Yahoo Tuning Groups Ultimate Backup tuning-math Naming temperaments

I've been thinking about names for linear temperaments and ways of classifying them; there's probably nothing much new here, but a summary might have some use.

We know that wedgies are useful for identifying linear temperaments (and any temperament in which two independent commas vanish, like 11-limit planar temperaments). But they have two serious drawbacks as names: they're not easily recognized or understood by looking at them. Even the 7-limit wedgies are pretty long, and higher limit wedgies are too long to use as names. Take this one for instance:

<<1, 9, -2, 12, -6, -30||

You've probably seen it, but do you recognize it? You can guess that it has a mapping of (0, 1, 9, -2) or (0, -1, -9, 2), which might be of some use, but then you might as well give the whole map, which isn't much longer:

[<1, 2, 6, 2|, <0, -1, -9, 2|]

Or you can give the mapping by steps, which tells you more directly something about the structure of the scale, but isn't as easily recognizable, and is even less suitable as a name.

[(22, 5), (35, 8), (51, 12), (62, 14)]

For most of the better temperaments, if you just give the mapping by steps of the octave (which is equivalent to a combination of two octave-based ET's), you can fill in the others by taking the best approximation of the primes in each of the ET's. So we could call this the 5&22 temperament.

But there's another alternative that looks somewhat interesting. Notice that the first thing in Graham Breed's temperament finder results is a fraction of an octave: in this case:

11/27, 489.3 cent generator

If you start with a generator like this, you end up with a closed temperament; it seems like there should be one best way to map this to a linear temperament, if the ET is consistent. You can figure out how many iterations to get to any interval by just adding 11 mod 27 until you end up with the best approximation, which will take a maximum of 26 iterations (subtract the result from 27 if it ends up more than 27/2 steps away). One drawback is that you also need to specify the period if it isn't an approximate octave; e.g., 1/6 (1/2) or 2/11 (1/2) for pajara.

The nice thing about calling this the "11/27-LT" is that none of the other methods give you the size of the generator, but this method gives you all that *plus* if you consult a Stern-Brocot tree you can find out which ET's it supports. It's basically giving it the name of a branch on the scale tree. You could even go up a branch and call it the "9/22-LT", since 22-ET is consistent; you don't actually need *two* consistent ETs to name it. If you want to be more precise and narrow it down, you could also call it 20/49-LT; all of these share the same mapping. Things get more complicated if the ET is inconsistent, but it should be manageable.

And of course the other way of naming these things is by giving them actual names. The big advantage is recognizability: if you've heard of "superpyth", you take one look at the name and say "Ah yes, *that* temperament". You might not immediately recognize "7/31" or "9&22", but probably anyone who's familiar with it knows the name "orwell". The disadvantage, of course, is that if you haven't heard of it, it doesn't mean anything. So you might as well use the name in combination with one or more of the other methods, especially if it's a less familiar one like 3/8 (1/2) 10&16 lemba <<6, -2, -2, -17, -20, 1||.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> I've been thinking about names for linear temperaments and ways of
> classifying them; there's probably nothing much new here, but a
summary
> might have some use.

Yes. This is good. Thanks Herman.

> For most of the better temperaments, if you just give the mapping
by
> steps of the octave (which is equivalent to a combination of two
> octave-based ET's), you can fill in the others by taking the best
> approximation of the primes in each of the ET's. So we could call
this
> the 5&22 temperament.

Well I'd prefer to call it the 22&27-LT. You can get the 5 by
subtraction and you can get an even better ET for it (49) by
addition. i.e. the two ETs should be two extremes of the generator
value such that you would actually consider using those ETs for the
LT. You wouldn't really use 5. The 5 just tells you that pentatonic
scales make sense in this LT. And as I say, you can get that by
subtraction.

> But there's another alternative that looks somewhat interesting.
Notice
> that the first thing in Graham Breed's temperament finder results
is a
> fraction of an octave: in this case:
>
> 11/27, 489.3 cent generator

Right. But why _is_ this 11/27 and not 20/49?

> One drawback is that you also need to specify the period if
> it isn't an approximate octave; e.g., 1/6 (1/2) or 2/11 (1/2) for
pajara.
>

That's a pretty serious drawback. At least with the two-ETs method
you can see if they are both even and know immediately that the
period is a half octave.

But you could just call Pajara "twin 4/46" or is it "twin 19/46",
or "twin 27/46", or should the denominator be some other ET?

> And of course the other way of naming these things is by giving
them
> actual names. The big advantage is recognizability: if you've
heard of
> "superpyth", you take one look at the name and say "Ah yes, *that*
> temperament". You might not immediately recognize "7/31"
or "9&22", but

Why wouldn't you immediately recognise one of these, if that's what
you've been used to seeing it called.

Superpyth isn't a good example of the problem, since it at least
suggests something related to pythagorean. The real problem is with
names like "keenan" or "porcupine" or "orwell".

> probably anyone who's familiar with it knows the name "orwell".

That's a tautology.

> The
> disadvantage, of course, is that if you haven't heard of it, it
doesn't
> mean anything.

Exactly.

> So you might as well use the name in combination with one
> or more of the other methods, especially if it's a less familiar
one
> like 3/8 (1/2) 10&16 lemba <<6, -2, -2, -17, -20, 1||.

So who needs the "lemba". It adds absolutely nothing, for me. For
some reason it suggests "unleavened bread" to me. Huh?

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Carl Lumma <ekin@lumma.org>

🔗Herman Miller <hmiller@IO.COM>

Dave Keenan wrote:

>>fraction of an octave: in this case:
>>
>>11/27, 489.3 cent generator
> > > Right. But why _is_ this 11/27 and not 20/49?

It could be either one: a 27-note scale with an approximate 11/27 generator or a 49-note scale with an approximate 20/49 generator. But a 27-note scale is simpler, and it's good to have an idea of the simplest useful scale associated with a temperament. Note that in the 11-limit these represent different temperaments:

11/27 [<1, 2, 6, 2, 1|, <0, -1, -9, 2, 6|]
20/49 [<1, 2, 6, 2, 10|, <0, -1, -9, 2, -16|]

To be more precise, since 27-ET isn't 11-limit consistent, you could call it 11/(27,43,63,76,93). But unless there turns out to be a lot of useful-looking temperaments that can't be notated any other way (which could very well be the case once we start dealing with the 11-limit), it would be simpler just to assume that 11/27 denotes a temperament based on the best mapping of each of the primes to degrees of 27-EDO, which is (27,43,63,76,93,100,110,115,122,131,134,...).

>>One drawback is that you also need to specify the period if >>it isn't an approximate octave; e.g., 1/6 (1/2) or 2/11 (1/2) for > > pajara.
> > > That's a pretty serious drawback. At least with the two-ETs method > you can see if they are both even and know immediately that the > period is a half octave.
> > But you could just call Pajara "twin 4/46" or is it "twin 19/46", > or "twin 27/46", or should the denominator be some other ET?

4/46 gives you [<2, 3, 5, 7|, <0, 1, -2, -8|], not pajara (see http://x31eq.com/diaschis.htm).

Pajara is historically associated with 22-ET, of course. But you can think of the denominator as representing the size of a typical MOS scale associated with the temperament, rather than an ET. In that case, the minimum is 10 steps, which matches Paul's decatonic scale.

>> if you've heard of >>"superpyth", you take one look at the name and say "Ah yes, *that* >>temperament". You might not immediately recognize "7/31" > > or "9&22", but > > Why wouldn't you immediately recognise one of these, if that's what > you've been used to seeing it called.

Combinations of numbers aren't especially easy to remember. It would be like using ZIP codes to refer to cities in the US, instead of names; they all look alike.

>>probably anyone who's familiar with it knows the name "orwell". > > > That's a tautology.

The pronoun "it" refers to the temperament represented by "7/31" and "9&22", which happens to be named "orwell". I think it's a fairly safe assumption that most people who've heard of this temperament will recognize that name. (Certainly "19/84" is more familiar, but it implies a greater degree of complexity, and could easily be overlooked by people who don't care for highly complex scales.)

>>So you might as well use the name in combination with one >>or more of the other methods, especially if it's a less familiar > > one > >>like 3/8 (1/2) 10&16 lemba <<6, -2, -2, -17, -20, 1||.
> > > So who needs the "lemba". It adds absolutely nothing, for me. For > some reason it suggests "unleavened bread" to me. Huh?

Anyone who's vaguely heard of it, but doesn't know much about it. You might not know how to recognize a beech tree if you see one, or how it differs from other trees, but you probably know that the word "beech" represents a kind of tree. I don't know the mapping of "nonkleismic" off the top of my head, and probably wouldn't recognize it if I saw it (it's [<1, -1, 0, 1|, <0, 10, 9, 7|]), but I do recall it being (in theory) a good temperament. "8/31" might give some idea of its usefulness, but doesn't distinguish it from the many other "n/31" temperaments. So even a questionable name like "nonkleismic" has some use. But "myna" is better because it links it with the starling family of temperaments. Along those lines, Gene's "Japanese monster" names also provide useful hints to similarities between temperaments.

Why does anything have a name? Why do we talk about "major thirds" when we could call them 5/4's? Language works by naming things; the problem is that the study of linear and higher-dimensional temperaments is so new that we haven't settled on the best names for things. So in the meantime, names which may end up being changed will have to be supplemented by numerical keys of one kind or another.

I'm leaning toward the fractional generator + period notation for unfamiliar temperaments, with wedgies for those few that can't easily be symbolized in this way. But I still find names easier to remember, and I don't want to discourage the naming of temperaments that look like they might be useful.

🔗Herman Miller <hmiller@IO.COM>

Dave Keenan wrote:

> Good summary Herman. But you left out my favourite method which > unlike some of the others does not treat ETs as if they were more > fundamental than LTs or JI, but instead relates to what this is all > about -- approximating JI.

That's the nice thing about tuning maps. You can interpret some of the other methods as detempering ET's, but there are other interpretations. 5/12 could be a 12-ET based scale, or it could be a 12-note MOS with a 5-step generator. Even 5&7 could be interpreted in MOS terms, as a scale with 5 steps of one size and 7 of another size.

> The name starts with a word for the number of periods per octave, if > more than one: twin, triple, quadruple, quintuple, 6-fold, 7-
> fold, ....
> > And then the generator is described in terms of the simplest n-odd-
> limit consonance (from the diamond) (or its octave inversion or > extension, as required). That is the one that takes the fewest > generators to approximate according to the LTs mapping. > > I use the following words if there is more than one generator to the > consonance: semi, tri, quarter, 5-part, 6-part, ....
> > Followed by the ratio or words for the consonance as given here:
> http://dkeenan.com/Music/Miracle/MiracleIntervalNam
> ing.txt
> > e.g. Miracle is "semi 7:8's" or "semi supermajor seconds". This could be potentially useful up to a point; certainly there's a mnemonic value in names like "semisixths". But I don't see how this can be generalized to the 7-limit and higher without being arbitrary. Which LT gets to be called "fourths" -- dominant (5&12), meantone (12&19), superpyth[agorean] (5&22), flattone (19&26), or schismic (12&29)? You could make good arguments at least for dominant, meantone, and schismic; then you need to figure out how to name the others. "Major thirds" could be either muggles (16&19) or magic (19&22), and so on.

> This is used up to some point where the LT is so complex you just > describe the generator in cents. e.g. What used to be called > Aritoxenean is the 12-fold 15 cent LT.
> > This at least works up to 11-limit.

But giving the generator in cents doesn't determine a unique mapping; you can derive one from a rational generator/period ratio if you make some assumptions, but an arbitrary value in cents could represent more than one temperament. An LT with a 316.5 cent generator can be mapped as [<1, 0, 1, 2|, <0, 6, 5, 3|] or [<1, 0, 1, -3|, <0, 6, 5, 22|]. With rational generators and the naming conventions I've described, you can unambiguously describe the first mapping as 5/19 and the second as 19/72.

🔗Graham Breed <graham@microtonal.co.uk>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >The name starts with a word for the number of periods per octave,
> >if more than one: twin, triple, quadruple, quintuple, 6-fold,
> >7-fold, ....
> >
> >And then the generator is described in terms of the simplest
> >n-odd-limit consonance (from the diamond) (or its octave inversion
> >or extension, as required). That is the one that takes the fewest
> >generators to approximate according to the LTs mapping.
> >
> >I use the following words if there is more than one generator to
> >the consonance: semi, tri, quarter, 5-part, 6-part, ....
>
> How do you choose a period/generator representation?

You don't have to. You just base it directly on the map -- any map
that's valid. i.e. the generator doesn't have to be in lowest
(period-reduced terms). And you don't have to decide on specific
optimum values of period and generator. That's the beauty of it.

For any of the ET/MOS/DE-based names you need to choose specific
values of period and generator. In most cases, different kinds of
optima do not change the period and generator enough to make much
difference, but I just found that while minimax and RMS versions of
5-limit Diminished can be described as 12&16-LT or 12&28-LT, the TOP
version cannot. It could be described as 12&20-LT or 12&32-LT but 20
and 32 are not 5-limit consistent, so the best you can do is 8&12-LT.

Similarly the minimax and RMS versions can be called 4/16-oct, 8/28-
oct, 12/40-oct, ... but the TOP version cannot. It has to be 4/12-
oct, 8/20-oct, 12/32-oct, .... Notice that I have not reduced these
fractions to lowest terms. This lets you extract the number of
periods per octave as the GCD of numerator and denominator.

Back to the map-based method.

The map tells you how many periods to the octave. That's all you
need to know about the period to know whether the temperament is
twin or triple etc.

For the rest of it, lets look at the simplest case first -- an LT
with one period to the octave, and one generator to some prime. i.e
there's a "1" (or a "-1") there staring at you from one of the
generators-per-prime slots.

If the 1 is in the prime-3 slot then it's "fourths". I'll say more
later about differentiating multiple temperaments having the same
name of generator.

If the 1 is in the prime-5 slot then it's "major thirds" or
just "thirds" (assuming the convention that if it's not explicitly
called minor or neutral or anything else, then it's major).

If the 1 is in the prime-7 slot then it's "supermajor seconds".
If the 1 is in the prime-11 slot then it's "super fourths".

If there is 1 or -1 generators to more than one prime then you give
them both, as in "fourth thirds".

If there is no 1 or -1 _directly_ as entries in the generator
mapping, then you look for two entries which differ by 1. e.g.
Kleismic has <0 6, 5].

If the difference of 1 generator is between 3's and 5's then
it's "minor thirds";
between 5's and 7's it's "augmented fourths";
between 3's and 7's it's "subminor thirds";
between 3's and 11's it's "neutral seconds";",
between 5's and 11's it's "narrow neutral seconds";
between 7's and 11's it's "narrow supermajor thirds".

It starts to get a bit hairy with those "narrow"s and "supermajor"s
and we might prefer to just give the approximated ratio. Or you
might prefer to give that every time.

If we're looking at 9 or 11-limit temps then we also have to double
the number in the 3's slot and see if anything differs from that by
1.

If the difference of 1 is between 9's and prime 5's it's "narrow
major seconds", although we could probably drop the "narrow" since
it unlikely we'd ever have a generator that approximates an 8:9;
between 9's and prime 7's it's "supermajor thirds";
between 9's and prime 11's it's "neutral thirds".

Again if there's more than one, list them all, e.g. "minor major
thirds".

If you can't find any 1's or differences of 1 then go looking for
2's or differences of 2, in exactly the same way, and put "semi" in
front of whatever you find. If no 2's then 3's and put "tri"
or "tripartite" or "3-part" (I used to say "tertia") in front. If
the n-limit diamond ratio with the fewest generators has 4
generators then put "quarter" in front. After that "5-part", "6-
part" etc.

But you also have to check that you're describing the correct octave
inversion or octave extension of the diamond ratio, so that when you
divide it into however many equal parts you really do get something
that is a valid generator. For example if you have 1 generator to
the prime-3 then you could call it fourths or fifths, but with 2
generators to the prime 3 then you have to check whether your
generator is a semifourth or semififth. Only one of those will be
correct. And with 4 generators to the prime-3, the generator might
even be a quarter eleventh or quarter twelfth.

When it comes to LTs with more than one period to the octave, you
have to be a little more careful. You the have to look at the
periods per prime as well as the generators per prime. You have to
ensure that, as well as having the minimum number of generators, the
diamond ratio being approximated has a number of periods which
corresponds to an integral number of octaves, i.e. that comes to
zero when taken modulo the number of periods in the octave.

I note that Erv Wilson uses this kind of terminology for LTs, at
least "semifourths" and "semififths".

🔗Carl Lumma <ekin@lumma.org>

At 06:37 PM 7/16/2004, you wrote:
>--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
>> >The name starts with a word for the number of periods per octave,
>> >if more than one: twin, triple, quadruple, quintuple, 6-fold,
>> >7-fold, ....
>> >
>> >And then the generator is described in terms of the simplest
>> >n-odd-limit consonance (from the diamond) (or its octave inversion
>> >or extension, as required). That is the one that takes the fewest
>> >generators to approximate according to the LTs mapping.
>> >
>> >I use the following words if there is more than one generator to
>> >the consonance: semi, tri, quarter, 5-part, 6-part, ....
>>
>> How do you choose a period/generator representation?
>
>You don't have to. You just base it directly on the map -- any map
>that's valid. i.e. the generator doesn't have to be in lowest
>(period-reduced terms).

So there'll be multiple names for each temperament?

>For any of the ET/MOS/DE-based names you need to choose specific
>values of period and generator. In most cases, different kinds of
>optima do not change the period and generator enough to make much
>difference, but I just found that while minimax and RMS versions of
>5-limit Diminished can be described as 12&16-LT or 12&28-LT, the TOP
>version cannot. It could be described as 12&20-LT or 12&32-LT but 20
>and 32 are not 5-limit consistent, so the best you can do
>is 8&12-LT.

I think that's why Gene is proposing to use ETs that represent the
extreme ranges of the generator.

>Back to the map-based method.
>
>The map tells you how many periods to the octave. That's all you
>need to know about the period to know whether the temperament is
>twin or triple etc.

What about temperaments that map 2 through a combination of both
the "period" and generator?

>For the rest of it, lets look at the simplest case first -- an LT
>with one period to the octave, and one generator to some prime. i.e
>there's a "1" (or a "-1") there staring at you from one of the
>generators-per-prime slots.
>
>If the 1 is in the prime-3 slot then it's "fourths". I'll say more
>later about differentiating multiple temperaments having the same
>name of generator.

Hmm... I thought one could refactor these maps is several
annoying ways. Thus, the reason for something called hermite
normal form -- whatever that is.

-Carl

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> > Right. But why _is_ this 11/27 and not 20/49?
>
> It could be either one: a 27-note scale with an approximate 11/27
> generator or a 49-note scale with an approximate 20/49 generator.
But a
> 27-note scale is simpler, and it's good to have an idea of the
simplest
> useful scale associated with a temperament. Note that in the 11-
limit
> these represent different temperaments:
>
> 11/27 [<1, 2, 6, 2, 1|, <0, -1, -9, 2, 6|]
> 20/49 [<1, 2, 6, 2, 10|, <0, -1, -9, 2, -16|]

It's not only different limits that cause this sort of problem, it's
also different optimisation criteria at the same limit. TOP and
minimax-beat-rate are two extremes that will sometimes give
generators so different as to correspond to different ET/MOS/DEs.
Any time the generator happens to be very close to some small-
denominator fraction of an octave this will be a problem, e.g. with
5-limit Diminished, as I mentioned in another post in this thread.

Is there any way we are ever going to agree on which octave fraction
is most representative of the temperament. It seemed like Gene was
nailing that down somewhat with p-optimal, but then along came TOP
(or was it copoptimal? -- I have no idea what that is).

> 4/46 gives you [<2, 3, 5, 7|, <0, 1, -2, -8|], not pajara (see
> http://x31eq.com/diaschis.htm).

Oops. Sorry. That should have been 8/46. But even so, that's only
good for "5-limit pajara" or diaschismic.

My question was whether we all agree we should use the smallest
possible value of the generator ( the one that's less than half the
period) in these octave-fraction-type names? I note that Paul is not
doing this in his paper when he gives generators in cents. He is
using whatever falls out of a simple algorithm for deriving the
mapping from a set of vanishing commas.

I note that the number of periods per octave can be obtained as the
GCD of numerator and denominator, since we won't be reducing an
octave fraction like 8/46.

> Pajara is historically associated with 22-ET, of course. But you
can
> think of the denominator as representing the size of a typical MOS
scale
> associated with the temperament, rather than an ET. In that case,
the
> minimum is 10 steps, which matches Paul's decatonic scale.

As I've said elsewhere, in the two-ET/MOS/DE method (the two-
cardinalities method?) of naming, I'd like the two numbers to give
the denominators of two convergents (or semi-convergents) of the
generator as an octave fraction, such that one is near the minimum
useful generator size and the other is near the maximum.

It would be ideal if you could also obtain the typical MOS/DE
cardinality by subtracting these two numbers, and obtain a good
approximation of an optimum generator by adding them.

For example, calling meantone the "12&19-LT" works perfectly. 12-ET
and 19-ET are very near the extreme generator values re "harmonic
waste". The typical MOS cardinality is 19-12 = 7, and a near optimum
occurs at 12+19 = 31-ET.

I think at least one of the two numbers should be a convergent, i.e.
it should give the cardinality of a Rothenberg-proper MOS/DE for
most optimum generator sizes.

> >> if you've heard of
> >>"superpyth", you take one look at the name and say "Ah yes,
*that*
> >>temperament". You might not immediately recognize "7/31"
> >
> > or "9&22", but
> >
> > Why wouldn't you immediately recognise one of these, if that's
what
> > you've been used to seeing it called.
>
> Combinations of numbers aren't especially easy to remember. It
would be
> like using ZIP codes to refer to cities in the US, instead of
names;
> they all look alike.

Don't you recognise your own zipcode and those of people you
regularly send mail to, and similarly phone numbers, although these
have many more digits than we're talking about here. It's more like
Australian postcodes, which are only four digits. But not even as
bad as that.

They are generally pairs of only 2 digit numbers taken from a small
set of n-limit consistent ETs, whose numbers already have many
associations for us. So it's really just associating pairs of
already familiar things that we're already used to representing as
numbers. And they have the enormous advantage that they are not
totally opaque jargon to a newcomer, as are names like sensipent,
orson, amity, subchrome, wurschmidt, compton.

At least when I see a postcode I've never seen before, I can
immediately tell what state it's in, and sometimes I can figure out
some towns I know that it must be near.

> >>probably anyone who's familiar with it knows the name "orwell".
> >
> > That's a tautology.
>
> The pronoun "it" refers to the temperament represented by "7/31"
and
> "9&22", which happens to be named "orwell". I think it's a fairly
safe
> assumption that most people who've heard of this temperament will
> recognize that name.

Only if they have been reading the tuning lists, and even then they
may only know that they seen the name but have no way of picking it
out of the jargon-diarrhoea that we're swimming in.

Imagine if we hadn't learnt that George Secor originally
discovered "Miracle", and George Secor turned up on the list for the
firs time now. How long would it take him to realise we were talking
about his temperament every time we write "miracle". compared to if
we were instead calling it the "31&41-LT" or the "7/72-oct-LT"?

> (Certainly "19/84" is more familiar, but it implies
> a greater degree of complexity, and could easily be overlooked by
people
> who don't care for highly complex scales.)

Yes. 84-ET is outside of most people's familiarity zone. But suppose
someone who had independently discovered that temperament turned up
on the lists. How long would it take them to figure out "orwell" as
opposed to "subminor thirds".

Gene complains that some of these descriptive names are "a
mouthfull". So what? How many times a day do you find yourself
having to say or type them? Anyway, what's a few extra keystrokes
for one person in exchange for a whole lot of extra understanding on
the part of a whole lot of readers.

> >>So you might as well use the name in combination with one
> >>or more of the other methods, especially if it's a less familiar
> >
> > one
> >
> >>like 3/8 (1/2) 10&16 lemba <<6, -2, -2, -17, -20, 1||.
> >
> >
> > So who needs the "lemba". It adds absolutely nothing, for me.
For
> > some reason it suggests "unleavened bread" to me. Huh?
>
> Anyone who's vaguely heard of it, but doesn't know much about it.
You
> might not know how to recognize a beech tree if you see one, or
how it
> differs from other trees, but you probably know that the
word "beech"
> represents a kind of tree. I don't know the mapping
of "nonkleismic" off
> the top of my head, and probably wouldn't recognize it if I saw it
(it's
> [<1, -1, 0, 1|, <0, 10, 9, 7|]), but I do recall it being (in
theory) a
> good temperament. "8/31" might give some idea of its usefulness,
but
> doesn't distinguish it from the many other "n/31" temperaments. So
even
> a questionable name like "nonkleismic" has some use. But "myna" is
> better because it links it with the starling family of
temperaments.

I have been looking at "myna" in Paul's table and I never once
associated it with "starling". To me it was just another random
assembly of syllables. And even if I had, the name starling gives me
no clues as to the identity of _that_ temperament.

> Along those lines, Gene's "Japanese monster" names also provide
useful
> hints to similarities between temperaments.

Sure, it's good to indicate similarities, but it still remains just
an isolated clump of related somethings, with no clue as to what
they are.

The point is, we can do a lot better. We can actually have names
that give someone a clue, even when they have not been initaiated
into the Smith-Erlich-Miller mysteries.

And in a situation where the names are a priori meaningless in
musical terms, and so any one is good as another, why the heck do
you guys have the need to keep changing them!!!!? It's tempting to
assume it's sheer arrogance or egotism, such as I was (more or less)
accused of when I wanted to use descriptive names in my
Microtempered Guitar article for Xenharmonikon. So I included the
cryptic/meaningless names as well, and now half of them are probably
obsolete.

> Why does anything have a name? Why do we talk about "major thirds"
when
> we could call them 5/4's?

In that case, I assume it's historical. But notice that in both
cases the term doesn't come from outer-space, but is descriptive,
using words/symbols with something like their existing meanings.

> Language works by naming things; the problem
> is that the study of linear and higher-dimensional temperaments is
so
> new that we haven't settled on the best names for things. So in
the
> meantime, names which may end up being changed will have to be
> supplemented by numerical keys of one kind or another.

I totally disagree. I don't see any point to using musically-
meaningless names or eponyms that may have to be changed, except for
the few most commonly discussed or used temperaments.

Paul Erlich recently pointed out that we use names for colours, not
wavelength numbers. And I responded that, while most people can
distinguish thousands of colours, they only use simple words for
about 20, and use combinations of these, and adjectives like dark
light etc.

> I'm leaning toward the fractional generator + period notation for
> unfamiliar temperaments, with wedgies for those few that can't
easily be
> symbolized in this way. But I still find names easier to remember,
and I
> don't want to discourage the naming of temperaments that look like
they
> might be useful.

OK. Well It seems we aren't that far apart in our thoughts on this,
but based on the colours thing, I wouldn't like to see non-
descriptive names for more than about the best 20. We've already got
more than 50, and we've only got to the 7-limit.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> At 06:37 PM 7/16/2004, Dave Keenan wrote:
> >> How do you choose a period/generator representation?
> >
> >You don't have to. You just base it directly on the map -- any
map
> >that's valid. i.e. the generator doesn't have to be in lowest
> >(period-reduced terms).
>
> So there'll be multiple names for each temperament?

No. The name ends up the same.

> >For any of the ET/MOS/DE-based names you need to choose specific
> >values of period and generator. In most cases, different kinds of
> >optima do not change the period and generator enough to make much
> >difference, but I just found that while minimax and RMS versions
of
> >5-limit Diminished can be described as 12&16-LT or 12&28-LT, the
TOP
> >version cannot. It could be described as 12&20-LT or 12&32-LT but
20
> >and 32 are not 5-limit consistent, so the best you can do
> >is 8&12-LT.
>
> I think that's why Gene is proposing to use ETs that represent the
> extreme ranges of the generator.

Sure, I like that idea, but what you consider extreme, depends on
what you consider optimal.

> >The map tells you how many periods to the octave. That's all you
> >need to know about the period to know whether the temperament is
> >twin or triple etc.
>
> What about temperaments that map 2 through a combination of both
> the "period" and generator?

Aw c'mon Carl. Gimme a break. :-)

None of the other methods mentioned in this thread (except giving
the full wedgie or a full mapping) can handle that either.

> Hmm... I thought one could refactor these maps is several
> annoying ways. Thus, the reason for something called hermite
> normal form -- whatever that is.

Sure, but it makes no difference, assuming we agree to use the
smallest of the equivalent generators (the one that's less than half
the period) and describe that generator as a fraction of the diamond
ratio that requires the fewest (absolute number of) generators (and
zero periods). Just as we describe the period as a fraction of the
octave.

But I certainly am assuming that the map has no generators in prime-
2, as is the case for 99.9% of maps we've ever talked about.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> Dave Keenan wrote:
> > e.g. Miracle is "semi 7:8's" or "semi supermajor seconds".
>
> This could be potentially useful up to a point; certainly there's
a
> mnemonic value in names like "semisixths". But I don't see how
this can
> be generalized to the 7-limit and higher without being arbitrary.
> Which
> LT gets to be called "fourths" -- dominant (5&12), meantone
(12&19),
> superpyth[agorean] (5&22), flattone (19&26), or schismic (12&29)?
You
> could make good arguments at least for dominant, meantone, and
schismic;
> then you need to figure out how to name the others. "Major thirds"
could
> be either muggles (16&19) or magic (19&22), and so on.

Right. This is where we use adjectives like wide and narrow applied
to the generator (but only where these don't imply a different n-
limit ratio entirely).

And the fallback method is to rank them by some badness measure
(probably most reasonable badness measures will agree on the ranking
of temperaments having the same ratio approximated by their
generator), and then the best one gets to have no adjective and the
others are called "complex", "supercomplex" or "inaccurate", "super-
inaccurate", as the case may be. Is their a shorter word
for "inaccurate"?

> > This is used up to some point where the LT is so complex you
just
> > describe the generator in cents. e.g. What used to be called
> > Aritoxenean is the 12-fold 15 cent LT.
> >
> > This at least works up to 11-limit.
>
> But giving the generator in cents doesn't determine a unique
mapping;
> you can derive one from a rational generator/period ratio if you
make
> some assumptions, but an arbitrary value in cents could represent
more
> than one temperament. An LT with a 316.5 cent generator can be
mapped as
> [<1, 0, 1, 2|, <0, 6, 5, 3|] or [<1, 0, 1, -3|, <0, 6, 5, 22|].
With
> rational generators and the naming conventions I've described, you
can
> unambiguously describe the first mapping as 5/19 and the second as
19/72.

I only proposed using cents when the temp is so complex that the
diamond ratio with the fewest generators has say 5 or more
generators in it. That's getting pretty complex. How many 12-fold 15-
cent temperaments do you know of?

🔗Herman Miller <hmiller@IO.COM>

Dave Keenan wrote:

> For any of the ET/MOS/DE-based names you need to choose specific > values of period and generator. In most cases, different kinds of > optima do not change the period and generator enough to make much > difference, but I just found that while minimax and RMS versions of > 5-limit Diminished can be described as 12&16-LT or 12&28-LT, the TOP > version cannot. It could be described as 12&20-LT or 12&32-LT but 20 > and 32 are not 5-limit consistent, so the best you can do is 8&12-LT.

The notation "12&16" doesn't need to be restricted to tunings with exact octaves; it's just a convenient shorthand for a temperament with map [<4, 6, 9|, <0, 1, 1|]. Certainly, to get from the name "12&16" to the map [<4, 6, 9|, <0, 1, 1|], you need to adopt the convention of exact octave ET's, but the values of the period and generator can be any values that are consistent with the map. But it seems that what you're saying is that 12+16 isn't an MOS scale in TOP diminished; it skips from 12+8 to 12+20. So I was clearly wrong about the MOS interpretation in this case. Well, once you get up to 12 notes TOP diminished has a better 2/1 at (3, 3), so you're dealing with an inconsistent temperament in any case; you might as well close it at 12 notes and call it a well- temperament.

> Similarly the minimax and RMS versions can be called 4/16-oct, 8/28-
> oct, 12/40-oct, ... but the TOP version cannot. It has to be 4/12-
> oct, 8/20-oct, 12/32-oct, .... Notice that I have not reduced these > fractions to lowest terms. This lets you extract the number of > periods per octave as the GCD of numerator and denominator.

"4/12" is a nice convention; I think I'll adopt it.

Apparently the generator/period ratio of 5-limit TOP diminished is somewhere around 0.33985, which puts it on an entirely different branch of the scale tree from the ET-based version of diminished with the largest g/p ratio, which is 12&16 (g/p = 0.285714). So essentially there are two different kinds of scales that fit the same temperament map; the one with a better JI approximation has a different scale structure from the pure octave-based one. I'm wondering if these different scale structures are similar enough to be given the same name....

At least I can see that it would be misleading to include TOP diminished in the category of "4&12" or "4/16"....

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@attglobal.net>

hi Dave and Herman (and everyone else),

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

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

> As I've said elsewhere, in the two-ET/MOS/DE method
> (the two-cardinalities method?) of naming, I'd like the
> two numbers to give the denominators of two convergents
> (or semi-convergents) of the generator as an octave fraction,
> such that one is near the minimum useful generator size
> and the other is near the maximum.
>
> <snip most of a long and interesting post>
>
> > I'm leaning toward the fractional generator + period
> > notation for unfamiliar temperaments, with wedgies for
> > those few that can't easily be symbolized in this way.
> > But I still find names easier to remember, and I
> > don't want to discourage the naming of temperaments
> > that look like they might be useful.
>
> OK. Well It seems we aren't that far apart in our
> thoughts on this, but based on the colours thing,
> I wouldn't like to see non-descriptive names for more
> than about the best 20. We've already got more than 50,
> and we've only got to the 7-limit.

i'm convinced that those of you who are putting forth
the argument against the cute verbal names are just
frustrated at the inability to keep up with the pace
of developments in tuning theory recently.

verbal names are really easy to remember, and short
concise numerical descriptions convey a lot of data,
so why not just use both?

i've always been a huge fan of redundant coding.
it makes life easy. so what if there is already one
name for something? natural linguistic processes are
always coining new names for old things.

i pursue the goal of including individual entries for
all these different temperaments and numerical descriptions
in the Encyclopaedia. that way a reader can simply look
up any name that's unfamiliar. i'm just buried with
work and haven't kept up with the lists for a long time
until recently, and would have to study a bit to be
able to write those pages.

if anyone else would like to contribute to this project,
please just post your efforts here and i'll make the
posts into webpages. i'll start the project by sending
a bunch of posts with the temperament names in the
subject lines.

if no-one objects to this, then by all means please
feel free to contribute more names as individual threads.
doing this will also coalesce a lot of related data
together for the archives of this list.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Herman Miller <hmiller@IO.COM>

Dave Keenan wrote:

>>11/27 [<1, 2, 6, 2, 1|, <0, -1, -9, 2, 6|]
>>20/49 [<1, 2, 6, 2, 10|, <0, -1, -9, 2, -16|]
> > > It's not only different limits that cause this sort of problem, it's > also different optimisation criteria at the same limit.

This isn't a "problem"; these really are different temperaments. It's only a problem when two different g/p ratios give the same temperament and you need to decide between them.

>>4/46 gives you [<2, 3, 5, 7|, <0, 1, -2, -8|], not pajara (see >>http://x31eq.com/diaschis.htm).
> > > Oops. Sorry. That should have been 8/46. But even so, that's only > good for "5-limit pajara" or diaschismic.

Ah; "pajara" and "diaschismic" are different 7-limit temperaments; I was under the impression that the 5-limit one has only been called "diaschismic" and not "pajara". But since pajara looks like the simplest (0, 1, -2) scale, it does make sense to use the same name for the 5-limit version.

> My question was whether we all agree we should use the smallest > possible value of the generator ( the one that's less than half the > period) in these octave-fraction-type names? I note that Paul is not > doing this in his paper when he gives generators in cents. He is > using whatever falls out of a simple algorithm for deriving the > mapping from a set of vanishing commas.

I think this is a reasonable convention.

> As I've said elsewhere, in the two-ET/MOS/DE method (the two-
> cardinalities method?) of naming, I'd like the two numbers to give > the denominators of two convergents (or semi-convergents) of the > generator as an octave fraction, such that one is near the minimum > useful generator size and the other is near the maximum. The issue here is quantifying "useful" in some easily defined / non-arbitrary way.

> It would be ideal if you could also obtain the typical MOS/DE > cardinality by subtracting these two numbers, and obtain a good > approximation of an optimum generator by adding them. > > For example, calling meantone the "12&19-LT" works perfectly. 12-ET > and 19-ET are very near the extreme generator values re "harmonic > waste". The typical MOS cardinality is 19-12 = 7, and a near optimum > occurs at 12+19 = 31-ET.
> > I think at least one of the two numbers should be a convergent, i.e. > it should give the cardinality of a Rothenberg-proper MOS/DE for > most optimum generator sizes.

Is that the "step size ratio <= 2:1" criterion?

> Don't you recognise your own zipcode and those of people you > regularly send mail to, and similarly phone numbers, although these > have many more digits than we're talking about here. It's more like > Australian postcodes, which are only four digits. But not even as > bad as that. I know my own ZIP code, but that's just 5 digits (if I live in one place long enough, I'll eventually get to the point where I can remember the full 9 digits). I always have to look up other ZIP codes and phone numbers. I keep my own phone number on a slip of paper so I'll have it if I need it; I don't need to fill it in on forms often enough to have it memorized yet. The point is that it's easier to recognize names, not that numbers are impossible to learn. The only ZIP codes I have memorized are the ones where I've lived.

> They are generally pairs of only 2 digit numbers taken from a small > set of n-limit consistent ETs, whose numbers already have many > associations for us. So it's really just associating pairs of > already familiar things that we're already used to representing as > numbers. And they have the enormous advantage that they are not > totally opaque jargon to a newcomer, as are names like sensipent, > orson, amity, subchrome, wurschmidt, compton.
> > At least when I see a postcode I've never seen before, I can > immediately tell what state it's in, and sometimes I can figure out > some towns I know that it must be near.

You could probably do the same with ZIP codes if you have large numbers of them memorized. You can roughly tell how far west a place is, at any rate. Or at least that seems to be the pattern. But I know that Topeka is in Kansas; I don't have a clue what its ZIP code might be.

>>assumption that most people who've heard of this temperament will >>recognize that name.
> > > Only if they have been reading the tuning lists, and even then they > may only know that they seen the name but have no way of picking it > out of the jargon-diarrhoea that we're swimming in.

It's possible that someone independently discovered the temperament we're calling "orwell", but would they be more likely to recognize it by the wedgie or the map? These conventions are also pretty much limited to the tuning lists and web sites of tuning list members. If anything, the generator size might give them the best clue. But we managed to figure out that Erv Wilson's "meta-mavila" was the same as what we'd been calling "pelogic"; differences in terminology aren't necessarily a problem.

> Gene complains that some of these descriptive names are "a > mouthfull". So what? How many times a day do you find yourself > having to say or type them? Anyway, what's a few extra keystrokes > for one person in exchange for a whole lot of extra understanding on > the part of a whole lot of readers.

I don't have any objection to supplementing the names with descriptions for those who are unfamiliar with the names. But if the descriptions themselves are used as a substitute for names, they could end up being confusingly similar.

> I have been looking at "myna" in Paul's table and I never once > associated it with "starling". To me it was just another random > assembly of syllables. And even if I had, the name starling gives me > no clues as to the identity of _that_ temperament.

The "Japanese monster" names weren't immediately apparent to me either, but once pointed out, they have mnemonic value.

> The point is, we can do a lot better. We can actually have names > that give someone a clue, even when they have not been initaiated > into the Smith-Erlich-Miller mysteries.
> > And in a situation where the names are a priori meaningless in > musical terms, and so any one is good as another, why the heck do > you guys have the need to keep changing them!!!!? It's tempting to > assume it's sheer arrogance or egotism, such as I was (more or less) > accused of when I wanted to use descriptive names in my > Microtempered Guitar article for Xenharmonikon. So I included the > cryptic/meaningless names as well, and now half of them are probably > obsolete.

There are so many temperaments described in these huge lists of wedgies and maps that it's easy to forget (or be unaware) that one of them already has a name; I suspect this might be what happened with "bug" vs. "beep". Sometimes the name wasn't a very good one to begin with, and a better one comes along, as with "pelogic" vs. "mavila". Sometimes there never really was any agreement on a name to begin with, as in the case of 12&60. Then the whole issue of whether 5-limit and 7-limit temperaments should have the same name came up. But other times a perfectly good name like "tripletone" seems to have been set aside in favor of one that isn't really any better. I'd just as soon keep "tripletone".

> OK. Well It seems we aren't that far apart in our thoughts on this, > but based on the colours thing, I wouldn't like to see non-
> descriptive names for more than about the best 20. We've already got > more than 50, and we've only got to the 7-limit.

I agree that most temperaments don't need specific names. It's probably not a good idea to name a temperament without even hearing it. I probably shouldn't have named "grackle" (12&77); I don't even know if it's any good. But I do think that anything that looks promising should at least get a provisional name of some kind to distinguish it from the crowd of identical-looking but very different temperaments.

In my case, I'm also trying to build a music theory for my fictional culture, which has been developing scales and temperaments for thousands of years. I don't intend for any of these fictional names to replace existing ones, but in the case of "lemba", I don't know of an existing name for this temperament, so I just use the Yasaro name.

Can you tell which of these are good temperaments?

[<1, 2, 2, 3|, <0, -4, 3, -2|]
[<1, 2, 3, 3|, <0, -2, -3, -1|]
[<1, 2, 1, 5|, <0, -1, 3, -5|]
[<3, 5, 7, 9|, <0, -1, 0, -2|]
[<1, 1, 2, 4|, <0, 2, 1, -4|]
[<2, 3, 5, 6|, <0, 1, -2, -2|]
[<2, 4, 5, 6|, <0, -2, -1, -1|]

I think that even short lists like this are confusing without some way to recognize individual temperaments. I can only keep track of them by consulting a list with names and searching for them. I've ignored many posts full of numbers because it's too much trouble to find out if there's anything interesting in there.

🔗Herman Miller <hmiller@IO.COM>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> i'm convinced that those of you who are putting forth
> the argument against the cute verbal names are just
> frustrated at the inability to keep up with the pace
> of developments in tuning theory recently.

Frustrated? Yes. At the pace of developments in tuning theory? No.

New algorithms and optimisation schemes for finding good linear
temperaments are advances in tuning theory. But using those
algorithms to spew hundreds of temperaments onto the list and giving
most of them musically-meaningless names, is not.

I think Graham had the right idea. Catalog the most interesting ones
(35 of them up to 15-limit) and put the software up on the web so
anyone can find others to fit their own specifications.
http://x31eq.com/catalog.htm
http://x31eq.com/temper/

I don't see any of the names since Graham's catalog as being an
improvement.

> verbal names are really easy to remember,

Sure, I can remember some of the musically-meaningless names. But I
haven't a clue what they refer to. But verbal names don't have to be
musically-meaningless.

> and short
> concise numerical descriptions convey a lot of data,
> so why not just use both?

That would be wonderful. But won't Gene just complain that that's
even more of a "mouthful" than just using descriptive verbal names
like "subminor thirds"?

Monz,

Even when you put together a list explaining all of the cute names,
it won't help someone joining the list cold. Imagine if Erv Wilson
decided to start reading the list. Apart from names that we've
adopted from his work, and the few descriptive ones that have
somehow survived (although Paul's working on remedying that
situation, with nonsense like sensipent, dimipent, srutal, semaphore
and augene to replace semisixths, diminished, diaschismic,
semifourths and augmented), Erv would have trouble even realising
that we were talking about linear temperaments, let alone know where
to look them up, unless everyone agrees to include a link to your
page in every post (or at least the first post of every thread) that
uses musically-meaningless names.

It doesn't have to be that way. Names can be verbal _and_ musically-
meaningful.