Temperament notation

This Peppermint notation business raises again the question of how, in general, you two are proposing to deal with notating temperaments. Is is to be a cut down version of JI, or do you relate it to ets supporting the temperament, or something else?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> This Peppermint notation business raises again the question of how,
in general, you two are proposing to deal with notating temperaments.
Is is to be a cut down version of JI, or do you relate it to ets
supporting the temperament, or something else?

It looks as if the method will be:

1) Identify an ET that closely resembles the temperament;

2) Use the ET notation for that temperament; and

3) Identify any instances in which symbols apart from that ET's
standard set might be more harmonically meaningful, and make
appropriate substitutions.

For example, if you want to notate the meantone temperament extended
to include ratios of 7, you could use the 31-ET notation. But the
standard 31-ET symbol set uses the 11-comma (32:33) symbol for a
single degree, and you're not interested in ratios of 11. So you
would probably want to use the 7-comma symbol (63:64) instead.

But if you're using the 11 limit in extended meantone, you could even
use both symbols (in different places) if you wanted to indicate
harmonic function (as long as it didn't confuse the player). This
brings up the question of whether it would be advisable to do this in
an ET as well. We have gone out of our way to define standard symbol
sets that would require the minimum amount of memorization and
confusion and maximum utility, so our policy is to discourage
departure from standard sets. But we're not going to be the notation
police, either, so a composer will be free to experiment with things
like this to see what's helpful and what isn't.

--George

At 08:43 AM 26/12/2002 -0800, George Secor wrote:
>--- In tuning-math@yahoogroups.com, "Gene Ward Smith
><genewardsmith@j...>" <genewardsmith@j...> wrote:
> > This Peppermint notation business raises again the question of how,
>in general, you two are proposing to deal with notating temperaments.
>Is is to be a cut down version of JI, or do you relate it to ets
>supporting the temperament, or something else?
>
>It looks as if the method will be:
>
>1) Identify an ET that closely resembles the temperament;

This is the tricky bit. There will generally be more than one such ET. Which one should we use.

For an octave-based linear temperament, if we know the exact size of the generator and period then we can be a bit more specific.

(a) The cardinality of the ET must be (octave/period times) the denominator of a convergent of generator/period, and

(b) The best fifth (approx 2:3) in the ET must be the same as the fifth calculated by applying the temperament's mapping to the best approximation of the generator (and period) in that ET, and

(c) The cardinality of the ET must not be so small that the desired length of generator chains wrap around or run into each other. So the cardinality of the ET must be greater (and may need to be a lot greater) than the cardinality of the finite scale being used in the temperament, and

(d) The cardinality of the ET must not be so large that it cannot be notated in sagittal notation, or cannot be notated without resorting to more unusual or complex sagittal symbols than necessary.

(e) The cardinality of the ET must not be so large that its validity according to criterion (a) above is overly sensitive to the exact value of the generator.

Now criteria (a) (b) and (c) above are completely defined, but (d) and (e) are fuzzy. One way around this is to replace (d) and (e) with

(f) The ET must be the smallest one that satisfies (a) (b) and (c).

The only problem with this is that different size scales within a temperament may end up with completely different notations. This could be turned on its head and instead a popular temperament might be used as an argument for determining the standard sagittal symbol set for some ET, so that the central parts of the generator chains agree between the various convergent ETs.

e.g. for meantone 12, 19, 31, 50
miracle 31, 41, 72
But it isn't possible to make this work with all temperaments and their ETs.

>2) Use the ET notation for that temperament; and
>
>3) Identify any instances in which symbols apart from that ET's
>standard set might be more harmonically meaningful, and make
>appropriate substitutions.
>
>For example, if you want to notate the meantone temperament extended to
>include ratios of 7, you could use the 31-ET notation. But the
>standard 31-ET symbol set uses the 11-comma (32:33) symbol for a single
>degree, and you're not interested in ratios of 11. So you would
>probably want to use the 7-comma symbol (63:64) instead.

I really don't see why it should change from the standard set of the ET. Someone using 31-ET without reference to any temperament may also not be interested in ratios of 11 and yet should probably just accept that /|\ is the symbol for both the 11-diesis and the 7-comma in 31-ET, because of its stronger association as a half-apotome symbol.

>But if you're using the 11 limit in extended meantone, you could even
>use both symbols (in different places) if you wanted to indicate
>harmonic function (as long as it didn't confuse the player). This
>brings up the question of whether it would be advisable to do this in
>an ET as well. We have gone out of our way to define standard symbol
>sets that would require the minimum amount of memorization and
>confusion and maximum utility, so our policy is to discourage departure
>from standard sets. But we're not going to be the notation police,
>either, so a composer will be free to experiment with things like this
>to see what's helpful and what isn't.

Agreed. I'm just saying that I severely doubt it will be helpful. One major reason for having standards in the first place, is so we can communicate with each other. If everyone invents their own sagittal notation for an ET, they may find it personally "helpful" right up until they want someone else to read it.

Whether the standards proposed by we two, will be thought acceptable by a wider community is another question, but there will only be a very brief period (starting now) when it will be possible to negotiate a change to any standard.

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

--- In tuning-math@yahoogroups.com, David C Keenan <d.keenan@u...> wrote:

> Whether the standards proposed by we two, will be thought acceptable by a
> wider community is another question, but there will only be a very brief
> period (starting now) when it will be possible to negotiate a change to any
> standard.

Whatever you do, I suggest you keep in mind that notating linear temperaments is just as important as notating ets, if not more so.
Note that what we have now is a linear temperament notation adapted for use as a 12-et notation, and not the other way around.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:

> Whatever you do, I suggest you keep in mind that notating linear
temperaments is just as important as notating ets, if not more so.

Good point.

> Note that what we have now is a linear temperament notation adapted
for use as a 12-et notation, and not the other way around.

Good point.

Thanks to the efforts of you and Graham Breed and others on this list
we now have agreed lists of the most important 5-limit and 7-limit
linear temperaments which we can attempt to notate.

We also have Graham's lists of LTs for higher limits up to 21,
http://x31eq.com/temper.html
maybe you could check that his algorithm hasn't missed any important
ones, starting with 9-limit.

We should try to develop rules for notating them which do not depend
on predetermined ET notations. Ideally the notation would depend only
on the LT's mapping from gens and periods to primes, and not on any
particular value for the generator.

The problem is that we can't specify a notation that will work for
unlimited length chains of an LT while only allowing one or two
symbols against each note. And what's more, for even moderate length
chains we will typically need to use symbols for primes that do not
appear in the LT's map. For example with a chain of more than 21 notes
in 5-limit meantone, we wish to have a single-symbol alternative to
double-sharps and double-flats, e.g. Fx may be written G^ so that in
pitch order the letters can remain monotonic. This ^ symbol must
relate to a prime greater than 5. In this case either 7 or 11 will do.
For longer chains we may need to use symbols for 13-commas.

So even though it might only be used for 5-limit, the notation ends up
being for a particular 7, 11 or 13 limit mapping of meantone, when in
fact the range of generator sizes that are of interest for 5-limit
meantone, may encompass more than one 7 or 11 or 13-limit mapping. How
do we choose which one to use?

--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:

> We should try to develop rules for notating them which do not depend
> on predetermined ET notations. Ideally the notation would depend only
> on the LT's mapping from gens and periods to primes, and not on any
> particular value for the generator.

I don't know how you expect to do that. The most obvious approach seems to me to pick nominals for a MOS with 26 steps or fewer, pick the generator which has the lowest height in the correct p-limit, and use something like sharps and flats. That is completely at odds with what you are doing. If I understand how Graham's decimal notation works, it would be a generalization of that.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
>
> > We should try to develop rules for notating them which do not
depend
> > on predetermined ET notations. Ideally the notation would depend
only
> > on the LT's mapping from gens and periods to primes, and not on
any
> > particular value for the generator.
>
> I don't know how you expect to do that. The most obvious approach
seems to me to pick nominals for a MOS with 26 steps or fewer, pick
the generator which has the lowest height in the correct p-limit, and
use something like sharps and flats. That is completely at odds with
what you are doing. If I understand how Graham's decimal notation
works, it would be a generalization of that.

Yes, it is completely at odds. Yes, it is a generalisation of
multi-sharp/flat meantone notation and Graham's decimal notation for
miracle temperament, and the 4, 7 and 8 natural notations for kleismic
described in my "Chain of minor thirds" article.

I have nothing against these temperament-specific notations. Very
early in the "Common notation ..." thread I said the same, and one
could certainly use sagittal symbols consistently for this purpose.

By the way, although there are 26 letters in our alphabet, there are
good reasons, relating to human cognition, why one should aim to have
between 5 and 9 naturals if such a proper MOS exists for the
temperament, and otherwise keep it as close to 9 as possible, more
than 12 is probably useless. I think that even 4 would be better than
13 or more, if such a choice were available.

But as you say, this is not what George and I are trying to do. When
one learns one such temperament-specific notation there is almost
nothing one can carry over to an unrelated temperament, particularly
if it involves a different number of nominals. Nor is there anything
much one can use from one's knowledge of conventional notation.

George and I are attempting to design an _evolution_ for those who are
unlikely to be interested in such a _revolution_, and we'd love your
help.

We want notations where C:G is always the temperament's approximation
of a 1:3 (if it has one) and C:E* is always it's approximation of
a 1:5, where * stands for a single saggital symbol or none at all, and
so on up the primes. But we want more than this. We want it so the
notation for any reasonable-sized MOS in the temperament can be
arranged in pitch order with monotonic letters, e.g. one is never
forced to have any C* higher in pitch than any D*, and one is not
forced to use more than one symbol with any letter.

As one extends the chains of generators in any temperament, there
eventually comes a proper MOS which is so close to an equal
temperament that there is little point in adding more notes. There
seems no disagreement that this happens at 72 for miracle, but for
meantone it is unclear to me whether it is 31 or 50. It seems to be 50
in the 5-limit optimised case, and 31 for 7 and 11 limit.

For schismic is it 41 or 53?
For kleismic is it 53 or 72?
etc.

>>I don't know how you expect to do that. The most obvious approach
>>seems to me to pick nominals for a MOS with 26 steps or fewer,
>>pick the generator which has the lowest height in the correct
>>p-limit, and use something like sharps and flats. That is
>>completely at odds with what you are doing. If I understand how
>>Graham's decimal notation works, it would be a generalization of
>>that.
>
>Yes, it is completely at odds. Yes, it is a generalisation of
>multi-sharp/flat meantone notation and Graham's decimal notation
>for miracle temperament, and the 4, 7 and 8 natural notations for
>kleismic described in my "Chain of minor thirds" article.

I believe that the simplest way to notate 'diatonic' music is to
put the transposition in the fingers and have the scale degrees
make sense on paper. Handing the mind scale degrees is
indispensible pre-processing for working with 'diatonic' music.
The generalized keyboard would reduce the number of fingerings for
each scale -- the musician could learn twelve tunings, reading
from 'diatonic' notation in each case, with the same amount of
effort needed to learn to read standard notation on the piano.

But George's thoughts go a long way with me... could it be that
for a strict performer, who had to cover lots of tunings,
'transpositionally invarient' notation is the way to go?

>By the way, although there are 26 letters in our alphabet, there
>are good reasons, relating to human cognition, why one should aim
>to have between 5 and 9 naturals if such a proper MOS exists for
>the temperament, and otherwise keep it as close to 9 as possible,

Dave's right, Gene. Where did you get 26 from?

>more than 12 is probably useless. I think that even 4 would be
>better than 13 or more, if such a choice were available.

I'd hate to compare like this. 4 is seriously too small. If we're
allowed to write music that allows the listener to subset melodies,
then 13 would be far better. If we're forced to write tone rows
then probably 13 would be worse.

>But as you say, this is not what George and I are trying to do.
>When one learns one such temperament-specific notation there is
>almost nothing one can carry over to an unrelated temperament,
>particularly if it involves a different number of nominals.

I disagree (see above). I imagine that once the fingerings are
learned, the mind could transform them to scale degrees and back
fairly easily, and that much of the ability to extract scale degree
motion from one 'diatonic' notation would work on other 'diatonic'
notations (it's quite graphic, after all). Again, I'd take very
seriously any input from George on this matter.

>Nor is there anything much one can use from one's knowledge of
>conventional notation.

I would think that learning some new fingerings would be easier
than keeping track of all the bizare accidental motion if you
force decatonic music (say) onto 7 nominals.

>We want notations where C:G is always the temperament's
>approximation of a 1:3 (if it has one) and C:E* is always it's
>approximation of a 1:5, where * stands for a single saggital
>symbol or none at all, and so on up the primes.

Easy! You just use 7-et and slap on as many accidentals as needed.

>But we want more than this. We want it so the notation for any
>reasonable-sized MOS in the temperament can be arranged in pitch
>order with monotonic letters, e.g. one is never forced to have
>any C* higher in pitch than any D*,

? You mean 7 monotonic letters?

If you're forcing 7 nominals, you'll have to give up the idea that
accidentals represent chromatic uvs.

>and one is not forced to use more than one symbol with any letter.

Well, you can always get that by just adding more flags. Not sure
if such flag profusion is any better or worse that stacking a
single flag... in standard notation, double-sharp has a dedicated
flag, while double-flat does not... IOW, flags are independent of
accidentals.

-Carl

--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
> ...
> As one extends the chains of generators in any temperament, there
> eventually comes a proper MOS which is so close to an equal
> temperament that there is little point in adding more notes. There
> seems no disagreement that this happens at 72 for miracle, but for
> meantone it is unclear to me whether it is 31 or 50. It seems to be
50
> in the 5-limit optimised case, and 31 for 7 and 11 limit.

It depends on what method you're using to optimize. The 5-limit
minimax generator is a 1/4-comma fifth, which would also favor 31.
In order to get the slow-beating minor third of 50-ET you also get a
slow-beating major third, plus a faster-beating fifth that is not
acceptable to some. So I always thought that 31 was the clear choice
for meantone.

> For schismic is it 41 or 53?

How about 94?

> For kleismic is it 53 or 72?

I'll have to taken a better look at this one.

> etc.

etc.

--George

--- In tuning-math@yahoogroups.com, "gdsecor <gdsecor@y...>"
<gdsecor@y...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
> <d.keenan@u...> wrote:
> > ...
> > As one extends the chains of generators in any temperament, there
> > eventually comes a proper MOS which is so close to an equal
> > temperament that there is little point in adding more notes.
There
> > seems no disagreement that this happens at 72 for miracle, but
for
> > meantone it is unclear to me whether it is 31 or 50. It seems to
be
> 50
> > in the 5-limit optimised case, and 31 for 7 and 11 limit.
>
> It depends on what method you're using to optimize. The 5-limit
> minimax generator is a 1/4-comma fifth, which would also favor 31.
> In order to get the slow-beating minor third of 50-ET you also get
a
> slow-beating major third, plus a faster-beating fifth that is not
> acceptable to some. So I always thought that 31 was the clear
choice
> for meantone.
>
> > For schismic is it 41 or 53?
>
> How about 94?

Now that I've more time to look at this, I would say definitely 94,
for two reasons:

1) If you're including the 7th harmonic, then you might as well take
this to the 11 limit (since the minimax generator for both is the
same). The 11th harmonic occurs in the series of fifths in 41, 53,
and 94 in the +23 position, but in 41 it is also closer -- in the -18
position, which is not typical for the schismic family of
temperaments. So I eliminate 41 as my choice.

2) The 7 and 11-limit minimax generator is ~702.193c (7:9 being
exact) giving a maximum error of ~4.331c (for 5:7 and 5:9). The 53-
ET fifth is ~701.887c (max. error ~12.681c), but the 94-ET fifth is
much closer to the ideal: ~702.178c (max. error ~4.722c), and the
same may be said for the 13 and 15-limit minimax generator
(~702.109c, 13:14 being exact). So I eliminate 53, leaving 94 as my
choice.

> > For kleismic is it 53 or 72?
>
> I'll have to taken a better look at this one.

I've done that and I conclude that, unless you are sticking with a 5
limit, the choice is clearly 72 over 53. It is best to put the
figures in a table to show this:

Generator Size Max. error Exact

5-minimax ~316.993c ~1.351c 2:3
53-ET ~316.981c ~1.408c
72-ET ~316.667c ~2.980c
125-ET ~316.800c ~2.314c

7,9-minimax ~316.765c ~2.732c 4:7
53-ET ~316.981c ~6.167c
72-ET ~316.667c ~3.910c
125-ET ~316.800c ~3.088c

11-minimax ~316.745c ~2.976c 9:11
53-ET ~316.981c ~12.681c
72-ET ~316.667c ~3.910c
125-ET ~316.800c ~4.892c

I threw 125 in there also, since it does slightly better than 72 at
the 7 and 9 limit (and also at the 13 and 15 limit, which has the
same minimax generator as for the 7 and 9 limit). But since the 11
limit is the highest you can go while keeping the max error under 4
cents, that's the limit I would use, and 72 has the advantage.

Something else I noticed about the choice of 72 as the notation for
both the Miracle and kleismic temperaments: the progression of
sagittal symbols for a 72-ET panchromatic scale (one passing through
all the tones) is the same as that for a sequence of tones differing
by the generating interval in both temperaments. To illustrate:

72-ET: C C\! C!) C\!/ B|) B/| B
Miracle: C B\! Bb!) A\!/ G|) F#/| F
kleismic: C A\! F#!) Eb\!/ B|) G#/| F

The pattern then repeats. I believe that this is a useful property
that provides a further justification for basing the kleismic
notation on 72.

--George

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> >>I don't know how you expect to do that. The most obvious approach
> >>seems to me to pick nominals for a MOS with 26 steps or fewer,
> >>pick the generator which has the lowest height in the correct
> >>p-limit, and use something like sharps and flats. That is
> >>completely at odds with what you are doing. If I understand how
> >>Graham's decimal notation works, it would be a generalization of
> >>that.
> >
> >Yes, it is completely at odds. Yes, it is a generalisation of
> >multi-sharp/flat meantone notation and Graham's decimal notation
> >for miracle temperament, and the 4, 7 and 8 natural notations for
> >kleismic described in my "Chain of minor thirds" article.
>
> I believe that the simplest way to notate 'diatonic' music is to
> put the transposition in the fingers and have the scale degrees
> make sense on paper. Handing the mind scale degrees is
> indispensible pre-processing for working with 'diatonic' music.
> The generalized keyboard would reduce the number of fingerings for
> each scale -- the musician could learn twelve tunings, reading
> from 'diatonic' notation in each case, with the same amount of
> effort needed to learn to read standard notation on the piano.
>
> But George's thoughts go a long way with me... could it be that
> for a strict performer, who had to cover lots of tunings,
> 'transpositionally invarient' notation is the way to go?
>
> >By the way, although there are 26 letters in our alphabet, there
> >are good reasons, relating to human cognition, why one should aim
> >to have between 5 and 9 naturals if such a proper MOS exists for
> >the temperament, and otherwise keep it as close to 9 as possible,
>
> Dave's right, Gene. Where did you get 26 from?
>
> >more than 12 is probably useless. I think that even 4 would be
> >better than 13 or more, if such a choice were available.
>
> I'd hate to compare like this. 4 is seriously too small. If we're
> allowed to write music that allows the listener to subset melodies,
> then 13 would be far better. If we're forced to write tone rows
> then probably 13 would be worse.
>
> >But as you say, this is not what George and I are trying to do.
> >When one learns one such temperament-specific notation there is
> >almost nothing one can carry over to an unrelated temperament,
> >particularly if it involves a different number of nominals.
>
> I disagree (see above). I imagine that once the fingerings are
> learned, the mind could transform them to scale degrees and back
> fairly easily, and that much of the ability to extract scale degree
> motion from one 'diatonic' notation would work on other 'diatonic'
> notations (it's quite graphic, after all). Again, I'd take very
> seriously any input from George on this matter.

I've been thinking this over while in the process of answering other
things. I'm not familiar with the problems of notating some of the
esoteric ETs, for example, those for which Blackwood wrote etudes
(though I have a recording), so my reply is going to be from a
somewhat limited perspective. So take it for what it's worth.

Let me offer an example from my viewpoint that may shed some light on
some of the issues involved. Carl, you seem to be looking at the
problem of notation from a keyboardist's viewpoint (from your mention
of fingerings), but one's understanding and framework for a notation
must be broader than and/or independent of that. But even if we
stick to a keyboard application for the moment, let me give an
example of how a "transpositionally invariant" notation (if I
understand the term correctly) could pose more problems than it would
solve.

Let's consider the Miracle tuning notated using Graham's decimal
notation, which I presume you would consider transpositionally
invariant. Suppose you write something for Blackjack or Canasta in
decimal notation, and I want to play it on my Scalatron -- with a
Bosanquet generalized rather than a decimal keyboard. My only option
is to map Canasta into a 31-tone octave (41 would also be possible if
I had that many program boards on the instrument, but that's a more
difficult fingering pattern and another issue entirely). If I am
doing Blackjack, then I have that as a subset -- so far, so good.

Now I already know how to read 31-ET in Fokker's notation or sagittal
notation -- both with 7 nominals and the same semantics, differing
only in the cosmetics. And I can also read 72-ET sagittal notation
and can easily convert it at sight to 31-ET -- the rules are simple:
the 5 comma vanishes and the 7 comma is translated to an 11 comma.
So I would have no problem reading Miracle mapped to either 31 or 72
sagittal. But what value is the decimal notation to me, and what
incentive would I have to learn it without a decimal keyboard?

Now reverse the circumstances. Give me a decimal keyboard and
sufficient time to learn it, and then give me something in Miracle to
read in sagittal notation. I should do just fine with it, because I
can translate the notes into pitches and the pitches into keyboard
locations.

Now give me the same decimal keyboard with Partch's 11-limit JI
mapped onto it (observe that this was the reason that I originally
came up with the layout). Again, I should do just fine with 72-ET
sagittal notation, assuming that I am proficient with the keyboard.
So what is the point of learning a decimal notation, if it will have
a slight advantage only with a specific keyboard and/or a specific
tuning geometry? If I decide to use the Miracle temperament instead
of 11-limit JI for Partch's music, is there any advantage in using
the decimal notation with 10 nominals for this purpose over a 7-
nominal sagittal notation?

We're all familiar with a 7-nominal / heptatonic / diatonic notation,
and that's what Dave and I have been building on to produce what I
call a "generalized" notation -- one that is semantically independent
of any particular tonal geometry or division of the octave. It may
not be the best notation for everything, but it will do a lot of
different things and will do many of them extremely well.

--George

Would someone explain "minimax" generator (I understand rms generator)

Thanks

"gdsecor
<gdsecor@yahoo.co To: tuning-math@yahoogroups.com
m>" <gdsecor cc: (bcc: Paul G Hjelmstad/US/AMERICAS)
Subject: [tuning-math] Re: Temperament notation
12/31/2002 01:03
PM
Please respond to
tuning-math

--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
> ...
> As one extends the chains of generators in any temperament, there
> eventually comes a proper MOS which is so close to an equal
> temperament that there is little point in adding more notes. There
> seems no disagreement that this happens at 72 for miracle, but for
> meantone it is unclear to me whether it is 31 or 50. It seems to be
50
> in the 5-limit optimised case, and 31 for 7 and 11 limit.

It depends on what method you're using to optimize. The 5-limit
minimax generator is a 1/4-comma fifth, which would also favor 31.
In order to get the slow-beating minor third of 50-ET you also get a
slow-beating major third, plus a faster-beating fifth that is not
acceptable to some. So I always thought that 31 was the clear choice
for meantone.

> For schismic is it 41 or 53?

How about 94?

> For kleismic is it 53 or 72?

I'll have to taken a better look at this one.

> etc.

etc.

--George

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Hi George,

>Let me offer an example from my viewpoint that may shed some
>light on some of the issues involved. Carl, you seem to be
>looking at the problem of notation from a keyboardist's
>viewpoint (from your mention of fingerings), but one's
>understanding and framework for a notation must be broader than
>and/or independent of that.

Everything I said should apply to any instrument. I used
keyboarding as a 'worst-case' example.

>But even if we stick to a keyboard application for the moment,
>let me give an example of how a "transpositionally invariant"
>notation (if I understand the term correctly) could pose more
>problems than it would solve.

That was a bad choice of terminology. The issue that Gene and
I are raising is: should notation be based on the melodic scale
being used, or should they be based on the 7-tone meantone
diatonic scale musicians are already familiar with?

The issue sort-of assumes the notation will be used to write
what I've been calling "diatonic music" -- music that takes
melodies and primary harmonies from the same small scale, as
90% of Western music does.

>Let's consider the Miracle tuning notated using Graham's decimal
>notation, which I presume you would consider transpositionally
>invariant.

Let's nix that term, but yes, Graham's decimal notation is what
I'd advocate for Miracle.

>But what value is the decimal notation to me, and what incentive
>would I have to learn it without a decimal keyboard?

() It gives you an invaluable tool for understanding the music.

() The re-learning won't be as bad as you fear on a decimal
keyboard.

() Since your meantone generalized keyboard is at root a planar
hexagonal tiling, many mappings exist to make it more 'decimal'.
It may be that none reflect the secor:octave cycles in the
correct way (as far as the distance of the keys from the player,
etc.), but there should be a way to get the ten nominals of the
decimal scale under the fingers.

>Now give me the same decimal keyboard with Partch's 11-limit JI
>mapped onto it (observe that this was the reason that I originally
>came up with the layout). Again, I should do just fine with 72-ET
>sagittal notation, assuming that I am proficient with the
>keyboard.

You seem to be saying that it's easier to learn to find pitches
on a keyboard than it is to learn to find pitches in a notation...
For me, it's the opposite.

>If I decide to use the Miracle temperament instead of 11-limit JI
>for Partch's music, is there any advantage in using the decimal
>notation with 10 nominals for this purpose over a 7-nominal
>sagittal notation?

Let's ask it this way: take the well-tempered clavier and re-
write it with 6 nominals. Is that only a slight disadvantage?

Yes, my way means more work for the person interested in learning
multiple scales. But:

() If you use a generalized keyboard, you could learn 12 scales
with the same amount of work as people spend on learning the
diatonic scale on the piano, excluding the work it takes to
find the pitches in a tuning from blobs on paper... which I seem
to think is easier than you do...

>We're all familiar with a 7-nominal / heptatonic / diatonic
>notation, and that's what Dave and I have been building on to
>produce what I call a "generalized" notation -- one that is
>semantically independent of any particular tonal geometry or
>division of the octave. It may not be the best notation for
>everything, but it will do a lot of different things and will
>do many of them extremely well.

Definitely a worthwhile project.

-Carl

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:
>
> Would someone explain "minimax" generator (I understand rms generator)

Let's take meantone for an example. If we define the three linear functions

u1 = x - l2(3)

u2 = 4x - 4 - l2(5)

u3 = 3x -4 - l2(5/3)

using "l2" to mean log base 2, then the rms generator can be found
by minimizing u1^2+u2^2+u3^2. On the other hand, we can minimize
|u1| + |u2| + |u3| instead (which gives 1/4 comma meantone.) In fact,
for any p>1, we can minimize |u1|^p + |u2|^p + |u3|^p; if for instance
p is 4, this gives us essentially 1/3 comma meantone (.33365 comma.)
If we consider

(|u1|^p + |u2|^p + |u3|^p)^(1/p)

as p tends to infinity, in the limit we get

max(|u1|,|u2|,|u3|). Minimizing this gives us the minimum maximum, or
minimax--in this case, it is again 1/4 comma meantone.

Minimax generators can also be calculated with Scala by the
"calculate/minimax" command. It shows the least squares optimum
at the same time, so you don't need to enter everything twice
if you want that also.

The next version still to come will have a new dialog to support
the easy calculation of equal beating temperaments, in the Tools
menu.

Manuel

--- In tuning-math@yahoogroups.com, "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:
> >
> > Would someone explain "minimax" generator (I understand rms generator)
>
> Let's take meantone for an example. If we define the three linear functions
>
> u1 = x - l2(3)
>
> u2 = 4x - 4 - l2(5)
>
> u3 = 3x -4 - l2(5/3)
>
> using "l2" to mean log base 2, then the rms generator can be found
> by minimizing u1^2+u2^2+u3^2. On the other hand, we can minimize
> |u1| + |u2| + |u3| instead (which gives 1/4 comma meantone.) In fact,
> for any p>1, we can minimize |u1|^p + |u2|^p + |u3|^p; if for instance
> p is 4, this gives us essentially 1/3 comma meantone (.33365 comma.)
> If we consider

It turns out this is so high because of round off error. I redid the calculation using 100 digits of accuracy, and got:

p=2 7/26 comma meantone

p=4 7/26 comma meantone again

p=6 .26295498 comma meantone, about 5/19 comma

p=8 .25942006 comma meantone, about 7/27 comma

p=10 .25739442 comma meantone, about 9/35 comma

p=1 and p=infinity, 1/4 comma meantone

--- In tuning-math@yahoogroups.com, "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:

> p=2 7/26 comma meantone
>
> p=4 7/26 comma meantone again
>
> p=6 .26295498 comma meantone, about 5/19 comma
>
> p=8 .25942006 comma meantone, about 7/27 comma
>
> p=10 .25739442 comma meantone, about 9/35 comma
>
> p=1 and p=infinity, 1/4 comma meantone

To this you may add

p=3 (7+sqrt(11))/38 comma meantone, .271490126 comma, a little flatter than the rest of them, but still far from 1/3 comma. If you want an irrational meantone, this seems to be at least as well motivated as golden meantone, and more so that Lucy tuning. Maybe I should claim it and pester everyone to adopt it.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> Hi George,
>
> >Let me offer an example from my viewpoint that may shed some
> >light on some of the issues involved. Carl, you seem to be
> >looking at the problem of notation from a keyboardist's
> >viewpoint (from your mention of fingerings), but one's
> >understanding and framework for a notation must be broader than
> >and/or independent of that.
>
> Everything I said should apply to any instrument. I used
> keyboarding as a 'worst-case' example.
>
> >But even if we stick to a keyboard application for the moment,
> >let me give an example of how a "transpositionally invariant"
> >notation (if I understand the term correctly) could pose more
> >problems than it would solve.
>
> That was a bad choice of terminology. The issue that Gene and
> I are raising is: should notation be based on the melodic scale
> being used, or should they be based on the 7-tone meantone
> diatonic scale musicians are already familiar with?

The issue that I raise is: how many different notations can you
expect a person to learn?

> The issue sort-of assumes the notation will be used to write
> what I've been calling "diatonic music" -- music that takes
> melodies and primary harmonies from the same small scale, as
> 90% of Western music does.
>
> >Let's consider the Miracle tuning notated using Graham's decimal
> >notation, which I presume you would consider transpositionally
> >invariant.
>
> Let's nix that term, but yes, Graham's decimal notation is what
> I'd advocate for Miracle.

Of course.

> >But what value is the decimal notation to me, and what incentive
> >would I have to learn it without a decimal keyboard?
>
> () It gives you an invaluable tool for understanding the music.

Okay, but only as long as the music is written in the Miracle
temperament, yes? (More about this below.)

> () The re-learning won't be as bad as you fear on a decimal
> keyboard.

Oh, I have no problem with the idea of learning a new keyboard.

But I don't expect that I will ever have a decimal keyboard (since
it's *getting* a new keyboard that is the biggest problem, and I've
already been through that once). And even if I did, I expect that
most of the things that I would play on it wouldn't even be in the
Miracle temperament, in which case I pose the question: should
everything playable on this keyboard be in decimal notation? (More
about this below.)

> () Since your meantone generalized keyboard is at root a planar
> hexagonal tiling, many mappings exist to make it more 'decimal'.
> It may be that none reflect the secor:octave cycles in the
> correct way (as far as the distance of the keys from the player,
> etc.), but there should be a way to get the ten nominals of the
> decimal scale under the fingers.

The tones would be on a diagonal row of keys that (ascending in
pitch) would go off the near edge of the keyboard; but they could be
picked up at the far edge, so yes, it can be done without
extraordinary effort.

> >Now give me the same decimal keyboard with Partch's 11-limit JI
> >mapped onto it (observe that this was the reason that I originally
> >came up with the layout). Again, I should do just fine with 72-ET
> >sagittal notation, assuming that I am proficient with the
> >keyboard.
>
> You seem to be saying that it's easier to learn to find pitches
> on a keyboard than it is to learn to find pitches in a notation...
> For me, it's the opposite.

Actually I do find it easier to perceive the pitch relationships on a
generalized keyboard (of whatever sort) than from a notation, but
that's not what I was trying to say.

The broader point that I was trying to make seems to have gotten lost
in all of the details of the discussion. I was trying to show that
there is no particular advantage in using decimal notation to notate
music that is *not* based on the Miracle geometry (e.g., Partch's
music, which is better understood in reference to an 11-limit
tonality diamond), but for which the tones may still be very suitably
mapped onto a decimal keyboard. The advantage of decimal notation
comes into effect only when and if you are using the Miracle
temperament itself, i.e., exploiting the tonal relationships that are
unique to Miracle. Likewise, if I play something in 31, 41, or 72-ET
on a decimal keyboard that was composed by someone utterly ignorant
of Miracle as an organizing principle for tonality (as I believe
*all* of us were up until a couple of years ago -- myself included),
is the decimal notation going to benefit me in any way if the
composition which I am playing was not conceived as being decatonic?
I think not. These three divisions can be treated as *either*
heptatonic *or* decatonic, and I could even show you a unidecatonic
MOS subset of 31-ET (with a very useful tetrad that occurs in 5
places). But it would be unrealistic to expect anyone to learn three
different notations for one tonal system, according to which tonal
relationships are exploited in a given piece. (Or suppose that a
piece is heptatonic in one place and decatonic in another. Do we
switch notations in the middle of the page?)

My point is that alternate tunings often do not tie us down to
specific tonal organizations, so the choice of a tuning is often not
enough to determine how many nominals would be "best" for its
notation. Since we are already acquainted with a notation that uses
7 nominals, and if that works reasonably well for many alternative
tunings, then why not have a generalized notation that builds on that?

So I would therefore require a microtonal musician to learn no more
than one new notation. A composer may wish to do otherwise when
composing, but a translation would be provided for the player.

> >If I decide to use the Miracle temperament instead of 11-limit JI
> >for Partch's music, is there any advantage in using the decimal
> >notation with 10 nominals for this purpose over a 7-nominal
> >sagittal notation?
>
> Let's ask it this way: take the well-tempered clavier and re-
> write it with 6 nominals. Is that only a slight disadvantage?

I was about to say no, but only because 6 nominals will hardly work
well with anything, even for Partch (since 6:7:8:9:10:11 isn't a
constant structure). But if you can think in terms of a 6+6
keyboard, maybe it isn't that bad after all (until you decide that
maybe sharps and flats should be different in pitch).

I'm not really arguing against specialized notations with other than
7 nominals, but I don't think that we can expect very many players to
learn them.

> Yes, my way means more work for the person interested in learning
> multiple scales. But:
>
> () If you use a generalized keyboard, you could learn 12 scales
> with the same amount of work as people spend on learning the
> diatonic scale on the piano, excluding the work it takes to
> find the pitches in a tuning from blobs on paper... which I seem
> to think is easier than you do...

The notes on the paper give you a two-dimensional visual orientation -
- placement on the staff (first dimension, but with non-uniform
steps) plus altering symbols (second dimension). A generalized
keyboard gives you two visual dimensions with uniform steps, plus
sound, plus a tactile or kinesthetic experience. But the two
together are even better, or lacking a real keyboard, a keyboard
diagram labeled with the notation is still good. (I often refer to
the decimal keyboard diagram labeled with ratios and 72-ET sagittal
notation that I put here in the files section.)

I have no doubt that the decimal keyboard and notation together would
be easy and even fun to learn, but I wonder whether very many persons
would ever have the opportunity to do it.

> >We're all familiar with a 7-nominal / heptatonic / diatonic
> >notation, and that's what Dave and I have been building on to
> >produce what I call a "generalized" notation -- one that is
> >semantically independent of any particular tonal geometry or
> >division of the octave. It may not be the best notation for
> >everything, but it will do a lot of different things and will
> >do many of them extremely well.
>
> Definitely a worthwhile project.

I have been wondering whether I should temporarily put the portion of
my XH18 article that explains the basics of the sagittal notation in
the tuning-math files section so that some of those who have been
reading these discussions can take a look and offer comments -- maybe
even *use* it.

--George

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
>
> Would someone explain "minimax" generator (I understand rms generator)
>
> Thanks

Frankly I think "minimax" is a silly term. I prefer to call it the
max-absolute (MA) generator. Obviously we're trying to minimise the
error measure in both cases, but we don't say "miniRMS". In one case
we minimise (I'd prefer to say "optimise") the Root of the Mean of the
Squares of the errors and in the other it is the Maximum of the
Absolute-values of the errors. RMS and MA.

>>That was a bad choice of terminology. The issue that Gene and
>>I are raising is: should notation be based on the melodic scale
>>being used, or should they be based on the 7-tone meantone
>>diatonic scale musicians are already familiar with?
>
>The issue that I raise is: how many different notations can you
>expect a person to learn?

As many as he needs to play the music he wants to.

Since learning septimal notation gives access to 300 years of
Western music, the potential payoff for each new one shouldn't
be grounds for complaint.

>>>But what value is the decimal notation to me, and what incentive
>>>would I have to learn it without a decimal keyboard?
>>
>> () It gives you an invaluable tool for understanding the music.
>
>Okay, but only as long as the music is written in the Miracle
>temperament, yes? (More about this below.)

Only if the music is written in a decatonic scale, yes. There
are many ways to supply the chromatic pitches outside of Miracle
temperament.

>The tones would be on a diagonal row of keys that (ascending in
>pitch) would go off the near edge of the keyboard; but they could
>be picked up at the far edge, so yes, it can be done without
>extraordinary effort.

I'd say there's nothing far worse about such a setup than in
using the Halberstadt for 12-equal.

>>>Now give me the same decimal keyboard with Partch's 11-limit
>>>JI mapped onto it (observe that this was the reason that I
>>>originally came up with the layout). Again, I should do just
>>>fine with 72-ET sagittal notation, assuming that I am proficient
>>>with the keyboard.
>>
>>You seem to be saying that it's easier to learn to find pitches
>>on a keyboard than it is to learn to find pitches in a notation...
>>For me, it's the opposite.
>
>Actually I do find it easier to perceive the pitch relationships
>on a generalized keyboard (of whatever sort) than from a notation,

Well, that's an important statement. Since I don't have any
experience playing a generalized keyboard of any sort, I have
nothing to offer. I can say that learning to read music was
easier for me than learning to finger the piano. I really have
no idea if this relationship would remain when learning a new
keyboard/notation pair.

>The broader point that I was trying to make seems to have gotten
>lost in all of the details of the discussion. I was trying to
>show that there is no particular advantage in using decimal
>notation to notate music that is *not* based on the Miracle
>geometry (e.g., Partch's music, which is better understood in
>reference to an 11-limit tonality diamond), but for which the
>tones may still be very suitably mapped onto a decimal keyboard.
>The advantage of decimal notation comes into effect only when and
>if you are using the Miracle temperament itself, i.e., exploiting
>the tonal relationships that are unique to Miracle. Likewise, if
>I play something in 31, 41, or 72-ET on a decimal keyboard that
>was composed by someone utterly ignorant of Miracle as an
>organizing principle for tonality (as I believe *all* of us were
>up until a couple of years ago -- myself included), is the decimal
>notation going to benefit me in any way if the composition which
>I am playing was not conceived as being decatonic?

No! But there are other temperaments with interesting decatonic
scales besides 31, 41, and 72, and all of them would get ten
nominals under my pen, and I suspect the differences in
accidentals would be easy for performers to learn.

>But it would be unrealistic to expect anyone to learn three
>different notations for one tonal system, according to which
>tonal relationships are exploited in a given piece.

Perhaps we'll have to agree to disagree.

>(Or suppose that a piece is heptatonic in one place and decatonic
>in another. Do we switch notations in the middle of the page?)

Absolutely! Just like switching clefs or key signatures.

>My point is that alternate tunings often do not tie us down to
>specific tonal organizations, so the choice of a tuning is often
>not enough to determine how many nominals would be "best" for its
>notation.

Indeed, any time we leave "diatonic" writing, the need to show
scale intervals in the notation disappears. Then what I meant
by 'transpositionally invariant' notation would be optimal -- a
notation in which acoustic intervals always look the same. In
diatonic notation it is scale intervals (2nds, 3rds, etc.) that
always look the same.

You mention Partch's music, which doesn't really use any fixed
melodic scale. I would think transpositionally invariant
notation would be optimal, for the scores at least.

But Partch had the right idea... since his instruments played
different scales (tonality diamond, microchromatic scales,
ancient melodic scales, etc.), he notated differently for each
of them.

>Since we are already acquainted with a notation that uses 7
>nominals, and if that works reasonably well for many alternative
>tunings, then why not have a generalized notation that builds on
>that?

I am genuinely interested to see how it looks. You should debut
it with sample music, both original and classical.

>So I would therefore require a microtonal musician to learn no
>more than one new notation. A composer may wish to do otherwise
>when composing, but a translation would be provided for the player.

A distinct posibility.

>>Let's ask it this way: take the well-tempered clavier and re-
>>write it with 6 nominals. Is that only a slight disadvantage?
>
>I was about to say no, but only because 6 nominals will hardly
>work well with anything,

They'd work spledidly for wholetone music. It's fortunate that
12-equal supports the wholetone scale without collisions. But
if you look at octatonic music, it would be much better notated
with 8 nominals. I would go so far as to suggest that this held
back octatonic music in the last century.

>even for Partch (since 6:7:8:9:10:11 isn't a constant structure).

The diatonic scale in 12-equal isn't a constant structure either.
Actually, it's strict propriety that's important, and the
violations aren't too bad here, and I think 6 nominals would be
ideal. But Partch doesn't really stick to this scale, so...

>I'm not really arguing against specialized notations with other
>than 7 nominals, but I don't think that we can expect very many
>players to learn them.

They'll learn them if there's cool music written in them.

Learning any notation is a tremendously difficult problem
for humans, but they never cease to amaze me.

>I have no doubt that the decimal keyboard and notation together
>would be easy and even fun to learn, but I wonder whether very
>many persons would ever have the opportunity to do it.

Did you see any of the virtual keyboard projector posts I've
made to the main list over the past year?

-Carl

At 12:12 AM 4/01/2003 +0000, Dave Keenan <d.keenan@uq.net.au> wrote:
>--- In tuning-math@yahoogroups.com, "Gene Ward Smith
><genewardsmith@j...>" <genewardsmith@j...> wrote:
>--- In tuning-math@yahoogroups.com, "gdsecor <gdsecor@y...>"
><gdsecor@y...> wrote:
>
> > Since the 5-limit optimal generator size depends on the optimizing
> > method, then it's a tossup between 31 and 50.
>
>One rather rough and ready rule along these lines would be to pick the
>last division which occurs as a denominator of a convergent for both
>the rms and minimax generator.

That sounds like an excellent idea!

>For Miracle, we get 72 in both the 7 and 11 limits, and for meantone,
>we get 31 in both the 5 and 7 limits (11 limit too, obviously; I
>didn't bother to compute it since it was clear what the result would
>be.) For the 7-limit schismic, we have 94, but for 5-limit, it runs
>all the way up to 289. I suppose 53, 118, 171 and 289, all very
>similar from a 5-limit schismic point of view, would look completely
>different when notated sagitally?

As a matter of fact there is a sagittal notation that agrees with 118, 171 and 289 and is also compatible with 53 and 94. In the two-symbol form, it strongly resembles a linear-temperament-specific notation with 12 nominals, like the decimal notation for miracle. This is possible because the generator happens to be a fifth and the chroma happens to be the 5-comma (80;81).

A chain of 60 notes looks like this.

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

The reason I say it's only "compatible" with 53 and 94 is because the standard sets for these do not use the 25-comma symbol //|

I suggest this notation (or its single-symbol counterpart) should be used for all open schismics of up to 60 notes.

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

Notice that this sagittal notation for schismic depends only on the temperament's mapping from generators to primes and not on any particular range of generator sizes. The fact that it only uses 5-limit comma symbols (if //| is taken as a double 5-comma rather than a 25 comma) means that it is valid for schismic at all odd limits.

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

This points the way to similar multi-symbol sagittal notations for other linear temperaments (LTs). As well as being as independent of generator size as possible and hence independent of any particular ET (which may not be fully acheivable when the generator is not a fifth) it will make the sagittal notation for an LT resemble as closely as possible the ideal notation where more than 7 nominals are allowed.

You first decide how many nominals there should be. Call this N. Then examine the available symbol commas in order of popularity, applying the LT's primes-to-generators mapping, until a symbol is found that corresponds to a chain of N generators (the chroma), also possibly 2N, 3N etc.

Naming the notes of the central chain is another matter which I haven't worked out in general, except by using an ET.

For example, in the case of Miracle, we would want 10 nominals and we find that |) as the 7-comma 63;64 corresponds to the chroma.

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

The following is perhaps a more natural sagittal notation (based on 72-ET), but makes no attempt to look like it has 10 nominals.

C Db/| D|) E\!/ F!) F#\! G Ab/| A|) B\!/
C!) C#\! D Eb|\ E|) F/|\ G!) G#\! A Bb/|
B|) C/|\ D!) D#!/ E F/| Gb|) G/|\ A!) A#\!
B
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

At 02:32 AM 4/01/2003 +0000, wallyesterpaulrus <wallyesterpaulrus@yahoo.com> wrote:
>dave,
>
>MAD means "Mean Absolute Deviation"
>
>which is another error criterion yet.
>
>so MA would breed confusion.

OK. I admit defeat on the acronym. But I'd still prefer to call it max-absolute rather than minimax.

>--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
><d.keenan@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > >
> > > Would someone explain "minimax" generator (I understand rms
>generator)
> > >
> > > Thanks
> >
> > Frankly I think "minimax" is a silly term. I prefer to call it the
> > max-absolute (MA) generator. Obviously we're trying to minimise the
> > error measure in both cases, but we don't say "miniRMS". In one case
> > we minimise (I'd prefer to say "optimise") the Root of the Mean of
>the
> > Squares of the errors and in the other it is the Maximum of the
> > Absolute-values of the errors. RMS and MA.

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> >>That was a bad choice of terminology. The issue that Gene and
> >>I are raising is: should notation be based on the melodic scale
> >>being used, or should they be based on the 7-tone meantone
> >>diatonic scale musicians are already familiar with?
> >
> >The issue that I raise is: how many different notations can you
> >expect a person to learn?
>
> As many as he needs to play the music he wants to.

I'm talking about microtonal performers, who (I imagine) would want
to learn as few systems of notation as possible -- I think most would
settle for one, if it will handle most everything.

> Since learning septimal notation gives access to 300 years of
> Western music, the potential payoff for each new one shouldn't
> be grounds for complaint.

What, exactly, do you mean by "septimal notation"? The notation Dave
and I have been working on includes septimal intervals -- and a lot
more besides. One major problem with learning different notations is
that a given symbol could mean one thing in one notation and
something else in another, and even without this you start almost
from scratch with each new notation. Better to have a single
versatile (albeit complicated) notation for the performer that
requires knowledge of only a small subset of symbols for a given
application; another application may include other symbols, but there
would be commonality (and no conflict) between them.

> >>>But what value is the decimal notation to me, and what incentive
> >>>would I have to learn it without a decimal keyboard?
> >>
> >> () It gives you an invaluable tool for understanding the music.
> >
> >Okay, but only as long as the music is written in the Miracle
> >temperament, yes? (More about this below.)
>
> Only if the music is written in a decatonic scale, yes. There
> are many ways to supply the chromatic pitches outside of Miracle
> temperament.
>
[GS, re mapping the Miracle temperament on a Bosanquet generalized
keyboard:]
> >The tones would be on a diagonal row of keys that (ascending in
> >pitch) would go off the near edge of the keyboard; but they could
> >be picked up at the far edge,

This would be true of a 31 mapping; a 41 mapping (ascending) would go
off the far edge. But in either case my conclusion holds:

> so yes, it can be done without
> >extraordinary effort.
>
> I'd say there's nothing far worse about such a setup than in
> using the Halberstadt for 12-equal.

I'd be the first to say that it's far better than the Halberstadt for
12-ET, because like intervals occur in like patterns, so you need
only a single fingering pattern for a given interval, chord, scale,
etc., regardless of the key. (However, like patterns do not
guarantee like intervals, as would be the case with a decimal
keyboard, so you must take care to observe in which keys this will
work, just as you would need to avoid the wolf of meantone
temperament on a Halberstadt keyboard.)

> >>>Now give me the same decimal keyboard with Partch's 11-limit
> >>>JI mapped onto it (observe that this was the reason that I
> >>>originally came up with the layout). Again, I should do just
> >>>fine with 72-ET sagittal notation, assuming that I am proficient
> >>>with the keyboard.
> >>
> >>You seem to be saying that it's easier to learn to find pitches
> >>on a keyboard than it is to learn to find pitches in a notation...
> >>For me, it's the opposite.
> >
> >Actually I do find it easier to perceive the pitch relationships
> >on a generalized keyboard (of whatever sort) than from a notation,
>
> Well, that's an important statement. Since I don't have any
> experience playing a generalized keyboard of any sort, I have
> nothing to offer. I can say that learning to read music was
> easier for me than learning to finger the piano. I really have
> no idea if this relationship would remain when learning a new
> keyboard/notation pair.

That involved acquiring a certain amount of manual dexterity
(including independent movement of hands and fingers) that you
already possess when learning a new keyboard. I could play simple
things immediately the first time I sat down at a Bosanquet
keyboard. A decimal keyboard would not immediately be as familiar,
but things come quickly if you learn to do things by learning
interval patterns.

> >The broader point that I was trying to make seems to have gotten
> >lost in all of the details of the discussion. I was trying to
> >show that there is no particular advantage in using decimal
> >notation to notate music that is *not* based on the Miracle
> >geometry (e.g., Partch's music, which is better understood in
> >reference to an 11-limit tonality diamond), but for which the
> >tones may still be very suitably mapped onto a decimal keyboard.
> >The advantage of decimal notation comes into effect only when and
> >if you are using the Miracle temperament itself, i.e., exploiting
> >the tonal relationships that are unique to Miracle. Likewise, if
> >I play something in 31, 41, or 72-ET on a decimal keyboard that
> >was composed by someone utterly ignorant of Miracle as an
> >organizing principle for tonality (as I believe *all* of us were
> >up until a couple of years ago -- myself included), is the decimal
> >notation going to benefit me in any way if the composition which
> >I am playing was not conceived as being decatonic?
>
> No! But there are other temperaments with interesting decatonic
> scales besides 31, 41, and 72, and all of them would get ten
> nominals under my pen, and I suspect the differences in
> accidentals would be easy for performers to learn.
>
> >But it would be unrealistic to expect anyone to learn three
> >different notations for one tonal system, according to which
> >tonal relationships are exploited in a given piece.
>
> Perhaps we'll have to agree to disagree.
>
> >(Or suppose that a piece is heptatonic in one place and decatonic
> >in another. Do we switch notations in the middle of the page?)
>
> Absolutely! Just like switching clefs or key signatures.

Hey, you were supposed to say "no, of course not!" Suppose it goes
back and forth from heptatonic to decatonic every measure, or the
right hand does something decatonic, while the left hand is
heptatonic, etc. Or there may come a point where the music can get
so sophisticated that it would be difficult to tell what-tonic the
composer had in mind. Let's try not to confuse the poor performer
(or the *good* performer either).

> ...
> You mention Partch's music, which doesn't really use any fixed
> melodic scale. I would think transpositionally invariant
> notation would be optimal, for the scores at least.
>
> But Partch had the right idea... since his instruments played
> different scales (tonality diamond, microchromatic scales,
> ancient melodic scales, etc.), he notated differently for each
> of them.

Help! That's what I'm trying to avoid -- insofar as possible, I want
the pitch notation to be independent of the instrument. How can you
understand vertical harmony in a score if the notation is different
in different lines?

> >Since we are already acquainted with a notation that uses 7
> >nominals, and if that works reasonably well for many alternative
> >tunings, then why not have a generalized notation that builds on
> >that?
>
> I am genuinely interested to see how it looks. You should debut
> it with sample music, both original and classical.

My paper has a couple of samples, plus examples of the 17-limit
consonances in the key of C. These at least give you an idea of how
things look on paper. This, along with the explanation of the
notation, would be enough to get you started. (I'll have to get part
of that posted soon, after I go through it and make a few small
changes.)

> ...
> >>Let's ask it this way: take the well-tempered clavier and re-
> >>write it with 6 nominals. Is that only a slight disadvantage?
> >
> >I was about to say no, but only because 6 nominals will hardly
> >work well with anything,
>
> They'd work spledidly for wholetone music. It's fortunate that
> 12-equal supports the wholetone scale without collisions. But
> if you look at octatonic music, it would be much better notated
> with 8 nominals. I would go so far as to suggest that this held
> back octatonic music in the last century.
>
> >even for Partch (since 6:7:8:9:10:11 isn't a constant structure).
>
> The diatonic scale in 12-equal isn't a constant structure either.

That's true only as a technicality -- the augmented fourth and
diminished fifth in the diatonic scale just happen to be the same
size in 12-ET, but since they are functionally different, this isn't
a liability. Partch's hexads make more sense if you consider them
heptatonically (with a vacanct position between 6 and 7).

> Actually, it's strict propriety that's important, and the
> violations aren't too bad here, and I think 6 nominals would be
> ideal. But Partch doesn't really stick to this scale, so...
>
> >I'm not really arguing against specialized notations with other
> >than 7 nominals, but I don't think that we can expect very many
> >players to learn them.
>
> They'll learn them if there's cool music written in them.

This year I intend to spend more time composing music than writing
papers! -- now that I have a notation that will do everything I need
it to.

> ...
> Did you see any of the virtual keyboard projector posts I've
> made to the main list over the past year?

No, I haven't. I've been spending so much time on the notation
project that I usually only have time to read quickly (or scan)
through postings on the main list, so there are a lot of things I
have spent less time on than I would have liked. Would you refer me
to an example or two?

--George

>>Since learning septimal notation gives access to 300 years of
>>Western music, the potential payoff for each new one shouldn't
>>be grounds for complaint.
>
>What, exactly, do you mean by "septimal notation"?

A notation with 7 nominals!

>But in either case my conclusion holds:
>
>>>so yes, it can be done without
>>>extraordinary effort.
>>
>>I'd say there's nothing far worse about such a setup than
>>in using the Halberstadt for 12-equal.
>
>I'd be the first to say that it's far better than the
>Halberstadt for 12-ET, because like intervals occur in like
>patterns, so you need only a single fingering pattern for a
>given interval, chord, scale, etc., regardless of the key.

We agree, then.

>(However, like patterns do not guarantee like intervals, as
>would be the case with a decimal keyboard, so you must take
>care to observe in which keys this will work, just as you
>would need to avoid the wolf of meantone temperament on a
>Halberstadt keyboard.)

I don't follow this. It is possible to make a transpositionally
invariant decimal keyboard, so that like patterns do guarantee
like intervals.

>>>(Or suppose that a piece is heptatonic in one place and
>>>decatonic in another. Do we switch notations in the middle
>>>of the page?)
>>
>>Absolutely! Just like switching clefs or key signatures.
>
>Hey, you were supposed to say "no, of course not!" Suppose it
>goes back and forth from heptatonic to decatonic every measure,
>or the right hand does something decatonic, while the left hand
>is heptatonic, etc.

It's just a judgement call on the part of the composer, as it
is with clefs and key signatures.

>>You mention Partch's music, which doesn't really use any fixed
>>melodic scale. I would think transpositionally invariant
>>notation would be optimal, for the scores at least.
>>
>>But Partch had the right idea... since his instruments played
>>different scales (tonality diamond, microchromatic scales,
>>ancient melodic scales, etc.), he notated differently for each
>>of them.
>
>Help! That's what I'm trying to avoid -- insofar as possible, I
>want the pitch notation to be independent of the instrument. How
>can you understand vertical harmony in a score if the notation is
>different in different lines?

Oh, you have to have unified notation for a score, as I say,
"for scores at least". But parts are a different matter.

>>I am genuinely interested to see how it looks. You should debut
>>it with sample music, both original and classical.
>
>My paper has a couple of samples, plus examples of the 17-limit
>consonances in the key of C. These at least give you an idea of
>how things look on paper. This, along with the explanation of
>the notation, would be enough to get you started. (I'll have to
>get part of that posted soon, after I go through it and make a
>few small changes.)

Great! Please let me know with XH 18 (or is it 19?) comes out.

>>>... 6:7:8:9:10:11 isn't a constant structure).
>>
>>The diatonic scale in 12-equal isn't a constant structure either.
>
>That's true only as a technicality -- the augmented fourth and
>diminished fifth in the diatonic scale just happen to be the same
>size in 12-ET, but since they are functionally different, this
>isn't a liability.

And in harmonics 6-12, the aug 3rd and dim 4th don't function
differently?

>Partch's hexads make more sense if you consider them
>heptatonically (with a vacanct position between 6 and 7).

I'd imagine I'd mainly stick to triads if I were to write
'diatonic' music with harmonics 6-12.

>>Did you see any of the virtual keyboard projector posts I've
>>made to the main list over the past year?
>
>No, I haven't. I've been spending so much time on the notation
>project that I usually only have time to read quickly (or scan)
>through postings on the main list, so there are a lot of things
>I have spent less time on than I would have liked. Would you
>refer me to an example or two?

Msg. #s 35809 and 41680.

-Carl

() I would like to distinguish between two types of notation.

....() Systems in which a given acoustic interval always
covers the same distance on the staff, in terms of lines
and spaces. ("trans. invariant notations")

....() Systems in which a given scale interval (2nd, 3rd,
etc.) always covers the same distance on the staff.
("diatonic" notations)

() For non-diatonic music, the former type of notation
is preferable.

() Regarding the former type, the number of nominals is
more-or-less irrelevant.

() For diatonic music, the latter type of notation is
preferable.

() Regarding the latter type, the number of nominals is
crucial.

() A single notation of the latter type has occupied
generations of musicians in our culture.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> >>Since learning septimal notation gives access to 300 years of
> >>Western music, the potential payoff for each new one shouldn't
> >>be grounds for complaint.
> >
> >What, exactly, do you mean by "septimal notation"?
>
> A notation with 7 nominals!

What I would prefer to call a heptatonic notation.

> >(However, like patterns do not guarantee like intervals, as
> >would be the case with a decimal keyboard, so you must take
> >care to observe in which keys this will work, just as you
> >would need to avoid the wolf of meantone temperament on a
> >Halberstadt keyboard.)
>
> I don't follow this. It is possible to make a transpositionally
> invariant decimal keyboard, so that like patterns do guarantee
> like intervals.

I'm sorry, but evidently I didn't make myself clear. I was speaking
of what occurs with either a 31 or 41 mapping of Miracle onto a
*Bosanquet* generalized keyboard. For Miracle on a decimal keyboard
you have an ideal situation, such as you describe.

> ...
> >>You mention Partch's music, which doesn't really use any fixed
> >>melodic scale. I would think transpositionally invariant
> >>notation would be optimal, for the scores at least.
> >>
> >>But Partch had the right idea... since his instruments played
> >>different scales (tonality diamond, microchromatic scales,
> >>ancient melodic scales, etc.), he notated differently for each
> >>of them.
> >
> >Help! That's what I'm trying to avoid -- insofar as possible, I
> >want the pitch notation to be independent of the instrument. How
> >can you understand vertical harmony in a score if the notation is
> >different in different lines?
>
> Oh, you have to have unified notation for a score, as I say,
> "for scores at least". But parts are a different matter.

As a practical expedient, a part will have to be in whatever notation
the player is comfortable with. This might make communication
between a conductor or music director (reading the score) and a
player more difficult, but conductors will need to become astute and
versatile in these situations.

> >>I am genuinely interested to see how it looks. You should debut
> >>it with sample music, both original and classical.
> >
> >My paper has a couple of samples, plus examples of the 17-limit
> >consonances in the key of C. These at least give you an idea of
> >how things look on paper. This, along with the explanation of
> >the notation, would be enough to get you started. (I'll have to
> >get part of that posted soon, after I go through it and make a
> >few small changes.)
>
> Great! Please let me know with XH 18 (or is it 19?) comes out.

It will be XH18, and I have no idea how long it will be (which is why
I'll try to put some of the explanation about the notation in the
tuning-math files soon).

> >>>... 6:7:8:9:10:11 isn't a constant structure).
> >>
> >>The diatonic scale in 12-equal isn't a constant structure either.
> >
> >That's true only as a technicality -- the augmented fourth and
> >diminished fifth in the diatonic scale just happen to be the same
> >size in 12-ET, but since they are functionally different, this
> >isn't a liability.
>
> And in harmonics 6-12, the aug 3rd and dim 4th don't function
> differently?

I was interpreting the phrase "diatonic scale in 12-equal" to mean 5-
limit with no chromatic intervals, but the harmonic minor scale
slipped my mind. To address your question, the aug 3rd and dim 4th
*do* function differently in the scale, and the larger interval
subtends fewer diatonic degrees -- yet we are not disoriented! It is
interesting to contemplate that if we used a notation with 12
nominals for 12-ET that we would be unable to observe a distinction
on the printed page. So you have a point.

> >>Did you see any of the virtual keyboard projector posts I've
> >>made to the main list over the past year?
> >
> >No, I haven't. I've been spending so much time on the notation
> >project that I usually only have time to read quickly (or scan)
> >through postings on the main list, so there are a lot of things
> >I have spent less time on than I would have liked. Would you
> >refer me to an example or two?
>
> Msg. #s 35809 and 41680.

Thanks! I'll have to take a look.

--George

> > >What, exactly, do you mean by "septimal notation"?
> >
> > A notation with 7 nominals!
>
> What I would prefer to call a heptatonic notation.

Of course; much better.

>>And in harmonics 6-12, the aug 3rd and dim 4th don't function
>>differently?
//
>It is interesting to contemplate that if we used a notation
>with 12 nominals for 12-ET that we would be unable to observe
>a distinction on the printed page.

Exactly.

>>Msg. #s 35809 and 41680.
>
>Thanks! I'll have to take a look.

Promising technology for microtonalists, to be sure. For
infinite flexibility we loose velocity and aftertouch, so
we'd be stuck to organ-type patches if we wanted them to
sound good.

We also loose tactile feedback, which would probably make
sight reading impossible. Notation could be replaced on
these instruments by a 'follow the lights' approach, in
which the whole key can light up, if you like! Graham,
are you listening?

Certainly exciting that technology exists to bring the holy
grail of an infinitely configurable, extremely portable
keyboard within reach of the consumer (indeed: cheap!).

In the year between msg. 35809 and 41680, it went from
trade-show demo to at least two companies providing OEM
kits. Assuming there's a flexible inferface in there
somewhere, a microtonal keyboard software project (perhaps
two projectors would be needed for a full keyboard) might
not be too difficult...

-Carl

Thanks. I've read Monz's summary of Woolhouse's book which I've found very
helpful. I was just wondering, though, why he doesn't take the square root
when calculating Root-Mean-Square. Is it unneccessary? I also see a little
problem with the final calculation for rms for meantone. If multiplication
is commutative, you could take the results to ALSO mean (3/2)/(7/26)
^(81/80) which is obviously a wrong result. Clarification, anyone? Thanks.

"Gene Ward Smith
<genewardsmith@ju To: tuning-math@yahoogroups.com
no.com>" cc: (bcc: Paul G Hjelmstad/US/AMERICAS)
<genewardsmith Subject: [tuning-math] Re: Minimax generator

01/03/2003 12:56
PM
Please respond to
tuning-math

--- In tuning-math@yahoogroups.com, "Gene Ward Smith <genewardsmith@j...>"
<genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> >
> > Would someone explain "minimax" generator (I understand rms generator)
>
> Let's take meantone for an example. If we define the three linear
functions
>
> u1 = x - l2(3)
>
> u2 = 4x - 4 - l2(5)
>
> u3 = 3x -4 - l2(5/3)
>
> using "l2" to mean log base 2, then the rms generator can be found
> by minimizing u1^2+u2^2+u3^2. On the other hand, we can minimize
> |u1| + |u2| + |u3| instead (which gives 1/4 comma meantone.) In fact,
> for any p>1, we can minimize |u1|^p + |u2|^p + |u3|^p; if for instance
> p is 4, this gives us essentially 1/3 comma meantone (.33365 comma.)
> If we consider

It turns out this is so high because of round off error. I redid the
calculation using 100 digits of accuracy, and got:

p=2 7/26 comma meantone

p=4 7/26 comma meantone again

p=6 .26295498 comma meantone, about 5/19 comma

p=8 .25942006 comma meantone, about 7/27 comma

p=10 .25739442 comma meantone, about 9/35 comma

p=1 and p=infinity, 1/4 comma meantone

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--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
>
> Thanks. I've read Monz's summary of Woolhouse's book which I've
found very
> helpful. I was just wondering, though, why he doesn't take the
square root
> when calculating Root-Mean-Square. Is it unneccessary?

if you´ve minimized the mean-square, you´ve also minimized the root-
mean-square.

I also see a little
> problem with the final calculation for rms for meantone. If
multiplication
> is commutative, you could take the results to ALSO mean (3/2)/(7/26)
> ^(81/80) which is obviously a wrong result. Clarification, anyone?
Thanks.

i don´t have the woolhouse in front of me, but the correct formula is

(3/2)/(81/80)^(7/26)

>
>
>

> "Gene Ward
Smith

> <genewardsmith@ju To: tuning-
math@yahoogroups.com

> no.com>" cc: (bcc: Paul
G Hjelmstad/US/AMERICAS)
> <genewardsmith Subject: [tuning-
math] Re: Minimax generator
>

> 01/03/2003
12:56

>
PM

> Please respond
to

> tuning-
math

>

>

>
>
>
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>"
> <genewardsmith@j...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > >
> > > Would someone explain "minimax" generator (I understand rms
generator)
> >
> > Let's take meantone for an example. If we define the three linear
> functions
> >
> > u1 = x - l2(3)
> >
> > u2 = 4x - 4 - l2(5)
> >
> > u3 = 3x -4 - l2(5/3)
> >
> > using "l2" to mean log base 2, then the rms generator can be found
> > by minimizing u1^2+u2^2+u3^2. On the other hand, we can minimize
> > |u1| + |u2| + |u3| instead (which gives 1/4 comma meantone.) In
fact,
> > for any p>1, we can minimize |u1|^p + |u2|^p + |u3|^p; if for
instance
> > p is 4, this gives us essentially 1/3 comma meantone (.33365
comma.)
> > If we consider
>
> It turns out this is so high because of round off error. I redid the
> calculation using 100 digits of accuracy, and got:
>
> p=2 7/26 comma meantone
>
> p=4 7/26 comma meantone again
>
> p=6 .26295498 comma meantone, about 5/19 comma
>
> p=8 .25942006 comma meantone, about 7/27 comma
>
> p=10 .25739442 comma meantone, about 9/35 comma
>
> p=1 and p=infinity, 1/4 comma meantone
>
>
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--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
>
> Thanks. I've read Monz's summary of Woolhouse's book which I've
found very
> helpful. I was just wondering, though, why he doesn't take the
square root
> when calculating Root-Mean-Square. Is it unneccessary?

If you only want to find an optimum generator it is unnecessary, in
fact you don't even need to take the mean, just minimise the sum of
the squares of the errors. But if you want to express the resulting
overall error in a perceptually meaningful fashion, e.g. for comparing
different temperaments, then you need to take the root of the mean of
the squares, so it has dimensions of log-of-frequency-ratio, and
therefore (usually) units of cents.

> I also see a little
> problem with the final calculation for rms for meantone. If
multiplication
> is commutative, you could take the results to ALSO mean (3/2)/(7/26)
> ^(81/80) which is obviously a wrong result. Clarification, anyone?
Thanks.

I find it easier to follow when expressed in the logarithmic frequency
domain (e.g. in cents) as 702.0c - 21.5c * 7/26

Thanks. How would one calculate the rms generator for say, Kleismic? Would
it be possible to get the final formula that calculates 317.0796753c? How
about for a whole bunch of these, like Gene's list (Message 4554). If I
could just get a final formula I think I could use Woolhouse to figure out
how it is calculated:)

"Dave Keenan
<d.keenan@uq.net. To: tuning-math@yahoogroups.com
au>" <d.keenan cc: (bcc: Paul G Hjelmstad/US/AMERICAS)
Subject: [tuning-math] Re: Minimax generator
01/08/2003 06:40
PM
Please respond to
tuning-math

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
>
> Thanks. I've read Monz's summary of Woolhouse's book which I've
found very
> helpful. I was just wondering, though, why he doesn't take the
square root
> when calculating Root-Mean-Square. Is it unneccessary?

If you only want to find an optimum generator it is unnecessary, in
fact you don't even need to take the mean, just minimise the sum of
the squares of the errors. But if you want to express the resulting
overall error in a perceptually meaningful fashion, e.g. for comparing
different temperaments, then you need to take the root of the mean of
the squares, so it has dimensions of log-of-frequency-ratio, and
therefore (usually) units of cents.

> I also see a little
> problem with the final calculation for rms for meantone. If
multiplication
> is commutative, you could take the results to ALSO mean (3/2)/(7/26)
> ^(81/80) which is obviously a wrong result. Clarification, anyone?
Thanks.

I find it easier to follow when expressed in the logarithmic frequency
domain (e.g. in cents) as 702.0c - 21.5c * 7/26

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--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:
>
> Thanks. How would one calculate the rms generator for say, Kleismic?

Kleismic tells us that

3 ~ k^6
5 ~ 2 k^5
6/5 ~ k

where k is the generator. If we take log base 2 of both sides and
call x=log2(k), we get

u1 = 6*x-log2(3) ~ 0
u2 = 5*x+1-log2(5) ~ 0
u3 = x-log2(6/5) ~ 0.

The least squares solution to this is found by finding the minimum of
the polynomial function

F(x) = u1^2 + u2^2 + u3^2

which we may easily do by taking the derivative and setting F'(x)=0.

If you have a program which solves linear programming problems, you can set up the standard minimization problem with objective function
y and constraint set

{y>=u1, y>=u2, y>=u3, y>=-u1, y>=-u2, y>=-u3, x>=0, y>=0}

and solve for the minimax generator x.