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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>

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> To unsubscribe from this group, send an email to:

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>

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> Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

--- 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.