Mt. Meru

>

Hello Graham~
Each of the diagrams shows the recurrent sequence in which is generated by the diagonal. Possibly an example will
illustrate more than anything else. Taking the slendro that i use
page 5 and 6
http://anaphoria.com/meruone.PDF
I happen to have tuned which now amounts to a small orchestra to ta 12 tone sequence starting at 7 and going through
200 (harmonics). obviously this is further out than the indonesian do , but if one has 12 tone instruments to retune and
the scale makes a good MOS (actually a constant structure at this point) it seemed unavoidable. I have with my marion
Prosynth taken it out to 37 places just to hear

one can see how the differance tones of the pitches
generates tones within the scale 200-114= 86, 151-86=65 and so forth. one could just as easily have them generated by
the sum tone also 7+9=16, 12+16=21 etc. One need not confine oneself to the numbers that come off Mt. Meru. One can
'seed' this formula how ever one wishes. For instance instead of using 7,9,12 we could start with 10,13,17 or any three
harmonics one wishes (or subharmonics if one wished). In this way each person can generate their own slendro in the same
way each village does now. True they will all evindually converge on the same point, but ihave noticed with most of
these scales the most interesting points are where the series hasn't quite gotten there. it also gives each subset its
own flavor.

Each of the diagonals can be used in a simular fashion. Interestingly after your classic fibonacci series the first
two that come off Mt. Meru are scales close to Slendro and Pelog. This is the only place where i have seen such a a
relationship between the two. Before Erv's discovery of these in Mt. Meru, i compared the slendros within Kunst's book
and found a this recurrent sequence with slight error average leaning toward 2 vibrations off, but we only have a one
octave sample. i can't seem to find these papers atthe moment, but had sent a copy to Lou Harrison. In return Lou had
sent me his paper "Slippery Slendro". The first two version that lou suggest in this paper are the two that are very
close to those come out of Mt Meru in the same order. first his 7,9,12,16,21,28 and as he has a 49 next this occurs two
steps up but in Mt. meru the pentatonic uses a 37 instead of a 36. thereby having the the equal beating triad
32,37,42. The 49 giving you 42, 49, 56. this last triad being the first in this particular manifestation where the
differance tone is a pitch in the scale.

>
> From: Graham Breed <graham@microtonal.co.uk>
>
>
> kraig grady wrote:
>
> >both these scales are easily explained as recurrent series of differance tones, along with slendro. In fact they are
> >the 2nd 3rd, and fourth scales that Mt. Meru Produces. his is a very high coincidence in itself. The latter Chopi
> >scales i have exchanged data with Andrew Tracy, hugh's son who collected the data, and he was quite impressed with
> >how well the numbers fell into place. How this happens isthe big mystery now. but differance tones are as easily to
> >follow and tune to as the femoving of beats.
> >
> >
> Great! Can you explain it, then? The links to difference tones and Mt
> Meru. From the Wilson Archive, Mt Meru seems to involve generalizations
> of Pascal's triangle and continued fraction expansions, but no musical
> scales or difference tones.
>
> Graham
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> >
>
> Hello Graham~
> Each of the diagrams shows the recurrent sequence in which is
generated by the diagonal. Possibly an example will
> illustrate more than anything else. Taking the slendro that i use
> page 5 and 6
> http://anaphoria.com/meruone.PDF
> I happen to have tuned which now amounts to a small orchestra to
ta 12 tone sequence starting at 7 and going through
> 200 (harmonics). obviously this is further out than the indonesian
do , but if one has 12 tone instruments to retune and
> the scale makes a good MOS (actually a constant structure at this
point) it seemed unavoidable. I have with my marion
> Prosynth taken it out to 37 places just to hear
>
>
> one can see how the differance tones of the pitches
> generates tones within the scale 200-114= 86, 151-86=65 and so

this seems akin to how jacques dudon conceptualizes scales (jacques,
you still with us?)

>True they will all evindually converge on the same point,

i like that word.

>this last triad being the first in this particular manifestation
>where the
> differance tone is a pitch in the scale.

kraig grady wrote:

> Each of the diagrams shows the recurrent sequence in which is generated by the diagonal. Possibly an example will
>illustrate more than anything else. Taking the slendro that i use
> page 5 and 6
> http://anaphoria.com/meruone.PDF
> I happen to have tuned which now amounts to a small orchestra to ta 12 tone sequence starting at 7 and going through
>200 (harmonics). obviously this is further out than the indonesian do , but if one has 12 tone instruments to retune and
>the scale makes a good MOS (actually a constant structure at this point) it seemed unavoidable. I have with my marion
>Prosynth taken it out to 37 places just to hear
> >
Well, page 5 is slicing Pascal's triangle to get the recursive sequence

C_n = C_n-3 + C_n-2

starting with 1, 1, 1. Then it shows adjacent terms converge to 1.32... and on page 6 the continued fraction expansion of the logarithm to base 2 is given. Which presumably means that your taking MOSs getting closer to 0.406... octaves as the generator. So how is this slendro, other than being close to 5-equal?

I thought every MOS was a constant structure.

>one can see how the differance tones of the pitches
>generates tones within the scale 200-114= 86, 151-86=65 and so forth. one could just as easily have them generated by
>the sum tone also 7+9=16, 12+16=21 etc. One need not confine oneself to the numbers that come off Mt. Meru. One can
>'seed' this formula how ever one wishes. For instance instead of using 7,9,12 we could start with 10,13,17 or any three
>harmonics one wishes (or subharmonics if one wished). In this way each person can generate their own slendro in the same
>way each village does now. True they will all evindually converge on the same point, but ihave noticed with most of
>these scales the most interesting points are where the series hasn't quite gotten there. it also gives each subset its
>own flavor.
> >
I can see how those arithmetical equations are true, but not what they have to do with slendro. When you say "Mt. Meru" you mean Pascal's triangle, right? It looks like all it's doing is supplying a few numbers, so you may as well dispense with it altogether. I've never seen any evidence of a village slendro being constructed as an MOS.

> Each of the diagonals can be used in a simular fashion. Interestingly after your classic fibonacci series the first
>two that come off Mt. Meru are scales close to Slendro and Pelog. This is the only place where i have seen such a a
>relationship between the two. Before Erv's discovery of these in Mt. Meru, i compared the slendros within Kunst's book
>and found a this recurrent sequence with slight error average leaning toward 2 vibrations off, but we only have a one
>octave sample. i can't seem to find these papers atthe moment, but had sent a copy to Lou Harrison. In return Lou had
>sent me his paper "Slippery Slendro". The first two version that lou suggest in this paper are the two that are very
>close to those come out of Mt Meru in the same order. first his 7,9,12,16,21,28 and as he has a 49 next this occurs two
>steps up but in Mt. meru the pentatonic uses a 37 instead of a 36. thereby having the the equal beating triad
>32,37,42. The 49 giving you 42, 49, 56. this last triad being the first in this particular manifestation where the
>differance tone is a pitch in the scale.
> >
All you have is a relationship between a scale that's a bit like 5-equal, and one that's a bit like 7-equal. Your "pelog" seems to be the pelogic 4/9 MOS, which has little relationship with any observed pelog, even if it matches the theory.

The first to versions of what follow this pattern?

Ah, "differance" again!

Graham

--- In tuning@yahoogroups.com, Graham Breed <graham@m...> wrote:
> kraig grady wrote:
>
> > Each of the diagrams shows the recurrent sequence in which is
generated by the diagonal. Possibly an example will
> >illustrate more than anything else. Taking the slendro that i use
> > page 5 and 6
> > http://anaphoria.com/meruone.PDF
> > I happen to have tuned which now amounts to a small orchestra to
ta 12 tone sequence starting at 7 and going through
> >200 (harmonics). obviously this is further out than the indonesian
do , but if one has 12 tone instruments to retune and
> >the scale makes a good MOS (actually a constant structure at this
point) it seemed unavoidable. I have with my marion
> >Prosynth taken it out to 37 places just to hear
> >
> >
> Well, page 5 is slicing Pascal's triangle to get the recursive
sequence
>
> C_n = C_n-3 + C_n-2
>
> starting with 1, 1, 1. Then it shows adjacent terms converge to
1.32...
> and on page 6 the continued fraction expansion of the logarithm to
base
> 2 is given. Which presumably means that your taking MOSs

if i recall correctly, we already had this whole conversation on the
specmus list, and no, kraig's not taking MOSs.

> be the
> pelogic 4/9 MOS, which has little relationship with any observed
>pelog,
> even if it matches the theory.

matches what theory? why would you say it has little relationship
with any observed pelog -- don't you just mean that it's not exactly
matched by any observed pelog? it's certainly a better fit than any
other so simply defined scale.

>I thought every MOS was a constant structure.

Since when? Isn't the diatonic scale in 12-equal a
counterexample?

>Ah, "differance" again!

Oh no! :)

-Carl

>
> From: Graham Breed <graham@microtonal.co.uk>
> Subject: Re: Mt. Meru
>
> ucture at this point) it seemed unavoidable. I have with my marion
> >Prosynth taken it out to 37 places just to hear
> >
> >
> Well, page 5 is slicing Pascal's triangle to get the recursive sequence
>
> C_n = C_n-3 + C_n-2
>
> starting with 1, 1, 1. Then it shows adjacent terms converge to 1.32...
> and on page 6 the continued fraction expansion of the logarithm to base
> 2 is given. Which presumably means that your taking MOSs getting closer
> to 0.406... octaves as the generator. So how is this slendro, other than
> being close to 5-equal?

well it is actually called meta-slendro. First you can listen to it and tell me if it sounds like slendro to you. One
balinese wanted to know what village it was copied from. secoundly you can seed it with any ratios or fractions that you want.
Unlike other versions of slendro generated by other means , this produces variation in the size of the intervals, which is
what you find from actual field measurements. 5-equal has mnothing to do with Slendro. there are no slewndro of equal size
intervals

>
>
> I thought every MOS was a constant structure.

it is , what is the confusion.

>
>
> >one can see how the differance tones of the pitches
> >generates tones within the scale 200-114= 86, 151-86=65 and so forth. one could just as easily have them generated by
> >the sum tone also 7+9=16, 12+16=21 etc. One need not confine oneself to the numbers that come off Mt. Meru. One can
> >'seed' this formula how ever one wishes. For instance instead of using 7,9,12 we could start with 10,13,17 or any three
> >harmonics one wishes (or subharmonics if one wished). In this way each person can generate their own slendro in the same
> >way each village does now. True they will all evindually converge on the same point, but ihave noticed with most of
> >these scales the most interesting points are where the series hasn't quite gotten there. it also gives each subset its
> >own flavor.
> >
> >
> I can see how those arithmetical equations are true, but not what they
> have to do with slendro. When you say "Mt. Meru" you mean Pascal's
> triangle, right?

Mt. Meru is the older name predating Pascal by maybe 2,000 years. There are figures of mt. meru in india and representations
of the mythical mountain of that name in indonesia. there are no records of Pascal's triangle in that part of the world until
recently.

> It looks like all it's doing is supplying a few
> numbers, so you may as well dispense with it altogether.

why

> I've never seen
> any evidence of a village slendro being constructed as an MOS.

the structure is a recurrent sequence and a constant structure if even the term could apply since you are talking about every
interval a different size.

>
>
> > Each of the diagonals can be used in a simular fashion. Interestingly after your classic fibonacci series the first
> >two that come off Mt. Meru are scales close to Slendro and Pelog. This is the only place where i have seen such a a
> >relationship between the two. Before Erv's discovery of these in Mt. Meru, i compared the slendros within Kunst's book
> >and found a this recurrent sequence with slight error average leaning toward 2 vibrations off, but we only have a one
> >octave sample. i can't seem to find these papers atthe moment, but had sent a copy to Lou Harrison. In return Lou had
> >sent me his paper "Slippery Slendro". The first two version that lou suggest in this paper are the two that are very
> >close to those come out of Mt Meru in the same order. first his 7,9,12,16,21,28 and as he has a 49 next this occurs two
> >steps up but in Mt. meru the pentatonic uses a 37 instead of a 36. thereby having the the equal beating triad
> >32,37,42. The 49 giving you 42, 49, 56. this last triad being the first in this particular manifestation where the
> >differance tone is a pitch in the scale.
> >
> >
> All you have is a relationship between a scale that's a bit like
> 5-equal, and one that's a bit like 7-equal.

i think one needs to listen to it. if you take it out to 12 places you can use the scale just like they do with the Wayang
kulit. With one pentatonic tuned slightly higher than another. So 10 tones of my series could be used , and is used in just
this musical way.
We know that Slendro comes from China and we know that the chinese has been pouring over this triangle for centuries. If it is
a coincidence , it is a great one. On the other hand we have no examples of chinese attempting to create 5 of 7 equal.
hypothetically, this scales allow also for all types of variations. slendro is many scales, just like this series produce.
Possibly the first slendroes might have been tuned thus way with time allowing even further variations into what we have
today. Like i mentioned , i found that i got really good answers from kunst's measurements along these lines.
Quite a few people have pointed out as Pelog being a cycle of 9 tones with usually only 7 tuned up. Dan Wolf has one in
one of the Xenharmonikons.

>
>
> Graham
>

North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, Graham Breed <graham@m...> wrote:
> kraig grady wrote:

> Well, page 5 is slicing Pascal's triangle to get the recursive
sequence
>
> C_n = C_n-3 + C_n-2

What does this have to do with Pascal's triangle? It's a cubic
recursion.

Paul Erlich wrote:

>if i recall correctly, we already had this whole conversation on the >specmus list, and no, kraig's not taking MOSs.
> >
Oh yes, I see you tried hard to get to the bottom of it. As those archives aren't public, and there's no concise explanation anyway, I'll try and explain it as I now understand it.

Pascal's Triangle (Mt Meru) is only used to get what Mathworld seems to call a linear recurrence equation:

http://mathworld.wolfram.com/LinearRecurrenceEquation.html

These always have X[n] depending on two previous entries with coefficients of 1 or -1. So the first is the usual Fibonacci sequance of A[n] = A[n-2] + A[n-1]. Then B[n] = B[n-3] + B[n-1] which must be the Chopi scale, C[n] = C[n-3] + C[n-2] which is supposed to be slendro and D[n] = D[n-4] + D[n-1] which is sort of pelogic (it converges the other side of 9-equal).

The significance of these series may be that they specify frequency ratios for a scale to be tuned up. In which case the equation is specifying that the difference tone between two particular notes in the scale gives another note in the scale. Hence, with the general form

X[n] = X[n-a] + X[n-b]

that can be rearranged as

X[n] - X[n-b] = X[n-a]

so the difference tone of the fundamentals of two notes b steps apart gives the note a steps below the higher one. For this to make sense, you have to have 0<b<a, so I don't think there's any loss of generality. You could consider more terms, but it'd mean more complex patterns of difference tones. You can also start with different seeds, which I think Kraig is aware of but doesn't give as examples.

Each sequence approaches an MOS as the terms approach infinity. That means that adjacent terms approach a constant ratio. This result doesn't depend on the seeds. Going back to the general form

X[n] = X[n-a] + X[n-b]

the converging generator is a solution to the polynomial

x = 1/x**a + 1/x**b

where ** is exponentiation, or equivalently

x**a = 1 + x**(a-b)

If you consider octaves as an equivalence interval, than you can introduce powers of 2 as coefficients. There are a lot more choices here, but the obvious example is metameantone:

x**4 = 2x - 2

which can also be given as

X[n] = 2X[n-3] + 2X[n-4]

Some examples are

1, 1, 1, 1, 4, 4, 4, 10, 16, 16, 28, 52, 64, 88, 160, 232, 304, 496, 784, 1072, 1600, ...

0, 0, 0, 1, 0, 0, 2, 2, 0, 4, 8, 4, 8, 24, 24, 24, 64, 96, 96, 176, 320, 384, 544, 992, ...

8, 12, 18, 27, 40, 60, 90, 134, 200, 300, 448, 668, 1000, 1496, 2232, 3336, 4992 ...

The last one is seeded with JI fifths.

>>be the >>pelogic 4/9 MOS, which has little relationship with any observed >>pelog, >>even if it matches the theory.
>> >>
>
>matches what theory? why would you say it has little relationship >with any observed pelog -- don't you just mean that it's not exactly >matched by any observed pelog? it's certainly a better fit than any >other so simply defined scale.
> >
The pelogic one, reversing large and small steps from a diatonic scale. I mean it isn't any closer in general than an arbitrary 7 from 9 tuning, and if somebody has an especially close match they can come forward with it. If no other so simply defined scale is so close a fit, then pelog isn't so simply defined. Kraig said he had pelog, not a simple approximation to pelog.

Graham

kraig grady wrote:

> well it is actually called meta-slendro. First you can listen to it and tell me if it sounds like slendro to you. One
>balinese wanted to know what village it was copied from. secoundly you can seed it with any ratios or fractions that you want.
>Unlike other versions of slendro generated by other means , this produces variation in the size of the intervals, which is
>what you find from actual field measurements. 5-equal has mnothing to do with Slendro. there are no slewndro of equal size
>intervals
> >
It sounds close to 5-equal, like other slendros.

Any other version of slendro except 5-equal itself will give variation in the size of the intervals. Like slendric, where the generator is approximately 8:7. Or taking 5-equal and moving each note by a small, random amount.

5-equal has everything to do with slendro. The deviations from one slendro to another are greater than their deviations from 5-equal, and nobody's found any other pattern. Saying that none of them are exactly the same doesn't negate this at all.

>>I thought every MOS was a constant structure.
>> >>
>
> it is , what is the confusion.
> >
Your phrasing ("actually a constant structure at this point") implied there are other points where it isn't a constant structure. It would have been clearer, and also shorter, if you'd said "and thus a constant structure".

>Mt. Meru is the older name predating Pascal by maybe 2,000 years. There are figures of mt. meru in india and representations
>of the mythical mountain of that name in indonesia. there are no records of Pascal's triangle in that part of the world until
>recently.
> >
As it's commonly known in English as Pascal's triangle, it'd ease understanding if you mentioned this, as "Mt Meru" is quite an obscure alternative.

I see you've avoided answering the more important question, but I think I've worked it out -- see my other message.

>>It looks like all it's doing is supplying a few
>>numbers, so you may as well dispense with it altogether.
>> >>
>
>why
> >
Occam's razor.

>>I've never seen
>>any evidence of a village slendro being constructed as an MOS.
>> >>
>
>the structure is a recurrent sequence and a constant structure if even the term could apply since you are talking about every
>interval a different size.
> >
It may be a constant structure -- I'm not sure what that would mean in the context. But there are plenty of constant structures that aren't MOSs. Is "recurrent sequence" the same as "linear recurrence equation"? What evidence do you have of a fit there?

>i think one needs to listen to it. if you take it out to 12 places you can use the scale just like they do with the Wayang
>kulit. With one pentatonic tuned slightly higher than another. So 10 tones of my series could be used , and is used in just
>this musical way.
>
If you take what out to 12 places? What scale? How does it relate to Wayang Kulit? References I've found only mention pelog 7 and slendro 5.

>We know that Slendro comes from China and we know that the chinese has been pouring over this triangle for centuries. If it is
>a coincidence , it is a great one. On the other hand we have no examples of chinese attempting to create 5 of 7 equal.
>
Where's your evidence for slendro coming from China?

AFAICT, the Chinese seem to have used Pascal's triangle for combinatorics, the same as everybody else. In which case they wouldn't have been looking at the same diagonals as you, let along giving them significance. If any of this is relevant, it's more likely they were tuning by the difference tones, or maybe playing with recursive sequences.

It isn't much of a coincidence at all. The number 5 isn't a statistically significant match.

The Thais, who are roughly between China and Java, tune close to 7-equal.

>hypothetically, this scales allow also for all types of variations. slendro is many scales, just like this series produce.
>Possibly the first slendroes might have been tuned thus way with time allowing even further variations into what we have
>today. Like i mentioned , i found that i got really good answers from kunst's measurements along these lines.
>
If you mean any section of any sequence, there are so many variations that you can probably get a match for anything you like around 5-equal.

> Quite a few people have pointed out as Pelog being a cycle of 9 tones with usually only 7 tuned up. Dan Wolf has one in
>one of the Xenharmonikons.
> >
I meant "a bit like 9-equal" at one point, rather than 7-equal.

Graham

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> Mt. Meru is the older name predating Pascal by maybe 2,000 years.

Pascal's triangle was discovered numerous times. The name for it is
still Pascal's triagle, and you really should use it.

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

> These always have X[n] depending on two previous entries with
> coefficients of 1 or -1.

What makes you say that? That defines a quadratic recurrence of a
special kind.

So the first is the usual Fibonacci sequance
> of A[n] = A[n-2] + A[n-1].

Qudratic, with characteristic polynomial x^2 - x - 1

This is
Then B[n] = B[n-3] + B[n-1] which must be
> the Chopi scale, C[n] = C[n-3] + C[n-2] which is supposed to be
slendro
> and D[n] = D[n-4] + D[n-1] which is sort of pelogic (it converges
the
> other side of 9-equal).

None of these are two-term recurrences, since you have left off the
coefficients of zero.

B[n] = 1 * B[n-1] + 0 * B[n-2] + 1 * B[n-3]
x^3 - x^2 - 1

C[n] = 0 * C[n-1] + 1 * C[n-2] + 1 * C[n-3]
x^3 - x - 1

D[n] = 1 * D[n-1] + 0 * D[n-2] + 0 * D[n-3] + 1 * D[n-4]
x^4 - x^3 - 1

> x**a = 1 + x**(a-b)

Or if c = a-b, even easier it is

x^a - x^c - 1

as a characteristic polynomial.

>
> If you consider octaves as an equivalence interval, than you can
> introduce powers of 2 as coefficients. There are a lot more
choices
> here, but the obvious example is metameantone:
>
> x**4 = 2x - 2

I've been calling this Wilson meantone; metameantone being the
recurrence relationship meantones--I think.

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

> http://mathworld.wolfram.com/LinearRecurrenceEquation.html

Here's something else from Weisstein which comes up in this
connection:

http://mathworld.wolfram.com/Pisot-VijayaraghavanConstant.html

The adventurous are invited to consider the musical uses of the likes
of

x^20-2x^19+x^17-x^16+x^14-x^13+x^11-x^9+x^7-x^6+x^4-x^3+x-1

as a characteristic polynomial for a linear recurrence.

>
>
> Pascal's Triangle (Mt Meru) is only used to get what Mathworld seems to
> call a linear recurrence equation:
>
> http://mathworld.wolfram.com/LinearRecurrenceEquation.html

interesting, we can find no referance to other diagonals besides the fibonacci series coming off Mt. Meru or pascals triangle.
if you happen to run accross one, please inform. otherwise, if nothing else, an attribute of this triangle has been discovered.

>
> The pelogic one, reversing large and small steps from a diatonic scale.
> I mean it isn't any closer in general than an arbitrary 7 from 9 tuning,
> and if somebody has an especially close match they can come forward with
> it. If no other so simply defined scale is so close a fit, then pelog
> isn't so simply defined. Kraig said he had pelog, not a simple
> approximation to pelog.

slip of words. it should be meta pelog. Erv has an older sequence for Pelog he discovered before
the other diagonals of which i hope to put up soon also.

>
>
> Graham
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 9
> Date: Thu, 28 Aug 2003 22:43:55 +0100
> From: Graham Breed <graham@microtonal.co.uk>
> Subject: Re: Mt. Meru
>
> kraig grady wrote:
>
> > well it is actually called meta-slendro. First you can listen to it and tell me if it sounds like slendro to you. One
> >balinese wanted to know what village it was copied from. secoundly you can seed it with any ratios or fractions that you want.
> >Unlike other versions of slendro generated by other means , this produces variation in the size of the intervals, which is
> >what you find from actual field measurements. 5-equal has mnothing to do with Slendro. there are no slewndro of equal size
> >intervals
> >
> >
> It sounds close to 5-equal, like other slendros.
>
> Any other version of slendro except 5-equal itself will give variation
> in the size of the intervals.

there is no examples of a 5 equal slendro. nor do the indonesians claim they are making a 5 equal. Mc Phee has plently written a
bit on this.

> Like slendric, where the generator is
> approximately 8:7. Or taking 5-equal and moving each note by a small,
> random amount.

In the range i have tuned Meta slendro up, you will find quite a few 8/7 intervals

>
>
>
> >
> Your phrasing ("actually a constant structure at this point") implied
> there are other points where it isn't a constant structure. It would
> have been clearer, and also shorter, if you'd said "and thus a constant
> structure".

yes sometimes i am not clear

>
> >
> >
> As it's commonly known in English as Pascal's triangle, it'd ease
> understanding if you mentioned this, as "Mt Meru" is quite an obscure
> alternative.

the point is Mt. Meru is apart of the same culture as Slendro and Pelog, it makes sense to use it

>
>
> >
> >
> It may be a constant structure -- I'm not sure what that would mean in
> the context. But there are plenty of constant structures that aren't
> MOSs.

correct but i am not understanding this line of thought

> Is "recurrent sequence" the same as "linear recurrence
> equation"?

it appears so by Paul's post

> What evidence do you have of a fit there?

with slendro. Measurements in Kunst, but i worked these out possibly 7 or 8 years ago

>
>
> >i think one needs to listen to it. if you take it out to 12 places you can use the scale just like they do with the Wayang
> >kulit. With one pentatonic tuned slightly higher than another. So 10 tones of my series could be used , and is used in just
> >this musical way.
> >
> If you take what out to 12 places?

Meta Slendro

> What scale? How does it relate to
> Wayang Kulit?

Wayang kulit has two set of instruments tuned to two pentatonics one slightly higher than the other. I can extract 10 pitches
out of 12 and get just such an animal. there is no way to get at such animal by e equal

> References I've found only mention pelog 7 and slendro 5.
>
> >We know that Slendro comes from China and we know that the chinese has been pouring over this triangle for centuries. If it is
> >a coincidence , it is a great one. On the other hand we have no examples of chinese attempting to create 5 of 7 equal.
> >
> Where's your evidence for slendro coming from China?

the balinese says it was brought by a chinese princess

>
>
> AFAICT, the Chinese seem to have used Pascal's triangle for
> combinatorics, the same as everybody else. In which case they wouldn't
> have been looking at the same diagonals as you, let along giving them
> significance.

It is hard to say what the have not look at

> If any of this is relevant, it's more likely they were
> tuning by the difference tones, or maybe playing with recursive sequences.

I agree this is less spectulative

> It isn't much of a coincidence at all. The number 5 isn't a
> statistically significant match.

look at the scales in kunst

>
>
> The Thais, who are roughly between China and Java, tune close to 7-equal.

this is approximated by the next series, as are the Chopi scales. By the way i do have one up which shows the clossness of the
differance tones with deviations in VPS. since cents is really not something we here as just that.
http://anaphoria.com/chopi.gif
the other older measurements of Hugh Tracy fit much better but the problem is that his measurements were done comparing the
instruments against tuning forks 4 vibrations apart. This is the most acturate measurement available of any chopi instrument sent
to me by Andrew tracy quite a few years ago. I have tried to be tough on the measurements above, but tuning such short decay
instruments more acturately would be quite hard. That upper range tones would be 10 vibrations off might be the limit of what one
could do. As i stated with the Kunst measurements , i found a consistent deviation of 2 vibrations throughout. which implies that
it is also quite possible that such slight deviations are deliberate for musical reasons , not known to us, or is influenced by
the timbre which migh also have some bearing on the above.

>
>
> >hypothetically, this scales allow also for all types of variations. slendro is many scales, just like this series produce.
> >Possibly the first slendroes might have been tuned thus way with time allowing even further variations into what we have
> >today. Like i mentioned , i found that i got really good answers from kunst's measurements along these lines.
> >
> If you mean any section of any sequence, there are so many variations
> that you can probably get a match for anything you like around 5-equal.

Then i will pose this question. Tuners are tuning to something . this something would would imagine would be some form of
acoustical phenomenon. What are they tuning too then?

>
>
> > Quite a few people have pointed out as Pelog being a cycle of 9 tones with usually only 7 tuned up. Dan Wolf has one in
> >one of the Xenharmonikons.
> >
> >
> I meant "a bit like 9-equal" at one point, rather than 7-equal.

Like i said i should have used the term metapelog. It appears that pelog is quite a bit more complecated in it being a wider
family of scales ad might be constructed in more than one way.

>
>
> Graham
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 10
> Date: Thu, 28 Aug 2003 22:14:30 -0000
> From: "Gene Ward Smith" <gwsmith@svpal.org>
> Subject: Re: Mt. Meru
>
> --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> > Mt. Meru is the older name predating Pascal by maybe 2,000 years.
>
> Pascal's triangle was discovered numerous times. The name for it is
> still Pascal's triagle, and you really should use it.

why when he invented 2,000 years after the people of india. if it had been invenrtewd by a white person 2,00 years before pascal,
no one would suggest such a thing. Also we are not talking about Pascal cause he is outside the culture we are talking aout.

>
>
> _
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
>
> >why would you say it has little relationship
> >with any observed pelog -- don't you just mean that it's not
exactly
> >matched by any observed pelog? it's certainly a better fit than
any
> >other so simply defined scale.
> >
> >
> The pelogic one, reversing large and small steps from a diatonic
scale.
> I mean it isn't any closer in general than an arbitrary 7 from 9
tuning,

what do you mean? like LLsssss, for example??

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

> > it is , what is the confusion.
> >
> >
> Your phrasing ("actually a constant structure at this point")
implied
> there are other points where it isn't a constant structure.

it didn't imply that to me. the context was

"the scale makes a good MOS (actually a constant structure at this
point)"

which implied to me that full MOS-hood had note yet been reached,
since we're still in the finite stages of the recursion.

> It would
> have been clearer, and also shorter, if you'd said "and thus a
constant
> structure".

no.

> But there are plenty of constant structures that aren't
> MOSs.

yes, such as these.

> AFAICT, the Chinese seem to have used Pascal's triangle for
> combinatorics, the same as everybody else. In which case they
wouldn't
> have been looking at the same diagonals as you, let along giving
them
> significance. If any of this is relevant, it's more likely they
were
> tuning by the difference tones, or maybe playing with recursive
sequences.
>
> It isn't much of a coincidence at all. The number 5 isn't a
> statistically significant match.
>
> The Thais, who are roughly between China and Java, tune close to 7-
equal.

graham, i lose your logic here, as i often do.

Paul Erlich wrote:

>it didn't imply that to me. the context was
>
>"the scale makes a good MOS (actually a constant structure at this >point)"
>
>which implied to me that full MOS-hood had note yet been reached, >since we're still in the finite stages of the recursion.
> >
Ah, yes, that makes some sense.

>>The Thais, who are roughly between China and Java, tune close to 7-
>> >>
>equal.
>
>graham, i lose your logic here, as i often do.
> >
I meant that, if it's relevant that the Chinese didn't have 7-equal, it's also worth mentioning that the Thais roughly do. If a scale can make its way from China to Java and Bali, it can get there from Thailand much more easily. Not that it matters, as 7-equal is only a red herring here.

Graham

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

> >>The Thais, who are roughly between China and Java, tune close to
7-
> >>
> >>
> >equal.
> >
> >graham, i lose your logic here, as i often do.
> >
> >
> I meant that, if it's relevant that the Chinese didn't have 7-
equal,
> it's also worth mentioning that the Thais roughly do.

if you say so. i lost your logic for the whole paragraph i quoted,
though, not just this last part.

Paul Erlich wrote:

>if you say so. i lost your logic for the whole paragraph i quoted, >though, not just this last part.
> >
Oh, well, I'll pick through it then

> AFAICT, the Chinese seem to have used Pascal's triangle for > combinatorics, the same as everybody else. In which case they wouldn't
> have been looking at the same diagonals as you, let along giving them
> significance. If any of this is relevant, it's more likely they were
> tuning by the difference tones, or maybe playing with recursive sequences.

The reference for the history of Pascal's triangle and combinatorics is

http://math.truman.edu/~thammond/history/YangHui.html

Having a use for Pascal's triangle doesn't mean you'll end up finding these particular sequences in it. It's easier to find them from scratch.

If somebody were tuning a scale by fourths or fifths, they might notice that some of the difference tones were close to notes in the scale. If they altered the generators to make the match exact, then they'd end up with the pelog and slendro variants in question. If this pattern of scales has a significance, that's likely to be the reason.

> It isn't much of a coincidence at all. The number 5 isn't a > statistically significant match.

The meta-slendro is only like slendro because it's like 5-equal. The convergent generator is 1.0284 steps of 5-equal. As any random generator must fit 5-equal to within half a step, there's a 5.7% chance of getting this close by accident. If you randomly choose 8 generators, there's a 37% chance that one of them will be this close. So it isn't much of a coincidence that a method for producing generators should produce something like 5-equal. If you don't get one prominently, all you have to do is try a different method, or give a significance to some other generator.

For pelogic, you probably want a generator between 3/7 and 5/11 octaves. As all generators are equivalent within a tritone, this covers 2*(5/11-3/7) = 2*(35-33)/77 = 4/77. So there's around a 5% chance a randomly chosen generator will give something like pelogic. Out of 7 randomly chosen generators, there's a 31% chance one of them will be within this range. If you come up with three different ways of producing 8 generators that include something like 5-equal, you can expect one of them to also have a pelogic. Better than that if you can skew it so that they're more likely to be around a tritone than around an octave or unison (I don't know if that's the case here). It still isn't much of a coincidence. Around 12% of sets of 8 randomly chosen generators will have both pelogic and something like 12-equal.

Graham

Paul Erlich wrote:

>what do you mean? like LLsssss, for example??
> >
I mean there seems to be a 9 note template behind it, and the 7 notes given are in the Pelogic pattern (like 1 1 2 1 1 1 2 or 1 1 2 1 1 2 1 or something else). But the intervals of 1 step may be larger or smaller than 1/9 octaves, and the intervals of 2 steps may be larger or smaller than 2/9 octaves. So there's no consistency that suggets the underlying 9 note scale is itself a pelogic.

Anyway, here are the pelogs from the Scala archive, expressed in 9-equal to one decimal place:

Observed Javanese Pelog scale 1.0 2.3 1.0 0.8 1.0 2.1 0.8

Gamelan Saih pitu from Ksatria, Den Pasar (South Bali). 1/1=312.5 Hz
1.1 1.2 1.8 1.2 1.1 1.6 1.1

Bamboo gambang from Batu lulan (South Bali). 1/1=315 Hz
1.1 1.3 1.2 1.5 1.1 1.2 1.5

Gamelan Gong from Padangtegal, distr. Ubud (South Bali). 1/1=555 Hz 1.1 1.3 2.7 1.0 2.8

Hindu-Jav. demung, excavated in Banjarnegara. 1/1=427 Hz 1.4 1.1 1.4 1.1 1.6 1.0 1.4

Gamelan Kyahi Munggang (Paku Alaman, Jogja). 1/1=199.5 Hz 1.1 1.1 1.8 1.1 1.2 1.2 1.4

Gamelan Semar pegulingan, Ubud (S. Bali). 1/1=263.5 Hz 1.0 1.5 1.5 1.1 0.9 3.1

Gamelan Kantjilbelik (kraton Jogja). Measured by Surjodiningrat, 1972. 0.9 1.0 2.0 1.1 0.9 1.1 2.0

from William Malm: Music Cultures of the Pacific, the Near East and Asia. 0.9 1.1 2.2 0.8 0.9 1.2 1.9 1.1 1.1 2.1 1.0 0.9 1.5 1.7

Subset of 24-tET (Sumatra?) 1.5 1.1 1.1 1.5 1.5 1.1 1.1

Pelog, average class A. Kunst 1949 0.9 1.1 2.2 0.8 0.8 1.2 1.9

"Normalised Pelog", Kunst, 1949. Average of 39 Javanese gamelans 0.9 1.1 2.0 1.0 0.9 1.2 1.9

Pelog, average class B. Kunst 1949 0.9 1.0 2.0 1.0 0.8 1.3 1.9

Pelog, average class C. Kunst 1949 0.9 1.1 1.8 1.2 0.8 1.2 1.9

Chalmers' Pelog/BH Slendro
0.0 1.7 0.0 -0.2 1.9 -1.1 2.9 0.0 0.0 1.7 -0.9 2.9

"Blown fifth" pelog, von Hornbostel, type a. 1.2 1.2 1.2 1.6 1.2 1.2 1.6

"Primitive" Pelog, step of blown semi-fourths, von Hornbostel, type b. 0.8 1.2 2.0 1.2 0.8 1.2 2.0

Bill Alves JI Pelog, 1/1 vol. 9 no. 4, 1997. 1/1=293.33
1.7 0.6 1.2 1.7 0.8 1.2 1.7

Gamelan Kyahi Kanyut Mesem pelog (Mangku Nagaran). 1/1=295 Hz 0.9 1.1 1.9 1.2 0.7 1.3 1.8

Gamelan Kyahi Bermara (kraton Jogja). 1/1=290 Hz 0.8 1.0 2.0 1.2 0.7 1.3 2.0

Gamelan Kyahi Pangasih (kraton Solo). 1/1=286 Hz 1.0 1.1 2.0 0.9 0.9 1.4 1.7

New mixed gender Pelog 1.2 1.2 1.6 1.2 1.2 1.2 1.6

"Primitive" Pelog, Kunst: Music in Java, p. 28 0.8 1.2 2.0 1.2 0.8 1.2 2.0

Modern Pelog designed by Dan Schmidt and used by Berkeley Gamelan 1.2 1.1 2.0 0.9 0.8 1.5 1.4

Gamelan selunding from Kengetan, South Bali (Pelog), 1/1=141 Hz 0.9 2.9 1.0 1.2 3.0 0.9 2.9 1.0 1.2 3.0 0.9

JI Pelog with stretched 2/1 and extra tones between 2-3, 6-7. Wolf, XH 11, '87 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

Obviously some of these are modern fabrications. I don't know how reliable any of them are. The first one does fit the pattern, although in a different mode. The second one is the right mode. The third one seems to be veering towards 7-equal. "New mixed gender Pelog" is perfect, but there's nothing to say where it comes from. Also the "Blown fifth" whatever that means. The "Subset of 24-tET (Sumatra?)" is an MOS, but the superscale is 10 notes, not 9. The rest of them are all over the place. Pelogic would be 1.2 1.2 1.5 1.2 1.2 1.2 1.5
Meru 2 is 0.9 0.9 2.2 0.9 0.9 0.9 2.2
Meru 4 is 0.6 0.6 2.9 0.6 0.6 0.6 2.9

Actually, this is a better match than you'd expect by random, so there might be something to it ;)

Graham

kraig grady wrote:

>there is no examples of a 5 equal slendro. nor do the indonesians claim they are making a 5 equal. Mc Phee has plently written a
>bit on this.
> >
So what?

>> Like slendric, where the generator is
>>approximately 8:7. Or taking 5-equal and moving each note by a small,
>>random amount.
>> >>
>
> In the range i have tuned Meta slendro up, you will find quite a few 8/7 intervals
> >
Yes, but that's hardly the same as an MOS generated by 8/7, is it?

>>As it's commonly known in English as Pascal's triangle, it'd ease
>>understanding if you mentioned this, as "Mt Meru" is quite an obscure
>>alternative.
>> >>
>
>the point is Mt. Meru is apart of the same culture as Slendro and Pelog, it makes sense to use it
> >
You're pushing it a bit for India and Indonesia to be the same culture.

>>What evidence do you have of a fit there?
>> >>
>
> with slendro. Measurements in Kunst, but i worked these out possibly 7 or 8 years ago
> >
How likely would it be to get the same matches with random data?

>>>i think one needs to listen to it. if you take it out to 12 places you can use the scale just like they do with the Wayang
>>>kulit. With one pentatonic tuned slightly higher than another. So 10 tones of my series could be used , and is used in just
>>>this musical way.
>>>
>>> >>>
>>If you take what out to 12 places?
>> >>
>
> Meta Slendro
>
So you're using a 12 note scale to get a 10 note scale?

>>What scale? How does it relate to
>>Wayang Kulit?
>> >>
>
> Wayang kulit has two set of instruments tuned to two pentatonics one slightly higher than the other. I can extract 10 pitches
>out of 12 and get just such an animal. there is no way to get at such animal by e equal
> >
Any MOS close 5-equal (of which there are only 3 types) can be extended to give a 10 note scale like this. It's hardly a complicated operation to do with an arbitrary scale anyway.

>>Where's your evidence for slendro coming from China?
>> >>
>
> the balinese says it was brought by a chinese princess
> >
That doesn't sound like a very credible story.

> look at the scales in kunst
> >
Isn't Kunst supposed to be the least reliable source for these things?

> this is approximated by the next series, as are the Chopi scales. By the way i do have one up which shows the clossness of the
>differance tones with deviations in VPS. since cents is really not something we here as just that.
> http://anaphoria.com/chopi.gif
>the other older measurements of Hugh Tracy fit much better but the problem is that his measurements were done comparing the
>instruments against tuning forks 4 vibrations apart. This is the most acturate measurement available of any chopi instrument sent
>to me by Andrew tracy quite a few years ago. I have tried to be tough on the measurements above, but tuning such short decay
>instruments more acturately would be quite hard. That upper range tones would be 10 vibrations off might be the limit of what one
>could do. As i stated with the Kunst measurements , i found a consistent deviation of 2 vibrations throughout. which implies that
>it is also quite possible that such slight deviations are deliberate for musical reasons , not known to us, or is influenced by
>the timbre which migh also have some bearing on the above.
> >
You get a generator of 2.014/7 octaves, which is good. There's only a 21% chance of getting so close to 7-equal by chance.

That's 2 vibrations per second? It could be they're optimizing beats to be around there. But I don't see the consistency. You have two that match perfectly, three within 1 vps, and only 2 between 1 and 3 vbps.

>Then i will pose this question. Tuners are tuning to something . this something would would imagine would be some form of
>acoustical phenomenon. What are they tuning too then?
> >
I don't know. William Sethares showed a fairly good match to the timbres they were using. Daniel Wolf seems to know more detail going by the SpecMus discussion, but I didn't follow it.

>why when he invented 2,000 years after the people of india. if it had been invenrtewd by a white person 2,00 years before pascal,
>no one would suggest such a thing. Also we are not talking about Pascal cause he is outside the culture we are talking aout.
>
No one would suggest what? Weren't Tartaglia and Gualtieri white? Pascal never claimed to invent it. It happens to be named after him in English. As we're writing in English, it eases understanding to use the English name. Should we move the whole discussion to Sanskrit if we happen to mention somewhere in contact with India?

Graham

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

> You're pushing it a bit for India and Indonesia to be the same culture.
>

Not the same culture, but cultures sharing a Hindu/Buddhist substrate.
Bali and parts of Java (Tengger) are still Hindu. And Javanese culture
still has many Hindu-Buddhist elements, for example the shadow plays
and dances based on Mahabarata and Ramayana stories.

> >>What scale? How does it relate to
> >>Wayang Kulit?
> >>

Maybe Kraig Grady means the three pathet?

> >>Where's your evidence for slendro coming from China?
> >>

I don't know about the transmission direction, but it seems okay to
say that a anhemitonic pentatonic is a common thread to China (5 in
12), slendro (5 in 5), and the Thai scales (5 in 7).

> >>
> Should we move the whole discussion to Sanskrit if we
> happen to mention somewhere in contact with India?
>

The metapelog and metaslendro scales were invented by Erv Wilson. He
happened to use Pascal's triangle to invent them. To recognize the
similiarity to traditional Javanese scales, he called them "the scales
of Mt Meru", the form of P's triangle old javanese or balinese would
have known. Naming is an aesthetic decision and the right to name is
associated with priority. Erv Wilson has the right to name his scales
(and groups of scales) as he likes.

Gabor (with English help from Daniel Wolf)

>
> From: Graham Breed <graham@microtonal.co.uk>
> Subject: Re: Mt. Meru
>
> kraig grady wrote:
>
> >there is no examples of a 5 equal slendro. nor do the indonesians claim they are making a 5 equal. Mc Phee has plently written a
> >bit on this.
> >
> So what?

you brought up 5- equal as being related to slendro, not i

> >> Like slendric, where the generator is
> >>approximately 8:7. Or taking 5-equal and moving each note by a small,
> >>random amount.
> >>
> >>
> >
> > In the range i have tuned Meta slendro up, you will find quite a few 8/7 intervals
> >
> >
> Yes, but that's hardly the same as an MOS generated by 8/7, is it?

yes it is but if you look at the sl patterns of actual slendro you will notice that some size of fourths makes more sense that a chain of 8/7's. also you might notice in my Kunst example the lowest notes are spaced in 'fourths'

>
>
> >>As it's commonly known in English as Pascal's triangle, it'd ease
> >>understanding if you mentioned this, as "Mt Meru" is quite an obscure
> >>alternative.
> >>
> >>
> >
> >the point is Mt. Meru is apart of the same culture as Slendro and Pelog, it makes sense to use it
> >
> >
> You're pushing it a bit for India and Indonesia to be the same culture.

every shadow play opens up with a tree sitting on top of Mt .Meru

>
>
> >>What evidence do you have of a fit there?
> >>
> >>
> >
> > with slendro. Measurements in Kunst, but i worked these out possibly 7 or 8 years ago
> >
> >
> How likely would it be to get the same matches with random data?

pick 5 pitches at random and you are more likely to get something closer to pelog than slendro. pick out random junk and play on it it is funny how you get little pelog like melodic fragments

>
>
> >>>i think one needs to listen to it. if you take it out to 12 places you can use the scale just like they do with the Wayang
> >>>kulit. With one pentatonic tuned slightly higher than another. So 10 tones of my series could be used , and is used in just
> >>>this musical way.
>
> >
> So you're using a 12 note scale to get a 10 note scale?

in this case yes

>
>
> >>What scale? How does it relate to
> >>Wayang Kulit?
> >>
> >>
> >
> > Wayang kulit has two set of instruments tuned to two pentatonics one slightly higher than the other. I can extract 10 pitches
> >out of 12 and get just such an animal. there is no way to get at such animal by e equal
> >
> >
> Any MOS close 5-equal (of which there are only 3 types) can be extended
> to give a 10 note scale like this. It's hardly a complicated operation
> to do with an arbitrary scale anyway.

but do the beat patterns line up like they do in this case. One has to listen to them

>
>
> >>Where's your evidence for slendro coming from China?
> >>
> >>
> >
> > the balinese says it was brought by a chinese princess
> >
> >
> That doesn't sound like a very credible story.

the story is she brought it as a wedding gift to her balinese husband. The story is widely known in Bali and why would one dispute it
or at least consider it as having some bearing on it history

>
>
> > look at the scales in kunst
> >
> >
> Isn't Kunst supposed to be the least reliable source for these things?

not the most perfect in that we know that he got rid of stretched octaves. still it appears to be more reliable than any other when the subject came up on the gamelan list.

>
>
> > this is approximated by the next series, as are the Chopi scales. By the way i do have one up which shows the clossness of the
> >differance tones with deviations in VPS. since cents is really not something we here as just that.
> > http://anaphoria.com/chopi.gif
> >the other older measurements of Hugh Tracy fit much better but the problem is that his measurements were done comparing the
> >instruments against tuning forks 4 vibrations apart. This is the most acturate measurement available of any chopi instrument sent
> >to me by Andrew tracy quite a few years ago. I have tried to be tough on the measurements above, but tuning such short decay
> >instruments more acturately would be quite hard. That upper range tones would be 10 vibrations off might be the limit of what one
> >could do. As i stated with the Kunst measurements , i found a consistent deviation of 2 vibrations throughout. which implies that
> >it is also quite possible that such slight deviations are deliberate for musical reasons , not known to us, or is influenced by
> >the timbre which migh also have some bearing on the above.
> >
> >
> You get a generator of 2.014/7 octaves, which is good. There's only a
> 21% chance of getting so close to 7-equal by chance.
>
> That's 2 vibrations per second? It could be they're optimizing beats to
> be around there. But I don't see the consistency. You have two that
> match perfectly, three within 1 vps, and only 2 between 1 and 3 vbps.
>
> >Then i will pose this question. Tuners are tuning to something . this something would would imagine would be some form of
> >acoustical phenomenon. What are they tuning too then?
> >
> >
> I don't know. William Sethares showed a fairly good match to the
> timbres they were using.

the instruments used for pelog and slendro are rthe same design so influence of timbre is questionable.

> Daniel Wolf seems to know more detail going by
> the SpecMus discussion, but I didn't follow it.

i understand he saw it as as a form of tempering, which i have yet to see evidence that such an idea could be theirs as much as it is ours

> >why when he invented 2,000 years after the people of india. if it had been invented by a white person 2,00 years before pascal,
> >no one would suggest such a thing. Also we are not talking about Pascal cause he is outside the culture we are talking aout.
> >
> No one would suggest what? Weren't Tartaglia and Gualtieri white?

i ado not know who these individuals are

>
> Pascal never claimed to invent it. It happens to be named after him in
> English.

what did he call it . do you know?

> As we're writing in English, it eases understanding to use the
> English name. Should we move the whole discussion to Sanskrit if we
> happen to mention somewhere in contact with India?

If it helps to understand where the tuning comes from yes. When looking at the tuning of other cultures it is far more reliable to base it on things found with in the culture. What ever our own beliefs are, tunings are more often than not, religious objects. we gain no insight into how they come about by pulling them out of it. For instance in
Martin Braun Recent case where you had 6 and 9 tone scales, without knowing that these are the main male and female numbers, one can not figure out just why they stopped at these places.
There is allot of untranslated math from China and it still seems a possibility that in it we will find an examination of all the diagonals. It seems quite amazing that out side of the single fibonacci diagonal found by Thomas Green in the Fibonacci Quarterly, It appears that Erv is the only one to spot these others. I say this cause i know
there are those on this list who possibly could find others and possibly this might trick you in to proving this wrong:). Erv believes that the Chinese are too smart to overlook such a thing and since they were pouring over this figure for thousands of years, someone you would think have noticed it.

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

>Then i will pose this question. Tuners are tuning to something.

Not necessarily. One can just tune. Then one can even remember
what one did and repeat it.

Or one can tune until it 'sounds good'. Here again we are out
of the realm of acoustics, and into the realm of taste, which
we know there's 'no accounting for'.

>> I don't know. William Sethares showed a fairly good match to
>> the timbres they were using.
>
> the instruments used for pelog and slendro are rthe same design
> so influence of timbre is questionable.

One can still optimize for a timbre within each of two different
scale gestalts.

I haven't looked at Sethares' gamelan analysis in many years, but
at the time it seemed a bit fishy.

-Carl

--- In tuning@yahoogroups.com, "alternativetuning"

/tuning/topicId_46559.html#46647

> The metapelog and metaslendro scales were invented by Erv Wilson. He
> happened to use Pascal's triangle to invent them. To recognize the
> similiarity to traditional Javanese scales, he called them "the
scales
> of Mt Meru", the form of P's triangle old javanese or balinese would
> have known. Naming is an aesthetic decision and the right to name is
> associated with priority. Erv Wilson has the right to name his
scales
> (and groups of scales) as he likes.
>
> Gabor (with English help from Daniel Wolf)

***It's nice to have some backup here for Kraig Grady, who tends to
know what he's talking about in these matters...

J. Pehrson

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

> The meta-slendro is only like slendro because it's like 5-equal.

are you sure there's no similarity beyond that?

> The
> convergent generator is 1.0284 steps of 5-equal.

right, but kraig wasn't using the convergent generator, he was using
particular scales from the sequence before it converges (which only
really happens after an infinite number of iterations).

--- In tuning@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
>
> >what do you mean? like LLsssss, for example??

let me quote the previous message in the thread:

[paul]
> >why would you say it has little relationship
> >with any observed pelog -- don't you just mean that it's not
exactly
> >matched by any observed pelog? it's certainly a better fit than
any
> >other so simply defined scale.
> >
[graham]
> The pelogic one, reversing large and small steps from a diatonic
scale.
> I mean it isn't any closer in general than an arbitrary 7 from 9
tuning,

ok, back to now

> I mean there seems to be a 9 note template behind it, and the 7
notes
> given are in the Pelogic pattern (like 1 1 2 1 1 1 2 or 1 1 2 1 1 2
1 or
> something else).

well that's really all i was saying in this thread. and isn't
this "reversing large and small steps from a diatonic
scale" as you yourself put it??

> But the intervals of 1 step may be larger or smaller
> than 1/9 octaves, and the intervals of 2 steps may be larger or
smaller
> than 2/9 octaves. So there's no consistency that suggets the
underlying
> 9 note scale is itself a pelogic.

unless all the small steps are similar to one another and all the
large steps are similar to one another. in which case "pelogic" (in
my sense of the word) is in evidence, whether there's an underlying 9-
note scale or no.

> Anyway, here are the pelogs from the Scala archive,

how did you determine that all these qualify as "pelogs"?

> Chalmers' Pelog/BH Slendro
> 0.0 1.7 0.0 -0.2 1.9 -1.1 2.9 0.0 0.0 1.7 -0.9 2.9

what's up with these zero and negative figures?

> Gamelan selunding from Kengetan, South Bali (Pelog), 1/1=141
Hz
> 0.9 2.9 1.0 1.2 3.0 0.9 2.9 1.0 1.2 3.0 0.9

it's hard to deny that this one, and several of the others, suggest a
simplifying specification in terms of 9-equal quite strongly.
dividing an octave into 9 equal parts is something many western
musicians can easily comprehend, particularly those who are familiar
with these precise divisions via 72-equal.

> The first one does fit the pattern, although
> in a different mode. The second one is the right mode.

what do you consider "the right mode"??? why?

> The third one
> seems to be veering towards 7-equal.

or 16-equal, or 23-equal?

> "New mixed gender Pelog" is
> perfect, but there's nothing to say where it comes from.

anyone?

> Also the
> "Blown fifth" whatever that means.

hornbostel believed that the fifth derived from overblowing on bamboo
tubes was ~678 cents, which is how he got 23-equal.

> Pelogic would be 1.2 1.2 1.5 1.2 1.2 1.2 1.5

meaning? are you talking about the rms-optimal 5-limit linear
temperament that i call pelog?

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

> That's 2 vibrations per second? It could be they're optimizing
beats to
> be around there.

i don't think that's kraig's point -- rather, if all the frequencies
are shifted by 2 vibrations per second, all the difference tones that
coincided with scale tones before will still coincide with scale
tones after.

Paul Erlich wrote:

>>The meta-slendro is only like slendro because it's like 5-equal. >> >>
>
>are you sure there's no similarity beyond that?
> >

No, it's impossible to prove a negative. I don't know of any other defining features of slendro, and am willing to be educated. Kraig suggests a bias towards coincident difference tones, but his argument is weak and I haven't gone into the figures. Neither is it really a priority to do so -- I think I know what the Mt Meru stuff is about now, and it's interesting regardless of connections with existing music.

>>The >>convergent generator is 1.0284 steps of 5-equal.
>> >>
>
>right, but kraig wasn't using the convergent generator, he was using >particular scales from the sequence before it converges (which only >really happens after an infinite number of iterations).
> >
If the series converges close to 5-equal, the scales you get along the way will also be close, but not as close.

Graham

Me:

>>I mean there seems to be a 9 note template behind it, and the 7 >> >>
>notes > >
>>given are in the Pelogic pattern (like 1 1 2 1 1 1 2 or 1 1 2 1 1 2 >> >>
>1 or > >
>>something else).
>> >>

Paul:

>well that's really all i was saying in this thread. and isn't >this "reversing large and small steps from a diatonic
>scale" as you yourself put it??
> >
It depends on how you interpret the diatonic scale, which is also unclear to me. This is the analog of a 7 from 12 scale. However, Pythagorean and meantone tunings give a general s s L s s s L pattern where s and L have consistent sizes and consistent deviations from 12-equal. I don't know what a random sampling of European instruments through history would show. A 5-limit JI scale is closer to meantone than most of the pelogs I looked at to a more general 7 from 9 MOS.

No, that's getting convoluted, isn't it? To one decimal place accuracy, 9/8 is 2.0 steps of 12-equal and 10/9 is 1.8 steps. There's only one size of semitone (16/15). I'm not going through the list again, but aside from the obviously suspicious ones, I think it's only that "New mixed gender" that has consistently equal large/small steps, and a consistent deviation from 9-equal.

>unless all the small steps are similar to one another and all the >large steps are similar to one another. in which case "pelogic" (in >my sense of the word) is in evidence, whether there's an underlying 9-
>note scale or no.
> >
Yes, that's what the data don't show.

>>Anyway, here are the pelogs from the Scala archive,
>> >>
>
>how did you determine that all these qualify as "pelogs"?
> >
The filename begins with "pelog"

>>Chalmers' Pelog/BH Slendro
>>0.0 1.7 0.0 -0.2 1.9 -1.1 2.9 0.0 0.0 1.7 -0.9 2.9
>> >>
>
>what's up with these zero and negative figures?
> >
It looks like it's two scales in one, and so not monotonically increasing pitch. Here's the original file:

! pelog_jc.scl
!
Chalmers' Pelog/BH Slendro
12
!
1/1
8/7
8/7
9/8
64/49
6/5
3/2
3/2
3/2
12/7
8/5
2/1

Yes, 64/49 > 6/5.

64/49 * 5/6 = 32*5 / (49*3) = 160/147

> >
>>Gamelan selunding from Kengetan, South Bali (Pelog), 1/1=141 >> >>
>Hz > >
>>0.9 2.9 1.0 1.2 3.0 0.9 2.9 1.0 1.2 3.0 0.9
>> >>
>
>it's hard to deny that this one, and several of the others, suggest a >simplifying specification in terms of 9-equal quite strongly. >dividing an octave into 9 equal parts is something many western >musicians can easily comprehend, particularly those who are familiar >with these precise divisions via 72-equal.
> >
Yes, 0.2 steps of 9-equal still come out at 27 cents, but given it comes from a small sample this is probably too close to be a coincidence (I haven't worked out how unlikely it is to be chance).

>>The first one does fit the pattern, although >>in a different mode. The second one is the right mode.
>> >>
>
>what do you consider "the right mode"??? why?
> >
There's a reference in Sethares to a particular mode, but it doesn't seem to be consistent. I was working out the patterns as I wrote that message.

>>The third one >>seems to be veering towards 7-equal.
>> >>
>
>or 16-equal, or 23-equal?
> >
In terms of 9-equal again:
1.1 1.3 1.2 1.5 1.1 1.2 1.5

7-equal:
0.9 1.0 0.9 1.2 0.9 1.0 1.1

16-equal:
2.0 2.3 2.2 2.7 2.0 2.2 2.6

23-equal:
2.9 3.3 3.1 3.9 2.9 3.2 3.8

The closest match is to 7-equal, and it deviates considerably from 16-equal. The small steps have a range of 0.4 steps of 23-equal, which isn't especially close. In 9-equal, there's no clear distinction between large and small steps.

>hornbostel believed that the fifth derived from overblowing on bamboo >tubes was ~678 cents, which is how he got 23-equal.
> >
Yes, so that's a mathematically constructed scale.

>>Pelogic would be 1.2 1.2 1.5 1.2 1.2 1.2 1.5
>> >>
>
>meaning? are you talking about the rms-optimal 5-limit linear >temperament that i call pelog?
> >
It's probably the minimax. Do you call that pelog? The two Balinese matches from this unscientific sample are both close to those intervals, so maybe that's a pattern.

Graham

Paul Erlich wrote:

>i don't think that's kraig's point -- rather, if all the frequencies >are shifted by 2 vibrations per second, all the difference tones that >coincided with scale tones before will still coincide with scale >tones after.
> >
Really? So 2 vps relative to what? I didn't think there was an absolute pitch for any of these.

Graham

>

Hello Graham. I really would have thought the example i posted from Kunst's book would show how acturate this fomula seems to be. I have no idea of what you want for an argument that isn't weak. Of course there isthe sound of Slendro also and i recxommend listening to
this and the other raw examples that exist.

>
> From: Graham Breed <graham@microtonal.co.uk>
> Subject:
>
>
> No, it's impossible to prove a negative. I don't know of any other
> defining features of slendro, and am willing to be educated. Kraig
> suggests a bias towards coincident difference tones, but his argument is
> weak and I haven't gone into the figures. Neither is it really a
> priority to do so -- I think I know what the Mt Meru stuff is about now,
> and it's interesting regardless of connections with existing music.
>
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

kraig grady wrote:

> Hello Graham. I really would have thought the example i posted from
Kunst's book would show how acturate this fomula seems to be. I have
no idea of what you want for an argument that isn't weak. Of course
there isthe sound of Slendro also and i recxommend listening to
> this and the other raw examples that exist.

Okay, let's try the same method on this simplest possible JI slendro

40 46 53 61 70 80 92 106 122 140 161 185 212 244 280 322 369 424 488
560 643 739 849 975 1120 1287 1478 1698 1950 2240

40+53 = 93 +1
46+61 = 107 +1
53+70 = 123 +1
61+80 = 141 +1
70+92 = 162 +1
80+106= 186 +1
92+122= 214 +2
106+140=246 +2
122+161=283 +3
140+185=325 +3

It consistently gives an error of 1Hz for a while, and then the
discrepancy starts to rise. It's actually more consistent than the
example you give, but isn't a recurrent sequence! I'll leave you to
work out how I generated it.

The match is good partly because it concerns the lowest, and so least
accurately measured notes. It breaks down with higher notes.

And another reason your example isn't convincing is that there's only
one of them -- it doesn't go anywhere to proving the general rule.

Given the poor quality of data we have to work with, perhaps there's
no way to provide a better argument, in which case you shouldn't make
grandiose claims. But if you want to try, give some kind of
statistical measure of how likely these coincidences are with a
randomly generated scale (I suggest a Gaussian perturbation relative
to 5-equal).

Graham

--- In tuning@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
>
> >i don't think that's kraig's point -- rather, if all the
frequencies
> >are shifted by 2 vibrations per second, all the difference tones
that
> >coincided with scale tones before will still coincide with scale
> >tones after.
> >
> >
> Really? So 2 vps relative to what?

relative to one of the mt. meru scales.

> I didn't think there was an
> absolute pitch for any of these.

surely you're free to assign absolute pitch as you please.

>

As i stated kunst's measurement are not acturate above the lower octices as he assume the octave were equivalent. You could also try listening to it. And by the way listen to 5 equal as see it it resembles slendro. You can also go through the rest of Kunst's
measurements, which i see no reason to prove. I gave the clearest example. Where does this slendro occur you have below?
i see a sequence of adding 6-7-8-9-10. but explain how someone can tune this by ear? you might care to look at the spacing of the lowestest tones in the actual tuning and how they are spaced.

>
> From: "Graham Breed" <graham@microtonal.co.uk>
> Subject: Re: Mt. Meru
>
> kraig grady wrote:
>
> > Hello Graham. I really would have thought the example i posted from
> Kunst's book would show how acturate this fomula seems to be. I have
> no idea of what you want for an argument that isn't weak. Of course
> there isthe sound of Slendro also and i recxommend listening to
> > this and the other raw examples that exist.
>
> Okay, let's try the same method on this simplest possible JI slendro
>
> 40 46 53 61 70 80 92 106 122 140 161 185 212 244 280 322 369 424 488
> 560 643 739 849 975 1120 1287 1478 1698 1950 2240
>
> 40+53 = 93 +1
> 46+61 = 107 +1
> 53+70 = 123 +1
> 61+80 = 141 +1
> 70+92 = 162 +1
> 80+106= 186 +1
> 92+122= 214 +2
> 106+140=246 +2
> 122+161=283 +3
> 140+185=325 +3
>
> It consistently gives an error of 1Hz for a while, and then the
> discrepancy starts to rise. It's actually more consistent than the
> example you give, but isn't a recurrent sequence! I'll leave you to
> work out how I generated it.
>
> The match is good partly because it concerns the lowest, and so least
> accurately measured notes. It breaks down with higher notes.
>
> And another reason your example isn't convincing is that there's only
> one of them -- it doesn't go anywhere to proving the general rule.
>
> Given the poor quality of data we have to work with, perhaps there's
> no way to provide a better argument, in which case you shouldn't make
> grandiose claims. But if you want to try, give some kind of
> statistical measure of how likely these coincidences are with a
> randomly generated scale (I suggest a Gaussian perturbation relative
> to 5-equal).
>
> Graham
>
> ________________________________

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- kraig grady wrote:

> As i stated kunst's measurement are not acturate above the lower
octices as he assume the octave were equivalent. You could also try
listening to it. And by the way listen to 5 equal as see it it
resembles slendro. You can also go through the rest of Kunst's
> measurements, which i see no reason to prove. I gave the clearest
example. Where does this slendro occur you have below?
> i see a sequence of adding 6-7-8-9-10. but explain how someone can
tune this by ear? you might care to look at the spacing of the
lowestest tones in the actual tuning and how they are spaced.

You haven't said in this thread why the lower octaves should be
particularly affected by an assumption of octave equivalence. And if
that is the case why is one of the octaves in your example 1290 cents?

I see you persist in your obsession with 5-equal. I assume you know
that the scale you gave from Kunst is remarkably close to it.

http://anaphoria.com/kunstslendro.jpeg

It is, in terms of 5-equal relative to 70 Hz,

-4.04 -1.74 0.00 1.01 2.01 2.99 4.01 5.00 6.01 7.01 8.01 9.01 10.00
11.01 12.01 13.01 14.01 15.38 16.01 17.01 18.01 19.01 20.38 21.01
22.01 23.01 24.01 25.38

Out of 28 notes, 23 are within 1.1% of 5-equal. As one of those is
fixed, the probability (according to a binomial distribution) of 22 or
more out of 27 events occurring when the probability of each is 2.2%
comes out at 0.0000000000000000000000000000025%. That's very unlikely
to be a concidence.

Of the 5 notes that don't fit, two are the lowest two recorded. And
the very lowest one fits the pattern as accurately as the data can
show. The three higher notes that don't fit are all related by
perfect octaves.

Assuming this is a garbled octave-equivalent scale, the chance of a
random match for 4 out of 4 notes is only 0.00002%.

So that establishes there's a correlation to 5-equal. And a much
better one than you have to coincident difference tones. That 1.1% is
2.64 cents, or 0.15 Hz at 100 Hz. Which, in this range (the one you
took your data from) is smaller than the resolution in which the data
are given. Yet you're happy to allow deviations of up to 2 Hz
relative to your own theory!

What good would listening to the scale do? We're talking about a
correlation to a mathematical sequence, not an aesthetic judgement.

No, I will not go through Kunst's measurements to disprove your
theory. I've said this before. If you think they show something, do
the work yourself. If you won't prove it, don't make claims in public.

If you don't recognise the scale I gave, I see no point in continuing
this discussion.

Graham

>
> Subject: Re: Mt. Meru

>
> If you don't recognise the scale I gave, I see no point in continuing
> this discussion.

I am so glad you that said that being the overly generous person you are. The proof will be in the pudding

>
>
> Graham
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

hello all,

i need a very complete database of ratios which
may be potential unison-vectors.

i realize that there's a good 5-limit list on my
own "equal-temperaments" Dictionary webpage, and
i'm pretty sure that i've seen lists of these up to
the 11-limit on the tuning-math list. has anything
been analyzed beyond the 11-limit? can someone
point me to links or text messages? thanks.

-monz

--- In tuning@yahoogroups.com, <monz@a...> wrote:
> hello all,
>
> i need a very complete database of ratios which
> may be potential unison-vectors.

Sounds like something to ask on tuning-math. But you'd better define
what you mean by "very complete" and how big a ratio can be and still
be considered an approximate unison.

> i realize that there's a good 5-limit list on my
> own "equal-temperaments" Dictionary webpage, and
> i'm pretty sure that i've seen lists of these up to
> the 11-limit on the tuning-math list. has anything
> been analyzed beyond the 11-limit? can someone
> point me to links or text messages? thanks.

I have a spreadsheet with about 360 31-limit ratios less than 0.5
cent, if that's any use. But I'm sure Gene or Graham or Paul could
generate exactly what you want if you define it sufficiently.

Joe,

You can generate them with "unison.cmd" in the command
file directory of Scala.

Manuel

--- In tuning@yahoogroups.com, <monz@a...> wrote:

> i realize that there's a good 5-limit list on my
> own "equal-temperaments" Dictionary webpage, and
> i'm pretty sure that i've seen lists of these up to
> the 11-limit on the tuning-math list. has anything
> been analyzed beyond the 11-limit? can someone
> point me to links or text messages? thanks.

I've got my own personal 13-limit list which I could send, and
superparticular commas (which becoming increasingly important as we
go to higher limits) were well surveyed by Chalmers.

--- In tuning@yahoogroups.com, <monz@a...> wrote:
>
> hello all,
>
>
> i need a very complete database of ratios which
> may be potential unison-vectors.
>
> i realize that there's a good 5-limit list on my
> own "equal-temperaments" Dictionary webpage,

http://www.sonic-arts.org/dict/eqtemp.htm

it's still in need of revision . . . a slightly more complete and
correct version of this, which applies better to the graphs on your
page, is at

/tuning/database?
method=reportRows&tbl=10&sortBy=5&sortDir=up

(be sure to scroll through the various pages)

((i know you knew but this is for the benefit of new members
following along))

by combining a pair of unison vectors from this list, and allowing
both unison vectors to vanish, you can derive any of a pretty vast
number of equal temperaments, or well-temperaments if you do the
tempering unequally . . . the graphs show these equal temperaments at
the intersections of the lines corresponding to the unison vectors.

7-limit and higher are of course of great interest too, but the
graphs would have to be three-dimensional, and would be pretty
impossible to view without the ability to rotate them around and view
them from various angles (hint hint) . . .

hi paul,

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning@yahoogroups.com, <monz@a...> wrote:
> >
> > hello all,
> >
> >
> > i need a very complete database of ratios which
> > may be potential unison-vectors.
> >
> > i realize that there's a good 5-limit list on my
> > own "equal-temperaments" Dictionary webpage,
>
> http://www.sonic-arts.org/dict/eqtemp.htm
>
> it's still in need of revision . . . a slightly more complete and
> correct version of this, which applies better to the graphs on your
> page, is at
>
> /tuning/database?
> method=reportRows&tbl=10&sortBy=5&sortDir=up
>
> (be sure to scroll through the various pages)
>
> ((i know you knew but this is for the benefit of new members
> following along))

actually, it's good that you gave me the link again too.

> by combining a pair of unison vectors from this list, and
> allowing both unison vectors to vanish, you can derive any
> of a pretty vast number of equal temperaments, or
> well-temperaments if you do the tempering unequally . . .
> the graphs show these equal temperaments at the intersections
> of the lines corresponding to the unison vectors.
>
> 7-limit and higher are of course of great interest too,
> but the graphs would have to be three-dimensional, and would
> be pretty impossible to view without the ability to rotate
> them around and view them from various angles (hint hint) . . .

no need for you to drop hints about that to *me*!
and it's no problem, either ... my software already
does 3-D rotation.

the big problem is that we poor humans can only visualize
up to 3 dimensions, so that we have to projecting any
tuning system which has 4 or more dimensions down into 3.

-monz

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

> the big problem is that we poor humans can only visualize
> up to 3 dimensions, so that we have to projecting any
> tuning system which has 4 or more dimensions down into 3.

oops ... i meant either "... we have to project any tuning
system which has 4 or more dimensions down into 3", or
"... we have to resort to projecting..." etc.

point is: we can only visualize >3-dimensional stuff in
terms of 3-D or less ... and my software already does that.

(PS for those who are itching to try it: the release of
version 1.0 is expected for February 2004.)

-monz