET and /|) for 2 steps, but these both represent the same interval in
>> beatles temperament since |) is tempered out.
>
> After choosing symbols for magic tripod notation ... and
> getting it wrong ... I've decided that I don't like the
> Athenian ET notations.  It's better to work with the JI
> logic and find a symbol of about the right size.  But there
> could be all kinds of subtleties I'm not aware of.

Another thing to consider is the size of fifths in beatles temperament.
The fifths at the far ends of the chain are far enough off that "B"
sounds sharp and "F" sounds flat. So B\! and F/| are actually reasonable
notations for those notes, while B|) and F!) are better representations
of the actual pitches from the chain of tempered fifths than just a
plain B or F would be.

>> So, one of these has to be replaced. /| turns out to be one step of the
>> beatles[27] MOS; the other step could be ~| or ~|)
>
> /| is the syntonic comma, isn't it?  That's too small to be
> a step from 27-equal or anything close to it.

Yes; in fact /|) is a much better match. If the conflict with 27-ET
notation can be ignored, it's a potential choice for notating (-5, +17).
There are so many criteria to consider though, like symbol arithmetic.

/| + ||\ = /||\
/|) + (|\ = /||\

Since there's not any good reason to use (|\ for (+1, -3) in place of
||\ , that's another reason in favor of /| .

I'm wondering though how the 27-ET notation in the Sagittal paper came
about. [2, 2, -1, -1> is about 1.1 steps of 27-ET, not anywhere near 2.
Better choices for (+2) of 27-ET include )||( [-3, -1, 2>, which happens
to match the 2-step (+6, -20) of beatles[27] that I mentioned.

> ~|) is large in the PDF, and so deemed important, but
> doesn't belong to any of the subsets.  Again, my opinion,
> but I'd rather ignore these non-subsetted characters
> altogether.  The full Sagittal set is too large.  Athenian
> symbols should be given priority, or Trojan ones where
> relevant (I prefer the Trojan logic to Athenian ETs).  A
> 7-bit font would be very convenient.

A~|) is 49/32 relative to D, which is a pretty useful ratio to be able
to notate in a 7-limit temperament. A simpler alternative is A.(|\ 14/9,
but that uses an accent and the alternative A|\\ is less well known. The
only other one that might be under consideration is A'(| 75/49.

> Do you care about the prime limit?  I think sometimes you
> have to use a symbol from outside the limit for a tempered
> notation, because it comes in between two intervals you're
> tempering to be the same.
>
> In this case, /|\ is one of the main symbols, the quarter
> tone.  And then )/|\ is an ugly symbol that isn't in any
> subsets ... but will look like /|\ with a bit of dirt on the
> screen so it'll be easily understood.

Actually, the 11-limit generator mapping that makes the most sense in
this case is <0, 2, -9, -4, 5] which does have /|\ as a notation for
(-2, +7), along with (|) )||~ and others. (|( shows up as an alternative
for ~|) but that's another one of those unfamiliar symbols.

Herman Miller wrote:
> I've started to take an interest in the beatles 7-limit temperament
>
> [<1, 1, 5, 4], <0, 2, -9, -4]>
> TOP-MAX P = 1197.104145, G = 354.720338
> TOP-RMS P = 1196.641985, G = 354.908182
>
> Since this is practically 27-ET, it would be nice to use 27-ET Sagittal
> notation for it. Unfortunately, Sagittal notation uses /| for 1 step of
> 27-ET and /|) for 2 steps, but these both represent the same interval in
> beatles temperament since |) is tempered out.

After choosing symbols for magic tripod notation ... and
getting it wrong ... I've decided that I don't like the
Athenian ET notations.  It's better to work with the JI
logic and find a symbol of about the right size.  But there
could be all kinds of subtleties I'm not aware of.

> So, one of these has to be replaced. /| turns out to be one step of the
> beatles[27] MOS; the other step could be ~| or ~|)

/| is the syntonic comma, isn't it?  That's too small to be
a step from 27-equal or anything close to it.

~|) is large in the PDF, and so deemed important, but
doesn't belong to any of the subsets.  Again, my opinion,
but I'd rather ignore these non-subsetted characters
altogether.  The full Sagittal set is too large.  Athenian
symbols should be given priority, or Trojan ones where
relevant (I prefer the Trojan logic to Athenian ETs).  A
7-bit font would be very convenient.

> One of the 2-step intervals is )||( which, when combined with the /|
> step results in ||\ for a 3-step interval. This fits well into the
> notation, where for instance E!!/ is a major third below G.
>
> The other 2-step interval is less obvious, but here's one that works.
>
>   ~|) + )/|\ = ||\

||\ is a good Spartan symbol so use it if it fits.

> There's another 3-step interval, although it's not as useful for actual
> notation. But for the sake of completeness here it is.
>
>   /| + )/|\ = ~~||
>
> So far we've got:
>
> (+3, -10)  ~|)   [-4, -1, 0, 2>
> (-5, +17)  /|    [-4, 4, -1>
> (+6, -20)  )||(  [-3, -1, 2>
> (-2,  +7)  )/|\  [-3, 4, 1, -2>
> (+1,  -3)  ||\   [-7, 3, 1>
> (-7, +24)  ~~||  [-7, 8, 0, -2>
>
> I haven't looked much into larger intervals, although /||\ would be a
> 4-step interval twice the size of )/|\ (-4, +14). It would be convenient
> for the notation to use /|\ in place of )/|\ (which would imply an
> 11-limit temperament).

Do you care about the prime limit?  I think sometimes you
have to use a symbol from outside the limit for a tempered
notation, because it comes in between two intervals you're
tempering to be the same.

In this case, /|\ is one of the main symbols, the quarter
tone.  And then )/|\ is an ugly symbol that isn't in any
subsets ... but will look like /|\ with a bit of dirt on the
screen so it'll be easily understood.


                    Graham

I've started to take an interest in the beatles 7-limit temperament

[<1, 1, 5, 4], <0, 2, -9, -4]>
TOP-MAX P = 1197.104145, G = 354.720338
TOP-RMS P = 1196.641985, G = 354.908182

Since this is practically 27-ET, it would be nice to use 27-ET Sagittal
notation for it. Unfortunately, Sagittal notation uses /| for 1 step of
27-ET and /|) for 2 steps, but these both represent the same interval in
beatles temperament since |) is tempered out.

So, one of these has to be replaced. /| turns out to be one step of the
beatles[27] MOS; the other step could be ~| or ~|)

One of the 2-step intervals is )||( which, when combined with the /|
step results in ||\ for a 3-step interval. This fits well into the
notation, where for instance E!!/ is a major third below G.

The other 2-step interval is less obvious, but here's one that works.

   ~|) + )/|\ = ||\

There's another 3-step interval, although it's not as useful for actual
notation. But for the sake of completeness here it is.

   /| + )/|\ = ~~||

So far we've got:

(+3, -10)  ~|)   [-4, -1, 0, 2>
(-5, +17)  /|    [-4, 4, -1>
(+6, -20)  )||(  [-3, -1, 2>
(-2,  +7)  )/|\  [-3, 4, 1, -2>
(+1,  -3)  ||\   [-7, 3, 1>
(-7, +24)  ~~||  [-7, 8, 0, -2>

I haven't looked much into larger intervals, although /||\ would be a
4-step interval twice the size of )/|\ (-4, +14). It would be convenient
for the notation to use /|\ in place of )/|\ (which would imply an
11-limit temperament).

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Paul H wrote:
>
> > Does anyone know alot about Myhills' Property? Mixing the black
> > pentachord and its subsets with the white diaton and its subsets
in a
> > controlled way could have mathematical significance....
>
> Myhill's property is another kind of MOS, but definitely
> assuming period=octave IIRC.  There's an obvious link
> between MOS scales and propriety which I gave proofs for here:
>
> http://x31eq.com/proof.html
>
>
>                    Graham

Thanks. I will study these proofs. Is it true that subsets of the
pentatonic and diatonic also have Myhill's property? Are there
other interesting proprietary scales besides Rothenberg-proper scales?

On another note, I am finding fascinating patterns studying black
and white keys (literally) in the Steiner System (Outer(Aut(S6)->
Mathieu(M12)) which is a S(5,6,12) system based on the
ProjectiveGeometry(3,2) with 15 points, 35 lines and 15 hyperplanes.

I didn't even have to transpose, and many things just fell in place.
I am mapping My Hexachord System onto this and am working to
map the 35 hexachord types against the 350 combinations found
from 3 black and 3 white keys. However, I don't have a complete
construction yet, but I know there is at least one solution.

The 35 lines of the Smallest Projective space might also map
on the 35 hexachords, using my hexachord extension.

Another fun fact is that there are 35 pentachord types also
based on interval vector, and M12 Symmetry group of (S(5,6,12) of
course relate hexads to pentads and is quintuply and sharply
transitive.

PGH

Paul H wrote:

> Does anyone know alot about Myhills' Property? Mixing the black
> pentachord and its subsets with the white diaton and its subsets in a
> controlled way could have mathematical significance....

Myhill's property is another kind of MOS, but definitely
assuming period=octave IIRC.  There's an obvious link
between MOS scales and propriety which I gave proofs for here:

http://x31eq.com/proof.html


                    Graham

--- In tuning-math@yahoogroups.com, "Paul H" <phjelmstad@...> wrote:
>
> I have analyzed the 5 Rothenberg Proprietary Scales.
>
> Diatonic - CDEFGAB
> Mel Minor - CDEbFGAB
> Harm Major - CDEFGAbB
> Harm Minor - CDEbFGBbB
> Locrian Major -CDEFGgAbBb
>
> Using the Hexachord lettering of My Hexachord System, here is an
> examination of the seven subsets of each scale
>
> N5 S5 N5 M5 R5 R5 M5
> V5 V5 N5 T5 K  T5 N5
> H  D  J5 B5 G  R5 S5
> G  B5 J5 D  H  T5 R5
> V1 W  V5 U  V5 W  V1
>
> Some facts - only the 5 side of the M5 symmetry is used, except
> for V1 V5. Also, O1/O5 is the only M5 couple not represented.
>
> In terms of the basic letters (D4 X S3 X S2) where S2 is
> complementation, and arranged by tritone count
>
> 1
> 6
> 6
> 1
>
> or 7 in the Upper Realm and 7 in the Lower Relam, where the total
> is 13 Upper and 13 Lower of the 26 Letters.
>
> Also, in terms of the 35 sets, arranged by Sisters groupings
> and tritone counts (my I-CHING system)
>
> M5
>
> B5N5T5, DR5, S5(S5)
>
> J5V1V5, GK, HW
>
> U
>
> Notice the symmetry in the Sisters groups and in the ATTIC and
CELLAR
>
> 1
>
> 3, 2, 2
> 3, 2, 2
>
> 1
>
> Now both sides of S5 (Z related set with complement) are used,
> otherwise only one side of a Z relation are used
>
> T5, DR5
> J5, GK
>
> Note the symmetry here also
>
> 1,2
> 1,2
>
> Now of course any hexachord could be found as the subset of some
> scale, the M1 sides are of course more chromatic. Adding some
> non proprietary scales, like the Hungarian, could probably help
> to canvass all 35 hexachord types, and more Z related complements.
>
> Now the fun is that in MathieuGroup(12), or M12, based on Steiner
> System (5,6,12) a block design, we can examine pentads and hexads
> and thus in music, pentachords and hexachords. Now of course
> septachords are merely complements of pentachords, thereby
> we consider seven notes scales and their subsets in the same manner
> as the Steiner system (5,6,12) just complement 5->7 and 6->6,
> which may be Z-related, and of course, that is the purpose of
> S2 in the D4 X S3 X S2 or D12 X S2 Group.
>
> Now construction of M12 from Outer(Aut(S6)-> M12 uses these
> hexad types B5CDEFGJ1J5KLM1N5QR1R5S1S5T1T5V1WYZ or 23 of the 35
> hexachord types, and BCDEFGJKLMNQRSTVWYZ Letters, so only
> IPU in the CELLAR and A in the ROOF are missing and also
> X and H thus:
>
> A
> X
> H
> UPI
>
> 1
> 1
> 1
> 3
>
> Filled out more
>
> AM5
> B1N1, X
> O1O5, H
> UPI
>
> Once again note the Symmetry!
>
> 2
> 2,1
> 2,1
> 3
>
> with the CELLAR a little bottom heavy in both cases.
>
> Used
>
> EMY
> BNT, DLR, CS
> JV, FZGK, QW
>
> (None)
>
> Filled out
>
> EM1Y
>
> B5N5T1T5, DLR1R5, CS1S5
> J1J5V1, FZGK, QW
>
> Excluding the ROOF there is symmetry between Picture A and Pictures
> B and C from the construction based on PG(3,2).
>
> (12)
> 40+20
> 40+20
>
> producing the 132 Steiner Hexads and many amusing patterns.
>
> PGH

I should stick to one subject in a post:)

Anyway, I realized that the reason some hexachords don't get
canvassed easily is due to the fact that they are "Echer-like"
scales/chords.

For example, P=(11,0,1,5,6,7) has B,C,E,F part of CM scale,
and C,Db,F,Gb part of DbM scale (other dissections are possible)
so it is like an ambiguous Escher print.

Now while normal scales like harmonic minor can use sharps and
flats together, going to two or more of each produces ambiguities.

The hexad subsets left out of the Rothenberg Proprietary scales are
like this:

1. The Hexatonic hexachord (A)
2. Sets with long chromatic chains (one side of M5 symmetry)
3. Sets that require forcing natural and accidental both

In fact that covers all that remain.

If we add the Gypsy scale (or Hungarian) we pick up a few more
hexachords

0,2,3,6,7,8,11

0,2,3,6,7,8: Z
0,2,3,6,7,11: D
0,2,3,7,8,11: D
0,2,6,7,8,11: F
0,3,6,7,8,11: D
2,3,6,7,8,11: D

Well, picks up FZ anyway.

C,E,I,L,O,Q,P,X,Y have two semitones (item 3 above)

If we blur these out we get pentads which are quite ordinary,
and fit easily into Rothenberg scales. (remove middle note in 0,1,2)

B1,J1,M1,N1,O1,R1,S1,T1 are in item 2 above

Some hexads are complements or Z-relations, these actually
have the same issues, either 2 or 3 above.


PGH

Does anyone know alot about Myhills' Property? Mixing the black
pentachord and its subsets with the white diaton and its subsets in a
controlled way could have mathematical significance....

--- In tuning-math@yahoogroups.com, "Paul H" <phjelmstad@...> wrote:
>
> Have been studying Myhill's Property and also Rothenberg Scales.
>
> Interesting that the diatonic and the pentatonic scales have
> Myhill's property. So I thought I would study hexachords based
> on black and white keys. Subsets of each have the property,
> combined together, i don't know....
>
> Here is how all 924 hexachords break down
>
> White Keys Black Keys Hexachords
>
> 6  0   7
> 5  1  105
> 4  2  350
> 3  3  350
> 2  4  105
> 1  5  7
>
> Hold C fixed
>
> 6  0  6
> 5  1  75
> 4  2  200
> 3  3  150
> 2  4  30
> 1  5  1
>
> Notice subtracting Column Three Lower from Column Three Upper
produces
>
> 1,30,150,200,75,6 which is just the lower set reversed. Also
> 1+30+200=6+75+150=231 which is exactly half of 462 and 1/4 of 924..
>
> Now I have found that all 80 hexachord types can be expressed
> with no more than 2 black keys.
>
> Now to study Rothenberg Scales and their hexachord subsets.
>
> PGH

Now, of course the combinations for white and black keys are based on
7 * 1, 21 * 5, 35 * 10, 35 * 10, 21 * 5, 7 * 1 which is simple
combinations. This gives

7,105,350,350,105,7. Now I was excited to find that the Steiner
System I mention in my other post (Rothenberg Scales) is literally
1/7 of this in every combination: 1,15,50,50,15,1.

I studied this a bit, and found that of course, of the 66 Steiner
Hexads and their complements, all five blacks are used in each row.
(If a hexad has 2 blacks, its complement will have 3, etc)

Therefore 66*5=330 blacks used. Now of course one would expect
equal usage, and its true, each black is used 66 times.

Now its possible to derive an algorithm which will split the 924
hexads into seven systems. (The first a Steiner System as given
and the last six systems a Sextuple Steiner System, and possibly
six independent systems?) Now for example with no blacks,
we have one in SS, and of course 7 possible.

With one black there are 15 sets, and of course each black is
used 3 times (3 3 3 3 3) and in the Septuple system (all hexads)
it's clear that each black is used 21 times, for each 21 white
5-sets (Binom(7,5)). One might expect that there are just 3 white
5-sets used, which is 1/7 of the 21 in the full system, so
that you obtain a one-to-one with each black, for each one, but
it is more subtle, there are 15 white 5-sets, and the 6 not
used determine the key. It gets more elaborate with 3W 3B but
the same principles apply.

Finding Seven systems will determine a matching to the 35
lines of PG(3,2), which is used to give the construction of the
Steiner System (5,6,12) using Picture A in 3 combinations
(15*3) and Pictures B and C (10 + 10) and a base set (012345)
this makes 66, with their complements makes 132 hexads.

I also feel that this will match up one to one with the 35
hexachord types based on interval vector, that is D12 X S2
where S2 is 2-complementation.

PGH

I have analyzed the 5 Rothenberg Proprietary Scales.

Diatonic - CDEFGAB
Mel Minor - CDEbFGAB
Harm Major - CDEFGAbB
Harm Minor - CDEbFGBbB
Locrian Major -CDEFGgAbBb

Using the Hexachord lettering of My Hexachord System, here is an
examination of the seven subsets of each scale

N5 S5 N5 M5 R5 R5 M5
V5 V5 N5 T5 K  T5 N5
H  D  J5 B5 G  R5 S5
G  B5 J5 D  H  T5 R5
V1 W  V5 U  V5 W  V1

Some facts - only the 5 side of the M5 symmetry is used, except
for V1 V5. Also, O1/O5 is the only M5 couple not represented.

In terms of the basic letters (D4 X S3 X S2) where S2 is
complementation, and arranged by tritone count

1
6
6
1

or 7 in the Upper Realm and 7 in the Lower Relam, where the total
is 13 Upper and 13 Lower of the 26 Letters.

Also, in terms of the 35 sets, arranged by Sisters groupings
and tritone counts (my I-CHING system)

M5

B5N5T5, DR5, S5(S5)

J5V1V5, GK, HW

U

Notice the symmetry in the Sisters groups and in the ATTIC and CELLAR

1

3, 2, 2
3, 2, 2

1

Now both sides of S5 (Z related set with complement) are used,
otherwise only one side of a Z relation are used

T5, DR5
J5, GK

Note the symmetry here also

1,2
1,2

Now of course any hexachord could be found as the subset of some
scale, the M1 sides are of course more chromatic. Adding some
non proprietary scales, like the Hungarian, could probably help
to canvass all 35 hexachord types, and more Z related complements.

Now the fun is that in MathieuGroup(12), or M12, based on Steiner
System (5,6,12) a block design, we can examine pentads and hexads
and thus in music, pentachords and hexachords. Now of course
septachords are merely complements of pentachords, thereby
we consider seven notes scales and their subsets in the same manner
as the Steiner system (5,6,12) just complement 5->7 and 6->6,
which may be Z-related, and of course, that is the purpose of
S2 in the D4 X S3 X S2 or D12 X S2 Group.

Now construction of M12 from Outer(Aut(S6)-> M12 uses these
hexad types B5CDEFGJ1J5KLM1N5QR1R5S1S5T1T5V1WYZ or 23 of the 35
hexachord types, and BCDEFGJKLMNQRSTVWYZ Letters, so only
IPU in the CELLAR and A in the ROOF are missing and also
X and H thus:

A
X
H
UPI

1
1
1
3

Filled out more

AM5
B1N1, X
O1O5, H
UPI

Once again note the Symmetry!

2
2,1
2,1
3

with the CELLAR a little bottom heavy in both cases.

Used

EMY
BNT, DLR, CS
JV, FZGK, QW

(None)

Filled out

EM1Y

B5N5T1T5, DLR1R5, CS1S5
J1J5V1, FZGK, QW

Excluding the ROOF there is symmetry between Picture A and Pictures
B and C from the construction based on PG(3,2).

(12)
40+20
40+20

producing the 132 Steiner Hexads and many amusing patterns.

PGH

Have been studying Myhill's Property and also Rothenberg Scales.

Interesting that the diatonic and the pentatonic scales have
Myhill's property. So I thought I would study hexachords based
on black and white keys. Subsets of each have the property,
combined together, i don't know....

Here is how all 924 hexachords break down

White Keys Black Keys Hexachords

6  0   7
5  1  105
4  2  350
3  3  350
2  4  105
1  5  7

Hold C fixed

6  0  6
5  1  75
4  2  200
3  3  150
2  4  30
1  5  1

Notice subtracting Column Three Lower from Column Three Upper produces

1,30,150,200,75,6 which is just the lower set reversed. Also
1+30+200=6+75+150=231 which is exactly half of 462 and 1/4 of 924..

Now I have found that all 80 hexachord types can be expressed
with no more than 2 black keys.

Now to study Rothenberg Scales and their hexachord subsets.

PGH

I wrote:

>These are (val comma badness), where val is the non-torsional

Oh, and I'm assuming for rank 1 temperaments, removing torsion
is as simple as ignoring vals with GCD > 1.  Somebody please
correct me if that's not right.

-Carl

>I just uploaded a plot of notes/octave vs. simplest 7-limit
>comma, here:
>
>http://tech.groups.yahoo.com/group/tuning-math/files/carl/
>
>Looks like it might work.

Here are the best 3 ETs up to 100 ET for several different
limits using this scheme.

These are (val comma badness), where val is the non-torsional
val with the least TOP damage for the ET after the corresponding
TOP stretch has been applied, comma is the simplest vanishing
comma for that val which involves all the limit's prime factors
and where "simplest" is defined in terms of tenney height
(though I'm generating candidate commas by unweighted lattice
distance and there is a possibility that a comma lower tenney
height is outside my window), and badness is the tenney height
of this comma raised to the standard logflat exponent times the
top damage of the val.

5-limit:
   ((53 84 123) 15625/15552 45)
   ((65 103 151) 32805/32768 70)
   ((87 138 202) 15625/15552 73))
7-limit:
   ((99 157 230 278) 2401/2400 21)
   ((72 114 167 202) 225/224 25)
   ((94 149 218 264) 225/224 38))
11-limit:
   ((72 114 167 202 249) 441/440 23)
   ((99 157 230 278 343) 441/440 33)
   ((94 149 218 264 325) 539/540 36))
13-limit:
   ((94 149 218 264 325 348) 1715/1716 39)
   ((72 114 167 202 249 266) 1715/1716 39)
   ((87 138 202 244 301 322) 4235/4212 48))
17-limit:
   ((72 114 167 202 249 266 294) 715/714 30)
   ((94 149 218 264 325 348 384) 715/714 31)
   ((80 127 186 225 277 296 327) 715/714 36))

The results are biased towards larger ETs, and I doubt the
results are logflat anymore.  There's clearly a penalty to
having more notes/octave, though I'll argue it's less important
than the penalty of greater Graham complexity in the embedded
rank 2 temperaments.  I was thinking maybe
	 (notes/octave * simplest comma)^logflat_expt
would be one compromise.  If anyone is following this, I'd
love to have your comments.

-Carl

I wrote:

>One might also disqualify commas that don't include all
>primes in the limit.  For example, the best 11-limit val
>above, <31 49 72 87 107|, tempers out 81/80.  But the
>simplest complete 11-limit comma it tempers out seems to
>be 441/440.

Simplest by Tenney height, that is.

I just uploaded a plot of notes/octave vs. simplest 7-limit
comma, here:

http://tech.groups.yahoo.com/group/tuning-math/files/carl/

Looks like it might work.

-Carl

Hello,

This email message is a notification to let you know that
a file has been uploaded to the Files area of the tuning-math
group.

   File        : /carl/ComplexityPlot.xls
   Uploaded by : clumma <carl@...>
   Description : Simplest comma vs. notes/octave complexity for ETs

You can access this file at the URL:
http://groups.yahoo.com/group/tuning-math/files/carl/ComplexityPlot.xls

To learn more about file sharing for your group, please visit:
http://help.yahoo.com/l/us/yahoo/groups/original/members/web/index.htmlfiles

Regards,

clumma <carl@...>

Kalle wrote:

>> when the error is Tenney-weighted, the sequence of 5-limit
>> ETs of decreasing badness is
>>
>> 1, 12, 53, 4296,...
>>
>> Of meantone ETs 12-equal has the lowest log-flat badness!
>
>Are there any other badness measures which are as non-ad hoc as
>log-flat but which would give more reasonable values for ETs?

I was playing with logflat badness the other night, generating
this list, where I look at ETs up to 1000 and down to the
number of primes (e.g. 4-ET is smallest 7-limit ET considered),
and print the top results such that 12-ET is their median.  The
val used for each ET is the one with the lowest TOP damage after
the TOP stretch is applied, and complexity is the ET number.

5-limit
  <53 84 123| 117
  <12 19 28|  148
  <3 5 7|     157
7-limit
  <99 157 230 278| 155
  <12 19 28 34|    169
  <31 49 72 87|    176
11-limit
  <31 49 72 87 107|    132
  <72 114 167 202 249| 135
  <22 35 51 62 76|     156
  <12 19 28 34 42|     170
  <5 8 12 14 18|       190
  <8 13 19 23 28|      201
  <41 65 95 115 142|   202
13-limit
  <72 114 167 202 249 266| 165
  <12 19 28 34 42 45|      170
  <8 13 19 23 28 30|       181
17-limit
  <72 114 167 202 249 266 294| 143
  <12 19 28 34 42 45 49|       156
  <46 73 107 129 159 170 188|  161

As I've done on a few occasions in the past, I tinkered
with the exponents on error and complexity, but didn't find
anything that seemed to work better than logflat.

Then today I think I hit on an explanation: the badness is
doing its job, it's just that ETs are unmusical.  Things
like 72 are not scales on which to base music, but rather
convenient tunings for rank 2 temperaments.  ETs in the
complexity range of musical scales aren't accurate enough,
and are too melodically simple.  Even 12-ET is not used
in music much -- the diatonic scale is.  This isn't a new
realization, but I'm trying it again for the first time. :)

For rank 2 temperaments, something like the length of the
continuous chain of generators needed to produce a
saturated chord makes more sense.  I think that's called
Graham complexity (right Graham?).  And I think it's the
same as the unweighted wedgie complexity if the footprint
of both generators and periods is considered (right Graham?).
Then there's scalar complexity... maybe somebody can tell
me what that is.

So one approach for ETs would be to report the complexity
of a rank 2 temperament it supports.  How to choose?  I'll
suggest using the complexity of the simplest comma tempered
out by the val under consideration.  This means one has to
go all the way to badness before picking a val for a number
of notes/octave, rather than just picking the one with the
lowest error.

One might also disqualify commas that don't include all
primes in the limit.  For example, the best 11-limit val
above, <31 49 72 87 107|, tempers out 81/80.  But the
simplest complete 11-limit comma it tempers out seems to
be 441/440.  I'd then take the log2 Tenney Height of this
comma as the val's weighted complexity (its taxicab
distance on a rectangular lattice would be its unweighted
complexity).

Has this been suggested before?  I haven't recoded my
search for it yet, but I've got to get to work before they
fire me.  Maybe somebody will beat me to it...

-Carl

This is really cool

http://glooscap.cs.dal.ca:8087/

-Carl

Graham wrote:

>I was finding the best mapping for TOP-RMS.  Here's the
>table using the best TOP-max mappings:
>
>  2: 33.0  77.3  77.3  77.3  77.6  77.6
>  3: 30.2  30.2  39.8  39.8  45.3  45.3
>  4: 33.0  33.0  33.0  37.5  37.5  37.5
>  5:  5.7  19.8  21.4  25.5  25.5  25.5
>  6: 30.2  30.2  30.2  30.2  30.2  30.2
>  7:  5.1   9.4  20.0  20.0  20.0  20.0
>  9: 11.2  14.2  14.2  14.2  14.2  14.7
> 10:  5.7  11.4  11.4  12.7  12.7  12.7
> 12:  0.6   3.6   6.1   7.6   8.6   8.6
> 15:  5.7   5.7   7.2   7.2   7.2   8.3
> 19:  2.3   2.3   3.8   6.3   6.3   6.4
> 22:  2.2   3.2   3.3   3.3   5.3   5.3
> 26:  3.1   3.7   3.8   4.1   4.1   4.1
> 31:  1.6   1.8   1.8   1.8   3.1   3.1
> 41:  0.2   1.4   1.4   1.9   2.4   2.7
> 46:  0.8   1.1   1.7   1.7   1.9   1.9
> 58:  0.5   1.5   1.5   1.5   1.5   1.6
> 72:  0.6   0.6   0.6   0.6   1.0   1.0
>
>                   Graham

My numbers agree, though this no longer shows all
the successive improvements in 17-limit damage.
Here's the complete list:

  (2 (33.0 77.3 77.3 77.3 77.6 77.6))
  (3 (30.2 30.2 39.8 39.8 45.3 45.3))
  (4 (33.0 33.0 33.0 37.5 37.5 37.5))
  (5 (5.7 19.8 21.4 25.5 25.5 25.5))
  (7 (5.1 9.4 20.0 20.0 20.0 20.0))
  (8 (15.0 15.0 15.0 15.0 15.0 15.0))
  (9 (11.2 14.2 14.2 14.2 14.2 14.7))
  (10 (5.7 11.4 11.4 12.7 12.7 12.7))
  (12 (0.6 3.6 6.1 7.6 8.6 8.6))
  (15 (5.7 5.7 7.2 7.2 7.2 8.3))
  (17 (1.2 8.0 8.0 8.0 8.0 8.0))
  (19 (2.3 2.3 3.8 6.3 6.3 6.4))
  (22 (2.2 3.2 3.3 3.3 5.3 5.3))
  (26 (3.1 3.7 3.8 4.1 4.1 4.1))
  (27 (2.9 2.9 2.9 3.8 3.8 3.8))
  (29 (0.5 3.5 3.5 3.5 3.5 3.5))
  (31 (1.6 1.8 1.8 1.8 3.1 3.1))
  (39 (1.8 2.9 2.9 2.9 2.9 2.9))
  (41 (0.2 1.4 1.4 1.9 2.4 2.7))
  (46 (0.8 1.1 1.7 1.7 1.9 1.9))
  (58 (0.5 1.5 1.5 1.5 1.5 1.6))
  (72 (0.6 0.6 0.6 0.6 1.0 1.0))

-Carl

2009/1/5 Carl Lumma <carl@...>:

> What's the primary advantage, as you see it, to TOP-RMS over
> TOP-Max?

That it's easier to work with the optimizations algebraically.  I
think the badness in the title is a very interesting function and
should be studied as pure mathematics.  But I haven't found it in that
context.


                           Graham

Graham wrote:

>Going by equation (5) in composite.pdf it looks like the straight RMS
>also only depends on the metric in equation (7).  So optimal errors
>aren't special here.  They happened to be the kind I was interested
>in.

OK.

>I haven't proven the limit to infinity but it obviously works.

OK.

>So then it's easy.  The matrix G is already diagonalized.  It gives
>you a set of prime weights for a TOP-RMS-like error.  And the higher
>the complexity the more it looks like TOP.

What's the primary advantage, as you see it, to TOP-RMS over
TOP-Max?

-Carl

2009/1/5 Carl Lumma <carl@...>:
> Graham wrote:

>>Any Tenney limit, or the intersection of any Tenney and prime limits,
>>is a particular set of prime weights for an RMS calculation.
>
> Yes, that's what I remember from your paper.  Except the other
> day when I was looking at it again, the tables seemed to show
> Farey limits intsead.

There are tables for both.

>>So the
>>optimal RMS over all intervals is the same as the optimal RMS for
>>given weights of the prime intervals.  These weights tend to the
>>Tenney weights the complexity tends to infinity,
>
> You've proven this then?  I don't know why you keep saying
> "optimal", since that to me means a tuning.  Hopefully what
> you're describing here is valid for any tuning I'd like to
> measure.

If you're looking for temperament classes you naturally compare them
with their optimal tunings.  The computation is the same order of
complexity regardless of how many intervals you started with.  It only
depends on the number of primes.

Going by equation (5) in composite.pdf it looks like the straight RMS
also only depends on the metric in equation (7).  So optimal errors
aren't special here.  They happened to be the kind I was interested
in.

I haven't proven the limit to infinity but it obviously works.

>>if you weight all the intervals equally.
>>
>>I think you still get Tenney weighting for a big family of weightings
>>as the complexity approaches to infinity.  It depends on intervals of
>>similar complexity having similar weight.
>
> I don't know how to parse this... you seem to be talking about
> two different but simultaneous sets of weightings.

Start with n intervals defined on d primes.  Put the weights in a
matrix called W, which is nxd or dxn whichever way you hold it up.
There's another matrix G which is dxd and defines the same problem
using only the prime intervals.

>>The reason it matters how you approach infinity is that Farey limits
>>are of a different form.  They can't be done with simple prime weights
>>at all.  You need a matrix to specify cross-weights.  But such a
>>matrix still doesn't add much to the complexity of the calculation.  I
>>don't know what the RMS converges to in this case.
>
> Yes, well Tenney limits are all I care about here.

So then it's easy.  The matrix G is already diagonalized.  It gives
you a set of prime weights for a TOP-RMS-like error.  And the higher
the complexity the more it looks like TOP.


                            Graham

Graham wrote:

>> The whole point of prime-based error is that it's a shortcut to
>> measuring all the intervals.  Ideally for Max-primes one uses an
>> error weighting that makes it come out equal to to Max-all.
>> Tenney weighting happens to do that.  For RMS-primes the natural
>> extension would be to use a weighting that makes it come out
>> equal to RMS-all.  But then there's the question of how to
>> define "all"... does the RMS converge?  I think you talk about
>> this in one of your papers.  Here I was thinking an upper bound
>> would be good, but maybe that doesn't make sense.
>
>Any Tenney limit, or the intersection of any Tenney and prime limits,
>is a particular set of prime weights for an RMS calculation.

Yes, that's what I remember from your paper.  Except the other
day when I was looking at it again, the tables seemed to show
Farey limits intsead.

>So the
>optimal RMS over all intervals is the same as the optimal RMS for
>given weights of the prime intervals.  These weights tend to the
>Tenney weights the complexity tends to infinity,

You've proven this then?  I don't know why you keep saying
"optimal", since that to me means a tuning.  Hopefully what
you're describing here is valid for any tuning I'd like to
measure.

>if you weight all the intervals equally.
>
>I think you still get Tenney weighting for a big family of weightings
>as the complexity approaches to infinity.  It depends on intervals of
>similar complexity having similar weight.

I don't know how to parse this... you seem to be talking about
two different but simultaneous sets of weightings.

>The reason it matters how you approach infinity is that Farey limits
>are of a different form.  They can't be done with simple prime weights
>at all.  You need a matrix to specify cross-weights.  But such a
>matrix still doesn't add much to the complexity of the calculation.  I
>don't know what the RMS converges to in this case.

Yes, well Tenney limits are all I care about here.

-Carl

2009/1/5 Carl Lumma <carl@...>:

> The whole point of prime-based error is that it's a shortcut to
> measuring all the intervals.  Ideally for Max-primes one uses an
> error weighting that makes it come out equal to to Max-all.
> Tenney weighting happens to do that.  For RMS-primes the natural
> extension would be to use a weighting that makes it come out
> equal to RMS-all.  But then there's the question of how to
> define "all"... does the RMS converge?  I think you talk about
> this in one of your papers.  Here I was thinking an upper bound
> would be good, but maybe that doesn't make sense.

Any Tenney limit, or the intersection of any Tenney and prime limits,
is a particular set of prime weights for an RMS calculation.  So the
optimal RMS over all intervals is the same as the optimal RMS for
given weights of the prime intervals.  These weights tend to the
Tenney weights the complexity tends to infinity, if you weight all the
intervals equally.

I think you still get Tenney weighting for a big family of weightings
as the complexity approaches to infinity.  It depends on intervals of
similar complexity having similar weight.

The reason it matters how you approach infinity is that Farey limits
are of a different form.  They can't be done with simple prime weights
at all.  You need a matrix to specify cross-weights.  But such a
matrix still doesn't add much to the complexity of the calculation.  I
don't know what the RMS converges to in this case.

So TOP-RMS, or weighted-prime RMS in general, is consistent with
Tenney limits.  It means intervals like 30:1 are treated equally with
6:5.  As long as you optimize the scale stretch that's probably good
enough.  It's the argument for not bothering with STD errors unless
they're simpler in a given context.  The applicability is that you
don't know exactly what you want so you guess some numbers that are
likely to work.


                            Graham

Graham wrote:
>>>> a^2 + 2ac + c^2      a^2 d^2 + c^2 b^2
>>>> ---------------  <=  -----------------
>>>> b^2 + 2bd + d^2           b^2 d^2
>>>>
>>>> (a^2 b^2 c^2) + 2(a^2 c^2 bd)  <=  (c^4 a^2) + (d^4 b^2) +
>>>> 2(c^3 a^2 d) + 2(d^3 b^2 c) + (b^2 c^2 d^2)
>>>>
>>>> Which seems quite true.  Unless I made a mistake, which I
>>>> probably did.  Does this make any sense?
>
>I get 2 a b^2 c d^2 <= a^2 d^4 + 2 a^2 b d^3 + b^4 c^2 + 2 b^3 c^2 d.
>That may still be wrong but I don't see how you can get a^2 b^2 c^2.

I used
http://xrjunque.nom.es/precis/polycalc.aspx

Now I'm getting
(a^2*b^2*d^2)+(b^2*c^2*d^2)+(b^2*d^2*2ac) <=
(a^4*d^2)+(c^4*b^2)+(a^3*d^2*2c)+(c^3*b^2*2a)+(a^2*b^2*c^2)+(a^2*c^2*d^2)

>>>I haven't followed it, but it should be.  The logic for RMS being
>>>less than the max value is that RMS is a kind of average.
>>
>> The above is an argument that, e.g. in the 5-limit, the RMS
>> of the Tenney-weighted errors of 3 & 5 would be *greater than*
>> or equal to the weighted error of 5/3 or 15/8.  That is,
>> RMS-primes would give an upper bound on the weighted errors of
>> these compound intervals.  Ideally this would be extended to
>> all intervals, as in Max-primes.  Otherwise, we might start
>> looking for a weighting (other than log Tenney Height) which
>> satisfied such an inequality for RMS-primes.
>
>Why?

The whole point of prime-based error is that it's a shortcut to
measuring all the intervals.  Ideally for Max-primes one uses an
error weighting that makes it come out equal to to Max-all.
Tenney weighting happens to do that.  For RMS-primes the natural
extension would be to use a weighting that makes it come out
equal to RMS-all.  But then there's the question of how to
define "all"... does the RMS converge?  I think you talk about
this in one of your papers.  Here I was thinking an upper bound
would be good, but maybe that doesn't make sense.

>Knowing the TOP-RMS is smaller than TOP-max means if you can find all
>the temperament classes in some range with TOP-RMS error below a given
>value, you know it also contains all classes in the same range with
>TOP-max below the same value.  Assuming you're interested in TOP-max
>for some reason, that's useful because it's easier to search for low
>TOP-RMS errors.

You're probably right.

-Carl

2009/1/5 Carl Lumma <carl@...>:
> Graham wrote:

>>> If so, it occurs to me that my other question about TOP-RMS that
>>> IIRC you said you still hadn't fully answered boils down to
>>> this: If a/b <= c/d, where a and c are errors and b and d are
>>> weights, then for TOP-max
>>>
>>> a/b <= (a+c)/(b+d) <= c/d
>>>
>>> in the worst case, when errors add.  TOP-RMS is then
>>>
>>> (a+c)/(b+d)  ??  sqrt((a/b)^2 + (c/d)^2)
>>>
>>> Let's assume ?? is <=, which is what we want.  So,
>>>
>>>             ( a^2 d^2 + c^2 b^2 )
>>> ...  <=  sqrt( ----------------- )
>>>             (      b^2 d^2      )
>>>
>>> (a+c)^2       a^2 d^2 + c^2 b^2
>>> --------  <=  -----------------
>>> (b+d)^2            b^2 d^2
>>>
>>> a^2 + 2ac + c^2      a^2 d^2 + c^2 b^2
>>> ---------------  <=  -----------------
>>> b^2 + 2bd + d^2           b^2 d^2
>>>
>>> (a^2 b^2 c^2) + 2(a^2 c^2 bd)  <=  (c^4 a^2) + (d^4 b^2) +
>>> 2(c^3 a^2 d) + 2(d^3 b^2 c) + (b^2 c^2 d^2)
>>>
>>> Which seems quite true.  Unless I made a mistake, which I
>>> probably did.  Does this make any sense?

I get 2 a b^2 c d^2 <= a^2 d^4 + 2 a^2 b d^3 + b^4 c^2 + 2 b^3 c^2 d.
That may still be wrong but I don't see how you can get a^2 b^2 c^2.

>>I haven't followed it, but it should be.  The logic for RMS being
>>less than the max value is that RMS is a kind of average.
>
> The above is an argument that, e.g. in the 5-limit, the RMS
> of the Tenney-weighted errors of 3 & 5 would be *greater than*
> or equal to the weighted error of 5/3 or 15/8.  That is,
> RMS-primes would give an upper bound on the weighted errors of
> these compound intervals.  Ideally this would be extended to
> all intervals, as in Max-primes.  Otherwise, we might start
> looking for a weighting (other than log Tenney Height) which
> satisfied such an inequality for RMS-primes.

Why?

>>The average is always going to be less than the worst case.  So
>>if you take the TOP-max tuning, the weighted RMS must be less
>>than the worst case, which is the TOP-max error.  The TOP-RMS
>>will be for a different tuning but that, also being optimal, can
>>only be smaller again.
>
> Right, but this sounds like it is not desirable.  Don't we
> want an *upper* bound to be quickly computable from the primes?

I don't know what you want.  You said this came from a question you
asked at some unspecified time.  If you want the worst error, you can
always calculate the worst error.

Knowing the TOP-RMS is smaller than TOP-max means if you can find all
the temperament classes in some range with TOP-RMS error below a given
value, you know it also contains all classes in the same range with
TOP-max below the same value.  Assuming you're interested in TOP-max
for some reason, that's useful because it's easier to search for low
TOP-RMS errors.


                              Graham