Questions about Regular Mapping & Rank-Two Temperaments

Question 1: for a rank-2 temperament, how does one arrive at a generator and period from a pair of commas to be tempered out?

Question 2: how are boundaries between temperament classes worked out?

Question 3: is there a simple way to calculate the largest MOS scale for a given temperament class?

Question 4: how complex is the math to calculate a mapping for a temperament?

Question 5: how do you name a temperament that exists on the boundary between two temperament classes (i.e. where for a single value of generator, there are two possible mappings of primes that have equal error)?

I really want to understand this stuff.

-Igs

no doubt the answers will help me as well

On Mon, Jun 14, 2010 at 1:55 PM, cityoftheasleep <igliashon@...>wrote:

>
>
> Question 1: for a rank-2 temperament, how does one arrive at a generator
> and period from a pair of commas to be tempered out?
>
> Question 2: how are boundaries between temperament classes worked out?
>
> Question 3: is there a simple way to calculate the largest MOS scale for a
> given temperament class?
>
> Question 4: how complex is the math to calculate a mapping for a
> temperament?
>
> Question 5: how do you name a temperament that exists on the boundary
> between two temperament classes (i.e. where for a single value of generator,
> there are two possible mappings of primes that have equal error)?
>
> I really want to understand this stuff.
>
> -Igs
>
>
>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Question 1: for a rank-2 temperament, how does one arrive at
> a generator and period from a pair of commas to be tempered out?

At the moment, Graham's code (available on his site) is about
the best answer to that.

> Question 2: how are boundaries between temperament classes
> worked out?

They're not. One of the biggest holes in the theory of regular
temperaments is how the different temperaments relate to one
another, especially at higher prime limits. A few concepts
have been developed and are in their infancy... I'm partial
to Gene's "comma sequences".

http://lumma.org/tuning/gws/commaseq.htm

> Question 3: is there a simple way to calculate the largest MOS
> scale for a given temperament class?

The MOS series is infinite.

> Question 4: how complex is the math to calculate a mapping for
> a temperament?

That depends which mapping you want. Usually we use the
mapping that corresponds to the Hermite normal form:

http://en.wikipedia.org/wiki/Hermite_normal_form

> Question 5: how do you name a temperament that exists on the
> boundary between two temperament classes (i.e. where for a
> single value of generator, there are two possible mappings of
> primes that have equal error)?

Two different temperaments will always have different wedgies.
The wedgies are the real 'genus and species' of the temperaments.
The 'common names' are usually thought up to be easy to remember.
If one or both of the temperaments in question don't have
common names, you can Christen them.

-Carl

Igs wrote:

> Question 1: for a rank-2 temperament, how does one arrive at a generator
> and period
> from a pair of commas to be tempered out?

There are several ways to do this and probably more effective ones than the
way I was doing it last year during my experiments; there are people who
could certainly suggest something better than me. But if you don't mind
about my method being "ahum ... A bit time-consuming", then you can try to
temper out the three "primary" commas separately and turn the result into a
single temperament. For example, if you want to temper out 126/125 and
225/224 at the same time, you're also tempering out all multiples or
divisions of these two. This makes it possible to find commas where one of
the primes is missing. So once you decide to temper out both 126/125 and
225/224, then you're also tempering out 81/80 (no 7s), 59049/57344 (no 5s),
3136/3125 (no 3s) -- in theory, there's even a "2-less" interval which also
vanishes but we exclude this as we want to use an equivalence interval of
2/1. As we know, tempering out 81/80 suggests approximating 5/4 by 4 fifths minus 2 octaves.
Then, tempering out 59049/57344 suggests approximating 7/4 by 10 fifths minus 5 octaves. Finally, tempering out 3136/3125 gives you a major second as a generator, of which there are two for approximating 5/4 and five for approximating 7/4. From this, you can spot that tempering out 126/125 and 225/224 at a time gives you a 7-limit version of meantone where the 7/4 approximation is represented by an augmented sixth.

> Question 2: how are boundaries between temperament classes worked out?

Most probably, I think, by finding MOS scales. For example, you can get 7-tone diatonic (5+2) both in meantone and in mavila, with the exception that they're 5 large + 2 small in meantone and 5 small + 2 large in mavila. Because of this, 7-equal can work both as meantone and as mavila. Similarly, you can get 12-tone chromatic (5+7) both in schismatic and in meantone, with the exception that they're 5 large + 7 small in schismatic and 7 large + 5 small in meantone. Therefore, 12-equal can work both as schismatic and as meantone. But you won't get mavila in 12-equal or schismatic in 7-equal.

> Question 3: is there a simple way to calculate the largest MOS scale for a
> given temperament class?

What do you mean? There isn't any limit on how large your scale is. For example, hanson makes a MOS with 11 tones, 15, 19, 34, 53, 72, 91 ... The list goes on.

> Question 4: how complex is the math to calculate a mapping for a
> temperament?

For finding the mapping from a combination of two EDOs, there are others who can explain how to do this. For finding the mapping from the vanishing interval, a good way is to take the three prime exponents (assuming the target intervals are 2/1, 3/1 and 5/1) and use that as our starting point. If we are interested only in the generator counts and don't care about the periods, we can discard the exponent for the prime which we use as our equivalence interval, change the sign in one of the two remaining numbers, and put them in reverse order. For example, the exponents for 81/80 are "-4, 4, -1"; so if we discard the first number, change the sign in one of the other two, and swap them, we get "1, 4" or "-1, -4", which tells us that 1 fifth (plus an octave) approximates 3/1 and 4 fifths approximate 5/1, or that -1 fourth (plus 2 octaves) approximates 3/1 and -4 fourths (plus 4 octaves) approximate 5/1. In cases where their GCD is not 1, we have to divide them by the GCD and use a period of "octave/GCD" instead of a full octave.
If we wish to get the proper generator size (and therefore the full mapping, including the periods), my favorite method is to break the three numbers into two sets in such a way that there's 0 at the 2nd position in one of them, there's 0 at the 3rd position one of them, and the number at the 1st position is divisible by the number at the 2nd or second position, respectively. For example, the exponents for 20000/19683 are "-5, 9, 4", which you can turn into "-9, 9, 0" and "4, 0, 4". This essentially tells you that 9 fifths should be the same as 4 major tenths, from which you can already figure out an appropriate generator size. Again, in cases where the GCD of the 2nd and 3rd number is not 1, the octave is split and then either the numbers are divided by the GCD (all of them, which makes the first a non-integer) or the second and third number are multiplied by the GCD (which results in a "squared" mapping if the GCD is 2).

> Question 5: how do you name a temperament that exists on the boundary
> between two temperament classes (i.e. where for a single value of > generator,
> there are two possible mappings of primes that have equal error)?

Again, there are other people who may have something to say on this.

Petr

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Question 1: for a rank-2 temperament, how does one arrive at a generator and period from a pair of commas to be tempered out?

It's only a pair of commas in the 7-limit, I'm afraid. Explaining how I actually do it wouldn't be much help, so I suggest instead you do a brute-force search for a pair of equal temperaments which temper out all of your commas. Let's say you end up with two vals,
<12 19 28 34| and <22 35 51 62|. Now, 22/12 = 11/6, and
6*<22 35 51 62| - 11*<12 19 28 34| = <0 1 2 2|. 11/6 has a convergent 9/5, and 5*<22 35 28 34| - 9*<12 19 28 34| = <2 4 3 4|. I can now do an optimization procedure on these two mappings, and get an optimized tuning to go with them, which might for example (it depends on how I optimize) be 600 cents with <2 4 3 4| and -491.186 cents with
<0 1 2 2|. I say "darn it" and switch the mapping to <0 -1 -2 -2| and now I have period (600 cents), generator (491.186 cents) and two mappings to go with them.

> Question 2: how are boundaries between temperament classes worked out?

They aren't exactly worked out, but certain equal temperaments define natural boundries between temperaments when they belong to both of them. For instance, 12edo is a natural boundry between meantone and garibaldi.

> Question 3: is there a simple way to calculate the largest MOS scale for a given temperament class?

No, because there isn't one unless you define a precise generator.

> Question 4: how complex is the math to calculate a mapping for a temperament?

How hard was the example I gave?

> Question 5: how do you name a temperament that exists on the boundary between two temperament classes (i.e. where for a single value of generator, there are two possible mappings of primes that have equal error)?

You name it "equal temperament".

I wrote:

> This essentially tells you
> that 9 fifths should be the same as 4 major tenths, from which you can
> already figure out an appropriate generator size.

I should have said: "which can also tell you that there are 9 generators in a major tenth and 4 in a fifth, from which you can already ...".

Petr

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> >
> > Question 1: for a rank-2 temperament, how does one arrive at a generator and period from a pair of commas to be tempered out?
>
> It's only a pair of commas in the 7-limit, I'm afraid. Explaining how I actually do it wouldn't be much help, so I suggest instead you do a brute-force search for a pair of equal temperaments which temper out all of your commas. Let's say you end up with two vals,
>

Okay, I had to look up "val" in the Tonalsoft encyclopedia but I think I understand it.

> <12 19 28 34| and <22 35 51 62|. Now, 22/12 = 11/6, and
> 6*<22 35 51 62| - 11*<12 19 28 34| = <0 1 2 2|. 11/6 has a convergent 9/5, and
> 5*<22 35 28 34| - 9*<12 19 28 34| = <2 4 3 4|.

I understand everything here except the "convergent 9/5". What does that mean and where does 9/5 come from? But I get (I think) why you're doing these mathematical operations: the vals for the two EDOs are in terms of a single degree of each EDO, and the operations produce vals in terms of the two generators for the temperament. However, the value of the generators are variable until you "optimize" them; and you must have equations that you use to solve for generators based on given mappings (or some such technique). Awesome. This is what I've been trying to understand for all these years.

I see also that by using vals in conjunction with monzos you can figure out how any JI interval is mapped in an equal temperament. Incredible! So if you were really bored, you could sit down with your favorite equal temperament and a bunch of JI intervals, figure out which JI intervals map to the same equal temperament intervals, and then work out all the commas that were tempered out. If you were REALLY bored, that is.

> > Question 2: how are boundaries between temperament classes worked out?
>
> They aren't exactly worked out, but certain equal temperaments define natural boundries between temperaments when they belong to both of them. For instance, 12edo is a natural boundry between meantone and garibaldi.
>
> > Question 3: is there a simple way to calculate the largest MOS scale for a given temperament class?
>
> No, because there isn't one unless you define a precise generator.
>
> > Question 5: how do you name a temperament that exists on the boundary between two temperament classes (i.e. where for a single value of generator, there are two possible mappings of primes that have equal error)?
>
> You name it "equal temperament".
>

Okay, that's what I *thought*. I've been spending a lot of time with Erv Wilson's scale tree, and I noticed that for MOS scales which all share the same period, the more notes you have in an MOS scale, the narrower the range of possible generator intervals is for that scale. For example: take the 7-note MOS scale of LLsLLLs, where you have a generator range from 2-of-5-EDO (2\5 in Andrew Heathwaite's Notation) to 3\7. When you extend all the generators in that range to their next moments of symmetry (i.e. to a 12-note MOS), you see a branching effect: you get a "node" at 5\12 where the 12-note MOS is an equal temperament, and any generator between 2\5 and 5\12 produces an MOS of 5L7s and any generator between 3\7 and 5\12 produces an MOS of 7L5s. Where there was one scale, there are now 3, 2 of which are unequal and thus can be extended to another (higher) MOS. This effect continues indefinitely, since there are an infinite number of equal temperaments. There are also "golden" generators in every range that will NEVER produce an EDO, but will subdivide the period infinitely and unequally, since the ratio of their large-step to their small-step is phi.

So the question I've been plagued with is, where do I draw the line to create a useful taxonomy of MOS scales for use in low-numbered equal temperaments (i.e. 36 or fewer notes per octave)? Clearly, there are going to be significant musical differences between scales that share low-number moments of symmetry but not higher ones, so I can't just lump all the 7-note 5L2s scales into one family. However, when two of these 7-note scales share the same 12-note MOS configuration, they seem to have more in common, musically-speaking. I mean, 19-EDO and 31-EDO both share the same 12-note "meantone" MOS, making them a bit more "similar" than "different". So it occurred to me that maybe branches on the scale tree might have some kind of correspondence with rank-two linear temperaments.

Now I'm wondering if it's possible to assert that scales which share the same n-note MOS configuration in a given period ALSO share the same prime-m-limit mapping. I.e., do all the 12-note MOS scales of 5L7s have the same 5-limit mapping, regardless of where the generator falls in the "allowable range"?

Thoughts?

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:

> > <12 19 28 34| and <22 35 51 62|. Now, 22/12 = 11/6, and
> > 6*<22 35 51 62| - 11*<12 19 28 34| = <0 1 2 2|. 11/6 has a convergent 9/5, and
> > 5*<22 35 51 62| - 9*<12 19 28 34| = <2 4 3 4|.
>
> I understand everything here except the "convergent 9/5". What does that mean and where does 9/5 come from?

Note that 11/6-9/5 = 1/30; this has a numerator of 1 and the denominator is the product of the denominators of 11/6 and 9/5. This is a useful property, though 9/5 is not unique in having it; if I take
2/1, then 2/1-11/6 = 1/6 works the same way, and I could have used
2*<12 19 28 34| - 1*<22 35 51 62| = <2 3 5 6| instead, leading to -130.594 as a generator.

One place to get it from is Scala. If I go to the "Tools" pull down menu, and open "convergents", I get a box ("Factor") I can put 11/6 in. Punch the "calculate" button at the bottom, and then look in the "Ratio" box. Hitting the "previous convergent" button brings up 9/5 and then 2/1.

>This effect continues indefinitely, since there are an infinite number of equal temperaments.

You can explore this sort of thing using the "convergents" machine on Scala, which also has "left child", "right child", "left parent", "right parent" buttons.

> So the question I've been plagued with is, where do I draw the line to create a useful taxonomy of MOS scales for use in low-numbered equal temperaments (i.e. 36 or fewer notes per octave)?

You could start with a somewhat arbitrary ukase to yourself to the effect that you are going to go by the 7-limit, and then let 7-limit temperaments sort it out for you, which they mostly will.

>I.e., do all the 12-note MOS scales of 5L7s have the same 5-limit mapping, regardless of where the generator falls in the "allowable range"?

MOS scales as such don't have mappings. You can start with something on the borderline of two 5-limit mappings, like 6/11, and note it is close to 17/31, which is not the best fifth of 31edo.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:

> > I understand everything here except the "convergent 9/5". What does that mean and where does 9/5 come from?
>
> Note that 11/6-9/5 = 1/30; this has a numerator of 1 and the denominator is the product of the denominators of 11/6 and 9/5. This is a useful property, though 9/5 is not unique in having it; if I take
> 2/1, then 2/1-11/6 = 1/6 works the same way, and I could have used
> 2*<12 19 28 34| - 1*<22 35 51 62| = <2 3 5 6| instead, leading to -130.594 as a generator.
>

Okay. After spending 45 minutes reading the wikipedia article on continued fractions, I've decided to just take you at your word here and rely on scala if ever I need a convergent.

> > So the question I've been plagued with is, where do I draw the line to create a useful taxonomy of MOS scales for use in low-numbered equal temperaments (i.e. 36 or fewer notes per octave)?
>
> You could start with a somewhat arbitrary ukase to yourself to the effect that you are going to go by the 7-limit, and then let 7-limit temperaments sort it out for you, which they mostly will.
>
> MOS scales as such don't have mappings. You can start with something on the borderline of two 5-limit mappings, like 6/11, and note it is close to 17/31, which is not the best fifth of 31edo.
>

Okay, I think I've answered my own question. I was trying to figure out if the generator ranges for families of MOS scales would reach 1-to-1 correspondence with the generator ranges for 7-limit temperaments if you defined "families" of MOS scales by sufficiently-large MOS scales (like 22 notes or something). But now I realize that ranges of generators for linear temperaments rely on an element of harmonic precision, which is by definition absent from MOS scales. I could take a "fifth" all the way out to 719.9999999...¢ and produce an MOS scale of 22 notes identical in configuration to one based on 710¢, but they'd be in very different temperament classes. Maybe if I went WAAAAY up to MOS scales of several hundred notes, I'd get enough specificity, but that would defeat the point.

Alright. I was hoping for a way to reduce temperament classes to MOS families, but I understand now that that is impossible. Classifying scales according to MOS properties alone isn't enough to tell you about the musical qualities of the scale, you have to look at the mapping. But of course, if you're only working with equal temperaments of fewer than 37 notes per octave, a "brute force" approach is also possible ;->. Maybe there are other ways I can classify scales in my alternative-EDO primer...I really don't want to overwhelm people with talk of commas, unison vectors, vals, wedgies, etc.

On 14 June 2010 21:55, cityoftheasleep <igliashon@...> wrote:
> Question 1: for a rank-2 temperament, how does one arrive at a generator and period from a pair of commas to be tempered out?

There are two steps here. The first is how you find the mapping from
the pair of commas. (You're assuming 7-limit in the question. For
higher limits there'd be more commas, which doesn't make it any harder
in principle.) The second step is finding the generator and period
from the mapping.

The first step is a null space or kernel problem. There is an algorithm here:

http://en.wikipedia.org/wiki/Null_space#Basis

Note that it may introduce contorsion. Contorsion is fairly easy to
remove from a rank 2 mapping but it's fiddly enough that I don't want
to look it up and explain it here.

There are other approaches to this. The easiest way is to start with
an equal temperament (rank 1 system). In the 7-limit, that means 3
linearly independent commas. If you know how to calculate the wedgie,
you're golden. The mapping of the equal temperament is the complement
of the wedge product of the commas.

If you don't know how to calculate the wedgie, you could try matrices.
I explained the process here several years ago:

http://x31eq.com/et.htm

There are a lot of tools around for calculating the inverse and
determinant of a matrix.

Once you know how to get the mapping for an equal temperament, you can
solve a rank 2 class by defining two equal temperaments. To do this,
you nominate two "phony" unison vectors, for which the best name is
still chromatic-unison vectors. Each of these intervals will define
an MOS. The canonical example is that the augmented prime (or
chromatic unison) of 25:24 is the chromatic unison vector for the
meantone diatonic. Treating the diatonic scale as an equal
temperament is like tempering out 25:24.

So, you take your two chromatic-unison vectors, and get the mappings
for two equal temperaments, which are MOS scales of the temperament
you're looking for. You may have to remove torsion. Put them
together and you have the mapping for a rank 2 class.

The second step, then, to get the generator and period from the
mapping. If you trust the two MOS scales you have the mappings for,
the generator/period ratio of their sum is an inverse modulo
operation. So, for meantone, you can start with 12 and 19 note MOS
scales and the inverse modulo will give you 13/31. That's fine if you
know what an inverse modulo is.

You can also find the generator/period mapping. For the normal case
where the octave is the interval of equivalence, that means reducing
the pair of mappings so that one starts with a zero. This is the
generator mapping. The other one may not be the correct period
mapping but you can sort that out later on. Then you need to optimize
the system to get the best period and generator. I gave some
equations for this at

http://x31eq.com/primerr.pdf

As a summary, note that TOP-RMS is a weighted linear least squares
system, so use your favorite linear least squares algorithm or
library. TOP-max (the original TOP) is a linear program. Use the
simplex algorithm or your favorite linear programming library.

Of course, the above is not that simple. There's not much we can do
about that. It's a middling-difficult mathematical problem. It
doesn't lead to any novel mathematics but it does require the
background of a first year science or engineering undergraduate.

> Question 2: how are boundaries between temperament classes worked out?

They don't have to be. The same tuning can work with two different
mappings. For example, the dominant seventh chord in meantone can be
treated as a 4:5:6:7, which gives dominant temperament. But meantone
has a better mapping of the 4:7 as an augmented sixth. You can use
both.

If you want to draw boundaries, I don't see any problems with
following MOS classes.

> Question 3: is there a simple way to calculate the largest MOS scale for a given temperament class?

This as been answered.

> Question 4: how complex is the math to calculate a mapping for a temperament?

Where are you starting from? Getting the higher-rank mappings from
equal temperament mappings is trivially simple. Getting the mappings
for equal temperaments is surprisingly complicated if you want to be
strict about it.

> Question 5: how do you name a temperament that exists on the boundary between two temperament classes (i.e. where for a single value of generator, there are two possible mappings of primes that have equal error)?

It has a rank one lower than the classes it's on the boundary between.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> There are two steps here. The first is how you find the mapping from
> the pair of commas. (You're assuming 7-limit in the question. For
> higher limits there'd be more commas, which doesn't make it any harder
> in principle.) The second step is finding the generator and period
> from the mapping.

Ah, so it's commas-->mapping-->generator, eh? No wonder I was confused. I thought it was commas-->generator-->mapping. This is helpful.

> The first step is a null space or kernel problem. There is an algorithm here:
>
> http://en.wikipedia.org/wiki/Null_space#Basis
>
> Note that it may introduce contorsion. Contorsion is fairly easy to
> remove from a rank 2 mapping but it's fiddly enough that I don't want
> to look it up and explain it here.

Woof. My mathematical education stopped a bit short of matrix operations like this.

> There are other approaches to this. The easiest way is to start with
> an equal temperament (rank 1 system). In the 7-limit, that means 3
> linearly independent commas. If you know how to calculate the wedgie,
> you're golden. The mapping of the equal temperament is the complement
> of the wedge product of the commas.

This is definitely preferable to matrices. The Tonalsoft encyclopedia is very helpful here.

> Once you know how to get the mapping for an equal temperament, you can
> solve a rank 2 class by defining two equal temperaments. To do this,
> you nominate two "phony" unison vectors, for which the best name is
> still chromatic-unison vectors. Each of these intervals will define
> an MOS. The canonical example is that the augmented prime (or
> chromatic unison) of 25:24 is the chromatic unison vector for the
> meantone diatonic. Treating the diatonic scale as an equal
> temperament is like tempering out 25:24.

But...you don't *really* temper out 25:24, right? Since you said "phony" unison vectors? And does this mean that 7-EDO tempers out 25:24?

> So, you take your two chromatic-unison vectors, and get the mappings
> for two equal temperaments, which are MOS scales of the temperament
> you're looking for. You may have to remove torsion. Put them
> together and you have the mapping for a rank 2 class.

Gotcha. Except for the torsion part, but that's okay. I just want a basic understanding of the process, I don't need to actually be able to *do* the math.

> You can also find the generator/period mapping. For the normal case
> where the octave is the interval of equivalence, that means reducing
> the pair of mappings so that one starts with a zero. This is the
> generator mapping. The other one may not be the correct period
> mapping but you can sort that out later on. Then you need to optimize
> the system to get the best period and generator. I gave some
> equations for this at
>
> http://x31eq.com/primerr.pdf
>

Well, that lost me right quick. But that's okay. Seeing what the math behind all this looks like is enough for me.

> Of course, the above is not that simple. There's not much we can do
> about that. It's a middling-difficult mathematical problem. It
> doesn't lead to any novel mathematics but it does require the
> background of a first year science or engineering undergraduate.

Wow, this is only first-year stuff? I have a new-found respect for math majors.

> > Question 2: how are boundaries between temperament classes worked out?
>
> They don't have to be. The same tuning can work with two different
> mappings. For example, the dominant seventh chord in meantone can be
> treated as a 4:5:6:7, which gives dominant temperament. But meantone
> has a better mapping of the 4:7 as an augmented sixth. You can use
> both.

Now this seems to me to be a result of the arbitrariness with which temperament classes are defined. It seems to me that if you defined a standard of accuracy in mapping a bit differently, it would no longer be possible to say one tuning can work with different mappings, or that a tuning could work worth more mappings than it is currently said to. This doesn't really answer my question, because I wanted to know at what point a tuning stops being compatible with a mapping. Clearly, there must be an error level at which a mapping becomes nonsense, and I want to know what that is.

> If you want to draw boundaries, I don't see any problems with
> following MOS classes.

I actually DO see a problem with that, though I didn't until today. By the current definitions of MOS classes, you can approach an equal temperament with an arbitrary level of closeness and still technically remain within a given MOS class. Consider the LLsLLLs "diatonic" scale where L = 239.6 cents and s = 1 cent. The average human couldn't distinguish that from 5-EDO, but it's still "technically" within the diatonic MOS class. Even extending it out to a 22-note MOS, it still behaves more like 5-EDO than 22-EDO. You need a much more complex system of classifying MOS scales to avoid such nonsense, and at the level of complexity necessary for that, it would be more pedagogically-efficient to take a "brute force" approach EDO-by-EDO or else teach linear temperament theory (math, jargon, and all). MOS classes are much more complex that I ever realized, and I actually think linear temperaments are simpler and more useful (if a bit vague on the boundary issue).

> > Question 5: how do you name a temperament that exists on the boundary between two temperament classes (i.e. where for a single value of generator, there are two possible mappings of primes that have equal error)?
>
> It has a rank one lower than the classes it's on the boundary between.

I figured as much...it's like being one branch closer to the root on Wilson's scale tree. But it still doesn't help with a taxonomic scheme. That's really where I'm stuck: coming up with a logical, systematic, and intuitive taxonomy of scales that lets me group MOS scales in various EDOs that have similar musical properties, which I can define without recourse to commas, unison vectors, wedgies, and such; and that I can "name" without having to dig through the archives of this list to explain "why" it's called something. I mean, you guys have been about as unsystematic as possible coming up with linear temperament names...sometimes it's a pun on the generator ("Father", "Amity"), sometimes it's a reference to an ethnic scale that the temperament approximates ("Pelogic", "Mavila", "Mohajira"), sometimes it's based on the degrees of some EDO that provides an optimal generator ("Orwell" [19\84], "Beatles" [19\64]), sometimes it's someone's name ("Blackwood", "Hanson"), sometimes it's named for an interval ("Kleismic", "Schismic"), sometimes it's historical ("Meantone"), and sometimes I don't even know where it came from ("Beep", "Bug", "Injera", "Ripple"). I mean, keeping those names straight is a challenge! Not that it's any less arbitrary than how biologists do it, and not that I have a better solution...but the thought of trying to *teach* this stuff to one of my musician-friends is ridiculous.

This is why I've been bugging you all: I want to know how to draw boundaries so I can lay out all the scales with "manageable" numbers of notes (i.e. the scales that function as scales and not gamuts/superscales) and figure out a systematic way of grouping and naming them, to make it easier to teach people how to use alternative equal temperaments. My attempts to do this according to MOS configuration failed, because the MOS method groups scales together which have drastically different musical properties. It doesn't seem like the regular mapping paradigm produces any clearer boundaries (or if it does, no one's explained it to me yet). So I'm still empty-handed.

Hi Igs,

> This is why I've been bugging you all: I want to know how to
> draw boundaries so I can lay out all the scales with "manageable"
> numbers of notes (i.e. the scales that function as scales and not
> gamuts/superscales) and figure out a systematic way of grouping
> and naming them, to make it easier to teach people how to use
> alternative equal temperaments. My attempts to do this according
> to MOS configuration failed, because the MOS method groups scales
> together which have drastically different musical properties.
> It doesn't seem like the regular mapping paradigm produces any
> clearer boundaries (or if it does, no one's explained it to
> me yet). So I'm still empty-handed.

The regular mapping paradigm is a machine that makes scales for
harmonious music. You put in what you consider harmonious and how
many notes/octave you want, turn the crank, and out pop scales.
They will all sound alike because they all approximate what you
consider harmonious, using roughly the same number of notes. It
doesn't seem that people can really hear the mappings (without
knowing what to listen for).

The only time you'll notice different musical properties (as far
as regular mapping is concerned) is if you write some music with
chord progressions and intertwined melodies and so on and then
switch the temperament. You'll get pitch drift unless both
temperaments happen to remove the comma(s) in question. So we
can put temperaments into families according to which commas they
remove (see the comma sequences link I posted earlier). However
it's only obviously audible in terms of pitch drifts.

Based on the things you were saying about consonance, and from
what I hear in your music, you may find melodic properties more
useful for classifyication. There you have things like the number
of notes/octave (or other equivalence interval), Rothenberg mean
variety, Rothenberg stability, and so on.

-Carl

I wrote:

> Based on the things you were saying about consonance, and from
> what I hear in your music, you may find melodic properties more
> useful for classifyication. There you have things like the number
> of notes/octave (or other equivalence interval), Rothenberg mean
> variety, Rothenberg stability, and so on.

For instance, this kind of brute-force approach:

For melody, we're really only interested in scales with 5-10 notes/octave. There are obviously 6 ETs in that range and each
one can be its own class.

Next up is rank 2, where there will be two kinds of 2nd: L & s.
For 7/oct, there are obviously 2^7 ways to arrange these two
steps. Except for periodic scales, we ignore circular
permutation. So it's really just the number of 2-color, 7-bead
"necklaces", which happens to only be 20.

Next we can restrict our gaze to Rothenberg-proper scales
(I think this is reasonable). This should give us a bound on
the ratio s/L for each pattern, and we can also enforce s < L.
Then we can ask where we switch Rothenberg rank-order matrices
as s/L is varied for each pattern. I believe the crossover
points are always when s/L is rational, and which rationals
should be deducible from the scale pattern.

Then we find a way to name each rank order matrix. Like,

"7.19.3/5.4/5"

would be the 7-note scale, 19th necklace (in alphabetical
order... yes there is a standard way to do this), with
3/5 < s/L < 4/5.

This is off the top of my head at 2am... but I think it might
work.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Igs,
>
> The regular mapping paradigm is a machine that makes scales for
> harmonious music. You put in what you consider harmonious and how
> many notes/octave you want, turn the crank, and out pop scales.
> They will all sound alike because they all approximate what you
> consider harmonious, using roughly the same number of notes. It
> doesn't seem that people can really hear the mappings (without
> knowing what to listen for).

I see. I've never heard it explained like that, and it makes more sense this way.

> The only time you'll notice different musical properties (as far
> as regular mapping is concerned) is if you write some music with
> chord progressions and intertwined melodies and so on and then
> switch the temperament. You'll get pitch drift unless both
> temperaments happen to remove the comma(s) in question. So we
> can put temperaments into families according to which commas they
> remove (see the comma sequences link I posted earlier). However
> it's only obviously audible in terms of pitch drifts.

But don't all these temperaments have wildly different scalar structures? Or are temperaments typically used with such large numbers of notes that their lower-order MOS-like structures become less important? I guess people using Orwell for a complete 7-limit temperament probably use more than 9 notes of it, now that I think of it.

> Based on the things you were saying about consonance, and from
> what I hear in your music, you may find melodic properties more
> useful for classifyication. There you have things like the number
> of notes/octave (or other equivalence interval), Rothenberg mean
> variety, Rothenberg stability, and so on.

Well, this isn't exactly about what *I* prefer aesthetically...I don't want to import my bias against "traditional" consonance into the primer I'm writing. However, I'm dealing with a lot of EDOs wherein traditional consonance is impossible, so in that sense, yes, melodic properties are probably more important.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> For instance, this kind of brute-force approach:
>
> For melody, we're really only interested in scales with 5-10 notes/octave. There are obviously 6 ETs in that range and each
> one can be its own class.
>
> Next we can restrict our gaze to Rothenberg-proper scales
> (I think this is reasonable). This should give us a bound on
> the ratio s/L for each pattern, and we can also enforce s < L.

How does Rothenberg propriety put a limit on the ratio s/L? Also, is this ratio expressing "size of s to size of L" or "number of s steps in scale to number of L steps in scale"?

> Then we can ask where we switch Rothenberg rank-order matrices
> as s/L is varied for each pattern. I believe the crossover
> points are always when s/L is rational, and which rationals
> should be deducible from the scale pattern.
>
> Then we find a way to name each rank order matrix. Like,
>
> "7.19.3/5.4/5"
>
> would be the 7-note scale, 19th necklace (in alphabetical
> order... yes there is a standard way to do this), with
> 3/5 < s/L < 4/5.
>
> This is off the top of my head at 2am... but I think it might
> work.

I'd like to hear more. If this is about setting ranges of "size of s to size of L", this will have the property I'm looking for, in that scales where s is vanishingly small or practically equal to L will be ruled out of families defined by number of s and number of L. That would be BRILLIANT!

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> > The only time you'll notice different musical properties (as far
> > as regular mapping is concerned) is if you write some music with
> > chord progressions and intertwined melodies and so on and then
> > switch the temperament. You'll get pitch drift unless both
> > temperaments happen to remove the comma(s) in question. So we
> > can put temperaments into families according to which commas they
> > remove (see the comma sequences link I posted earlier). However
> > it's only obviously audible in terms of pitch drifts.
>
> But don't all these temperaments have wildly different scalar
> structures?

Some do, some don't, but either way, regular mapping doesn't
address it. There's no input for what kind of scale structure
you want.

> Or are temperaments typically used with such large
> numbers of notes that their lower-order MOS-like structures
> become less important? I guess people using Orwell for a
> complete 7-limit temperament probably use more than 9 notes
> of it, now that I think of it.

Sure, but are they nominals, or accidentals? When you hear
classical music retuned to meantone chains of > 12 notes, do
you hear the scale as expanded? I don't.

> > Based on the things you were saying about consonance, and from
> > what I hear in your music, you may find melodic properties more
> > useful for classifyication. There you have things like the number
> > of notes/octave (or other equivalence interval), Rothenberg mean
> > variety, Rothenberg stability, and so on.
>
> Well, this isn't exactly about what *I* prefer aesthetically...
> I don't want to import my bias against "traditional" consonance
> into the primer I'm writing. However, I'm dealing with a lot
> of EDOs wherein traditional consonance is impossible, so in
> that sense, yes, melodic properties are probably more important.

In any case, I think the protocol I described is worth trying.
Unfortunately, while a guy like Graham or Gene could probably
do it in an hour, it takes a like me a full day. And whereas
a guy like Gene may be retired, I have two young boys to take
care of. And I'm terrible at pausing and restarting projects.
So it might be awhile.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> In any case, I think the protocol I described is worth trying.
> Unfortunately, while a guy like Graham or Gene could probably
> do it in an hour, it takes a like me a full day. And whereas
> a guy like Gene may be retired, I have two young boys to take
> care of. And I'm terrible at pausing and restarting projects.
> So it might be awhile.
>
Well I'm not retired either, but I'm a grad student with lots of free time so I'll see what I can figure out. I found this:

http://robertinventor.com/tuning-math/s___5/msg_4700-4724.html

An old tuning-math archive where you describe Rothenberg efficiency and stability. As a side-note, it seems you were explaining it to Gene Ward Smith...heartening for me to see that at some point, even Gene had to learn a thing or two about scales!

Anyway, I think this is the right direction and I'll see what I can work out.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> An old tuning-math archive where you describe Rothenberg efficiency and stability. As a side-note, it seems you were explaining it to Gene Ward Smith...heartening for me to see that at some point, even Gene had to learn a thing or two about scales!
>
My bad, it's Graham explaining those terms. Those old archives are confusing!

[view this with fixed-width font]

"cityoftheasleep" wrote:

> > Next we can restrict our gaze to Rothenberg-proper scales
> > (I think this is reasonable). This should give us a bound on
> > the ratio s/L for each pattern, and we can also enforce s < L.
>
> How does Rothenberg propriety put a limit on the ratio s/L?
> Also, is this ratio expressing "size of s to size of L" or
> "number of s steps in scale to number of L steps in scale"?

Size. If we have the pattern [s s s L], we get the interval
matrix

2nds 3rds 4ths 5ths
s 2s 3s 3sL
s 2s 2sL 3sL
s sL 2sL 3sL
L sL 2sL 3sL

where to be proper, we need 2s >= L, or s/L >= 1/2. Along
with the constraint that 3sL = your chosen equivalence
interval, we now have an equivalence class for scales --
Rothenberg's model says all scales meeting these conditions
should sound alike (melodically). For an FAQ on that, go
here
http://lumma.org/music/theory/TuningFAQ.txt
and scroll all the way down.

> > Then we find a way to name each rank order matrix. Like,
> >
> > "7.19.3/5.4/5"
> >
> > would be the 7-note scale, 19th necklace (in alphabetical
> > order... yes there is a standard way to do this), with
> > 3/5 < s/L < 4/5.
>
> I'd like to hear more. If this is about setting ranges of
> "size of s to size of L", this will have the property I'm
> looking for, in that scales where s is vanishingly small or
> practically equal to L will be ruled out of families defined
> by number of s and number of L. That would be BRILLIANT!

As I look at it in the light of day (after 5 hours of sleep),
it may be that for rank 2 scales we only get one bound on s/L.
The above interval matrix will fold down to different
rank-order matrices depending on the answers to max[2s, sL]
and max[3s, 2sL]. But since s < L, sL and 2sL will always
be greater, so we only have one rank-order matrix here.
I don't know if this is an artifact of the example I'm using,
or if it's generally true of rank 2 scales. But the name
of this one might be

5.2.1/2.x

That is, five notes, 2nd necklace, s/L lower-bound 1/2, and
no upper bound. To see why this is the 2nd necklace, go here:

http://www.wolframalpha.com/input/?i=necklace+5+beads+2+colors

-Carl

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> My bad, it's Graham explaining those terms. Those old archives
> are confusing!

You can get the original papers here:

http://lumma.org/tuning/rothenberg/AModelForPatternPerception.pdf

or the critical few excerpts here:

http://lumma.org/music/theory/RothenbergExcerpts.txt

-Carl

Oh man, thank you Carl!! This is going to take me some time to digest but I think this will do what I want it to. Very helpful!

-Igs

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
>
> > My bad, it's Graham explaining those terms. Those old archives
> > are confusing!
>
> You can get the original papers here:
>
> http://lumma.org/tuning/rothenberg/AModelForPatternPerception.pdf
>
> or the critical few excerpts here:
>
> http://lumma.org/music/theory/RothenbergExcerpts.txt
>
> -Carl
>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> > They will all sound alike because they all approximate what you
> > consider harmonious, using roughly the same number of notes. It
> > doesn't seem that people can really hear the mappings (without
> > knowing what to listen for).
>
> I see. I've never heard it explained like that, and it makes more sense this way.

It's not altogether true, however, and especially not for smaller scales. You get one kind of chord progression predominating over others depending on the generator, as for instance meantone likes relationships of a fifth. You get a lot of one kind of chord, not so many of another, and might not have any of a third. Again in smaller scales particularly, different melodic properties can be noted.

> But don't all these temperaments have wildly different scalar structures? Or are temperaments typically used with such large numbers of notes that their lower-order MOS-like structures become less important?

Most people don't use them with large numbers of notes, I'm kind of an exception, and even with me, it's the *differences*n between the temperaments which drive my thinking a lot.

I guess people using Orwell for a complete 7-limit temperament probably use more than 9 notes of it, now that I think of it.

Orwell[9] actually makes for good two-part harmony, and melodicaly is quite nice.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> That is, five notes, 2nd necklace, s/L lower-bound 1/2, and
> no upper bound. To see why this is the 2nd necklace, go here:
>
> http://www.wolframalpha.com/input/?i=necklace+5+beads+2+colors

Sorry, clipboard failure. Should be

http://www.wolframalpha.com/input/?i=necklace+4+beads+2+colors

The numbers are really manageable to create a complete catalog.
Only 58 rank 2 patterns for scales of 10 notes/period.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> I don't know if this is an artifact of the example I'm using,
> or if it's generally true of rank 2 scales. But the name
> of this one might be
>
> 5.2.1/2.x

I don't know that I like this much, so let me suggest another scheme. Let's say we want to subdivide 5L2s scales. Generators for these run from 4/7 to 3/5; in terms of the Minkowski "?" function that's 9/16 to
5/8. We can subdivide this into proper and improper, with the boundry at 9/16+1/32 = 19/32, and further subdivide by

9/16 < x < 9/16+1/64 = 37/64 smooth proper
37/64 < x < 37/64+1/64 = 19/32 lumpy proper
19/32 < x < 19/32+1/64 = 39/64 regular improper
39/64 < x < 39/64+1/64 = 5/8 irregular improper

I can translate these boundary values back to generators using the box function, getting

4/7 < x < 11/19 smooth proper
11/19 < x < 7/12 lumpy proper
7/12 < x < 10/17 regular improper
10/17 < x < 3/5 irregular improper

This is just the Stern-Brocot tree in action, and everything can be done without using ? and box, for instance using Scala. But to me at least it's clearer this way: all the regions are of the same size, and you know how deep you are in the tree by the denominator. However you do it, it seems to me this four-fold division makes for a pretty good systemization of scale types.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> I don't know that I like this much, so let me suggest another scheme. Let's say we want to subdivide 5L2s scales. Generators for these run from 4/7 to 3/5; in terms of the Minkowski "?" function that's 9/16 to
> 5/8. We can subdivide this into proper and improper, with the boundry at 9/16+1/32 = 19/32, and further subdivide by
>
> 9/16 < x < 9/16+1/64 = 37/64 smooth proper
> 37/64 < x < 37/64+1/64 = 19/32 lumpy proper
> 19/32 < x < 19/32+1/64 = 39/64 regular improper
> 39/64 < x < 39/64+1/64 = 5/8 irregular improper
>

And you could further subdivide as follows:
9/16 < x <9/16+1/128 = 73/128
73/128 < x < 73/128+1/128 = 37/64
37/64 < x < 37/64+1/128 = 75/128
And so on, right?

> I can translate these boundary values back to generators using the box function, getting
>
> 4/7 < x < 11/19 smooth proper
> 11/19 < x < 7/12 lumpy proper
> 7/12 < x < 10/17 regular improper
> 10/17 < x < 3/5 irregular improper
>

So the implied continuum here is from s=0 to s=L, with s=L/2 defining the boundary of proper and improper. s=3L/4 defines the boundary of smooth and lumpy, and s=L/3 defines the boundary between regular and irregular. One could theoretically create more groups by increasing the subdivisions in the Minkowski "?" form, and then combine selectively to form a group defined by (say) 4L/5<s<L/2, and another group as L/2<s<L/5. These groups will all have a one-to-one correspondence with sections of the scale tree.

So basically, it looks like to avoid the problem of MOS scales getting arbitrarily close to a low-numbered EDO, I can just "prune" the scale-tree of all the branches that get to close to an EDO. Oh my goodness. This is incredible! And so SIMPLE!

Thank you, Gene! One further question, though: in your above taxonomy, what would you call scales where x = 7/12, or 11/19, or 10/17? Answer me that, and my problem is SOLVED!!

-Igs

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> > I don't know if this is an artifact of the example I'm using,
> > or if it's generally true of rank 2 scales. But the name
> > of this one might be
> >
> > 5.2.1/2.x
>
> I don't know that I like this much,

What's not to like?

> so let me suggest another scheme. Let's say we want to
> subdivide 5L2s scales. Generators for these run from 4/7
> to 3/5; in terms of the Minkowski "?" function that's 9/16 to
> 5/8. We can subdivide this into proper and improper, with the
> boundry at 9/16+1/32 = 19/32, and further subdivide by
>
> 9/16 < x < 9/16+1/64 = 37/64 smooth proper
> 37/64 < x < 37/64+1/64 = 19/32 lumpy proper
> 19/32 < x < 19/32+1/64 = 39/64 regular improper
> 39/64 < x < 39/64+1/64 = 5/8 irregular improper
>
> I can translate these boundary values back to generators using
> the box function, getting
>
> 4/7 < x < 11/19 smooth proper
> 11/19 < x < 7/12 lumpy proper
> 7/12 < x < 10/17 regular improper
> 10/17 < x < 3/5 irregular improper
>
> This is just the Stern-Brocot tree in action, and everything
> can be done without using ? and box, for instance using Scala.
> But to me at least it's clearer this way: all the regions are
> of the same size, and you know how deep you are in the tree by
> the denominator. However you do it, it seems to me this four-
> fold division makes for a pretty good systemization of
> scale types.

What does this have to do with rank-order matrices?

-Carl

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> > 9/16 < x < 9/16+1/64 = 37/64 smooth proper
> > 37/64 < x < 37/64+1/64 = 19/32 lumpy proper
> > 19/32 < x < 19/32+1/64 = 39/64 regular improper
> > 39/64 < x < 39/64+1/64 = 5/8 irregular improper
> >
>
> And you could further subdivide as follows:
> 9/16 < x <9/16+1/128 = 73/128
> 73/128 < x < 73/128+1/128 = 37/64
> 37/64 < x < 37/64+1/128 = 75/128
> And so on, right?

Right.

> So the implied continuum here is from s=0 to s=L, with s=L/2 defining the boundary of proper and improper. s=3L/4 defines the boundary of smooth and lumpy, and s=L/3 defines the boundary between regular and irregular.

Right, except I suggest we use Blackwood's R = L/s, which is sort of standard. L=2 is 7/12, the mediant of 4/7 and 3/5, and the boundry between proper and improper. The mediant of 7/12 and 4/7 is 11/19, where R = 3/2, and it is the boundry between lumpy and smooth. The mediant between 7/12 and 3/5 is 10/17, R = 3, defining the boundry between regular and irregular.

One could theoretically create more groups by increasing the subdivisions in the Minkowski "?" form, and then combine selectively to form a group defined by (say) 4L/5<s<L/2, and another group as L/2<s<L/5.

Right; the next step in the process would be dividing at the mediant between 4/7 and 11/19, which is 15/26, and which has an R value of 4/3, and another division between 11/19 and 7/12, the mediant there being 18/31, and the R-value being 5/3, and so forth.

These groups will all have a one-to-one correspondence with sections of the scale tree.
>
> So basically, it looks like to avoid the problem of MOS scales getting arbitrarily close to a low-numbered EDO, I can just "prune" the scale-tree of all the branches that get to close to an EDO. Oh my goodness. This is incredible! And so SIMPLE!

I'm not sure what revelation has just struck you, but if you want total trouble avoidance you can always set R to be the golden ratio.

> Thank you, Gene! One further question, though: in your above taxonomy, what would you call scales where x = 7/12, or 11/19, or 10/17? Answer me that, and my problem is SOLVED!!

I dunno. 7/12 is the boundry between proper and improper, and Rothenberg calls that non-strict proper, but you could call it semiproper. 11/19 is the boundry between lumpy and smooth, so maybe it's semilumpy, and 10/17 is semiregular.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> What does this have to do with rank-order matrices?

Nothing I know of, but I didn't see how necklaces were going to help Igs, and I don't know what you are doing with rank-order matricies.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
>
> > What does this have to do with rank-order matrices?
>
> Nothing I know of, but I didn't see how necklaces were going to
> help Igs, and I don't know what you are doing with rank-order
> matricies.

The entire point of the exercise is to enumerate and name
rank order matrices, since that is what Rothenberg claims
are important.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > > What does this have to do with rank-order matrices?
> >
> > Nothing I know of, but I didn't see how necklaces were going to
> > help Igs, and I don't know what you are doing with rank-order
> > matricies.
>
> The entire point of the exercise is to enumerate and name
> rank order matrices, since that is what Rothenberg claims
> are important.

I thought the point of the exercise was to help classify MOS scales, which is a lot easier than looking at the kind of generality Rothenberg discusses.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> The entire point of the exercise is to enumerate and name
> rank order matrices, since that is what Rothenberg claims
> are important.

Speaking of Rothenberg, do you know a good source of biographical info?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> I thought the point of the exercise was to help classify
> MOS scales, which is a lot easier than looking at the kind of
> generality Rothenberg discusses.

Since I came up with the exercise, I think I should know
what the point of it is. -Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > I thought the point of the exercise was to help classify
> > MOS scales, which is a lot easier than looking at the kind of
> > generality Rothenberg discusses.
>
> Since I came up with the exercise, I think I should know
> what the point of it is. -Carl
>

I thought you were trying to help Igs out. As a general exercise in music theory, the idea has merit, and maybe it could be developed and incporporated into Scala or something. Scala already puts up a matrix when you invoke the "show interval matrix" command.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> > The entire point of the exercise is to enumerate and name
> > rank order matrices, since that is what Rothenberg claims
> > are important.
>
> Speaking of Rothenberg, do you know a good source of
> biographical info?

We met once in New York, and corresponded several times for
a few years after that. If he's still alive, he is very
advanced in age. He lived with his wife in a beautiful
apartment on the upper West side and had a company called
Inductive Inference, which as far as I could tell was his
way of working as a consultant. He wrote the original papers
on a grant from the Air Force I believe. He was quite
attracted to the idea of being a composer, but as far as I
know only one performance of his work was ever undertaken;
a trio for clarinet, trumpet and cello performed in the '80s
at Mills college. I've forgotten how I obtained a copy.
Johnny Reinhard may be the microtonalist with the closest
connection to him. I found out about him through an article
by John Chalmers in an old Xenharmonikon. As far as I know,
I am the foremost advocate of his model anywhere, having
undertaken a number of efforts to promote it, since it is
overall the best approach to melody theory I'm aware of.

-Carl

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> Scala already puts up a matrix when you invoke the "show
> interval matrix" command.

Yes, thanks in part to my constant prodding. But more relevant
is the "show interval ranking matrix" command. (Then there's "efficiency", reported in show data, thanks to my mailing copies
of unpublished Rothenberg papers to Manuel in 1999...)

-Carl

On 15 June 2010 21:14, cityoftheasleep <igliashon@...> wrote:

> But don't all these temperaments have wildly different scalar
> structures?  Or are temperaments typically used with such
> large numbers of notes that their lower-order MOS-like
> structures become less important?  I guess people using
> Orwell for a complete 7-limit temperament probably use
> more than 9 notes of it, now that I think of it.

There aren't enough people using novel regular temperaments that we
can really make generalizations. One approach is that you could plot
your music on the JI lattice, maybe using certain approximations, and
then convert it to a temperament. That seems to be what Carl's
talking about but I don't know anybody who works like that.

The more notes, yes, the more it's the odd limit (or whatever) that
drives the melodic structure. Any extended 7-limit system will let
you divide a 7:8 into 14:15:16, for example, although it's only in
Marvel temperaments that the two resulting steps are equal.

The scales I use, generally no more than 24 notes to the octave, do
restrict the melody. Magic (19&22) does have neutral thirds, but
there's only one on my keyboard. Dividing a fifth into two neutral
thirds requires a lot of notes in Magic. But in Miracle (31&41) it's
very simple. So if you like a lot of neutral thirds you're more
likely to start with Miracle. If you want to base everything on
neutral thirds, you end up with Mohajira of course, which can be
defined by a mapping.

Orwell (22&31) could be used for different reasons. Because of the 9
note MOS (which I noticed two of you have used as a subset of an equal
temperament), because it gives 11-limit harmony simply but
comprehensibly, or because it mixes 7-limit harmony with unequal
neutral thirds. There's no point making generalizations about how it
is used because the fact is it's barely being used at all.

What I have done is written music for Magic temperament but adapted it
to general Marvel temperament. The melodies I chose were still guided
by Magic, though.

> Well, this isn't exactly about what *I* prefer aesthetically...I
> don't want to import my bias against "traditional" consonance
> into the primer I'm writing.  However, I'm dealing with a lot of
> EDOs wherein traditional consonance is impossible, so in
> that sense, yes, melodic properties are probably more important.

The consonances you map don't have to be traditional. Any equal
temperament that isn't ridiculously small will approximate some
intervals or other. If you look for scales that approximate those
same intervals you'll probably end up back where you started.

Graham

Igs wrote (responding to Graham):

> Well, that lost me right quick. But that's okay. Seeing what the math > behind all this looks like is enough for me.

Ahhh, Igs, let me tell you something. I'm not that "skilled" in maths either but even so, I have found some sort of "my way" of viewing rank 2 tunings.
Long before I first read something about making 2D tunings from pairs of ETs, I had already found other ways to get them.
If you've read the message #90370 where I was answering your questions, there you could find me describing one of them -- if you were interested, I could possibly describe the other ones as well.

Petr