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Message: 3675 - Contents - Hide Contents

Date: Fri, 1 Feb 2002 22:30:31

Subject: Re: interval of equivalence, unison-vector, period

From: Graham Breed

Me:

>> Where am I going wrong?

Paul:

> I'm not saying you're wrong, only that your methods are different
> from Gene's -- most recently exemplified with the case that he
> considered "not a temperament" and you considered "22-tET".


I meant there where I was wrong with Gene's terminology.  I didn't call the 
thing he called "not a temperament" 22-tET.  I called it 
paultone/twintone/pajara.  Because that's what it is.  It's actually a 
mapping of 0-tET.  You could call it a paradox that something with no notes 
counts as a temperament.  You could then analyse the assumptions that led to 
it instead of shouting back "you're wrong" at the person who pointed it out.

I haven't yet seen that my methods are different from Gene's at all.  I've 
actually adopted wholesale his stuff about wedgies, so far as I understand 
it.  All we disagree on is interpretation.  What he's saying is pretty much 
where I started at <Linear temperaments from matrix formalism * [with cont.]  (Wayb.)> anyway.  Two 
generators (I didn't know the word then) which can be any size, and a number 
of commas which approximate to unisons.


                              Graham

Message: 3676 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 06:55:04

Subject: Re: interval of equivalence, unison-vector, period

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> 

>> And this could happen just as well for a group with a prime 
number of 
>> elements, such as {2, 25/24, 81/80} -> C7.
> 
> Yes, indeedy.
> 

>>> so this is rather different than a block with torsion elements.
>> 

>> Yes it is. Now we really need to revise the definition of 
torsion :(, 
>> and think of different names for these two things.
> 

> Why do we need to worry about it? 


For the sake of Monz' dictionary, perhaps?


>> Can you go into this in more detail, pretty please with sugar on 
top?
> 

> I'm not sure what you are asking for, so let's see if this does it:
> 
> The MT reduced basis for 22 et in the 7-limit is 
> {50/49, 64/63, 245/243}. If I take these in pairs and wedge them, I 

get three temperaments instead, which can also be thought of as a 
defining basis for 22-et:

> 
> 50/49^64/63 = [-2,4,4,-2,-12,11] -- twintone
> 
> 50/49^245/243 = [6,10,10,-5,1,2] with generators
> a = 3.0143/22 = 164.4176 cents and b = 1/2

Glassic
> 
> 64/63^245/243 = [1,9,-2,-30,6,12] with generators a = 8.9763/22 =
> 489.6152 cents and b = 1


"Big fifth" -- a unique facet of 22


>I can now wedge these with 2, and get triple wedge products. A 
>triple wedge product of three intervals will be a val, but it 
>doesn't have to be an equal temperament val.


What other kinds are there?

> 
> 50/49^64/63^2 = [0,2,-4,-4]
> 
> 50/49^245/243^2 = [0,-6,-10,-10]
> 
> 64/63^245/243^2 = [0,-1,-9,2]
> 
> This is giving us the non-octave part of the generator map. We 

could also wedge with other intervals of equivalence besides 2, and 
get what the corresponding temperament would be then; for instance

> 
> 50/49^64/63^3/2 = [-2,-2,-7,-8]
> 
> We can then use this mapping to primes (or [2,2,7,8], which seems 
>nicer and which a different order of the triple product would have 
>given us) to define a version of this temperament based on the fifth 
>as an interval of equivalence.


OK . . .


>>> This is *not* a temperament, or at least not one I'm interested 
in 
>>> hearing, so 2 is not acting as a unison, which is hardly a 
surprise.
> 

> Mapping 2 to 1, and both 5 and 7 to 1/9 does not strike me as much 

of a temperament.

Well . . . I'm lost . . . does this have anything to do with what you 
were once showing about your process, where for a "linear" or 2D 
temperament, you started off with two generators, but then found a 
different generator basis pair where you forced one member to be an 
octave?

Message: 3677 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 09:21:19

Subject: Re: new cylindrical meantone lattice

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> 

>> From: paulerlich <paul@s...>
>> To: <tuning-math@y...>
>> Sent: Friday, February 01, 2002 1:06 AM
>> Subject: [tuning-math] Re: new cylindrical meantone lattice
>> 
>> 

>>> So, how about a formula that plots 19-EDO as, literally,
>>> a close cousin to 1/3-comma meantone spiral?  How does
>>> take something that's roots of 2, and change it into
>>> "8ve"-equivalent fractional powers of 3 and 5?
>> 

>> Well, maybe there's another way to get the right spiral.
> 
> 

> I'm all ears.


Maybe Gene can help.

> 
>  
>> 

>>> Now about your other two objections:
>>> 
>>> 

>>>> (a) the density of points along the line, which doesn't
>>>> appear to be meaningful;
>>> 
>>> 

>>> I'm hoping that the post I just sent before this one,
>>> about composer choosing particular flavors of meantone,
>>> addresses this one.
>> 

>> Not at all -- I was referring to the fact that, for example,
>> in 5/18-comma meantone, the points on the spiral are rather
>> far apart from one another -- that doesn't seem particularly
>> meaningful.
> 
> 

> OK, the only way I can respond to this properly is to go ahead
> and create a 5/18-comma lattice and examine it.  That's not going
> to happen until tomorrow.


It's already on your meantone webpage applet!!


>> 

>>>> (b) the fact that you have to pin the spiral to a particular
>>>> "1/1" origin, which ruins the rotational symmetry of the
>>>> cylindrical meantone lattice
>>> 
>>> 

>>> I've already said elsewhere that the spiral doesn't have
>>> to be pinned to anything.  It can float anywhere the user
>>> wants it.  What's important is the angle of the spiral,
>>> as you've noted.
>> 

>> So maybe a set of arrows (say from every _true_ lattice point) 
>> pointing at that angle would be preferable to a spiral.
> 
> 

> Well, I think arrows are a good idea, sure.  But again,
> I'd leave the choice of spirals or arrows up to the user.

Cool!

Message: 3678 - Contents - Hide Contents

Date: Fri, 1 Feb 2002 22:33:35

Subject: Re: interval of equivalence, unison-vector, period

From: Graham Breed

Gene:

>> As long as 2 is represented, it seems to me any temperament is an
>> octave temperament. The basis I gave was for a fifth and a tritone
>> below a fifth, and I could if I wanted make the fifth a pure fifth,
>> but I could do that, and temper octaves, in the octave basis also.
>> 
>> There are three considerations: interval of equivalence of a scale
>> using a given temperament, a basis of generators for the
>> temperament, and the tuning of the temperament. This are independent.

Paul:

> So why did you say "this was not a temperament"? And isn't it true
> that, if you took it out to, say, 10 notes per approximate octave,
> and tuned the octaves pure, it would _not_ be an octave-repeating
> scale? This seems to be the point Graham is missing.


The thing he said wasn't a temperament has no notes to an octave, so you 
could say 2 isn't represented and so it isn't an octave temperament by that 
definition.

What *has* octave repetition got to do with anything?


                              Graham

Message: 3679 - Contents - Hide Contents

Date: Fri, 1 Feb 2002 01:24:11

Subject: Re: new cylindrical meantone lattice

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, February 01, 2002 1:15 AM
> Subject: [tuning-math] Re: new cylindrical meantone lattice
>
>
>> [me, monz]

>> Didn't do it that way at all.  Simply looked at the lattice
>> of the shifted-boundary Duodene PB and saw that 2/9-comma
>> slashed right across the middle of it.
> 

> Hmm . . . so you're not using only the consonant intervals, as you 
> said you were.



Did I say that?!

I suppose what I'm really doing is basing the position of the
meantone on the position of the defining unison-vectors.



> 
> Anyway, can you show me how it slashes right down the middle, which 
> some other meantone doesn't?
> 

>> Since the angle
>> of the meantone line on the flat lattice (and of the spiral
>> on the cylindrical) graphically shows the tempering of the
>> meantone in relation to the nearest JI pitches, I moved it
>> around until it was centered perfectly within the shifted PB,
>> and it seemed to distribute the error the most evenly.
> 

> Couldn't any other meantone do exactly the same thing?



Yeah, actually, I think you're right about that.  Too tired
to see it now ... I'll have to make several of these and compare
them.  So most likely I'll just keep adding more to that
Duodene webpage, and you can give me feedback from that
when you see them.



-monz


 



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Message: 3680 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 06:55:53

Subject: Re: Approximate consonances of Parch's 43 tone scale

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
>>> [.1383934690, 1/8]
>>> 

>>> a = 9.9643/72
>> 

>> Equivalently, about 1/72 oct. -- right?
> 

> Right. It suggests a temperament of 72 in terms of the 224-et, with 

a generator of 3/224 and another of 1/8.

> 

>>> = 31.0001/224 = 166.0721626
>>> 
>>> badness 147.3854996
>> 

>> This 11-limit badness is not directly comparable to 5-limit 
badness 
>> for 5-limit temperaments, is it?
> 

> Nope; it's similar in a way, because of the flatness condition.


Right, but is it directly comparable? Is 500 an equally "bad" score 
in both frameworks?

Message: 3681 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 22:48:42

Subject: Re: interval of equivalence, unison-vector, period

From: genewardsmith

--- In tuning-math@y..., graham@m... wrote:

> In-Reply-To: <a3d5gb+ldpj@e...>
> genewardsmith wrote:


All elements are 

>> torsion elements, and we have a finite group, so this is rather 
>> different than a block with torsion elements.
> 

> But it still involves torsion?


Certainly, but not in the sense of a torsion block with torsion, since it isn't a block.


>> This is *not* a temperament, or at least not one 
>> I'm interested in hearing, so 2 is not acting as a unison, which is 
>> hardly a surprise.
> 

> Of course it's a temperament.  It's twintone/paultone/pajara.


Is
pajara the new official name? I'd like to get this settled. As for
this val, which defines only one of two required generator mappings
being a temperament, that's only if you layer on some interpretation
and perform the extra calculations to find a good choice for the
second generator; taken by itself, it isn't one. It's telling us to
send the octave to a unison, and 5 and 7 both to 1/9; it's only after
you stick in half-octaves and send 7 to some tuning of 64/9 and 5 to a
half-octave below that that pajara emerges. Read literally as a
temperament, it sends 2 to 1 and 5 and 7 to 1/9, and I don't think
that qualifies.


> The octave is acting as a unison, but it's more complicated than
that.  As 
> it has torsion, it's actually half an octave that's acting as a
commatic 
> unison vector.


I would say it's acting as a generator, but if you make 2 a unison it
becomes a torsion element, since its square is an octave.


> (BTW, in an octave-equivalent system, half a unison is a half-octave
as 
> well as a unison.  This is obvious if you think of octave-equivalent

> frequency space as a Hilbert space, and remember that half the pitch
is 
> the same as the square root of the frequency.)


You get a real Hilbert space if you allow anything of the form 
3^e3 5^e5 ... which can have an infinite number of prime exponents so
long as e3^2 + e5^2 + ... converges. Is this what you mean? The result
isn't even guaranteed to be a real number, and I don't know what it
would be good for.


>>>> i0 = i2-h2
>>>> i0.basis

> [0, 1, -2, -2]
> 
> Hey, that's the same as g0 above!


And which I think hardly counts as a temperament. As I said, it's not
one I want to listen to.

Message: 3682 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 09:27:05

Subject: Re: new cylindrical meantone lattice

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> 

>> From: paulerlich <paul@s...>
>> To: <tuning-math@y...>
>> Sent: Friday, February 01, 2002 1:15 AM
>> Subject: [tuning-math] Re: new cylindrical meantone lattice
>> 
>> 
>>> [me, monz]

>>> Didn't do it that way at all.  Simply looked at the lattice
>>> of the shifted-boundary Duodene PB and saw that 2/9-comma
>>> slashed right across the middle of it.
>> 

>> Hmm . . . so you're not using only the consonant intervals, as 
you 
>> said you were.
> 
> 

> Did I say that?!
> 
> I suppose what I'm really doing is basing the position of the
> meantone on the position of the defining unison-vectors.


OK . . . then it doesn't depend on the intervals _or_ on the pitches 
in the scale. For there are other ways to "capture" the Duodene than 
with the parallelogram of the two unison vectors you've chosen.

Message: 3683 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 22:55:14

Subject: Re: Approximate consonances of Parch's 43 tone scale

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

>>> How many notes in contiguous, (equal-length?) chains of 
generators 
>>> does each of these need to encompass Partch's 'Genesis' scale?
> 

> As in Genesis of a Music?
> 

>> What's the answer for MIRACLE? Wasn't it an non-'Genesis' 43-tone 
>> scale that MIRACLE comprised in 45 consecutive notes in a chain of 
>> generators?
> 

> I don't know. I got the scale I analyzed from a web search; I didn't 

know there was more than one 43 tone Partch scale.

See Yahoo groups: /tuning/message/25575 * [with cont.] 

Message: 3684 - Contents - Hide Contents

Date: Fri, 1 Feb 2002 01:39:24

Subject: Re: new cylindrical meantone lattice

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, February 01, 2002 1:27 AM
> Subject: [tuning-math] Re: new cylindrical meantone lattice
>
>

>> I suppose what I'm really doing is basing the position of the
>> meantone on the position of the defining unison-vectors.
> 

> OK . . . then it doesn't depend on the intervals _or_ on the pitches 
> in the scale. For there are other ways to "capture" the Duodene than 
> with the parallelogram of the two unison vectors you've chosen.



Really?!  Do tell!

I had a hunch that there might be some hexagonal PBs that
define the Duodene as well, and upon looking now at
Internet Express - Quality, Affordable Dial Up... * [with cont.]  (Wayb.)
I can see how the hexagonal PB in the bottom graphic could
be shifted slightly to enclose the Duodene.

But are there any other unison-vectors that will enclose it?
Or is [4 -1],[0 -3],[-4 -2] the only set from which any two
will create the Duodene?



-monz


 



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Message: 3685 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 07:52:43

Subject: some omnitetrachordal systems

From: paulerlich

·05-pelog,,,,, as in 16-tET: 225 25
    limma=135:128 vanishes

·07-meantone,, as in 19-tET: 3332 332
    schisma=81:80 vanishes

·10-pajara,,,, as in 22-tET: 222223 2223
    "paultone" -- 50:49 and 64:63 vanish

·14-injera,,,, as in 26-tET: 22222221 222221
    "double diatonic" -- 50:49 and 81:80 vanish

·17-ankara,,,, as in 29-tET: 1221221222 1221222
    schisma=32805:32768 vanishes

·22-shruti,,,, as in 34-tET: 2121212121212 212121212              
    diaschisma=2048:2025 vanishes

Ankara should be clear (medieval Arabic schismic-17 system preserved 
in Modern Turkish theory), but why injera? It's that delicious 
Ethiopian bread and the Ethiopian alphabet, like ours, has 26 basic 
letters:

AncientScripts.com: Ethiopic Script * [with cont.]  (Wayb.)

Message: 3686 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 23:01:15

Subject: Re: interval of equivalence, unison-vector, period

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:


> So why did you say "this was not a temperament"? 


Because a "temperament" which sends

1-9/8--5/4--4/3--3/2--5/3--15/8 to

1--9--1/9--1/3--3--1/27--1/3

hardly seems worthy of the name. In any case, 2 isn't represented!

And isn't it true 

> that, if you took it out to, say, 10 notes per approximate octave, 
> and tuned the octaves pure, it would _not_ be an octave-repeating 
> scale? This seems to be the point Graham is missing.


We seem to be talking about different things--what is "it"? If you mean pajara, it's a temperament, not a scale.

Message: 3687 - Contents - Hide Contents

Date: Fri, 1 Feb 2002 23:41:12

Subject: Re: interval of equivalence, unison-vector, period

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, February 01, 2002 11:27 PM
> Subject: [tuning-math] Re: interval of equivalence, unison-vector, period
>
>

>>>> I would say it's acting as a generator, but if you
>>>> make 2 a unison it becomes a torsion element, since
>>>> its square is an octave.
>>> 

>>> This, along with my message to Monzo this morning,
>>> seems to show the very real problems with considering
>>> 2 a unison!
>> 
>> 

>> How is 2^2 an octave?  By definition, it's simply 2.
>> Now you guys have really lost me.
> 

> Dude, what exactly are you referring to? I thought this
> was amazingly clear, but I guess I'm wrong!



Oh, OK ... I think I get it.

If 2 = a unison, then 2^2 = an octave.  Yes?

But I'm still confused, because if 2 is a unison, then
essentially for purposes of tuning math 2=1.  So how does
squaring that get you to the octave?



-monz


 





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Message: 3688 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 07:53:02

Subject: Re: Approximate consonances of Parch's 43 tone scale

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:


>> Nope; it's similar in a way, because of the flatness condition.
> 

> Right, but is it directly comparable? Is 500 an equally "bad" score 
> in both frameworks?


More or less, to the extent the question even makes sense, I suppose.

Message: 3689 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 09:42:27

Subject: Re: new cylindrical meantone lattice

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:


> Really?!  Do tell!
> 
> I had a hunch that there might be some hexagonal PBs that
> define the Duodene as well, and upon looking now at
> Internet Express - Quality, Affordable Dial Up... * [with cont.]  (Wayb.)
> I can see how the hexagonal PB in the bottom graphic could
> be shifted slightly to enclose the Duodene.


There you go!


> But are there any other unison-vectors that will enclose it?
> Or is [4 -1],[0 -3],[-4 -2] the only set from which any two
> will create the Duodene?


Probably -- but you only used two of those. And anyway, we may have 
no reason for talking about this, because you seem to concur that 
perhaps any meantone would do what 2/9-comma did even for the block 
in question.

Message: 3690 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 23:07:43

Subject: Re: interval of equivalence, unison-vector, period

From: genewardsmith

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:

> Well, can you think of a word for something that acts like a unison vector 
> but isn't?  To cover the meanings of "unison vector", "generator", "period" 
> and "equivalence interval"?


What
about kernel element? Of course, a period is a kernel element only if
you make it one, by having a corresponding mapping, but that is the
case here. The same would be true of an equivalence interval--if we
send the half-octave to 1, it is a kernel element, but if we send 2 to
1 but not sqrt(2), then sqrt(2) is an element of order 2. One way we
get a cyclic group of order 11, the other way of order 22.

Message: 3691 - Contents - Hide Contents

Date: Fri, 1 Feb 2002 23:45:20

Subject: on-topic subject lines [was:: 171-EDO, Vogel (was: 7-limit MT...)]

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, February 01, 2002 11:29 PM
> Subject: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT reduced bases
for ets)
>
>
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

>> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>> 

>>>> 171: [2401/2400, 4375/4374, 32805/32768]
>>>> 
>>>> Wouldn't want to do that--look at those three
>>>> high-powered commas!
>> 

>>> (note that the title is an homage to Helmholtz)
>> 

>> Helmholtz liked the schismic temperament, and Vogel goes him
>> one better by combining schismic with ennealimmal, which the
>> above reduced basis shows is one way of thiking about 171-et.
>> You could temper either 53 tones or 72 tones with it, among
>> other things.
>> 
>> Since I am now writing a piece in 46-et and just finished one
>> in 53-et, I'll also add these:
>> 
>> 46: [126/125, 245/243, 1029/1024]
>> 53: [225/224, 1728/1715, 3125/3087]
>> 
>> I'm finding the 43-et set of commas quite useful.
>

> Do you really mean 43, or one of the above?



Guys, I try to be really diligent about changing the
subject line when the content of my post warrants it.

This one's apparently gone back to the one I diverted
it from.


-monz






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Message: 3692 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 08:07:40

Subject: Re: interval of equivalence, unison-vector, period

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:


>> This, I think, corresponds to how Graham thinks of things, and 
how I 
>> _used_ to think of things, before I understood torsion in the 
period-
>> is-1/2-or-1/9-or-1/N-octave sense.
> 
> 

> Paul, you're really good at explaining things.
> Please elaborate on this until I understand it.   :)


Oops -- I didn't mean that at all. I meant, before I understood 
torsion as it's defined in your dictionary. Thanks for pointing out 
my brain fart!

> 
> I don't recall anyone ever responding to the lattice diagram
> I made for the torsion definition:
> Definitions of tuning terms: torsion, (c) 2002... * [with cont.]  (Wayb.)
> 
> I thought that showing the pairs of pitches that are separated
> by two unison-vector candidates that are smaller than the
> actual unison-vectors defining the torsional-block might have
> been saying something significant about what a torsional-block
> is, or maybe at least something about this particular example.
> 
> Any thoughts?


Well, you're definitely doing something right in this case, since 
81:80 and 128:125 are definitely intervals that should represent 
equivalences here . . . but it won't necessarily be that case that 
smaller intervals in the parallelogram than the defining unison 
vectors fall into the "equivalent" category for every torsional block.


> The fog has still not cleared about the three items in the subject 
> line.


Really? OK, first of all, period is specific to MOS scales and the 
linear temperaments they come from.

Examples:

meantone temperament
unison vector: 81:80
interval of equivalence: octave
period: octave

MIRACLE temperament
unison vectors: 224:225, 385:384, 441:440
interval of equivalence: octave
period: octave

diminished/octatonic in 12-tET or 28-tET
unison vector: 648:625
interval of equivalence: octave
period: 1/4 octave

'paultone'
unison vectors: 50:49, 64:63
interval of equivalence: octave
period: 1/2 octave

Bohlen-Pierce
unison vectors: 245:243, 3087:3125
interval of equivalence: tritave (3:1)
period: tritave (3:1)

Message: 3693 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 09:42:56

Subject: Re: new cylindrical meantone lattice

From: paulerlich

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> you seem to concur that 
> perhaps any meantone would do what 2/9-comma did even for the block 
> in question.


I mean even for the parallelogram in question.

Message: 3694 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 23:49:47

Subject: Re: interval of equivalence, unison-vector, period

From: paulerlich

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:

> Me:

>>> The octave is acting as a unison, but it's more complicated 

that that.  As

>>> it has torsion, it's actually half an octave that's acting as a 
commatic
>>> unison vector.
> 
> Paul:

>> No offense, Graham, but could you at least invent some terminology
>> that makes sense for what you're talking about, instead of
>> misappropriating terminology that makes no sense the way you're 
using
>> it? Half an octave does not act a commatic unison vector here -- 
this
>> is very frustrating because I thought I had spent dozens of posts
>> explaining to you what a commatic unison vector is, and convincing
>> you that an octave isn't one and a fifth isn't one . . . did all 
that
>> arguing make no impression on you?
> 

> I'm fully aware that an octave is not a unison vector.  I've said 
so before 
> and I didn't say otherwise in that quote.  All I said is that it 
(or the 
> tritone) acts as a unison vector.  Which it does.  As far as the 
algebra's 
> concerned, it's exactly like a unison vector.


But why a "chromatic" unison vector? A chromatic unison vector 
indicates something that is actually tuned differently from 
an "equivalence".

> 
> Paul:

>> A chromatic unison vector is a generalized "augmented unison".
>> Nothing else.
> 

> Well, can you think of a word for something that acts like a unison 
vector 
> but isn't?


It does in your mechanics, but not in Gene's.


> Me:

>>> (BTW, in an octave-equivalent system, half a unison is a half- 
octave as
>>> well as a unison.  This is obvious if you think of octave-
equivalent
>>> frequency space as a Hilbert space, and remember that half the 
pitch is
>>> the same as the square root of the frequency.)
> 
> Paul:

>> Huh? So if the frequency is 6400 Hz, the square root of that is 
80,
>> and that's half the pitch??
> 

> Hmm, something wrong there.  I meant the square root of a frequency 
*ratio*.  
> Yes?  That seems to make sense.


But half the pitch? I think you mean half the interval, or half the 
pitch _difference_.

> 
> Me:

>>> octave a unison vector is like imposing octave equivalence.  
That's
>>> actually quite similar to something Fokker said.
> 
> Paul:

>> Please fill us in!
> 

> In <A.D. Fokker: Unison Vectors and Periodicity Bl... * [with cont.]  (Wayb.)>, "By common 
general 
> agreement all notes differing by an arbitrary number of octaves 
only, are 
> considered as unison, and as one and the same note."  I mentioned 
this last 
> time round as well.


Well, there are apparantly different ways of implementing this 
observation mathematically.

Message: 3695 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 08:09:36

Subject: Re: Approximate consonances of Parch's 43 tone scale

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 

>>> Nope; it's similar in a way, because of the flatness condition.
>> 

>> Right, but is it directly comparable? Is 500 an equally "bad" 
score 
>> in both frameworks?
> 

> More or less, to the extent the question even makes sense, I 
suppose.


Well, what if the question were phrased in terms of the density of  
temperaments that pass a "goodness" criterion in the vicinity of a 
given g (gens) value?

Message: 3696 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 09:59:08

Subject: Re: interval of equivalence, unison-vector, period

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:


>>> Glassic
>> 

>> Good name--where does it come from?
> 

> Sorry -- that's the wrong name. Glassic has b = 1 . . . my 
> piece "Glassic" uses it.


I recall that piece--I thought perhaps it was named for a temperament.


>> Another type of val of interest are the maps of generators to primes.


> What's the dual to that kind of val?


Generators.
For the period matrix (pair of vals) for twintone, it would be two
intervals, the first mapped to one and the other to zero, and the
second to zero and then one--the simplest example being
4/3 and 7/5.


>> I can do the same sort of thing starting from
>> [-2,-2,-7,-8], where I end up with
>> 
>> [-2 2]
>> [-2 3]
>> [-7 5]
>> [-8 6]
>> 
>> as a mapping from generators to primes; here "b" is a wide fifth 
>> and "a" is a tritone below that.
> 

> Wouldn't that just be a non-octave ET, approximately 11 tones per 
> octave?


No, it's two generators for twintone, only now instead of making one
of them an octave or a fraction of an octave, I've made it a fifth or
a fraction of a fifth--in this case, the full fifth. If you wanted an
et for it, 22 springs to mind. 11 can't work, because twintone needs a
tritone (in this case, the difference between the two generators.)

Message: 3697 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 23:52:16

Subject: Re: interval of equivalence, unison-vector, period

From: paulerlich

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:

> Me:

>>> Where am I going wrong?
> 
> Paul:

>> I'm not saying you're wrong, only that your methods are different
>> from Gene's -- most recently exemplified with the case that he
>> considered "not a temperament" and you considered "22-tET".
> 

> I meant there where I was wrong with Gene's terminology.  I didn't 
call the 
> thing he called "not a temperament" 22-tET.  I called it 
> paultone/twintone/pajara.  Because that's what it is.


Not really, because paultone/twintone/pajara repeat themselves every 
octave, while I don't think Gene's construction does -- that's why he 
said it's "not a temperament", I believe.


> It's actually a mapping of 0-tET. You could call it a paradox that 

something with no notes 

> counts as a temperament.  You could then analyse the assumptions 

that led to 

> it instead of shouting back "you're wrong" at the person who 

pointed it out.

Good point. But you're not going to help anyone understand this stuff 
by using misleading terminology. That's all I'm trying to say.

Message: 3698 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 08:11:12

Subject: Re: interval of equivalence, unison-vector, period

From: paulerlich

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:


> Bohlen-Pierce
> unison vectors: 245:243, 3087:3125
> interval of equivalence: tritave (3:1)
> period: tritave (3:1)


Well, what I meant was the BP linear temperament (generated by 7:3, 
with interval of equivalence 3:1), so 3087:3125 doesn't belong there. 
Should be:

Bohlen-Pierce linear temperament (Stearns/Benson/Keenan)
unison vectors: 245:243
interval of equivalence: tritave (3:1)
period: tritave (3:1)

Message: 3699 - Contents - Hide Contents

Date: Fri, 01 Feb 2002 10:04:56

Subject: Re: interval of equivalence, unison-vector, period

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:


>>> Another type of val of interest are the maps of generators to 
primes.
> 

>> What's the dual to that kind of val?
> 
> Generators.


That's what I was going to guess . . .


>For the period matrix (pair of vals) for twintone, it would be two 
>intervals, the first mapped to one and the other to zero, and the 
>second to zero and then one


Whoa -- this is a very confusing sentence. Can you clarify?


>--the simplest example being
> 4/3 and 7/5.


Not surprising, as these are normally taken as the period and the 
generator of twintone. But there are other possibilities, if you 
don't assume octave-equivalence?


>> Wouldn't that just be a non-octave ET, approximately 11 tones per 
>> octave?
> 
> No,


Well . . . I revised the question in the next message, which I hope 
you get to see.

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