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Message: 3001 - Contents - Hide Contents

Date: Sun, 06 Jan 2002 10:42:39

Subject: Still more 72-et scale types

From: genewardsmith

Here are ones derived from 2401/2400~1; while I give these as 72-et
types, they can be used for much more accurate 7-limit tunings; such
as 171, 270, 441, or 612; the 72-et in the 7-limit can be defined as
Miracle+Ennealimmal, and this is from the Ennealimmal side.

6 tones

[12, 9, 14]
[1, 2, 3]

7 tones

[3, 9, 14]
[1, 3, 3]

[12, 21, 2]
[2, 2, 3]

9 tones

[9, 2, 12]
[2, 3, 4]

[2, 12, 19]
[5, 2, 2]

[9, 12, 5]
[5, 1, 3]

10 tones

[9, 7, 5]
[5, 1, 4]

[9, 3, 5]
[6, 1, 3]

[9, 11, 3]
[3, 3, 4]

11 tones

[2, 7, 12]
[5, 2, 4]

[5, 9, 2]
[5, 5, 1]

13 tones

[9, 2, 3]
[6, 3, 4]

[3, 6, 11]
[7, 3, 3]

[2, 10, 17]
[9, 2, 2]

14 tones

[4, 12, 5]
[5, 1, 8]

15 tones

[7, 2, 5]
[6, 5, 4]

[5, 4, 7]
[9, 5, 1]

[2, 7, 10]
[9, 2, 4]

[4, 8, 5]
[6, 1, 8]

16 tones

[2, 5, 7]
[6, 5, 5]

[6, 3, 5]
[6, 7, 3]

[4, 2, 5]
[5, 1, 10]

[4, 4, 5]
[7, 1, 8]

[5, 4, 3]
[9, 6, 1]

[6, 8, 3]
[3, 3, 10]

19 tones

[6, 2, 3]
[6, 3, 10]

[2, 7, 3]
[9, 6, 4]

21 tones

[2, 2, 5]
[6, 5, 10]

22 tones

[5, 1, 3]
[9, 6, 7]

24 tones

[4, 4, 1]
[9, 7, 8]

25 tones

[2, 4, 3]
[9, 6, 10]

Message: 3002 - Contents - Hide Contents

Date: Sun, 06 Jan 2002 10:53:34

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

>> No. It just assumes that the overtones are pretty close to 
harmonic,
>> because they will then lead to the same ratio-intepretations for 
the
>> fundamentals as the fundamentals by themselves. If they're 50 cents
>> from harmonic, they will lead to a larger s value for the resulting
>> harmonic entropy curve, but that's about it.
> 

> s represents the blur of the spectral components coming in.  How
> could an inharmonic timbre change that?


When we're dealing with a dyad consisting of complex tones, and 
trying to apply harmonic entropy to that dyad, s is decreased below 
the value that sine waves in place of the complex tones would imply. 
The more inharmonic the timbre, the less s is decreased below the 
sine-wave case.


> 

>> You can synthesize inharmonic sounds, yes?
> 

> No, that's the problem.

Oops.


>>> Yes, to me, pelog sounds like a I and a III with a 4th in the
>>> middle.  But the music seems to use a fixed tonic, with not
>>> much in the way of triadic structure.
>> 

>> How about 5-limit intervals?
> 

> Not sure what you're asking.


Not much in the way of 5-limit intervals?


>>> Okay, let's take a
>>> journey...
>>> 
>>> "Instrumental music of Northeast Thailand"
>>> 
>>> Characteristic stop rhythm.  Harmonium and marimba-sounding
>>> things play major pentatonic on C# (A=440) or relative minor
>>> on A#.
>> 

>> This is clearly not a pelog tuning!
> 

> Right, it's the chinese pentatonic.  I threw it in for
> completeness.


Completeness of what?


>>> I still say there's nothing here that would turn up an optimized
>>> 5-limit temperament!
>> 

>> Forget the optimization. All you need is the mapping -- that
>> chains of three fifths make a major third and that chains of
>> four fifths make a minor third. This seems to be a definite
>> characteristic of pelog! Just as much as the "opposite" is a
>> characteristic of Western music, regardless of whether strict
>> JI, optimized meantone, 12-tET, or whatever is used.
> 

> Western music uses progressions of four fifths and expects to
> wind up on a major third.


These don't have to be triadic, harmonic progression.


> I didn't notice anything like this
> for the [1 -3] map (right?) on the cited discs.


[3 1]. It's not something you should expect to hear as a triadic 
harmonic progression. It's simply the way the 5-limit intervals fit 
together in the scale. If they didn't, the scale, and the music that 
depends on it, wouldn't work.


>>> I guess it all depends if you consider these tonic changes
>>> or just points of symmetry in a melisma (sp?).
>> 

>> Why does that matter?
> 

> One's a harmonic device, the other melodic.


There are a lot of simultaneities going on, regardless of whether you 
consider them to constitute "tonic changes".


> But I think a lot of the
> other stuff that goes along with harmonic music is missing
> from this music.  Western music requires meantone.  The pelog
> 5-limit map is far more extreme, but what suffers in this
> music as we change the tuning from 5-of- 7, to 23, to 16, all
> the way to strict JI?


23 and 16 give you the Pelog sound. 7 doesn't. Give me a strict JI 
scale to try.


> I think the tuning on these discs is
> closer to JI than 23-tET, and I don't hear them avoiding a
> disjoint interval.  Do you?


Avoiding a disjoint interval? You mean you hear it as 5-of-7? It 
modulates that much?? What exactly do you mean?


> Incidentally, I think Wilson agrees with your point of view
> here.  While he does caution against eager interps. of his
> ethno music theory, I think he thinks that harmonic mapping
> is inevitable, and atomic in music.  I'm not sure I agree.
> Not sure I disagree.


Well, the idea of this paper that Gene, Dave, Graham, and I are 
working on, at least it seems to me, is to start with the assumption 
that notes are connected to one another via simple-ratio intervals, 
explain periodicity blocks, show that an MOS results when you temper 
out all but one of the unison vectors, show that MOSs are linear, and 
present the "best" linear temperaments from this point of view. It's 
just a paper, not a manifesto, so there's nothing wrong with starting 
with a very simple and strong set of assumptions, and seeing where 
they lead.

Message: 3003 - Contents - Hide Contents

Date: Sun, 06 Jan 2002 10:57:22

Subject: Re: Math proof sought

From: genewardsmith

--- In tuning-math@y..., "tunerguy2002" <tunerguy2002@y...> wrote:


>> Is anyone aware of a proof of the general case? All leads 
>> appreciated.
> 

> This is a very simple consequence of the Fundamental Theorem of 
> Arithmetic. I'm sure Gene can give you the most concise proof of 
> this. You should ask the question at 


It
seems to me Paul has basically given the proof, which is to cite the
FTA. If you want the details, the FTA says that any positive rational
number has a *unique* representation as 2^e1 * 3^e2 * ...,
where the exponents ep are integers, all but a finite number being
zero. If you have positive rational numbers a and b, such that for
some prime p the exponent of p in the product representation of a,
which is called vp(a), the "valuation" at p of a, is not zero whereas
vp(b)=0, then vp(a^n) = n vp(a) > 0, but vp(b^m) = m*0 = 0; since they
are not equal in terms of the exponent of p, they cannot be equal by
the FTA. I think this covers the situation you had in mind; and in the
form I give here, can be generalized to situations where the
Fundamental Theorem itself does not apply.

Message: 3004 - Contents - Hide Contents

Date: Sun, 6 Jan 2002 18:39:07

Subject: Re: please simplify equation

From: monz

Hi Gene,


> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, January 06, 2002 4:00 PM
> Subject: [tuning-math] Re: please simplify equation
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 

>> 2^[ (8r+1) / (13r+3) ]
>> 
>> And Paul gave me these equivalent simplifications of it:
>> 
>> = 2^[ (2r-1) / (3r-1) ]
>> 
>> = 2^[ (3-r) / (4-r) ] 
>> 
>> 
>> I plotted the numbers of all three of the above formulas
>> into a graph, and can see how they're all related linearly.
>> Can you explain algebraically what's going on?  Please
>> be as detailed as possible.  Thanks.
>


 

> Not really. My (3r+1)/(5r+1) is (r+9)/19,
> your (8r+1)/(13r+3) is (r+18)/31, and
> Paul's (2r-1)/(3r-1) = (3-r)/(4-r) = (8-r)/11,
> so these are not the same.



Can you show me how you work this magic?


Here are my comments:

First, your (3r+1)/(5r+1) definitely isn't right anyway.
The exponent of 2 has to be ~0.580178728.  If r is PHI,
(3r+1)/(5r+1) = ~0.644003578 =/= ~0.580178728.


However,  for r = PHI = [1 + 5^(1/2)] / 2 ,

    (8r+1)/(13r+3)
  = (2r-1)/(3r-1)
  = (3-r)/(4-r)
  = (8-r)/11
  = ~0.580178728


but

   (3r+1)/(5r+1)  =/=  (r+9)/19   and
   
   (8r+1)/(13r+3) =/=  (r+18)/31 


So how do you get (8-r)/11 from (2r-1)/(3r-1) and
(3-r)/(4-r), and why are the other solutions incorrect?

  


> If you tell me what recurrence you are seeking the limit of,
> I'll tell you the answer.



Thanks for the offer, but... umm... I don't know what that means.

But these are the two things I'm looking for:


1)

Where r = PHI = [1 + 5^(1/2)] / 2 ,
my spreadsheet is calculating all three equations

      2^[ (8r+1) / (13r+3) ]
   =  2^[ (2r-1) / (3r-1) ]
   =  2^[ (3-r) / (4-r) ] 

to be the same to 9 decimal places.

I can see that they follow the general formula 2^x,
x = (ar+b)/(cr+d), where r = PHI = [1 + 5^(1/2)] / 2 .

I'm looking for the function which calculates a,b,c,d.


2)

I want to be able to describe some basic intervals of
golden meantone mathmatically, in terms of nothing but
PHI and numbers, as "ratios":

  v = 5th
  t = tone = major 2nd
  s = diatonic semitone = minor 2nd
  t^2 = major 3rd
  t*s = minor 3rd


I already have several equivalent expressions for v :

    2^[(8r+1)/(13r+3)]
  = 2^[(2r-1)/(3r-1)]
  = 2^[(3-r)/(4-r)]
  = 2^[(8-r)/11]
  
I'd like to have something like that final form
for t, s, t^2, and t*s as well.


We have these basic relationships, for r=PHI:

   v = (t^3)*s
   t = (v^2)/2  =  s*r
   s = (2^3)/(v^5) = t^(1/r)

I derived my 2^[(8r+1)/(13r+3)] by plugging the values
for t and s involving v, into the v equation.



-monz


 



_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.]  (Wayb.)

Message: 3005 - Contents - Hide Contents

Date: Sun, 06 Jan 2002 10:57:34

Subject: Re: tetrachordality

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

>>> So obviously, these two scales will come out
>>> the same.  But you've view -- and I remember
>>> doing some listening experiments that back you
>>> up (the low efficiency of the symmetrical
>>> version was the other theory there) -- is that
>>> the symmetrical version is not tetrachordal.
>>> 
>>> So what's going on here?  Where's the error
>>> in tetrachordality = similarity at transposition
>>> by a 3:2? 
>> 

>> An octave species is homotetrachordal if it has identical melodic
>> structure within two 4:3 spans, separated by either a 4:3 or a 3:2.
>> In the pentachordal scale, _all_ of the octave species are
>> homotetrachordal (some in more than one way). In the symmetrical
>> scale, _none_ of the octave species are homotetrachordal.
> 

> That's the def. in your paper.  But:
> 
> () I never understood how it reflects symmetry at the 3:2.


4:3 more clearly than 3:2. However, you could look at 3:2 spans if 
you wished, and still see a large gulf between the pentachordal and 
symmetrical decatonic scales.


> () "homotetrachordal" is a new term on me.  Are there precise
> defs. of homo- vs. omni- around?


Were those not precise enough for you?


> How did you choose these
> prefixes?


Homo = same -- two 4:3 spans that are the same
Omni = all -- all octave species are homotetrachordal.


> () We agreed a bit ago that 'the number of notes that change
> when a scale is transposed by 3:2 index its omnitetrachordality',
> right?


We did? I don't see transposition as coming into this -- rather, it's 
a property of the _untransposed_ scale, heard in its full, 
unmodulating glory.


> My current approach is just a re-scaling of this.  So
> do we want to revise this agreement?


I guess so!!

Message: 3006 - Contents - Hide Contents

Date: Sun, 06 Jan 2002 11:09:45

Subject: Re: Optimal 5-Limit Generators For Dave

From: paulerlich

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


> How do you mean?  The two meantones fit snugly on the two different 
> keyboards, and chords in the enharmonic genus typically alternate 
between 
> them.  As most chords are consonances, there's no other way of 
getting the 
> enharmonic melodies right.  For you to ask this question suggests 
either I 
> didn't understand you, or you don't have a copy of Vicentino's book.

I don't.

> It 
> is worth reading.


I'll have to look for it.


> I thought you had it because you recommended it to 
> somebody else.

I did?

> 
> Me:
>>> There's 

>>> also a half-octave system, [(2, 0), (3, 1), (4, 4)].  That's 
the 
>> one my 

>>> program would deduce from the octave-equivalent mapping [2 8].
> 
> Paul:

>>> From that unison vector? If so, I think you're confusion torsion 
>> with "contorsion".
> 

> This has nothing directly to do with unison vectors.


Then what do you mean by "the octave-equivalent mapping [2 8]"?

> 
> Me:

>>> If I had 
>>> such a program.  If anybody cares, is it possible to write 
one?  
>> Where 

>>> torsion's present, we'll have to assume it means divisions of 
the 
>> octave 
>>> for uniqueness.
> 
> Paul:

>> Huh? Clearly this doesn't work in the Monz sruti 24 case.
> 

> No, that can't be expressed in this particular octave equivalent 
system.  


Can it be expressed in any?

> 

>>> Gene said it isn't possible, but I'm not convinced.  How 
>>> could [1 4] be anything sensible but meantone?
>> 

>> Not sure what the connection is.
> 

> [1 4] is a definition of meantone: 4 fifths are equivalent to a 
major 
> third.  Is that a unique definition, or do we have to add "plus two 
> octaves"?


To be completely clear, yes.

> 

>>> Perhaps the first step is to find an interval that's only one 
>> generator 

>>> step, take the just value, period-reduce it and work everything 
>> else out 
>>> from that.
>> 

>> If the half-fifth is the generator, what's the just value?
> 

> Well, it could be either 5:4 or 6:5.


We've already mapped these to other intervals.


> Or 11:9 or 27:22.  Or 49:40 or 
> 60:49.


You can't just bring in 11 or 7 like that -- then you would have a 7-
limit or 11-limit system, with the associated mappings and all, which 
you could work out in the normal way.


> But if you mean the case where all consonances are specified in 
> terms of fifths, but the generator is a half-fifth, I thought I 
defined 
> those out of existence above.


Defined those out of existence? I thought you were saying this was 
the Vicentino enharmonic case.


> If not, you can take the square root.


That's not a just interval.


> Me:

>>> But there may be some cases where the optimal value should 
>>> cross a period boundary.
> 
> Paul:
>> ??
> 

> Say you have a system that divides the octave into two equal parts, 
and 
> 7:5 is a single generator steps.  It may happen that 7:5 
approximates best 
> to be larger than a half octave, so taking its just value for 
calculating 
> the mapping will get the wrong results.  This may be a real problem 
when 
> the octave is divided into 41 equal parts, like one of the higher-
limit 
> temperaments I came up with, and the generator is a fairly complex 
> interval.


Can you give a specific example?

> 
> Me:

>>> If you think it can't be done, show a counter-example: an 
>>> octave-equivalent mapping without torsion that can lead to two 
>> different 

>>> but equally good temperaments.
> 
> Paul:

>> Equally good? Under what criteria? Look, why do we care about the 
>> octave-equivalent mapping? Certainly we can't object to asking 
the 
>> mapping to be octave-specific, can we?
> 

> It should be fairly obvious if you get the mapping right because 
the 
> errors will be small.


Granted, but how can we object to asking the mapping to be octave-
specific? Wouldn't it be better to do that from the outset than to 
count on the errors being "small"?

> 
> You were the one originally pushing for octave-equivalent > 
calculations.  


I think Gene has convinced be that they won't work. The only way you 
can possibly distinguish cases of torsion correctly is with the 
octave-specific mapping.


> If you aren't bothered any more, I'm not; I was only trying to 
answer your 
> questions.  But it would be elegant to describe systems in the 
simplest 
> possible way, and one consistent with Fokker.  It's up to you if 
you don't 
> want the paper to cover that.


Fokker didn't run into any cases of torsion, but we have! The paper 
can cover Fokker's methods but doesn't need to be restricted to them.

Message: 3007 - Contents - Hide Contents

Date: Sun, 06 Jan 2002 11:11:01

Subject: Re: Some 9-tone 72-et scales

From: paulerlich

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


> Paul:

>> The 22-tET "Pythagorean diatonic" works exceptionally well.
> 

> You mean 4 4 1 4 4 4 1 ?

Yes.

> Isn't it proper


No: 4 + 4 + 4 > 1 + 4 + 4 + 1.

Message: 3008 - Contents - Hide Contents

Date: Sun, 06 Jan 2002 11:12:03

Subject: Re: please simplify equation

From: genewardsmith

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


> 
> v  =  10 ^ ( LOG( 1 / (2 ^ (9r - 1/r) ) )  /  ( -15r + 2/r - 1) )


I don't know what the base of the log is, presuming it is e, we get

b = ln(2)(3r+1)/(5r+1) for the exponent, and so v = 10^b.

Message: 3009 - Contents - Hide Contents

Date: Sun, 06 Jan 2002 11:16:40

Subject: Re: Some 12-tone, 2-step 46-et scales

From: paulerlich

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

> I was facinated to discover that the 7,5 system did a little better 

than the completely symmetrical 6,6 system.

> 
> [0, 4, 8, 12, 16, 20, 23, 27, 31, 35, 39, 43]
> [4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 3]
> edges   24   24   40   connectivity   3   3   6
> 
> [0, 4, 8, 12, 16, 20, 24, 27, 31, 35, 39, 43]
> [4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 3]
> edges   24   25   41   connectivity   3   3   6


This is very neat and important stuff. It's been claimed that the 
Indian scales derive from a second-order-maximally-even 7-out-of-12-
out-of-22 construction, which would imply the symmetrical 12-tone 
system above. However, the actual evidence supports the 
omnitetrachordal system. So you're saying one might explain this 
using some ratio of 7? Or did I misread this? I'd like to see/make 
lattices of these.

Message: 3010 - Contents - Hide Contents

Date: Sun, 06 Jan 2002 11:23:02

Subject: Re: Some 12-tone, 2-step 46-et scales

From: genewardsmith

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

So you're saying one might explain this 

> using some ratio of 7? Or did I misread this? 


I'm
not sure what your question is; what I was saying is that we get a
little better count of 7-limit intervals with the 7,5 system than with
the 6,6 system in the 46-et, which I did not expect.

I'd like to see/make 

> lattices of these.


I could send you a gif file from Maple's graph-drawing program,of the
sort I posted on the tuning list, but that would only be a starting
point.

Message: 3011 - Contents - Hide Contents

Date: Sun, 06 Jan 2002 11:24:58

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: genewardsmith

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


> Well, the idea of this paper that Gene, Dave, Graham, and I are 
> working on, at least it seems to me, is to start with the assumption 
> that notes are connected to one another via simple-ratio intervals, 
> explain periodicity blocks, show that an MOS results when you temper 
> out all but one of the unison vectors, show that MOSs are linear, and 
> present the "best" linear temperaments from this point of view.


How much of this is already published?

Message: 3012 - Contents - Hide Contents

Date: Sun, 06 Jan 2002 11:28:59

Subject: Re: Some 12-tone, 2-step 46-et scales

From: paulerlich

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

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> So you're saying one might explain this 

>> using some ratio of 7? Or did I misread this? 
> 

> I'm not sure what your question is; what I was saying is that we 
>get a little better count of 7-limit intervals with the 7,5 system 
>than with the 6,6 system in the 46-et, which I did not expect.

Right.


> I'd like to see/make 

>> lattices of these.
> 

> I could send you a gif file from Maple's graph-drawing program,of 
>the sort I posted on the tuning list, but that would only be a 
>starting point.


Sure. Now are there some 46-tET commas you did not take into account? 
You didn't answer my "hyper-torus" point on the tuning list yet . . .

Message: 3013 - Contents - Hide Contents

Date: Sun, 06 Jan 2002 11:32:18

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: paulerlich

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

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

>> Well, the idea of this paper that Gene, Dave, Graham, and I are 
>> working on, at least it seems to me, is to start with the 
assumption 
>> that notes are connected to one another via simple-ratio 
intervals, 
>> explain periodicity blocks, show that an MOS results when you 
temper 
>> out all but one of the unison vectors, show that MOSs are linear, 
and 
>> present the "best" linear temperaments from this point of view.
> 

> How much of this is already published?


The proof that MOSs are linear might be said to be published. The 
periodicity block concept was of course published by Fokker, though 
the explanation of periodicity blocks might better take off from this 
starting point, which you are all welcome to suggest changes to:

A gentle introduction to Fokker periodicity bl... * [with cont.]  (Wayb.)

As for the rest, I'm fairly certain it's entirely new work.

Message: 3014 - Contents - Hide Contents

Date: Sun, 06 Jan 2002 11:37:59

Subject: Re: My top 5--for Paul

From: paulerlich

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

> #1
> 
> 2^-90 3^-15 5^49
> 
> This is not only the the one with lowest badness on the list, it is 

the smallest comma, which suggests we are not tapering off, and is 
evidence for flatness.

> 
> Map:
> 
> [ 0  1]
> [49 -6]
> [15  0]
> 
> Generators: a = 275.99975/1783 = 113.00046/730; b = 1
> 
> I suggest the "Woolhouse" as a name for this temperament, because 

of the 730. Other ets consistent with this are 84, 323, 407, 1053 and 
1460.

> 
> badness: 34
> rms: .000763
> g: 35.5
> errors: [-.000234, -.001029, -.000796]
> 
> #2 32805/32768 Schismic badness=55
> 
> #3 25/24 Neutral thirds badness=82
> 
> #4 15625/15552 Kleismic badness=97
> 
> #5 81/80 Meantone badness=108
> 
> It looks pretty flat so far as this method can show, I think.


How well do these results back up my now-famous (I hope) heuristic, 
which involves only the size of the numbers in, and the difference 
between numerator and denominator of, the unison vector? How might we 
weight the gens and/or cents measures so that the heuristic will work 
perfectly?

Message: 3015 - Contents - Hide Contents

Date: Sun, 6 Jan 2002 14:29 +00

Subject: Re: Optimal 5-Limit Generators For Dave

From: graham@xxxxxxxxxx.xx.xx

Me:

>> This has nothing directly to do with unison vectors.

Paul:

> Then what do you mean by "the octave-equivalent mapping [2 8]"?


3:2 is 2 generators and 5:4 is 8 generators, all octave reduced.

Paul:

>>> Huh? Clearly this doesn't work in the Monz sruti 24 case.

Me:

>> No, that can't be expressed in this particular octave equivalent 
> system.  

Paul:

> Can it be expressed in any?


You could list all notes in the periodicity block as octave-equivalent vectors.  
How else were you expecting it to work?  Do you have an octave-equivalent algorithm 
for getting the periodicity block from the unison vectors?

Me:

>> But if you mean the case where all consonances are specified in 
>> terms of fifths, but the generator is a half-fifth, I thought I 
> defined 

>> those out of existence above.

Paul:

> Defined those out of existence? I thought you were saying this was 
> the Vicentino enharmonic case.


Yes, and it can't be unambiguously expressed as an octave-equivalent mapping.  It 
has torsion.  I said we weren't considering such systems yet.


>> If not, you can take the square root.
> 

> That's not a just interval.

So?


Paul (on systems where the just and tempered generators octave reduce differently):

> Can you give a specific example?


No, because I haven't coded anything up.  If you have code that works, I've been 
collecting test cases and I expect some of them will throw up this problem.

If they're allowed, the [2 8] systems are an example, because 350 and 850 give 
different octave-specific systems, but optimise to the same meantone.  You could 
differentiate them by saying that [2 8] means to divide the fifth, and [-2 -8] to 
divide the fourth, but that would still break the one to one relationship between 
mappings and temperaments.


Me:

>> It should be fairly obvious if you get the mapping right because 
> the 

>> errors will be small.

Paul:

> Granted, but how can we object to asking the mapping to be octave-
> specific? Wouldn't it be better to do that from the outset than to 
> count on the errors being "small"?


If nobody's objecting to the mapping being octave-specific there's no problem.  
Even so, there's nothing special about errors needing to be small in an 
octave-equivalent system.  A period-equivalent system is fully defined by it's 
mapping and the period.  Different generators will give different octave-specific 
mappings, but that's a relationship between the systems, not an inherent problem 
with one system.

The problem with period-equivalent systems (what the octave-equivalent route tends 
to lead to) is that they're harder to optimise.  When a particular interval 
approximates to an exact number of periods, you'll get a local maximum so steepest 
descent methods won't work.  The RMS error by generator isn't a quadratic equation, 
so that optimisation won't work.  A number of different generator sizes can make 
the same interval just, so minimax is harder.  I'm sure all these problems can be 
overcome, but they are problems.

Paul:

> I think Gene has convinced be that they won't work. The only way you 
> can possibly distinguish cases of torsion correctly is with the 
> octave-specific mapping.


I haven't seen that proven yet.  Let's get an algorithm first, and see if it 
doesn't work.  Where do you think torsion is a problem?  An octave-equivalent 
mapping can do everything a wedge product can.  You can add a parameter if you want 
to distinguish torsion from equal divisions of the octave.  In going from unison 
vectors to a mapping, torsion might show up as a common factor in the adjoint where 
it's a problem.  I haven't even got round to checking yet.  Pairs of ETs with 
torsion don't work with wedge products either.  It may be that the sign of the 
mapping can be used to disambiguate them.  Otherwise, give the range of generators 
as part of the definition.



> Fokker didn't run into any cases of torsion, but we have! The paper 
> can cover Fokker's methods but doesn't need to be restricted to them.


Wouldn't it be nice to say whether or not Fokker's methods would have worked if he 
had run into torsion?


                        Graham

Message: 3016 - Contents - Hide Contents

Date: Sun, 6 Jan 2002 15:36 +00

Subject: Paper (was Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVEE

From: graham@xxxxxxxxxx.xx.xx

Paul wrote:


> The proof that MOSs are linear might be said to be published. The 
> periodicity block concept was of course published by Fokker, though 
> the explanation of periodicity blocks might better take off from this 
> starting point, which you are all welcome to suggest changes to:
> 
> A gentle introduction to Fokker periodicity bl... * [with cont.]  (Wayb.)
> 
> As for the rest, I'm fairly certain it's entirely new work.


C Karp's "Analyzing Musical Tuning Systems" from Acustica Vo.54 (1984) 
should be considered.  He uses octave-specific, 5-limit matrices, 
including some inverses.  He does say, p.212, "... the temperament vector 
of any interval (a, b, c)_t, is associated with the c/b comma division 
temperament" and works through examples for fractional meantones.

Brian McLaren sent me a copy, in the days when he deigned to recognize 
mathematical theory.  It acknowledges one "Bob Marvin, who devised the 
matrix representation of tuning systems used here, and introduced it to 
the author."


                           Graham

Message: 3017 - Contents - Hide Contents

Date: Sun, 6 Jan 2002 13:22:31

Subject: Re: please simplify equation

From: monz

Hi Gene,



> From: genewardsmith <genewardsmith@xxxx.xxx>
> : <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, January 06, 2002 3:12 AM
> Subject: [tuning-math] Re: please simplify equation
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
>> 

>> v  =  10 ^ ( LOG( 1 / (2 ^ (9r - 1/r) ) )  /  ( -15r + 2/r - 1) )
> 

> I don't know what the base of the log is, presuming it is e, we get
> 
> b = ln(2)(3r+1)/(5r+1) for the exponent, and so v = 10^b.



Thanks for doing that.  Paul and I had a long online chat last
night in which I showed him what I had derived and he simplified
things for me.  

But I tried plugging your equation into my spreadsheet, and
got the wrong results.  The base of the log is 10, but that's
irrelevant now anyway, because I see now how I'm really looking
for an exponent that goes with base 2.


This formula expresses the golden meantone "5th",
where "r" is PHI = [1 + 5^(1/2)] / 2 . 

By plugging in (1/r) = (r-1), my equation reduces to:

  2^[ (8r+1) / (13r+3) ]

And Paul gave me these equivalent simplifications of it:

= 2^[ (2r-1) / (3r-1) ]

= 2^[ (3-r) / (4-r) ] 


I plotted the numbers of all three of the above formulas
into a graph, and can see how they're all related linearly.
Can you explain algebraically what's going on?  Please
be as detailed as possible.  Thanks.


-monz

 



_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.]  (Wayb.)

Message: 3018 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 01:29:46

Subject: Re: tetrachordality

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:


> The interval pattern stuff (the L-L-s of conventional theory) is
> a relative pitch thing... ?


Not sure what you mean.

Message: 3019 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 04:45:13

Subject: Re: Optimal 5-Limit Generators For Dave

From: genewardsmith

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


>> Wouldn't it be nice to say whether or not Fokker's methods would 

> have worked if he 

>> had run into torsion?


> I'm pretty sure the answer is no. Gene?


I don't know they are. What would he have done in the case of the 24-note business which was our first example?

Message: 3020 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 07:42:55

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: paulerlich

I wrote,


> I think it's very difficult for ears with Western=trained 
categorical 
> perception not to hear it as different.


That is, because they're the 4th and the maj. 7th, and we're _used_ 
to hearing these as a characteristic dissonance. 523 is far enough 
from 500 that our Western mind can categorize the entire pentatonic 
scale as root, M3, p4, p5, M7.

Message: 3021 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 02:02:33

Subject: Re: tetrachordality

From: clumma

>> >he interval pattern stuff (the L-L-s of conventional theory) is
>> a relative pitch thing... ?
> 

>Not sure what you mean.


I'm trying to understand the psychoacoustical basis for the
version in your paper, and recent posts about ethnic scales
on the main list (x+x+y, etc.).  And I'm trying to understand
the lack, if any, of a psychoacoustical basis for my stuff
(absolute pitches being transposed by a 3:2).

-Carl

Message: 3022 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 04:50:42

Subject: Re: Some 12-tone, 2-step 46-et scales

From: genewardsmith

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


> Sure. Now are there some 46-tET commas you did not take into account? 
> You didn't answer my "hyper-torus" point on the tuning list yet . . .


I
wasn't clear what you meant, but there are topological considerations
which come into graph theory. A graph can be a planar graph, for
instance, or a graph on a quotient (cylinder or torus), so it can have
a genus--it might be a graph on something with negative curvature. I
plan on reading some graph theory and seeing if anything I run across
suggests some application.

Message: 3023 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 08:59:59

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: clumma

>> >'ve never heard a voice
> 
> Voice?


As in, part or parts in the music sharing the same rhythm.
 
-Carl

Message: 3024 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 02:08:10

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: clumma

>>> >o. It just assumes that the overtones are pretty close to
>>> harmonic, because they will then lead to the same ratio-
>>> intepretations for the fundamentals as the fundamentals by
>>> themselves. If they're 50 cents from harmonic, they will
>>> lead to a larger s value for the resulting harmonic entropy
>>> curve, but that's about it.
>> 

>> s represents the blur of the spectral components coming in.
>> How could an inharmonic timbre change that?
>

>When we're dealing with a dyad consisting of complex tones, and 
>trying to apply harmonic entropy to that dyad, s is decreased
>below the value that sine waves in place of the complex tones
>would imply.  The more inharmonic the timbre, the less s is
>decreased below the sine-wave case.


Still doesn't explain how.  You need a way for data from the
combination-sensitive stuff to improve the spectral stuff coming
off the cochlea.  I don't think it works that way.  The "accuracy"
of the "fundamental" is improved as the spectral components get
closer to just, as the harmonic entropy calc. itself correctly
models.  But to change s in this way is a fudge, in my opinion.
With harmonic timbres, h.e. on the fundamentals is a good
approximation of things, but with inharmonic timbres, all spectral
components need to be put in to the h.e. calculation.  Jacking up
s may approximate this, but it would be a fudge.

Anyway, there is now psychoacoustic evidence for harmonic entropy.
In fact, it looks like it perfectly models what happens in
populations of "combination-sensitive" neurons in the inferior
colliculus.  At least in bats.  I plan on posting to
harmonic_entropy on this as soon as I can get the citations
together.


>>>> Yes, to me, pelog sounds like a I and a III with a 4th in
>>>> the middle.  But the music seems to use a fixed tonic, with
>>>> not much in the way of triadic structure.
>>> 

>>> How about 5-limit intervals?
>> 

>> Not sure what you're asking.
>

>Not much in the way of 5-limit intervals?


I think the large 2nd approximates a 5:4, and the perfect
4th a 3:2, with some tempering to reduce the roughness of
these intervals on the instrumentation used (as opposed to
tempering to improve the consonance of these interval in
different modes, to distribute any commas, etc.).


>> Right, it's the chinese pentatonic.  I threw it in for
>> completeness.
> 

>Completeness of what?


Of the journey from North to South, and of the survey of
pentatonic scales in motivic ethnic music of southeast asia.
And it was informative; nothing about the music changed as
we went from Pelog, to the hybrid, to the chinese pentatonic
_except_ the scale.  You could transcribe the notes and wind
up with the same stuff, more or less.


>> Western music uses progressions of four fifths and expects
>> to wind up on a major third.
> 

>These don't have to be triadic, harmonic progression.


I guess not.  But there's a big difference in how this stuff
is used.  The Indonesia music is motivic, not modal.  At least,
I follow the pitches and their positions in the scale, not the
intervals of the scale and there relation to one another.  The
harmonic motion is used to render some consonance, and some
tension/release action, but that's it.  It's a backdrop to the
motivic material.


>> I didn't notice anything like this
>> for the [1 -3] map (right?) on the cited discs.
> 

>[3 1]. It's not something you should expect to hear as a triadic
>harmonic progression. It's simply the way the 5-limit intervals
>fit together in the scale. If they didn't, the scale, and the
>music that depends on it, wouldn't work.


[3 1]?  I thought these maps expressed each odd identity, from
three to the limit, increasing from left to right, in numbers
of generators.  Thus up one 3:2 for the 3:2, and down three 3:2s
for the 5:4.


>> But I think a lot of the other stuff that goes along with
>> harmonic music is missing from this music.  Western music
>> requires meantone.  The pelog 5-limit map is far more extreme,
>> but what suffers in this music as we change the tuning from
>> 5-of- 7, to 23, to 16, all the way to strict JI?
> 

>23 and 16 give you the Pelog sound. 7 doesn't.


By gods, you're right!  7-of-5 doesn't sound like pelog at all.


>Give me a strict JI scale to try.


1/1 5/4 4/3 3/2 15/8

Sounds like a fine pelog to me.


>> I think the tuning on these discs is closer to JI than 23-tET,
>> and I don't hear them avoiding a disjoint interval.  Do you?
> 

>Avoiding a disjoint interval? You mean you hear it as 5-of-7?
>It modulates that much??


Actually, it doesn't.  They seem to stick mostly to I, IV, and
III (diatonic) with the bass, if you consider those tonics.  But
the melodic stuff does center itself on every degree of the
scale -- it treats the "bad" 4ths the same as the perfect 4ths.

To rephrase the question one more time, in what sense are these
bass notes tonics?  Do they change anything about the melody?
That is, what used to be scale degree 4 is now 1?  I say they
don't.  What I hear is a fixed 1.  The melody is a very slow
series of scale degrees above that 1.  On each note of the melody,
a bunch of ornamentation is hung, which is made of scale arpeggio
bits.  The bass starts and ends on 1, and goes to 3, 2, and
sometimes 4 (I-IV-III-V diatonic), to provide a sense of
tension/resolution.


>> Incidentally, I think Wilson agrees with your point of view
>> here.  While he does caution against eager interps. of his
>> ethno music theory, I think he thinks that harmonic mapping
>> is inevitable, and atomic in music.  I'm not sure I agree.
>> Not sure I disagree.
> 

>Well, the idea of this paper that Gene, Dave, Graham, and I
>are working on, at least it seems to me, is to start with the
>assumption that notes are connected to one another via simple-
>ratio intervals, explain periodicity blocks, show that an MOS
>results when you temper out all but one of the unison vectors,
>show that MOSs are linear, and present the "best" linear
>temperaments from this point of view. It's just a paper, not a
>manifesto, so there's nothing wrong with starting with a very
>simple and strong set of assumptions, and seeing where they lead.


Of course!  (I already can't wait!)

-Carl

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