Yahoo Tuning Groups Ultimate Backup tuning Creating Scala scl files for rank two temperaments

Chris wrote:

> So.. we are talking about overlaying two EDOs - capital idea!

That's one possible view on 2D tunings, not actually one I prefer. Graham's scripts allow you to find one particular temperament either by combining two EDOs or by specifying which interval should vanish in that particular temperament, depending on whether you choose "Temperament class from ETs" or "Unison vector search". Defining a temperament by the vanishing interval is a unique definition, tells you more about its possible application in actual music, and doesn't meaninglessly suggest larger scales than what's needed to actually use the temperament (which the ET search does if torsion occurs).

For example, I can get meantone by combining 7-EDO and 12-EDO but I can get exactly the same by combining 12-EDO and 19-EDO so it's up to me whether I call it "7&12" or "12&19". What's more, this hardly tells me anything else than the fact that 7-EDO and 12-EDO are both usable as meantone (unless I had such good imagination to understand what 7-EDO and 12-EDO have in common).

If I wanted to define meantone, the most important fact for me (from a "musical" point of view) would probably be that a major third should be represented by 4 fifths minus 2 octaves or that a major sixth should be represented by 3 fifths minus an octave or something like that -- because, well, that's what makes it meantone. So, in 5-limit JI, a pure major third is 5/4, 4 pure fifths minus 2 octaves are 81/64, and therefore the interval which turns into unison in the resulting temperament is 81/80 -- i.e. (81/64) / (5/4). Similarly, a pure major sixth is 5/3, 3 fifths minus an octave are 27/16, and therefore the vanishing interval is again 81/80.
If you then figure out the interval size for the period and the generator (which is exactly what Graham's "Unison vector search" does after you enter the vanishing interval), you get a temperament in which this is true.

However, the fact that "a major third is the same as 4 fifths minus 2 octaves" is not a "given thing" which should always be true, it's something which people "agreed on" during the 16th century. So you can actually take any target interval and represent it by a number of other intervals, which is what offers you the possibility of new harmonic systems and so on. That's why I've given an example of hanson in my last message to show you what I mean. In hanson, the intervals which sounds like a "fifth" is represented by 6 "sounding minor thirds" minus an octave (I say "sounding" because their meanings in hanson are not quite the same). So, a fifth is 3/2, 6 minor thirds minus an octave are 23328/15625, and therefore the interval which turns into unison here is 15625/15552. If you go to Graham's "Unison vector search" and enter "15625/15552", you get a temperament which uses minor thirds as generators and which approximates 3/1 by stacking six of these.

The reason why I was asking you about 34-EDO was that I deliberately chose an EDO which has good approximations to JI but isn't usable as meantone. If you never get to know something else than JI or meantone, you'll only understand 34-EDO as "another way to approximate JI" since "it doesn't make meantone. But once you start discovering other temperaments, sooner or later, you notice that, for example, approximating 3/1 in 34-EDO takes exactly 6 times more steps than approximating 6/5 and therefore 34-EDO can be used as hanson. Interestingly enough, 19-EDO is the one which can be used both as meantone and as hanson.

Or another example.
Let's say you're trying to find a tuning where 4 "seconds" make a usable fifth and where one "second" is close to the 5-limit "narrow major second" of 10/9 (so that you could get a major sixth using 5 of them and a major tenth using 9 of them). So, a fifth is 3/2, 4 narrow major seconds are 10000/6561, and therefore the interval which vanishes in the resulting temperament is 20000/19683. If you put this into Graham's "Unison vector search", you get a temperament where this is true -- i.e. tetracot.
Interestingly enough, in 34-EDO, 3/2 is approximated by 4 times more steps than 10/9, which means that not only can 34-EDO be used as hanson but it can also be used as tetracot -- you know what, that's why I've picked an example of tetracot. :-D

In message #90305, I've written the shortest understandable description of "period&generator mapping" I could probably manage, so let's hope that helps.

Petr

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Chris wrote:
>
> > So.. we are talking about overlaying two EDOs - capital idea!
>
> That's one possible view on 2D tunings, not actually one I prefer.

For any rank of temperament, it has the advantage that you only need to stick in as many numbers as the rank. For five limit, which you talk about, that's still two whereas the comma approach only involves one comma, big advantage commas. For 7-limit temperaments they both involve sticking in two values. But now it gets ugly: three commas for the 11-limit, four for the 13-limit, five for the 17-limit, six for the 19-limit. And all the commas must be independent!

Defining a temperament by the vanishing interval is
> a unique definition, tells you more about its possible application in actual
> music, and doesn't meaninglessly suggest larger scales than what's needed to
> actually use the temperament (which the ET search does if torsion occurs).

If so, it's only because Graham wants to do it that way. I never allow this to happen when I do this sort of thing. In fact I normally route the whole thing though wedgies, which gets rid of torsion (or contorsion, as we call it, in this case) automatically.

🔗Petr Parízek <p.parizek@...>

Gene wrote:

> For any rank of temperament, it has the advantage that you only need to > stick in as many numbers as the rank.

Understood. But this method of finding 2D tunings seems to be more based on the "mistuning-like" point of view and it's highly debatable whether something like this can reveal their characteristic musical possibilities to a composer whose aim is to primarily find new harmonic progressions and not to retune existing stuff. If I were as "unfamiliar" as I was 4 years ago and you offered me 34-EDO and 19-EDO without suggesting anything, I think I would just see how "mistuned" their 2|3|5 approximations are and then simply add the numbers to get 53-EDO. But I would probably not realize what 34-EDO and 19-EDO have in common and therefore get hanson (unless I had some additional "mathematical thoughts" flowing in my head, which I certainly didn't have until I first tried to temper out something else than 81/80 within the 5-limit framework).

> For five limit, which you talk about, that's still two whereas the comma > approach only involves one comma,
> big advantage commas. For 7-limit temperaments they both involve sticking > in two values. But now it gets ugly:
> three commas for the 11-limit, four for the 13-limit, five for the > 17-limit, six for the 19-limit.

This brings up the question what's the best way to discover the proper meaning of particular temperaments from a composer's point of view. For me it's pretty logical that the more dimmensions I begin with (for the original untempered system), the more numbers I need for the mapping if I want to get a 2D result. If I don't stick to the requirement of full prime limit approximations, I can temper out 3136/3125 within the 2|5|7-limit space; if I do, I can either simply "clamp the 3 on that" and get a 3D result or find another comma to temper out and get a 2D result. But I think that some unison vectors best reveal their "musical possibilities" if you temper out only one at a time. For example, last year, I made a short piece in a 3D tuning which tempers out 4000/3993 within 2|3|5|11-limit JI. Now I'm glad I wasn't tempering out anything else back then. Similarly, octacot or miracle may be fine, but the actual 3D version of "breed" suggest an even more efficient way to use the "breedsma's" possibilities, which one may not think of in the 2D tunings in the first place. With 6144/6125 it's similar.

> And all the commas must be independent!

I don't see anything negative in this. It's like saying that you can get a 7-limit 3D temperament by combining 12 and 19 and 34-equal but not 12 and 19 and 50-equal.

Petr

🔗Chris Vaisvil <chrisvaisvil@...>

It seems perhaps you are changing programs midstream?

I follow your instructions and go here

http://x31eq.com/temper/net.html

and enter 22 and 7

I get this

2 3 5 7
[< 22 35 51 62 ]>

Tuning Map (cents)
<1198.197, 1906.223, 2777.640, 3376.738]

Adjusted Error 7.989953 cents
TOP-RMS Error 2.846079 cents/octave
TOP-RMS Stretch -1.802568 cents/octave

temperament finding scripts

And I do not find

In the "Generator Tunings (cents)" box, ..... Otherwise, go to where
it say "Reduced Mapping",

But I see something that resembles

and look at the top row; in this case it will be [<2 3 5 6]. ....

But the values are different.

May I suggest to include an example scala file to use as confirmation
that the instructions were followed correctly.

On Sat, Jun 12, 2010 at 8:37 PM, genewardsmith
<genewardsmith@...> wrote:
>
>
>
> See if this is clear enough:
>
> http://xenharmonic.wikispaces.com/Creating+Scala+scl+files+for+rank+two+temperaments

🔗genewardsmith <genewardsmith@...>

🔗Graham Breed <gbreed@...>

On 13/06/2010, Petr Parízek <p.parizek@...> wrote:
> Gene wrote:
>
>> For any rank of temperament, it has the advantage that you only need to
>> stick in as many numbers as the rank.
>
> Understood. But this method of finding 2D tunings seems to be more based on
> the "mistuning-like" point of view and it's highly debatable whether
> something like this can reveal their characteristic musical possibilities to
> a composer whose aim is to primarily find new harmonic progressions and not
> to retune existing stuff. If I were as "unfamiliar" as I was 4 years ago and
> you offered me 34-EDO and 19-EDO without suggesting anything, I think I
> would just see how "mistuned" their 2|3|5 approximations are and then simply
> add the numbers to get 53-EDO. But I would probably not realize what 34-EDO
> and 19-EDO have in common and therefore get hanson (unless I had some
> additional "mathematical thoughts" flowing in my head, which I certainly
> didn't have until I first tried to temper out something else than 81/80
> within the 5-limit framework).

Well, you can't please everybody, can you? A composer whose primary
aim is to find new melodic patterns will look at the MOS family first.
A pair of equal temperaments specifies that.

Why would anybody offer you two equal temperaments without suggesting
anything else? It sounds like a bizarre initiative test.

> This brings up the question what's the best way to discover the proper
> meaning of particular temperaments from a composer's point of view. For me
> it's pretty logical that the more dimmensions I begin with (for the original
> untempered system), the more numbers I need for the mapping if I want to get
> a 2D result. If I don't stick to the requirement of full prime limit
> approximations, I can temper out 3136/3125 within the 2|5|7-limit space; if
> I do, I can either simply "clamp the 3 on that" and get a 3D result or find
> another comma to temper out and get a 2D result. But I think that some
> unison vectors best reveal their "musical possibilities" if you temper out
> only one at a time. For example, last year, I made a short piece in a 3D
> tuning which tempers out 4000/3993 within 2|3|5|11-limit JI. Now I'm glad I
> wasn't tempering out anything else back then. Similarly, octacot or miracle
> may be fine, but the actual 3D version of "breed" suggest an even more
> efficient way to use the "breedsma's" possibilities, which one may not think
> of in the 2D tunings in the first place. With 6144/6125 it's similar.

If you want to do that, you can do that. You can also treat 24&31 as
a contorted 5-limit system, like Vicentino did, and add higher limits
in later. There's no way to specify that with a unison vector, like
you can't specify torsional periodicity blocks with equal
temperaments.

>> And all the commas must be independent!
>
> I don't see anything negative in this. It's like saying that you can get a
> 7-limit 3D temperament by combining 12 and 19 and 34-equal but not 12 and 19
> and 50-equal.

With my web application, in fact, the commas do not need to be
independent. That's because it works by filtering equal temperaments.
Of course, you could put linearly dependent commas aside as well.

There are problems, though. There isn't a unique choice of unison
vectors -- exactly the same as there isn't a unique choice of equal
temperament mappings. The two views are dual to each other. It's
also difficult to get more than three unison vectors without torsion.
Which is to say, I still don't know how to do it.

So, yes, if you want to start with unison vectors you can do that.
But representing temperament classes by linearly independent sets of
unison vectors is a pain. Equal temperament mappings are easier to
work with.

Graham

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> >> And all the commas must be independent!
> >
> > I don't see anything negative in this. It's like saying that you can get a
> > 7-limit 3D temperament by combining 12 and 19 and 34-equal but not 12 and 19
> > and 50-equal.
>
> With my web application, in fact, the commas do not need to be
> independent. That's because it works by filtering equal temperaments.

They certainly do need to be independent if you expect to get a unique rank two temperament as a result.

It's
> also difficult to get more than three unison vectors without torsion.
> Which is to say, I still don't know how to do it.

One approach is to use wedgies and remove the GCD. Alteratively, Hermite reduce the unison vectors and divide each one by its GCD, removing powers.

🔗Graham Breed <gbreed@...>

On 14 June 2010 23:51, genewardsmith <genewardsmith@...> wrote:

>> >> And all the commas must be independent!

> They certainly do need to be independent if you expect to get a unique rank two temperament as a result.

Go to http://x31eq.com/temper/uv.html and put 243:242, 441:440,
385:384, 225:224, and 2401:2400 in the box. Press the button. See
how many rank two temperaments come out.

> It's
>> also difficult to get more than three unison vectors without torsion.
>> Which is to say, I still don't know how to do it.
>
> One approach is to use wedgies and remove the GCD.
> Alteratively, Hermite reduce the unison vectors and divide
> each one by its GCD, removing powers.

Yes, you've been talking about wedgies for years, but never explained
where the torsion goes.

How are you defining Hermite reduction? I've trawled the web looking
for clear, consistent, relevant definitions and failed. The best I
can come up with doesn't remove torsion. There's no guarantee each
vector has the right GCD to divide by.

Graham

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> > They certainly do need to be independent if you expect to get a unique rank two temperament as a result.
>
> Go to http://x31eq.com/temper/uv.html and put 243:242, 441:440,
> 385:384, 225:224, and 2401:2400 in the box. Press the button. See
> how many rank two temperaments come out.

That's five commas. I specified three commas.

> > One approach is to use wedgies and remove the GCD.
> > Alteratively, Hermite reduce the unison vectors and divide
> > each one by its GCD, removing powers.
>
> Yes, you've been talking about wedgies for years, but never explained
> where the torsion goes.

Where it goes?? It doesn't go anywhere other than away, obviously.

> How are you defining Hermite reduction? I've trawled the web looking
> for clear, consistent, relevant definitions and failed.

What's wrong with the discussion on Wikipedia?

The best I
> can come up with doesn't remove torsion. There's no guarantee each
> vector has the right GCD to divide by.

I think the chances would be much improved.

🔗Graham Breed <gbreed@...>

On 15 June 2010 10:05, genewardsmith <genewardsmith@...> wrote:

> That's five commas. I specified three commas.

You did, so that's two ways you got it wrong.

>> Yes, you've been talking about wedgies for years, but never explained
>> where the torsion goes.
>
> Where it goes?? It doesn't go anywhere other than away, obviously.

It doesn't go away. Every algorithm I have to get the unison vectors
out of the wedgie introduces torsion.

>> How are you defining Hermite reduction? I've trawled the web looking
>> for clear, consistent, relevant definitions and failed.
>
> What's wrong with the discussion on Wikipedia?

It wasn't there the last time I checked this and it contradicts other
definitions I found.

> The best I
>> can come up with doesn't remove torsion. There's no guarantee each
>> vector has the right GCD to divide by.
>
> I think the chances would be much improved.

I don't care about that. I want a reliable algorithm.

Graham

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 15 June 2010 10:05, genewardsmith <genewardsmith@...> wrote:
>
> > That's five commas. I specified three commas.
>
> You did, so that's two ways you got it wrong.

There are no doubt uses for the method of firing at random into the dark and hoping to hit what you are looking for, in this case a rank two temperament, but it's not something I would find useful for my purposes. If you do chose to do this, you might not get the sort of thing you are looking for, and in fact you might not get anything at all. If you do get something, it could have an accuracy lower than what you might expect from the commas you fed in.

In any case, my search philosophy is just a different philosophy than yours. That does not make it wrong, if you intended that comment seriously.

> >> Yes, you've been talking about wedgies for years, but never explained
> >> where the torsion goes.
> >
> > Where it goes?? It doesn't go anywhere other than away, obviously.
>
> It doesn't go away. Every algorithm I have to get the unison vectors
> out of the wedgie introduces torsion.

I can't imagine what you are doing that could possibly reintroduce torsion after taking out the GCD. What you complain of seems to be a mathematical impossibility. I suggested you show us an example of a wedgie which gives you a torsion problem.

🔗genewardsmith <genewardsmith@...>

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

> > It doesn't go away. Every algorithm I have to get the unison vectors
> > out of the wedgie introduces torsion.
>
> I can't imagine what you are doing that could possibly reintroduce torsion after taking out the GCD. What you complain of seems to be a mathematical impossibility. I suggested you show us an example of a wedgie which gives you a torsion problem.

Sorry, I didn't read this carefully. What you are saying is that you want a way of getting "unison vectors" (ie, commas of the temperament) out of the wedgie. You can do that by getting first what I've called both the subgroup commas or triprime commas of you wedgie. You now have a big pile of commas, and while an individual comma may be a power, the set as a whole will reduce to a list with no torsion problems.

🔗Graham Breed <gbreed@...>

On 15 June 2010 15:11, genewardsmith <genewardsmith@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>> > It doesn't go away. Every algorithm I have to get the unison vectors
>> > out of the wedgie introduces torsion.
>>
>> I can't imagine what you are doing that could possibly reintroduce torsion after taking out the GCD. What you complain of seems to be a mathematical impossibility. I suggested you show us an example of a wedgie which gives you a torsion problem.
>
> Sorry, I didn't read this carefully. What you are saying is that
> you want a way of getting "unison vectors" (ie, commas of the
> temperament) out of the wedgie. You can do that by getting
> first what I've called both the subgroup commas or triprime
> commas of you wedgie. You now have a big pile of commas,
> and while an individual comma may be a power, the set as a
> whole will reduce to a list with no torsion problems.

Yes, that's what I've been doing the whole time. It gives exactly the
same results as using the same "commas" without using wedgies once
you've removed torsion from the inputs. And it certainly does give
torsion problems. It's what I was doing before you showed up and
started claiming wedgies were the solution.

I remember Mystery caused some problems. In the 7-limit it gives an
interval-space wedgie of [[14, 46, -46, -29, 29, 0>>.

I think your triprime commas (with four primes) are

{[0, 14, 46, -46>, [-14, 0, -29, 29>, [-46, 29, 0, 0>, [46, -29, 0, 0>}

The first of those is a power, so

{[0, 7, 23, -23>, [-14, 0, -29, 29>, [-46, 29, 0, 0>, [46, -29, 0, 0>}

These are riddled with torsion. The 5 and 7 elements are always 29
except in the first one. The octave elements are always even numbers.
The first two (which are what I used to take) have a 7 in the 3
element.

Now, "reduce to a list" you say. If I take all these commas, and
apply my Hermite-like reduction, I get:

{[2, 3, 14, -14>, [0, 7, 23, -23>, [0, 0, 0, 0>, [0, 0, 0, 0>}

This gives the right result, so maybe it doesn't have torsion. And in
that case, maybe applying something like Hermite reduction to the
whole set of commas after removing powers is a valid algorithm. I'll
have to check it.

Graham

🔗genewardsmith <genewardsmith@...>

🔗Chris Vaisvil <chrisvaisvil@...>

Hi Gene,

Re: the xenwiki page for making scala files
http://xenharmonic.wikispaces.com/Creating+Scala+scl+files+for+rank+two+temperaments

I get to here

Otherwise, go to where it say "Reduced Mapping", and look at the top
row; in this case it will be [<2 3 5 6]. Take the first number, in
this case 2. This is the number of periods in an octave, call it "n".
If "P" is the first number, the period, and "G" is the second number,
the generator, then instead of P and G you may use 1200.0/n and
(1200.0G/nP) for the period and generator, which will give pure
octaves. In this case, we get 600.0 for the period and 107.48 for the
generator.

And this doesn't seem to get the approximately same result

This part is totally clear

Generator Tunings (cents)
[598.859, 106.844>

now in your example 1200/n = 600 - that makes sense

but then 1200(3/n2) - this becomes confusing - it appears the n=P ??
If "P" is the first number, the period

then correctly 1200*3/(2*2)) = 3600/4 = 900?

and n is the first number - Take the first number, in this case 2.
This is the number of periods in an octave, call it "n".

It seems the definition is combining both examples which are presented
as an "or" case?

Thanks Chris

🔗genewardsmith <genewardsmith@...>

🔗Graham Breed <gbreed@...>

On 16 June 2010 00:23, genewardsmith <genewardsmith@...> wrote:

> You are doing things backwards according to my way of working it, which is to reverse the monzos, then Hermite (or "Hermite-like", I suppose) reduce, and then reverse again. Gives better results. In this case, my "normal comma list" goes
>
> [<46 -29 0 0|, <10 -6 1 -1|]
>
> No torsion problems remain; the first is the 29-comma, and the second is 5120/5103, and together they are pretty informative about the temperament.

That's the way of getting simple ratios, yes. But I have a Tenney
reduction function that can take over once the torsion's removed.

And -- yes! -- this algorithm does give me torsion-free unison vectors
for every 7-prime limit class I could throw at it. Unfortunately I
only know how the "subgroup commas" are defined for this case.

It must depend on the algorithm used to do the reduction. It's
possible to use elementary row operations to remove the extra commas
and give torsion. So although it's nice to know my code's working I'd
like to know why.

Graham

🔗Chris Vaisvil <chrisvaisvil@...>

ok, but then what is the [<2 3 5 6] for?

On Tue, Jun 15, 2010 at 11:48 PM, genewardsmith <genewardsmith@...
> wrote:

>
>
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
>
> > This part is totally clear
> >
> > Generator Tunings (cents)
> > [598.859, 106.844>
> >
> > now in your example 1200/n = 600 - that makes sense
> >
> > but then 1200(3/n2) - this becomes confusing - it appears the n=P ??
> > If "P" is the first number, the period
>
> The first number in the box labeled "Generator Tunings (cents)" is
> 598.859, so that is "P". The second number is 106.844, which is "G".
>
>
>

🔗Petr Parízek <p.parizek@...>

Hi Chris.

Don't know how helpful this is but I'll try to jump in here.

> ok, but then what is the [<2 3 5 6] for?

Gene has given an example tuning which uses a half-octave as a period and a slightly wider minor second as a generator.
In this particular temperament, you need 2 periods (and no generators) to approximate the 2/1; you need 3 periods and 1 generator to approximate the 3/1; you need 5 periods minus 2 generators to approximate the 5/1; and you need 6 periods minus 2 generators to approximate the 7/1. Therefore, the respective period mapping is "2, 3, 5, 6" while the generator mapping is "0, 1, -2, -2". Or if you read the numbers by columns instead of lines, then you get a mapping of "2, 0" for 2/1, "3, 1" for 3/1, "5, -2" for 5/1, and "6, -2" for 7/1.

Does this answer your question?

Petr

🔗genewardsmith <genewardsmith@...>

🔗Chris Vaisvil <chrisvaisvil@...>

I'm thinking that perhaps it is unnecessary to include in the directions at
this point?
I get lost because the variable definitions become mixed - perhaps sticking
with just one set of numbers through as a first pass will be clearer - and
then giving variations after the first clear transition from rank two data
to scala is complete?

And to answer Petr

Reduced Mapping

2357
[<2356] <01-2-2]>
May I humbly suggest that for a page like this - which is auto-generated -
stating

Reduced Mapping

2357 - is this the prime limits for the tuning???

Periods = [<2356] number of generators to adjust corresponding periods by<01
-2-2]>
The bare numbers are difficult to comprehend because you have to know the
correct definitions and keep those definitions in mind - just like reading a
paper in another field than your own that uses a large number of acronyms.
This makes for a steeper learning curve.

I understand the need for the shorthand for the hard core tune smiths - on
the other hand life would be simpler for the rest of us in these situations
where it is only a one time "write it out in full".

Chris

On Wed, Jun 16, 2010 at 12:50 PM, genewardsmith <genewardsmith@...
> wrote:

>
>
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
> >
> > ok, but then what is the [<2 3 5 6] for?
>
> It's part of the mapping for the generator pair, which in this case I was
> only using to tell how many periods were in an octave.
>
>
>

🔗Petr Parízek <p.parizek@...>

Chris wrote:

> 2 3 5 7 - is this the prime limits for the tuning???

Yes, if you enter a single number into the "limit" field, the targets are successive primes.

> Periods = [<2 3 5 6]
> number of generators to adjust corresponding periods by <0 1 -2 -2]>

Correct.

> The bare numbers are difficult to comprehend because you have to know the
> correct definitions and keep those definitions in mind - just like reading > a
> paper in another field than your own that uses a large number of acronyms.
> This makes for a steeper learning curve.

In my case, I first discovered the unusually interesting and "prospective" phenomenon of 2D tempering without actually knowing the names of particular temperaments. Later I realized that I had actually rediscovered semisixths, hanson, orwell, and tetracot (before, I really thought I had found completely new temperaments, which then turned out to be false). Then some people here suggested Paul Erlich's "Middle Path" paper to me, then Graham mentioned the theses he wrote, Herman made an "introductory" webpage about rank 2 temperaments, then I managed to find some older conversation of Gene and Paul on in the "Tuning math archive", and finally I got so impressed by what I could do with it that I began to stay in this stuff day and night for more than a year.

Petr

Creating Scala scl files for rank two temperaments

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