Yahoo Tuning Groups Ultimate Backup tuning Re : Ethno2 microtunings

The last version of the rules of the Ethno2 microtonal demos competition mentions that "original tunings, when issued from recurrent series, can be modified by extension to other series of the same algorithms, as well as to more than 12 notes of the same series." I wanted to detail this point.
First I have to confess that I wrote the Ethno2 tunings in a total rush, each time I made a tuning I was happy with I passed to the next and I did not bother for further refinements.
Many of the tunings come from recurrent sequences, however the rational versions I gave for each of them are choices among an infinity of possibilities.
Some recurrent sequences are convergent, but some other diverge as they go up in the series, starting by a micro-wavering that amplifies itself until the series stops.
Depending on the context and the series, these divergences may either have musical interest, or can produce aberrations.
But in these cases one can always find more precise series, that will deteriorate within a higher number of terms.

Though perhaps no one noticed, the series of the SuperPyth scale I gave as an example on the 14th of april was ending with a strange very flat Bb, while it should have been around 7/4 ; this semi-fourth under 2/1 may have some melodic charm and is differentially-coherent like others, but the classic Bb you would expect instead for example in a Persian Shur Dastgah is missing.
So I am proposing here to fix it with a better series of the same recurrent sequence, and at the same time explaining how you can do it yourself with all recurrent sequences.
Instead of the 640/481 approximation of the fourth, the new one uses a more precise 1276/959.
Going up by fourths from A koron (half-b) to Eb- the series becomes :
2883
3836 (=959*2^2)
5104
6791
9036
12024
15995
21276
28352 (= 443*2^6 = C)
37727
497725
66760
= frequencies given by applying the Airos algorithm H(i+3) = 13H(i) - 8H(i+1)

And this is the new Scala file :

! soria12.scl
!
12 from a 17-notes cycle, equal-beating extended fifths (705.5685 c.) sequence
12
!
959/886
15995/14176
8345/7088
2259/1772
37727/28352
638/443
5319/3544
2883/1772
1503/886
12443/7088
6791/3544
2/1
! Airos recurrent sequence x^3 = 13 - 8x, Dudon 2008
! Eq-b of fourths, and septimal minor thirds with harmonic 7ths :
! 8(4 - 3x) = 3x^3 - 7 = (32x^3 - 56x)/13
! Airos fourths of 494.4315 c. (approx. 1276/959),
! cycle here starting from the neutral sixth (A koron = 2883/1772)
! -c with the 13th harmonic : (13 * 2883) - (32 * 959) = 6791, etc.

Both series (I would advise using rather this last one), can be extended to 17 tones without problem but more safely in reverse direction, by fourths under the starting tone (here 2883). In that direction the series converges, but through whole numbers that need to be divided by 13^n.
This is the Soria "superpyth" fifth algorithm : 13H(i+3) = 16H(i+2) + 8H(i)
(ex : 16 * 2883 + 8 * 1276 = 56336 and H(i+3) = 56336/13 )
One can also use the irrational solutions I give in cents in all these tunings as a generator, to create equal series just like you would do with any regular temperament.
My personal preference goes to rational series and therefore sligthly unequal scales, as I did in all these tunings, because of the multitude of modal options they offer, and also of their consequent synchronous beating properties.

- - - - - - -
Jacques

🔗genewardsmith <genewardsmith@...>

🔗Torsten Anders <torsten.anders@...>

Dear Jacques,

Thank you for this clarification and apologies that I don't quite
understand your recurrent series. Could you please be a bit more
specific.

Your Scala files state that, for example, the file N_America/
appalachian.scl is a recurrent series. The tuning is defined as
follows (I am copying the whole file including the comments here).

! appalachian.scl
!
Synchronous beating quasi-1/4 syntonic comma meantone temperament
12
!
4025/3852
3230/2889
128/107
5/4
1288/963
8075/5778
160/107
25/16
1610/963
5168/2889
200/107
2/1
! recurrent series : 864 1292 1932 2889 4320...
! Mezzo fractal sequence x^3 = (3/4)x^2 + 5/3
! (x = 1.495352392466 or 696.5826088 c.)
! Dudon 2006

You comments obviously give some hints how this scale was constructed,
but I am unable to understand them.

1. How did you create the stated recurrent series?
2. What is the relation between this series and the stated scale?

Sorry if this is all very obvious, I am just to dump to get it...

Best wishes,
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-586219
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

On 25.04.2010, at 18:02, Jacques Dudon wrote:

>
> The last version of the rules of the Ethno2 microtonal demos
> competition mentions that "original tunings, when issued from
> recurrent series, can be modified by extension to other series of
> the same algorithms, as well as to more than 12 notes of the same
> series." I wanted to detail this point.
> First I have to confess that I wrote the Ethno2 tunings in a total
> rush, each time I made a tuning I was happy with I passed to the
> next and I did not bother for further refinements.
> Many of the tunings come from recurrent sequences, however the
> rational versions I gave for each of them are choices among an
> infinity of possibilities.
> Some recurrent sequences are convergent, but some other diverge as
> they go up in the series, starting by a micro-wavering that
> amplifies itself until the series stops.
> Depending on the context and the series, these divergences may
> either have musical interest, or can produce aberrations.
> But in these cases one can always find more precise series, that
> will deteriorate within a higher number of terms.
>
> Though perhaps no one noticed, the series of the SuperPyth scale I> gave as an example on the 14th of april was ending with a strange
> very flat Bb, while it should have been around 7/4 ; this semi-
> fourth under 2/1 may have some melodic charm and is differentially-
> coherent like others, but the classic Bb you would expect instead
> for example in a Persian Shur Dastgah is missing.
> So I am proposing here to fix it with a better series of the same
> recurrent sequence, and at the same time explaining how you can do
> it yourself with all recurrent sequences.
> Instead of the 640/481 approximation of the fourth, the new one uses
> a more precise 1276/959.
> Going up by fourths from A koron (half-b) to Eb- the series becomes :
> 2883
> 3836 (=959*2^2)
> 5104
> 6791
> 9036
> 12024
> 15995
> 21276
> 28352 (= 443*2^6 = C)
> 37727
> 497725
> 66760
> = frequencies given by applying the Airos algorithm H(i+3) = 13H(i)
> - 8H(i+1)
>
> And this is the new Scala file :
>
> ! soria12.scl
> !
> 12 from a 17-notes cycle, equal-beating extended fifths (705.5685
> c.) sequence
> 12
> !
> 959/886
> 15995/14176
> 8345/7088
> 2259/1772
> 37727/28352
> 638/443
> 5319/3544
> 2883/1772
> 1503/886
> 12443/7088
> 6791/3544
> 2/1
> ! Airos recurrent sequence x^3 = 13 - 8x, Dudon 2008
> ! Eq-b of fourths, and septimal minor thirds with harmonic 7ths :
> ! 8(4 - 3x) = 3x^3 - 7 = (32x^3 - 56x)/13
> ! Airos fourths of 494.4315 c. (approx. 1276/959),
> ! cycle here starting from the neutral sixth (A koron = 2883/1772)
> ! -c with the 13th harmonic : (13 * 2883) - (32 * 959) = 6791, etc.
>
> Both series (I would advise using rather this last one), can be
> extended to 17 tones without problem but more safely in reverse
> direction, by fourths under the starting tone (here 2883). In that
> direction the series converges, but through whole numbers that need
> to be divided by 13^n.
> This is the Soria "superpyth" fifth algorithm : 13H(i+3) = 16H(i+2)
> + 8H(i)
> (ex : 16 * 2883 + 8 * 1276 = 56336 and H(i+3) = 56336/13 )
> One can also use the irrational solutions I give in cents in all
> these tunings as a generator, to create equal series just like you
> would do with any regular temperament.
> My personal preference goes to rational series and therefore
> sligthly unequal scales, as I did in all these tunings, because of
> the multitude of modal options they offer, and also of their
> consequent synchronous beating properties.
>
> - - - - - - -
> Jacques
>
>
>

🔗Jacques Dudon <fotosonix@...>

Gene wrote :

> Why in the world do you want to base a contest on scales > constructed from recurrent series, if the point of it is to demo > the Ethno2? Why not just let people use whatever scale they prefer?

Hi Gene, nice to see you back !
I never imagined a second to base a contest on scales constructed from recurrent series, I just offered a complement of understanding to the material I gave and besides it's only one part of the tunings.
Since I was asked to provide the microtunings for Ethno2, I grasped all what I felt was the best in what I worked on, know perfectly well and experienced for years, as deeply related to what I have been hearing since I came to this planet in the domain of traditional music.
If you read the competition rules you will see that people can use whatever other scale they want, but in addition also to a minimum of 3 minutes using some of the original tunings. Maximum of diversity, and also respect for the kids who want to find out what THEY will be able to play with the Ethno box - before they become microtonalists.
I offered to discuss all the rules and you could have done so, but apart from Marcel nobody discussed any of them and now I won't change them.
There are 2 boxes left and you're welcome to be a candidate !

- - - - - - -
Jacques

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Gene wrote :
>
> > Why in the world do you want to base a contest on scales
> > constructed from recurrent series, if the point of it is to demo
> > the Ethno2? Why not just let people use whatever scale they prefer?
>

> If you read the competition rules you will see that people can use
> whatever other scale they want, but in addition also to a minimum of
> 3 minutes using some of the original tunings.

Mostly those "original tunings" struck me as excessively conventional, loaded down with things like 12 note circulating temperaments and ethnic scales with a small number of notes, which was why I was interested when you suggested people might be able to use their own scales. If I understand correctly, the Ethno goes up to 24 notes to the octave, allowing a huge range of scales that, when I looked at them, your scale set didn't seem to be offering any of.

At minimum, it seems to me, there should have been MOS for linear temperaments aside from 12 notes of meantone.

> I offered to discuss all the rules and you could have done so, but
> apart from Marcel nobody discussed any of them and now I won't change
> them.

> There are 2 boxes left and you're welcome to be a candidate !

I'm not a good choice for someone using keyboard input.

🔗Jacques Dudon <fotosonix@...>

On Sun Apr 25, 2010, Torsten Anders wrote :

> Dear Jacques,

> Thank you for this clarification and apologies that I don't quite
> understand your recurrent series. Could you please be a bit more
> specific.
> Your Scala files state that, for example, the file N_America/
> appalachian.scl is a recurrent series. The tuning is defined as
> follows (I am copying the whole file including the comments here).

> 1. How did you create the stated recurrent series?
> 2. What is the relation between this series and the stated scale?

Hi Torsten,

fffff... Perhaps this harmonic temperament is not the very best example as a recurrent series, because it is made of several series :
864 1292 1932 2889 (...4320 6460 9660 ...14445) etc.
formed by the transpositions of the first four terms by 5 !

If you express the series this way :
x^0 = 864
x^1 = 1292
x^2 = 1932
x^3 = 2889
x^4 = 4320

you find that it verifies the Mezzo fractal sequence x^3 = (3/4)x^2 + 5/3 in :
(3/4) * 1932 + (5/3) * 864 = 2889,
(3/4) * 4320 + (5/3) * 1932 = 6460, and at other places, but
(3/4) * 2889 + (5/3) * 1292 = 4320.08333...
which I rounded to 5 * 864 = 4320 (as in a classical 1/4 syntonic comma meantone).
Of course with more and more very high numbers the series could have been continued after 4320.08333, but without much difference.
So, for the 1/4 syntonic comma meantone tuning (traditionally used in the fretting of the appalachian dulcimer), my choice was to keep it simple.
The difference with a 1/4 syntonic comma meantone is nano-small ; but Mezzo has better harmonic equal-beating than 5^(1/4) and can be considered in fact as THE equal-beating property of Young / Pietro Aaron.
This recurrent series is in fact based on the 320:321:322:323:324 harmonic division of the comma, which is considered more traditional than the one generated by 5^(1/4), as certainly Andreas Sparschuh and others would explain better than me.
Though this is not my favorite meantone, I personnally consider "Mezzo" as the best pertinent sequence for the 1/4 syntonic comma meantone on the acoustic level (the one you can't escape if you tune by ear, that's why I would risk to say "it is" the historical tuning), and 5^(1/4) as a complementary attractor. Both combined, what this tuning does, have a quadruple synchronous-beating property that is :
9x^2 - 6x^3 = (9/2)x^2 - 10 = 6x^3 - 20 = 6x^3 - 4x^4 where Mezzo shares the three first identities and 5^(1/4) the last two.

But if you want a better general explanation for my recurrent series, I suggest you to ask again with another one.
- - - - - - -
Jacques

🔗Jacques Dudon <fotosonix@...>

On Sun Apr 25, 2010, Gene wrote :

> Mostly those "original tunings" struck me as excessively
> conventional, loaded down with things like 12 note circulating
> temperaments and ethnic scales with a small number of notes

You are surprised to find "ethnic scales" in Ethno2 ??
Also it seems you are considering ethnic scales to be of poor value.
Personally I don't.
Think about it : Ethno1 was only in 12-ET, now Ethno2 offers ALL
possible tunings, and I am glad I could "load it down" as you say,
with a few ethnic scales. It is quite funny that I have to be in the
Tuning List to hear such critics !
And yes, there are a few totally new equal-beating versions of
several historic and useful temperaments, if you consider those to be"conventional" - but also many really unconventional tunings, I
think. You must have missed them. If you check for example in the
"multi-system" section, you will find some "double-face" or "hybrid"
tunings, that have for example one sequence on the white keys in
continuity with another sequence on black keys (EA7, NA6, SEA9), and
other unusual things, but always inspired by traditions.

> If I understand correctly, the Ethno goes up to 24 notes to the
> octave, allowing a huge range of scales that, when I looked at
> them, your scale set didn't seem to be offering any of.

At this moment it accepts from 5 to 12, plus 24 only, and I agree on
that point, I could have use some 24 tones/octave mappings, but I was
not certain of this and I had no way to verify. On the other hand, 24
tones playing is not of a common use for non-microtonalists, and
anyway user tunings have this possibility.

> At minimum, it seems to me, there should have been MOS for linear
> temperaments aside from 12 notes of meantone.

Again I am surprised how a man of science like you can be so quick to
criticize some works you haven't tried to understand nor even listen,
obviously.
What do you have in mind about linear temperaments that would be
missing, as for what concerns traditional scales ?
I avoided temperaments like Magic or Hanson and others on purpose,
simply because you won't see them used in traditional music. On the
contrary some subsets of Miracle are coherent to a different
traditions and such scales can be found in several of my tunings. The
point was not to display a collection of microtonal researches, but a
set of practical scales and realistic systems for traditional music
from the whole world, that's what it is and it's already more than that.
You say meantone is the only temperament you can find in Ethno2 ?
Here are a few examples of generators used here in rather bizarre
"meantones" :
Fong 169.38141189576 c. (SEA3, A12)
Ishku 184.36788075168 c. (ME12)
Slendra 234.86976848832 c. (A5, INDO2)
Baka 235.16875545672 c. (A5, INDO2, PER2)
Isrep 274.13603620368 c. (A20)
Chandrak 312.49618709916 c.(IN5)
Aksaka 325.86396379632 c. (A19)
Mohajira 348.91261178844 c. (ME4)
Arijaom 349.52343265716 c., or 850,47656734284 c. (ME3)
Buzurg 361.09738964 c. (PER2)
Ifbis 367.522672473 c. (PER10)
Amlak 405.27059311656c. (A13,IN8, IN10, NA5, NA11)
Leo 476.8073130162 c. (EA14)
Triseptra 483.35883814368 c. (A15)
Airos 494.4315 c. (PER1)
Melkis 499.11472 c. (ME2, EU12)
Then the lower range of my meantones start quite low with :
Gulu-nem 670.5060969 c. (INDO12)
While the superpyths (8 of them) go up to :
Myriadjerm 726.45631997364 c. (INDO13)

All of those were chosen because they make sense in systems from
specific cultures : Thaï, Burmese, Indonesian, Balkanic, Pygmy,
Mandinka, Ethiopian, Arabic, Persian, Indian, etc.
In fact, having a look on these recurrent sequences with the glasses
of linear temperaments is a very good idea and I would be curious to
know :
- if we have two different worlds here that will never meet ;
- or if some of these fractal generators would apply to known linear
temperaments ;
- or if others don't, but then could be new lines of linear
temperaments.
But I have more than a thousand of others, so where do I find a list
of all linear/or regular temperaments ?
I am astonished that Mohajira or Arijaom (348.91261178844 c. or
349.52343265716 c.), for example, have never been used as
temperaments, but may be I'm wrong.
- - - - - - -
Jacques

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> You are surprised to find "ethnic scales" in Ethno2 ??

No, but I don't see why you need to limit the possibilities. You say you are interested in scales which make sense in a particular culture, but personally I think everyone is in a particular culture, so that if a scale makes sense to someone, there you are.

> If you check for example in the
> "multi-system" section

There is no multi-system section in the set of scales I unzipped, but I did see some peculiar tunings of that sort. I much prefer more regular and comprehensible scale structures myself, but I'm all in favor of a wide variety.

> but always inspired by traditions.

You say both that you want to confine yourself to traditions, and that you are surprised to hear that anyone thinks your choices are overly confining and traditional, which seems like a contradiction to me.

> I avoided temperaments like Magic or Hanson and others on purpose,
> simply because you won't see them used in traditional music.

Same comment. You can't reasonably have it both ways.

> But I have more than a thousand of others, so where do I find a list
> of all linear/or regular temperaments ?

All of them is a big order, but

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

is something you could look at if that is your goal.

> I am astonished that Mohajira or Arijaom (348.91261178844 c. or
> 349.52343265716 c.), for example, have never been used as
> temperaments, but may be I'm wrong.

That looks suspiciously like semififths, the 11-limit
31&55 temperament with a neutral third generator.

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> You say meantone is the only temperament you can find in Ethno2 ?
> Here are a few examples of generators used here in rather bizarre
> "meantones" :
> Fong 169.38141189576 c. (SEA3, A12)
> Ishku 184.36788075168 c. (ME12)
> Slendra 234.86976848832 c. (A5, INDO2)
> Baka 235.16875545672 c. (A5, INDO2, PER2)
> Isrep 274.13603620368 c. (A20)
> Chandrak 312.49618709916 c.(IN5)
> Aksaka 325.86396379632 c. (A19)
> Mohajira 348.91261178844 c. (ME4)
> Arijaom 349.52343265716 c., or 850,47656734284 c. (ME3)
> Buzurg 361.09738964 c. (PER2)
> Ifbis 367.522672473 c. (PER10)
> Amlak 405.27059311656c. (A13,IN8, IN10, NA5, NA11)
> Leo 476.8073130162 c. (EA14)
> Triseptra 483.35883814368 c. (A15)
> Airos 494.4315 c. (PER1)
> Melkis 499.11472 c. (ME2, EU12)

I find a few of these names in my scale list, but where are the ones which are regularly generated? Where do these generators come from?

🔗Jacques Dudon <fotosonix@...>

On Tuesday Apr 27, 2010, Gene wrote :

> You say you are interested in scales which make sense in a > particular culture,

Personnally not exclusively, in the context of this Ethno2 output I was.

> but personally I think everyone is in a particular culture, so that > if a scale makes sense to someone, there you are.

Certainly, as long as your creations do not falsely pretend to be ethnomusicological, like lots of the "world music" or others do.
And I was well inspired to stick to that line, otherwise Motu would have presented any special thing I sent as "ethnic" anyway.

> There is no multi-system section in the set of scales I unzipped, > but I did see some peculiar tunings of that sort.

It's in the list by "scale systems" that should be in the archive.

> You say both that you want to confine yourself to traditions, and > that you are surprised to hear that anyone thinks your choices are > overly confining and traditional, which seems like a contradiction > to me.

It seems rather like an "clarification" of the context to me, than a contradiction.

> > You say meantone is the only temperament you can find in Ethno2 ?
> > Here are a few examples of generators used here in rather bizarre
> > "meantones" :
> > Fong 169.38141189576 c. (SEA3, A12)
> > Ishku 184.36788075168 c. (ME12)
> > Slendra 234.86976848832 c. (A5, INDO2)
> > Baka 235.16875545672 c. (A5, INDO2, PER2)
> > Isrep 274.13603620368 c. (A20)
> > Chandrak 312.49618709916 c.(IN5)
> > Aksaka 325.86396379632 c. (A19)
> > Mohajira 348.91261178844 c. (ME4)
> > Arijaom 349.52343265716 c., or 850,47656734284 c. (ME3)
> > Buzurg 361.09738964 c. (PER2)
> > Ifbis 367.522672473 c. (PER10)
> > Amlak 405.27059311656c. (A13,IN8, IN10, NA5, NA11)
> > Leo 476.8073130162 c. (EA14)
> > Triseptra 483.35883814368 c. (A15)
> > Airos 494.4315 c. (PER1)
> > Melkis 499.11472 c. (ME2, EU12)
>
> I find a few of these names in my scale list, but where are the > ones which are regularly generated? Where do these generators come > from?

I don't understand your question "where are the ones which are regularly generated ?". Do you mean among this list ?
All of them can be "regularly generated", or more precisely "linearly generated" I think, as far as what I understand about linear and regular temperaments. All of them come from my personal researches.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> > > Fong 169.38141189576 c. (SEA3, A12)
> > > Ishku 184.36788075168 c. (ME12)
> > > Slendra 234.86976848832 c. (A5, INDO2)
> > > Baka 235.16875545672 c. (A5, INDO2, PER2)
> > > Isrep 274.13603620368 c. (A20)
> > > Chandrak 312.49618709916 c.(IN5)
> > > Aksaka 325.86396379632 c. (A19)
> > > Mohajira 348.91261178844 c. (ME4)
> > > Arijaom 349.52343265716 c., or 850,47656734284 c. (ME3)
> > > Buzurg 361.09738964 c. (PER2)
> > > Ifbis 367.522672473 c. (PER10)
> > > Amlak 405.27059311656c. (A13,IN8, IN10, NA5, NA11)
> > > Leo 476.8073130162 c. (EA14)
> > > Triseptra 483.35883814368 c. (A15)
> > > Airos 494.4315 c. (PER1)
> > > Melkis 499.11472 c. (ME2, EU12)

> All of them can be "regularly generated", or more precisely "linearly
> generated" I think, as far as what I understand about linear and
> regular temperaments. All of them come from my personal researches.

If you researches are giving you eleven digits after the decimal place, then you are using a mathematical procedure to get these, and so I was asking what it was. How are they defined?

🔗Jacques Dudon <fotosonix@...>

On Wed Apr 28 Gene wrote :
(don't know why, but my precedent answer got some delay here...)

> > (Jacques) : All of them can be "regularly generated", or more > precisely "linearly
> > generated" I think, as far as what I understand about linear and
> > regular temperaments. All of them come from my personal researches.
>
> If you researches are giving you eleven digits after the decimal > place, then you are using a mathematical procedure to get these, > and so I was asking what it was. How are they defined?

I see. I thought you already knew by now that most of my material come from recurrent sequences ! (the Ethno2 Scala files gives them, with more less information, such as differential coherence /synchronous beating algorithms).
That was the reason of my precedent interrogations :
- do we have two different worlds here, that will never meet ?
- or if some of these fractal generators would apply to known linear temperaments ?
- or if others don't, but then could be new lines of linear temperaments ?
But I think we have already one answer with Mohajira (see my next message).

> > (Jacques) : where do I find a list of all linear/or regular > temperaments ?
>
> All of them is a big order, but
> http://x31eq.com/temper/regular.html
> is something you could look at if that is your goal.

Thanks, I know this tool but I what I am looking for is a list of all linear/or regular temperaments (or a number of them), with periods and generators.
- - - - - - -
Jacques

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> On Wed Apr 28 Gene wrote :

> > If you researches are giving you eleven digits after the decimal
> > place, then you are using a mathematical procedure to get these,
> > and so I was asking what it was. How are they defined?

> I see. I thought you already knew by now that most of my material
> come from recurrent sequences !

I guessed as much, and even thought of feeding them to an integer relations algorithm to reconstruct a polynomial, but somehow asking you seemed to make more sense.

(the Ethno2 Scala files gives them,
> with more less information, such as differential coherence /
> synchronous beating algorithms).

I only found a few of these; I seem to be missing something.

> That was the reason of my precedent interrogations :
> - do we have two different worlds here, that will never meet ?
> - or if some of these fractal generators would apply to known linear
> temperaments ?

Basically, they seem to me to be a way of tuning a linear temperament.

🔗Jacques Dudon <fotosonix@...>

Gene wrote :

> I guessed as much, and even thought of feeding them to an integer > relations algorithm to reconstruct a polynomial, but somehow asking > you seemed to make more sense.
>
> (the Ethno2 Scala files gives them,
> > with more less information, such as differential coherence /
> > synchronous beating algorithms).
>
> I only found a few of these; I seem to be missing something.

It is possible that in the rush I resumed the maths in some of the files, then I should complete them.
I notice that the values in cents I gave in my list for example, are often missing.
Then ME4 (Rast-Mohajira) does not even mentions the Mohajira algorithm, x^5 - x^4 = 1/2
And of course the series do not always appear at a first glance in the scales.
Apart from these kind of omissions, that I will fix, can you give a few examples of what is not clear ?

Fong 169.38141189576 c. (SEA3, A12)
Ishku 184.36788075168 c. (ME12)
Slendra 234.86976848832 c. (A5, INDO2)
Baka 235.16875545672 c. (A5, INDO2, PER2)
Isrep 274.13603620368 c. (A20)
Chandrak 312.49618709916 c.(IN5)
Aksaka 325.86396379632 c. (A19)
Mohajira 348.91261178844 c. (ME4)
Arijaom 349.52343265716 c., or 850,47656734284 c. (ME3)
Buzurg 361.09738964 c. (PER2)
Ifbis 367.522672473 c. (PER10)
Amlak 405.27059311656c. (A13, IN8, IN10, NA5, NA11)
Leo 476.8073130162 c. (EA14)
Triseptra 483.35883814368 c. (A15)
Airos 494.4315 c. (PER1)
Melkis 499.11472 c. (ME2, EU12)
Gulu-nem 670.5060969 c. (INDO12)
Simdek 676,48557456 c. (SEA5)
Nung-Phan 679.5604542 c. (SEA8)
Sapaan 680.015678 c. (SEA4)
Sireine 691.2348426 c. (OCE5, A2)
Cascade 695.31005796 c.(SEA10)
Myriadjerm 726.45631997364 c. (INDO13)

> > That was the reason of my precedent interrogations :
> > - do we have two different worlds here, that will never meet ?
> > - or if some of these fractal generators would apply to known linear
> > temperaments ?
>
> Basically, they seem to me to be a way of tuning a linear temperament.

That's what I think also, different paths to arrive to the same meadows,
with moments of rainbows when different methods can cumulate their respective lights.
- - - - - - -
Jacques

🔗Graham Breed <gbreed@...>

On 30 April 2010 02:23, Jacques Dudon <fotosonix@...> wrote:
> All of them is a big order, but

> http://x31eq.com/temper/regular.html
> is something you could look at if that is your goal.
>
> Thanks, I know this tool but I what I am looking for is a list of all linear/or regular temperaments (or a number of them), with periods and generators.

That's out of date.

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

is where you should be now. It gives names, but you have to click the
name to get the generators.

You could try here:

http://x31eq.com/cgi-bin/pregular.cgi?limit=7&error=5.0

Or the longer list here:

http://x31eq.com/cgi-bin/more.cgi?r=2&limit=7&error=5.0

and click every name to see the generators.

I'll have a look at your list tonight. What you're defining are MOS
families (I think that's what I called them before). A particular
number of notes gives an MOS. A mapping from primes gives a
temperament class. That's what we're interested in, and you should
understand it because you gave one for mohajira.

Graham

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

I looked again using a text editor, and it seems part of my problem has been that Scala had problems reading some of these and in any case I missed the marginal notes, which since I do that sort of thing myself I shouldn't have.

> Fong 169.38141189576 c. (SEA3, A12)

Let's start here. First, you don't use the names SEA3 or A12 in the list I have. There is "fong.scl", which is the following:

97/88, 97/88, 107/88, 107/88, 59/44, 65/44
1041/704, 287/176, 18/11, 633/352, 639/352, 2/1

The repeated notes make it hard for Scala to analyze the scale correctly, and strikes me as confusing and counterproductive anyway. If you want to use 12 keys, why not use 12 notes? Fong will still be in there. If you don't want to use 12 notes, then have the Scala file give fewer notes, it seems to me.

Removing the extra notes gives a nine note scale, some of the steps of which are close to your Fong generator of the real root of
x^3 - x - 1/8, and some of which are the result of peculiar interpolated comma steps of sizes 1041/1040 and 288/287. Getting rid of this bit of decoration produces the seven note scale

97/88, 107/88, 59/44, 1041/704, 287/176, 633/352, 2

This is, finally, the MOS we were looking for, but finding it was a lot of work. It could, of course, be extended to a 12 note scale with the same generator which would provide a lot more resources, even if they are not ones an ethnically minded musician would want. But why second guess what someone might want?

🔗genewardsmith <genewardsmith@...>

🔗Graham Breed <gbreed@...>

On 30 April 2010 17:42, genewardsmith <genewardsmith@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>> Why in the world don't you list 7-limit planar generators by comma?

Because I chose to do it this way.

> Some of these rank three babies have names, and some
> I've actually composed in--including breed, which Graham
> may feel would,be a bit much to stick on his own web site,
> I suppose. Anyway, there's

I have some of those, including breed, egocentric thought it is. Some
of them only apply in the 11-limit though. I'll check these offline
and upload them next time I'm in a net bar that allows me to do SFTP.

> starling 126/125 composed in
> marvel 225/224 composed in

That's the other 7-limit one I have now.

> breed 2401/2400 composed in
> hemifamity 5120/5103 composed in
> gamelan 1029/1024
> orwellian 1728/1715

That's this?

http://x31eq.com/cgi-bin/rt.cgi?ets=27_31_22&error=3.160&limit=7&invariant=3_-1_1_0_1_1_0_0_2

I called one Big Brother because it looks like Orwell. I think it's different:

http://x31eq.com/cgi-bin/rt.cgi?ets=14_31_22&error=5.0&limit=11&invariant=4_1_2_1_3_2_2_1_0_3_1_3

> ragismic 4375/4374
> landscape 250047/250000

As my tuning-math message seems not to have gone through, note that I
renamed Wonder to Jove.

Graham

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>

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

> As my tuning-math message seems not to have gone through, note that I
> renamed Wonder to Jove.

In keeping with the rest of the pantheon of rank 3 11-limit temperaments:

Jove. formerly Wonder (11-limit neutral-thirds lattice)
41&58&72, {243/242, 441/440}

Portent, 31&41&46, {385/384, 441/440}

Spetimal meantone plus 11, 7&12&19, {81/80, 126/125}

Freya, 31&41&&270 {2401/2400, 3025/3024}

Marvel, 22&31&41, {225/224, 385/384}

Zeus, 22&31&46, {121/120, 176/175}

Indra, 31&41&80, {540/539, 1375/1372}

Thrush, 31&46&58, {126/125, 176/175}

Prodigy, 29&31&41, {225/224, 441/440}

Minerva, 31&41&43, {99/98, 176/175}

Odin, 270&342&1578, {9801/9800, 151263/151250}

Thor, 46&72&80, {3025/3024, 4375/4374}

Baldur, 58&72&270, {2401/2400, 9801/9800}

7&12&19, {45/44, 81/80}

7&8&29, {250/243, 1331/1296}

🔗genewardsmith <genewardsmith@...>

🔗Herman Miller <hmiller@...>

genewardsmith wrote:
> There are a lot of potential generators on Jacques' list, I may look
> at more of them from time to time.

I'll see what I can dig up from my scale tree list.

Fong 169.38141189576 c. (SEA3, A12)
The generator is too large for porcupine, but within the range of a 7&71 temperament that I don't have a name for.

Ishku 184.36788075168 c. (ME12)
Nothing obvious, close to 2/13 of an octave.

Slendra 234.86976848832 c. (A5, INDO2)
As you mentioned, this could be rodan.

Baka 235.16875545672 c. (A5, INDO2, PER2)
This falls between rodan and a 5&56 temperament.

Isrep 274.13603620368 c. (A20)
This could be 22&35.

Chandrak 312.49618709916 c.(IN5)
Between myna and hanson.

Aksaka 325.86396379632 c. (A19)
Nothing obvious near this one.

Mohajira 348.91261178844 c. (ME4)
Semififths.

Arijaom 349.52343265716 c., or 850,47656734284 c. (ME3)
Also could be semififths.

Buzurg 361.09738964 c. (PER2)
Dicot is probably the best match for this one.

Ifbis 367.522672473 c. (PER10)
Nothing obvious near this one either.

Amlak 405.27059311656c. (A13, IN8, IN10, NA5, NA11)
Nothing obvious near this one either.

Leo 476.8073130162 c. (EA14)
Just misses vulture by a cent. Might possibly be 5&63.

Triseptra 483.35883814368 c. (A15)
Nothing obvious near this one either.

Airos 494.4315 c. (PER1)
Fairly close to a less common schismatic variety called schism.

Melkis 499.11472 c. (ME2, EU12)
Could be grackle (another schismatic temperament), but very close to 12-ET.

Gulu-nem 670.5060969 c. (INDO12)
An unnamed 9&25 temperament, or a variety of mavila.

Simdek 676,48557456 c. (SEA5)
Another variety of mavila, between 16- and 23-ET.

Nung-Phan 679.5604542 c. (SEA8)
Between mavila and 7-ET.

Sapaan 680.015678 c. (SEA4)
Between mavila and 7-ET.

Sireine 691.2348426 c. (OCE5, A2)
Between 7-ET and meantone.

Cascade 695.31005796 c.(SEA10)
A kind of meantone, between meanpop and flattone.

Myriadjerm 726.45631997364 c. (INDO13)
Nothing obvious near this one.

🔗genewardsmith <genewardsmith@...>

🔗Graham Breed <gbreed@...>

On 1 May 2010 03:56, genewardsmith <genewardsmith@...> wrote:
> There are a lot of potential generators on Jacques' list, I may look at more of them from time to time.
>
>> Slendra 234.86976848832 c. (A5, INDO2)
>
> This could certainly be Rodan, the 41&46 temperament. As with mohajira,
> if I had 24 notes I'd keep going until I reached all 24. Why not? Anyway
>, the mapping to primes <<3 17 -1 -13 -22 -20 ...||
> suggests you'll be missing a lot of the fun here if you don't. I know,
> of course, that this suggestion isn't ethnic, so shame on me.

It looks like Slendric to me. Which is fortunate. You'd almost think
there was some common inspiration behind the names.

http://x31eq.com/cgi-bin/rt.cgi?ets=5+26&limit=2.3.7

Rodan is here:

http://x31eq.com/cgi-bin/rt.cgi?ets=41+46&limit=17

I can make that name standard up to the 17-limit.

Now, as I believe these are following the same logic as Erv Wilson's
Mt Meru scales, that would be a place to look for equivalences.
Everything's here:

http://www.anaphoria.com/wilson.html

And this is the first one to read:

http://www.anaphoria.com/meruone.PDF

There are different generators in Jacques' list that cluster around
Mavila. The true Mavila would be the one that follows the same series
that Erv used.

Graham

🔗Graham Breed <gbreed@...>

Here's the generator for every rank 2 name in my database where the
period is an octave. That is, every TOP-RMS strictly linear
temperament based on consecutive primes. It tried to remove anything
obviously deviant, which would mean it was defined on a
non-consecutive prime limit. I may have missed some. Duplicate names
mean similar mappings for different limits.

38.413 Slender
45.139 Quartonic
77.191 Tertiaseptal
77.709 Valentine
77.881 Valentine
82.505 Nautilus
88.076 Octacot
98.670 Passion
100.838 Ripple
116.633 Miracle
116.675 Miracle
116.747 Miracle
125.608 Negrisept
125.755 Negripent
130.106 Mohajira
146.474 Bohpier
146.545 Bohpier
154.579 Nusecond
158.649 Hemikleismic
158.868 Hystrix
162.747 Porcupine
162.880 Porcupine
163.950 Porcupine
175.434 Sesquiquartififths
176.160 Tetracot
193.201 Luna
193.244 Hemithirds
193.898 Hemiwuerschmidt
228.334 Gorgo
230.336 Gamera
230.762 Gidorah
232.031 Cynder/Mothra
232.193 Cynder
233.930 Guiron
234.459 Rodan
239.977 Penta
251.881 Semaphore
252.635 Semaphore
259.952 Superpelog
260.388 Bug
271.107 Quasiorwell
271.426 Orwell
271.509 Orwell
271.546 Orwell
271.627 Orson
310.144 Myna
310.146 Myna
310.276 Myna
315.181 Parakleismic
315.240 Parakleismic
316.473 Keemun
316.732 Catakleismic
317.007 Hanson
317.121 Countercata
317.656 Keemun
321.847 Superkleismic
321.930 Superkleismic
339.519 Amity
348.119 Vicentino
348.415 Mohajira
348.477 Mohajira
348.558 Mohajira
348.594 Dicot
348.736 Mohajira
348.810 Mohajira
351.477 Hemififths
355.904 Beatles
378.479 Muggles
380.058 Magic
380.352 Magic
380.696 Magic
380.787 Magic
386.863 Grendel
387.383 Wuerschmidt
387.799 Wuerschmidt
425.942 Squares
425.957 Squares
426.276 Squares
427.208 Sidi
441.335 Clyde
443.058 Sensipent
443.383 Sensisept
443.626 Sensi
443.945 Sensisept
456.014 Father
464.845 Semisept
475.543 Vulture
475.636 Vulture
478.431 Mother
489.709 Superpyth
489.922 Superpyth
491.762 Quasisuper
496.746 Dominant
496.961 Undecental
497.384 Kwai
497.441 Cassandra
497.629 Schismatic
497.887 Alt. Cassandra
497.915 Garibaldi
498.243 Pontiac
498.264 Helmholtz
498.427 Dominant
498.444 Schism
498.761 Grackle
499.860 Sharptone
503.031 Meantone
503.358 Meantone
503.505 Meantone
503.566 Meanpop
503.761 Meantone
503.789 Meanpop
506.221 Flattone
516.694 Marvo
520.194 Mavila
526.003 Pelogic
567.594 Liese
568.865 Triton
580.267 Tritonic
580.286 Tritonic
582.452 Neptune

Graham

🔗Jacques Dudon <fotosonix@...>

Hi Herman, Gene, and Graham,

Thanks for your efforts with this "ethnic" selection of fractal
generators.
I am asking your attention on one point.
We are making here comparisons between octave-tempered and octavial
systems (my usual recurrent systems are based on pure 2/1).
In differential coherence or equal-beating systems the intervals are
very sensible, and it makes sense whatever octaves to consider the
precise values of the generators as important, though they would need
to be slightly retuned when transported in a stretched octave context.
But on the other hand, in the scales further developpement through
the various MOS and edos approached, the ratio period/generator would
be more pertinent, and in temperaments using heavy octave tempering
this makes a big difference.
For example with the Mavila temperament, whether I use the exact
generator or its cyclic ratio equivalent, it makes a difference of 4
cents, in which I have 8 different fractal species ; then there is
about just as much difference (again 8 different species) according
to my data between this last one and Samak (= Erv Wilson's Chopi-
Mavila).

Graham wrote :

> There are different generators in Jacques' list that cluster around
> Mavila. The true Mavila would be the one that follows the same series
> that Erv used.

It seems to me that what I've seen described as the "Mavila" linear
temperament does not match with Erv's "Mavila".
And I find very different values for it ; what are the best ones ? is
it identical to "Pelogic" ?.
BTW, just like with most other Mt Meru series, what Erv calls "Chopi-
Mavila" is one I found myself long ago I call "Samak" (and it is not
a 4th degree algorithm, but a 3rd degree, as I wrote to Kraig Grady
in this list - this may induce wrong series).
Anyway I didn't select Samak for the Ethno tunings, where I used in
the same range Sapaan, Nûng-Phan, and Simdek.
Here is a more complete list of the area :

Evan 686.2389753438 c. (SEA1)
Heptacle 685.23621617592 c. (SEA9)
Hawan 684.303509672 c.
Malika 684.23091460176 c.
Tareyh 684.06098909496 c.
Cybawan 683.91932969184 c.
Nueh 682.58974072152 c.
Shemag 681.72120226116 c.
Sapaan 680.015678 c. (SEA4)
Nung-Phan 679.5604542 c. (SEA7, SEA8)
Simdek 676.48557456 c. (SEA5)
Samak 676.33715352528 c.
Cylpud 675.89784547224 c.
- - - - - - - - - - - - - - - - - - - - - -
Jacques

> Gene Ward Smith wrote:
> > There are a lot of potential generators on Jacques' list, I may look
> > at more of them from time to time.

> Herman Miller wrote :
> I'll see what I can dig up from my scale tree list.
>
> Fong 169.38141189576 c. (SEA3, A12)
> The generator is too large for porcupine, but within the range of a
> 7&71
> temperament that I don't have a name for.
>
> Ishku 184.36788075168 c. (ME12)
> Nothing obvious, close to 2/13 of an octave.
>
> Slendra 234.86976848832 c. (A5, INDO2)
> As you mentioned, this could be rodan.
>
> Baka 235.16875545672 c. (A5, INDO2, PER2)
> This falls between rodan and a 5&56 temperament.
>
> Isrep 274.13603620368 c. (A20)
> This could be 22&35.
>
> Chandrak 312.49618709916 c.(IN5)
> Between myna and hanson.
>
> Aksaka 325.86396379632 c. (A19)
> Nothing obvious near this one.
>
> Mohajira 348.91261178844 c. (ME4)
> Semififths.
>
> Arijaom 349.52343265716 c., or 850,47656734284 c. (ME3)
> Also could be semififths.
>
> Buzurg 361.09738964 c. (PER2)
> Dicot is probably the best match for this one.
>
> Ifbis 367.522672473 c. (PER10)
> Nothing obvious near this one either.
>
> Amlak 405.27059311656c. (A13, IN8, IN10, NA5, NA11)
> Nothing obvious near this one either.
>
> Leo 476.8073130162 c. (EA14)
> Just misses vulture by a cent. Might possibly be 5&63.
>
> Triseptra 483.35883814368 c. (A15)
> Nothing obvious near this one either.
>
> Airos 494.4315 c. (PER1)
> Fairly close to a less common schismatic variety called schism.
>
> Melkis 499.11472 c. (ME2, EU12)
> Could be grackle (another schismatic temperament), but very close
> to 12-ET.
>
> Gulu-nem 670.5060969 c. (INDO12)
> An unnamed 9&25 temperament, or a variety of mavila.
>
> Simdek 676,48557456 c. (SEA5)
> Another variety of mavila, between 16- and 23-ET.
>
> Nung-Phan 679.5604542 c. (SEA8)
> Between mavila and 7-ET.
>
> Sapaan 680.015678 c. (SEA4)
> Between mavila and 7-ET.
>
> Sireine 691.2348426 c. (OCE5, A2)
> Between 7-ET and meantone.
>
> Cascade 695.31005796 c.(SEA10)
> A kind of meantone, between meanpop and flattone.
>
> Myriadjerm 726.45631997364 c. (INDO13)
> Nothing obvious near this one.

🔗Jacques Dudon <fotosonix@...>

On Fri Apr 30 Gene wrote :

> I looked again using a text editor, and it seems part of my problem
> has been that Scala had problems reading some of these and in any
> case I missed the marginal notes, which since I do that sort of
> thing myself I shouldn't have.

I also had the same type of problems with Scala on a Mac, however
Ethno 2 opens them all, luckily !

> > Fong 169.38141189576 c. (SEA3, A12)
>
> Let's start here. First, you don't use the names SEA3 or A12 in the
> list I have.

SEA3, A12, etc... are abbreviations for "SouthEast Asia", "Africa"
etc. and the same number in the list (appearing in Ethno) gives you
the name of the scl file.

> There is "fong.scl", which is the following :
>
> 97/88, 97/88, 107/88, 107/88, 59/44, 65/44
> 1041/704, 287/176, 18/11, 633/352, 639/352, 2/1
>
> The repeated notes make it hard for Scala to analyze the scale
> correctly, and strikes me as confusing and counterproductive
> anyway. If you want to use 12 keys, why not use 12 notes? Fong will
> still be in there. If you don't want to use 12 notes, then have the
> Scala file give fewer notes, it seems to me.

I have been using the full 12 notes for a variation in another Thaï
scale : evan_thai.scl, where the black keys follow a second spiral
cycle * x^7 / 16, and in others whenever it was making sense to me.
If I had reduced the mapping to 7 notes, the black keys would have
been silent and that would have been counterproductive. Instead here
they give the traditional pentatonic Thai subset, and allow for more
playing facilities.
But next time I will include more longer cycles, I promise !

> Removing the extra notes gives a nine note scale, some of the steps
> of which are close to your Fong generator of the real root of
> x^3 - x - 1/8, and some of which are the result of peculiar
> interpolated comma steps of sizes 1041/1040 and 288/287. Getting
> rid of this bit of decoration produces the seven note scale
>
> 97/88, 107/88, 59/44, 1041/704, 287/176, 633/352, 2

That's it, you're doing very well !
65 and 9 are not decorative, they are some of the differential tones
(639 - 2*287 and 97 - 88).
And 639 in option to 633, by shifting the big tone interval inverses
the scale !

> This is, finally, the MOS we were looking for, but finding it was a > lot of work. It could, of course, be extended to a 12 note scale
> with the same generator which would provide a lot more resources,
> even if they are not ones an ethnically minded musician would want.
> But why second guess what someone might want?

What I retain is that I should write the cents values of the root
generator and the series on a extra line to avoid some work to the
searchers and if possible longer !
Thanks a lot for your contribution.
- - - - - - - -
Jacques

🔗Chris <chrisvaisvil@...>

I noticed last night the ethno2 scl files I looked at do not have carriage returns or line feeds. While seemingly a small difference, depending on the program's parser this could be a problem.

The Appalachian tuning file was one formayed like this in specific.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: Jacques Dudon <fotosonix@wanadoo.fr>
Date: Sat, 1 May 2010 18:01:01
To: <tuning@yahoogroups.com>
Subject: [tuning] Re : Ethno2 microtunings

On Fri Apr 30 Gene wrote :

I also had the same type of problems with Scala on a Mac, however
Ethno 2 opens them all, luckily !

> > Fong 169.38141189576 c. (SEA3, A12)
>
> Let's start here. First, you don't use the names SEA3 or A12 in the
> list I have.

SEA3, A12, etc... are abbreviations for "SouthEast Asia", "Africa"
etc. and the same number in the list (appearing in Ethno) gives you
the name of the scl file.

> This is, finally, the MOS we were looking for, but finding it was a
> lot of work. It could, of course, be extended to a 12 note scale
> with the same generator which would provide a lot more resources,
> even if they are not ones an ethnically minded musician would want.
> But why second guess what someone might want?

🔗Jacques Dudon <fotosonix@...>

🔗Rustom Mody <rustompmody@...>

Hi Graham.

Thanks for that. Going up the links I see that http://x31eq.com/tuning.htm is quite a mine of info (and also fun to read)

I find one statement there (start.htm)

Most
Western Classical Music can be interpreted as a meantone approximation
of just intonation.

Can you point out some links to understand how/why this is so?

Thanks

Rustom
--- On Fri, 4/30/10, Graham Breed <gbreed@...> wrote:

> http://x31eq. com/temper/ regular.html

> is something you could look at if that is your goal.

> Thanks, I know this tool but I what I am looking for is a list of all linear/or regular temperaments (or a number of them), with periods and generators.

That's out of date.

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

is where you should be now. It gives names, but you have to click the

name to get the generators.

🔗Herman Miller <hmiller@...>

genewardsmith wrote:
> > --- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> >> Ishku 184.36788075168 c. (ME12) Nothing obvious, close to 2/13 of
>> an octave.
> > It could be that three steps give 11/8, and two a sharp 16/13. That
> tempers out 2^(-18) 11^2 13^3, and you could consider it a sort of
> 13&46 temperament using partial mappings. And if you do that, why not
> consider it the 7/46 generator instead? 3, 5, and 7 are now a bit
> more complex than 11, 13 or 17, but they are there.

7/46 <0, 17, -11, -21, 3, -2] has a decent 3, 5, and 7, yes, but with a 184.36788-cent generator these are way off (+32.30c, -14.36c, -40.55c). The 2/13 mapping <0, 4, 2, 5, 3, -2] is simpler and not much worse (+35.52c, -17.58c, -46.99c), but -1 is a slightly better mapping for the 7th harmonic (+46.81c), and -8 is a little more better (-43.77c). The generator is so close to 2 steps of 13 that it takes a long time to get a good approximation of 3 (108 steps to get closer than 10c) and even longer to approach 7.

🔗Herman Miller <hmiller@...>

Jacques Dudon wrote:
> Hi Herman, Gene, and Graham,
> > Thanks for your efforts with this "ethnic" selection of fractal generators.
> I am asking your attention on one point.
> We are making here comparisons between octave-tempered and octavial > systems (my usual recurrent systems are based on pure 2/1).
> In differential coherence or equal-beating systems the intervals are > very sensible, and it makes sense whatever octaves to consider the > precise values of the generators as important, though they would need to > be slightly retuned when transported in a stretched octave context.
> But on the other hand, in the scales further developpement through the > various MOS and edos approached, the ratio period/generator would be > more pertinent, and in temperaments using heavy octave tempering this > makes a big difference.
> For example with the Mavila temperament, whether I use the exact > generator or its cyclic ratio equivalent, it makes a difference of 4 > cents, in which I have 8 different fractal species ; then there is about > just as much difference (again 8 different species) according to my data > between this last one and Samak (= Erv Wilson's Chopi-Mavila).

Mavila temperament in particular has a wide range of generator sizes. In general, the more accurate a temperament, the more precise the size of the generator needs to be; since mavila tempers out 135/128 (a fairly large 92c interval), the tuning is not as sensitive. Graham's list of generator sizes had specific "optimal" sizes for the generator (which in this case are based on a 2/1 octave period), so the mavila generator for instance was given as 520.194 cents. For a tuning with stretched octaves, the TOP-RMS generator is 523.827 cents with a period of 1208.380 cents. So the generator/period ratio is the same.

The list I'm using is similar to Wilson's scale tree, with generators expressed as fractions of an octave (e.g. 4/9, 7/16, 10/23), except that I have a mapping associated with each entry, and I label them with the names of the corresponding temperaments (so 4/9, 7/16, and 10/23 all end up being labed as "mavila"). One problem with this is that some of these ET's are boundary cases, so I've got 3/7 labeled as "mavila" and 7/17 labeled as "helmholtz / schism". So Graham's list with the TOP-RMS generator sizes is probably a better reference.

> Graham wrote :
> >> There are different generators in Jacques' list that cluster around
>> Mavila. The true Mavila would be the one that follows the same series
>> that Erv used.
> > It seems to me that what I've seen described as the "Mavila" linear > temperament does not match with Erv's "Mavila".
> And I find very different values for it ; what are the best ones ? is it > identical to "Pelogic" ?.

Pelogic is one of two 7-limit versions of mavila, also known as hexadecimal.

> BTW, just like with most other Mt Meru series, what Erv calls > "Chopi-Mavila" is one I found myself long ago I call "Samak" (and it is > not a 4th degree algorithm, but a 3rd degree, as I wrote to Kraig Grady > in this list - this may induce wrong series).
> Anyway I didn't select Samak for the Ethno tunings, where I used in the > same range Sapaan, Nï¿½ng-Phan, and Simdek.
> Here is a more complete list of the area :
> > Evan 686.2389753438 c. (SEA1)
> Heptacle 685.23621617592 c. (SEA9)
> Hawan 684.303509672 c.
> Malika 684.23091460176 c.
> Tareyh 684.06098909496 c.
> Cybawan 683.91932969184 c.
> Nueh 682.58974072152 c.
> Shemag 681.72120226116 c.
> Sapaan 680.015678 c. (SEA4)
> Nung-Phan 679.5604542 c. (SEA7, SEA8)
> Simdek 676.48557456 c. (SEA5)
> Samak 676.33715352528 c.
> Cylpud 675.89784547224 c.
> - - - - - - - - - - - - - - - - - - - - - -

These are mainly in the mavila range or the transition range between mavila and meantone around 7-ET, with MOS scales of 7, 9, 16, and 23 steps except for Evan (which is on the meantone side of 7-ET).

🔗Graham Breed <gbreed@...>

On 1 May 2010 19:03, Jacques Dudon <fotosonix@...> wrote:

> But on the other hand, in the scales further developpement through
> the various MOS and edos approached, the ratio period/generator
> would be more pertinent, and in temperaments using heavy
> octave tempering this makes a big difference.

The generators I gave were really generator/period ratios for TOP-RMS
tuning, scaled up to be cents. They'll always be close to the optimal
pure octaves tuning. In so far as they differ, I don't take any of
these optima seriously. The point of a rank 2 temperament is that you
have freedom to vary the generator and (if you choose) period so long
as the approximations don't get too far out.

With Mohajira, you gave the mapping matrix from just primes to
tempered generators. Like this:

<0, 2, 8, -11, 5, -1, -10, -6]

At least, I think it originated with you and wasn't an unattributed
quote. It defines the temperament class. If you think about other
systems the same way, please give the mappings.

Graham

🔗Graham Breed <gbreed@...>

On 1 May 2010 22:10, Rustom Mody <rustompmody@...> wrote:

> Thanks for that. Going up the links I see that http://x31eq.com/tuning.htm is quite a mine of info (and also fun to read)

Thank you!

> I find one statement there (start.htm)
>
> Most Western Classical Music can be interpreted as a meantone approximation of just intonation.
>
> Can you point out some links to understand how/why this is so?

It's one of those broad, unsupported statements I threw out when I was
younger. Any beginning music theory text (that ignores serialism and
jazz) should make this clear. But it won't mention meantone. You
have to notice that you can spell everything consistently, and the
notation is really about meantone. It's a bit like travel guides not
mentioning that the amount of oxygen in the air is about the same
wherever you go.

Graham

🔗Jacques Dudon <fotosonix@...>

Chris Vaisvil wrote :

> I noticed last night the ethno2 scl files I looked at do not have > carriage returns or line feeds. While seemingly a small difference, > depending on the program's parser this could be a problem.

What do you mean, "do not have carriage returns" ? you had the whole scale in one line ?

> The Appalachian tuning file was one formayed like this in specific.

Was it different from others ?

I wrote my scala files on a Mac with TextEdit, and asked them to be tested on Mac and PC by the developpers.
Apparently they all work well with Ethno, as does any other Scala file I tried.

BTW I liked very much your last piece Chris, "In search of perfect consonance" -
the dissonances are refreshing and the sound is quite beautiful - what means 12+17 ?
- - - - - - -
Jacques

🔗Chris Vaisvil <chrisvaisvil@...>

Hi Jacques,

> What do you mean, "do not have carriage returns" ? you had the whole scale in one line ?

Yes exactly. And others files were the same. I thought it may have
been a Mac versus PC format issue.
If other programs have problem parsing the .scl file this might
contribute since a programmer could assume that the
"lf-cr" are there. (line feed - carriage return are archaic typewriter
- perhaps even telegraph left overs.)

> BTW I liked very much your last piece Chris, "In search of perfect consonance" -
> the dissonances are refreshing and the sound is quite beautiful - what means 12+17 ?

As for my piece - the idea was to use two tunings at the same time. So
along with my 12 TET guitar signal I also use fractal tune smithy to
"on the fly" retune by Roland GR-20 to Arabic 17 tone Pythagorean
tuning. The Roland is a guitar to midi converter plus synthesizer that
is played by the same guitar. So 12 + 17 means I am using 12 tet + the
17 tone tuning at the same time.

http://www.rolandus.com/products/productdetails.php?ProductId=592

Chris

🔗Jacques Dudon <fotosonix@...>

Hey, Thanks a lot Graham, that's a wonderful work !
And based on the octave, exactly what I wanted.

I'll take my time to explore, but just a question for now :

348.119 Vicentino
348.415 Mohajira
348.477 Mohajira
348.558 Mohajira
348.594 Dicot
348.736 Mohajira
348.810 Mohajira
351.477 Hemififths

= these different Mohajiras correspond to different limits ?
And I'm curious to know how "Vicentino" (5 limit ?) arrives to this generator.

Then what is :
130.106 Mohajira ? a generator producing similar divisions ?
This is close to 14/13, but I don't see direct relation.

Recurrent sequences for your Miracle temperaments were easy to find :
These have equal-beating properties, indicated by the algorithms (x = Miracle ratios) :
116.633 Miracle < 36 - 24x^6 = 15x - 16 or 15 - 10x^6 = 5x^13 - 8x^6 (= both 116.630 c.)
116.675 Miracle < 7x^6 - 8x7 = 12 - 8x^6 (= exactly 116.675 c.)
116.747 Miracle < 6x^9 - 11 = 9 - 6x^6 or 12 - 8x^6 = 7x^2 - 8 or 14x^8 - 24 = 7x^2 - 8 (= all three 116.737 c)

So I suppose you can say that the 16.675 c. is equal-beating ;
it has also the differential coherence property : 15x^6 - 12 = 8x^4 and his ratio is
1.06971726001893
- - - - - - - -
Jacques

> On Sat May 1 2010 Graham wrote :
>
> Here's the generator for every rank 2 name in my database where the
> period is an octave. That is, every TOP-RMS strictly linear
> temperament based on consecutive primes. It tried to remove anything
> obviously deviant, which would mean it was defined on a
> non-consecutive prime limit. I may have missed some. Duplicate names
> mean similar mappings for different limits.
>
> 38.413 Slender
> 45.139 Quartonic
> 77.191 Tertiaseptal
> 77.709 Valentine
> 77.881 Valentine
> 82.505 Nautilus
> 88.076 Octacot
> 98.670 Passion
> 100.838 Ripple
> 116.633 Miracle
> 116.675 Miracle
> 116.747 Miracle
> 125.608 Negrisept
> 125.755 Negripent
> 130.106 Mohajira
> 146.474 Bohpier
> 146.545 Bohpier
> 154.579 Nusecond
> 158.649 Hemikleismic
> 158.868 Hystrix
> 162.747 Porcupine
> 162.880 Porcupine
> 163.950 Porcupine
> 175.434 Sesquiquartififths
> 176.160 Tetracot
> 193.201 Luna
> 193.244 Hemithirds
> 193.898 Hemiwuerschmidt
> 228.334 Gorgo
> 230.336 Gamera
> 230.762 Gidorah
> 232.031 Cynder/Mothra
> 232.193 Cynder
> 233.930 Guiron
> 234.459 Rodan
> 239.977 Penta
> 251.881 Semaphore
> 252.635 Semaphore
> 259.952 Superpelog
> 260.388 Bug
> 271.107 Quasiorwell
> 271.426 Orwell
> 271.509 Orwell
> 271.546 Orwell
> 271.627 Orson
> 310.144 Myna
> 310.146 Myna
> 310.276 Myna
> 315.181 Parakleismic
> 315.240 Parakleismic
> 316.473 Keemun
> 316.732 Catakleismic
> 317.007 Hanson
> 317.121 Countercata
> 317.656 Keemun
> 321.847 Superkleismic
> 321.930 Superkleismic
> 339.519 Amity
> 348.119 Vicentino
> 348.415 Mohajira
> 348.477 Mohajira
> 348.558 Mohajira
> 348.594 Dicot
> 348.736 Mohajira
> 348.810 Mohajira
> 351.477 Hemififths
> 355.904 Beatles
> 378.479 Muggles
> 380.058 Magic
> 380.352 Magic
> 380.696 Magic
> 380.787 Magic
> 386.863 Grendel
> 387.383 Wuerschmidt
> 387.799 Wuerschmidt
> 425.942 Squares
> 425.957 Squares
> 426.276 Squares
> 427.208 Sidi
> 441.335 Clyde
> 443.058 Sensipent
> 443.383 Sensisept
> 443.626 Sensi
> 443.945 Sensisept
> 456.014 Father
> 464.845 Semisept
> 475.543 Vulture
> 475.636 Vulture
> 478.431 Mother
> 489.709 Superpyth
> 489.922 Superpyth
> 491.762 Quasisuper
> 496.746 Dominant
> 496.961 Undecental
> 497.384 Kwai
> 497.441 Cassandra
> 497.629 Schismatic
> 497.887 Alt. Cassandra
> 497.915 Garibaldi
> 498.243 Pontiac
> 498.264 Helmholtz
> 498.427 Dominant
> 498.444 Schism
> 498.761 Grackle
> 499.860 Sharptone
> 503.031 Meantone
> 503.358 Meantone
> 503.505 Meantone
> 503.566 Meanpop
> 503.761 Meantone
> 503.789 Meanpop
> 506.221 Flattone
> 516.694 Marvo
> 520.194 Mavila
> 526.003 Pelogic
> 567.594 Liese
> 568.865 Triton
> 580.267 Tritonic
> 580.286 Tritonic
> 582.452 Neptune
>
> Graham

🔗genewardsmith <genewardsmith@...>

🔗Graham Breed <gbreed@...>

On 2 May 2010 19:53, Jacques Dudon <fotosonix@...> wrote:
>
> Hey, Thanks a lot Graham, that's a wonderful work !
> And based on the octave, exactly what I wanted.
> I'll take my time to explore, but just a question for now :
> 348.119 Vicentino
> 348.415 Mohajira
> 348.477 Mohajira
> 348.558 Mohajira
> 348.594 Dicot
> 348.736 Mohajira
> 348.810 Mohajira
> 351.477 Hemififths
> = these different Mohajiras correspond to different limits ?

Yes.

> And I'm curious to know how "Vicentino" (5 limit ?) arrives to this generator.

Yes, it's 5-limit. It's how the numbers came out. You can see it's
roughly half of the 1/4 comma meantone fifth, which comes out as
348.289 cents.

> Then what is :
> 130.106 Mohajira ? a generator producing similar divisions ?
> This is close to 14/13, but I don't see direct relation.

Ah, that shouldn't be there. It's a case where the mapping doesn't
refer to consecutive primes, but the script to reverse engineer the
temperaments doesn't know that. So it treats the mapping as belonging
to a different prime limit and gives a random answer.

Graham

🔗Jacques Dudon <fotosonix@...>

Gene wrote :

> > (Jacques answering to Graham) : Recurrent sequences for your > Miracle temperaments were easy to find :
> > These have equal-beating properties, indicated by the algorithms
> > (x = Miracle ratios) :
> > 116.633 Miracle < 36 - 24x^6 = 15x - 16
> > or 15 - 10x^6 = 5x^13 - 8x^6 (= both 116.630 c.)
>
> Characteristic polynomials like these don't seem like good choices > for recurrent sequences, since they are loaded with roots with > greater absolute value than the positive real root. But we could > just use the positive real root as a generator, and then you'd get > the desired relationships between the notes.

These were just examples of how I can find eq-b recurrent sequences approaching about any temperament.
About their convergence, which is what I think you point,
some of the others you did not quote were convergent I think :

116.675 Miracle < 7x^6 - 8x4 = 12 - 8x^6 (= exactly 116.675 c.)
116.747 Miracle < 14x^8 - 24 = 7x^2 - 8 (= 116.737 c.)

and many other eq-b are such as :
116.5078 c. < 3x^9 = x^6 + 4
116.622 c. < 4x^13 = x^7 + 8, (close to the Meta-Miracle temperament)
116.668 c. < 8x^9 = 3x^3 + 11 (very close to 72-edo)
116.688 c. < 4x^15 = 4x^6 + 5
116.7114 c. < 7x^14 = 8x^6 + 6, etc.

But divergence in one direction is convergence on the other, so it is not such a problem.
Anyway I never thought of doing something else than finding eq-b generators here, as you said ;
not that I think it would be impossible, but they would result rapidly in high numbers and since only for Miracle I have more than a hundred of those, it would take the rest of my days.

> Except, alas, for the question of whether you can actually hear any > of this.

This question makes me worrying about you. You don't hear beats in tempered versions of 3/2, 8/5, 12/5, or 8/7 intervals ? :)
- - - - - - -
Jacques

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> But divergence in one direction is convergence on the other, so it is
> not such a problem.

Good point, but you've still not answered the question of why you need a recurrent series at all, instead of just using the generator.

> > Except, alas, for the question of whether you can actually hear any
> > of this.
>
> This question makes me worrying about you. You don't hear beats in
> tempered versions of 3/2, 8/5, 12/5, or 8/7 intervals ? :)

I've spent a lot of time playing around with equal beating, and find it nearly impossible to hear even under the best circumstances. Complicated relationships like the ones you posit would surely make matters worse. I asked for opinions from others, and Carl said he felt it was not important or, usually, even audible also. And, of course, if you do want such relationships you hardly need a linear recurrence to get them. I certainly have no objection to such scales, and in fact find them interesting, but insisting on them seems a little strange.

🔗Michael <djtrancendance@...>

Jacques>"This question makes me worrying about you. You don't hear beats in tempered versions of 3/2, 8/5, 12/5, or 8/7 intervals ? :)

Here's a weird "test" example.
Consider the ratios 8/5 (1.6), 21/13 (1.61538), and 13/8 (1.625). These are all very close, the top two being within about 13 cents of each other. But do we hear 21/13 as a 13/8 or a whole different interval?
Another example...3/2 vs. 50/33 vs. 40/27. These aren't as close but, even to my ears, it seems hard to tell of the are variations on the same tone or completely different tones in their own right. It also seems certain intervals have more "gravity" than others IE you can get more distance (in, say, cents) away from the tone while keeping the same general feel/mood.

3/2 for example, seems to have a nasty habit of making the ear want to consider any interval near it a sour version of it rather than a sweet/correct version of it's own individual note. 8/5, meanwhile, seems to give a bit more slack, but mostly to notes about it such as 13/8 and less so notes below it like 19/12 or 1.5833333 (perhaps because the "gravity of the 5th" sucks it in).

I (and I'm pretty sure a good few of you) would find it very useful to know how far you can get away from a strong/dominant interval like 3/2 to have a tone count as it's own tone rather than a sour version of another tone.

🔗Marcel de Velde <m.develde@...>

> I (and I'm pretty sure a good few of you) would find it very useful to know
> how far you can get away from a strong/dominant interval like 3/2 to have a
> tone count as it's own tone rather than a sour version of another tone.
>

In my opinion the way to do this is not to listen to dyads, but to listen to
chord progressions.
How is the ear to know things like the "root of the chord" etc and all else
that may come to play in how intervals are heard in actual music.
Even regular music theory doesn't reduce intervals to dyads etc and has
other concepts at play that determine the function of a chord etc.

So when doing JI, to listen simply to dyads, first 3/2, then 40/27 from
thesame groundtone, no wonder the 40/27 is going to sound strange.
But in actual music, a 40/27 can be perfectly ok (and a 3/2 would sound
wrong in such a location) in my opinion.
So one has to test it in a very different way, and "how far you can get away
from a strong/dominant interval like 3/2 to have a tone count as it's own
tone rather than a sour version of another tone" isn't how it works I think.
It's too simplistic.

Marcel

🔗Carl Lumma <carl@...>

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

> I've spent a lot of time playing around with equal beating, and
> find it nearly impossible to hear even under the best circumstances.
> Complicated relationships like the ones you posit would surely make
> matters worse. I asked for opinions from others, and Carl said he
> felt it was not important or, usually, even audible also. And, of
> course, if you do want such relationships you hardly need a linear
> recurrence to get them. I certainly have no objection to such
> scales, and in fact find them interesting, but insisting on them
> seems a little strange.

It's easy to demonstrate that difference tones and beat rates
are insignificant in 'normal' musical settings. However it's
always possible to create settings where they are significant,
and to be fair, Jacques' photonic disks may be such a setting
(I can't comment one way or the other). -Carl

🔗jacques.dudon <fotosonix@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@> wrote:
>
> > But divergence in one direction is convergence on the other, so it is
> > not such a problem.
>
> Good point, but you've still not answered the question of why you need a recurrent series at all, instead of just using the generator.

The same beating synchronicities + differential coherence relationships will occur in both. But my preference goes to rational series, even with high numbers because synchronous beatings will be polyrythmic, intervals will have more chances to integrate simple harmonic ratios, and will be unequal, thus the scales will present different modes (much like a well-temperament can be preferable to a 12-ET), and may be most important, because of the variety of versions rational series offer, with different harmonic content and at different degrees of divergence - in many cases some divergence is lucky, in other cases you may not want it (see for example the two versions I gave of soria.scl), and you can also always ultimately compare with the irrational solution.
>
> > > Except, alas, for the question of whether you can actually hear any
> > > of this.
> >
> > This question makes me worrying about you. You don't hear beats in
> > tempered versions of 3/2, 8/5, 12/5, or 8/7 intervals ? :)
>
> I've spent a lot of time playing around with equal beating, and find it nearly impossible to hear even under the best circumstances.

I don't believe you. You tried with sinus waves or what ? Have you tried with plain sawtooth waveforms ?

> Complicated relationships like the ones you posit would surely make > matters worse.

I can't follow what you say here. you call 3/2, 8/5, 12/5, or 8/7 like those I used in the Miracles complicated relationships ?

> I asked for opinions from others, and Carl said he felt it was not > important or, usually, even audible also.

It seems you mix up differential tones perception (Carl's answer) and equal beating phenomena here. I thought we were talking of the equal-beating qualities of my algorithms.

> And, of course, if you do want such relationships you hardly need a linear recurrence to get them.

If you want an isomorphism of the same equal-beating properties for all identical chords, certainly you do.

> I certainly have no objection to such scales, and in fact find them interesting, but insisting on them seems a little strange.

I never claimed such systems were the ultimately best solution, I don't have any belief on them and I don't think I am insisting on anybody to have a belief on them. I have done certain experiences in certain directions, I share them with who ever wants to experiment them and I just ask anyone to not project any preconceived idea on them before playing and making music with them and hearing the results.
- - - - -
Jacques.

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@...> wrote:

> > I asked for opinions from others, and Carl said he felt it
> > was not important or, usually, even audible also.
>
> It seems you mix up differential tones perception (Carl's answer)
> and equal beating phenomena here. I thought we were talking of
> the equal-beating qualities of my algorithms.

My answer is that neither is significant, unless you go out
of your way to make them significant. I probably shouldn't
have answered though, because I'm not going to take the time
to argue it again. It's been argued over MANY times on this
and the other lists in the past. Some people are convinced
difference tones or beat rates are all-important (even to the
point of explaining why simple rational intervals are consonant)
and in my experience, they simply aren't interested in learning
otherwise.

That said if you have a way of finding recurrence relations
that give generator sizes in the neighborhood of TOP or the
other error optimizations, there's no reason not to use them.

-Carl

🔗jacques.dudon <fotosonix@...>

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Jacques>"This question makes me worrying about you. You don't hear beats in tempered versions of 3/2, 8/5, 12/5, or 8/7 intervals ? :)
>
> Here's a weird "test" example.
> Consider the ratios 8/5 (1.6), 21/13 (1.61538), and 13/8 (1.625). These are all very close, the top two being within about 13 cents of each other. But do we hear 21/13 as a 13/8 or a whole different interval?

Hi Michael,
That's a very good question. First, we never hear intervals out of a certain context. In the right context (for example in some Buzurg mode), you will hear 21/13 as a 21/13, in others you might mistake it for 13/8.
Going back to my question to Gene, what would be the beatings of Phi then ? a complex mixture of its beats with 21/13, 13/8, and perhaps 8/5, all in Phi proportions but the reverse way ! That's another reason to say Phi is a "fractal" (and complex) ratio, compared to ratios like 3/2, 8/5, 12/5, or 8/7 who have not much ambiguity.
Note that a powers of Phi series is equal-beating ! compare these two beatings :
13 Phi^2 - 21 Phi and 8Phi - 13
- - - - - - -
Jacques

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@...> wrote:

> > I've spent a lot of time playing around with equal beating, and find it nearly impossible to hear even under the best circumstances.
>
> I don't believe you. You tried with sinus waves or what ? Have you tried with plain sawtooth waveforms ?

Soundfont sounds.

> > Complicated relationships like the ones you posit would surely make > matters worse.
>
> I can't follow what you say here. you call 3/2, 8/5, 12/5, or 8/7 like those I used in the Miracles complicated relationships ?

Sixteen times the frequency of a note minus fifteen times a secor up equals thirty-six times the note minus twenty-four times a fifth up isn't complicated? Plus, suppose the note is 440 Hz. Then the difference tone is 30.66 Hz, which is pretty low.

> It seems you mix up differential tones perception (Carl's answer) and equal beating phenomena here. I thought we were talking of the equal-beating qualities of my algorithms.

You need to perceive the difference tones for equal-beating to make a difference, don't you?

> > And, of course, if you do want such relationships you hardly need a linear recurrence to get them.
>
> If you want an isomorphism of the same equal-beating properties for all identical chords, certainly you do.

What in hell does this mean? Are you using the word "isomorphism" in some definable mathematical sense? Why does not simply using the generator in question give you "isomorphisms"?

🔗genewardsmith <genewardsmith@...>

🔗Michael <djtrancendance@...>

Carl>"Some people are convinced difference tones or beat rates are all-important (even to the
point of explaining why simple rational intervals are consonant) and in my experience, they simply aren't interested in learning otherwise."

I went through one point where I believed difference tones were nearly as important as periodicity. I also went through a period when I thought making all tones in a scale with a low o-tonal common denominator was perhaps the main key to consonance. Later I learned mirroring around intervals IE <using 4/3? 2 / (4/3) = 3/2 or <using 6/5> 3/2 * (6/5) = 9/5.

Looking back at everything I've tried, all I can say in general is the brain likes pattern. If it can get-to/calculate an interval or chord using JI in a scale, great. If it can get there using JI and mirroring, even better. If it can get there using mirroring, and proportionate beating (be it linear via JI or in symmetrical sections ALA Phi and other generators), also very good. If it can get the using all three of these techniques...even better. If it can get there using a circular temperament generation system in the above techniques...wow. Oh yeah, and helping the mind to find it via obeying critical band dissonance limits as much as possible certainly doesn't hurt either.

But seriously, I (like Jacques) have found difference tones as a valid optimizing tool. When I tried converting the "Silver Sections" scale to JI with all notes within 10 cents or so of the original scale...I got slightly audible changes. All yet somehow the non-JI version actually sounded no less stable just...stable in a different way. When Cameron did mirroring on an early-version Ptolemic JI scale I made he actually avoided "perfect" JI by a few cents to use mirroring...and the result was actually more consonant (it kept most of the JI purity and gained a whole new dimension on interval-mirrored purity).

In short I think JI, tempering using mirrored intervals around very periodic intervals, different tones, and a likely whole bunch of other "patternizing" systems definitely should have a valid place in tuning.

The seriously sad thing is just how many people seem to be brainwashed enough into thinking there is only one valid consonance theory, such as JI, and that therefore all others must be false. IMVHO, it's that kind of close-mindedness that often keeps the art of tuning from covering a lot of now ground.

-Michael
___

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗Michael <djtrancendance@...>

>"Wouldn't the logic of different consonance theories lead to different music."
Not necessarily because, in many cases, consonance theories intersect. You can use mirroring and JI and circular temperaments all in the same scale, for example. The one thing to remember though is often any time you include one theory you usually have to compromise another...but often the compromise is very little and the gain is tremendous if combined well.

I've seen intervals like 9/8, 6/5, 5/4,4/3,3/2,5/3,7/4, 9/5, 15/8...that are fairly common practice and virtually always work well. But I've also heard ones like 12/11. 11/10, 13/8, 11/9, 27/20, 13/10, 7/5, 22/15, 58/33, 11/6...that often work just as well and many of these seem to somewhat violate the rules for low-prime-limit JI chord-creating intervals. Not to mention the irrational intervals that can be used to estimate them. I'll say this about JI, it's pretty good for getting a quick estimate of what kinds of chords are likely possible...but it sure doesn't seem capable of explaining everything.

>"One theory may mean that perfection is possible, that there is reason, etc. I see that as the opposite of "sad" as you call it."
Admittedly, the flip side is the are ultimately many "perfect musics/scales/theories" as there are different artistic personalities making the music. I made the Silver Scale and eventually considered it too weird after not touching it for months and then coming back to it...but a song made with it became the most downloaded song on the SoOn online label. And now I've been working with mixed-JI (meaning follows some JI rules but not others) scales that go the complete opposite direction of the scales both in theory and practice and yet are doing very well (though not necessarily better).

Making a scale that uses one theory more than others is going to lean it toward use by a certain kind of musician...and certainly, say, leaning a theory all on JI is going to single out musicians who don't want the JI feel. Do you see any reason why all musician's preferences (new and old) can be singled down into one theory?

🔗genewardsmith <genewardsmith@...>

🔗Michael <djtrancendance@...>

Gene>"Judging my your example below you almost can (take any multiplication of ratios and call it a mirror)"
I don't think we're talking about the same type of "mirroring" in that case.
Example: Take 9/8 and 3/2 times 9/8. the latter is NOT a mirror as it as not the same distance from 3/2 as 9/8 is.
Second Example: 5/4 and 2/1 times 5/4. The latter is not a mirror because it is simply the same fraction (albeit an octave up). The mirror would be 2/1 over 5/4...because both 5/4 and its mirror would be the same distance from 2/1.
BTW, when I say mirror I mean around a very limited number of very simple intervals such as 2/1 and 3/2 and occasionally 4/3.

Gene>"Equal temperaments have more mirroring than anything else."
Sure it does...assuming you use all notes in the entire temperament as a scale (which I've seen rarely if "ever" happens).

Here's an example of why "common practice theory" 12TET scales aren't half as mirror-full as you'd expect given the whole 12TET tuning set of notes. In 12TET when you use any half step in combination with whole steps to make a scale within the tuning, you cancel out a whole bunch of "mirrors". For example...the distance from E to G in 12TET is not the distance from G to B or G to A...those do not mirror around 3/2 AKA "G".
You can easily say 12TET itself mirrors, say, around 3/2...for example G to F and G to A or G to E and G to A#...but at some point with doing this you run into tones that are simply out of key (IE A# is not in the key of C where G is the fifth).

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Gene>"Judging my your example below you almost can (take any multiplication of ratios and call it a mirror)"

> I don't think we're talking about the same type of "mirroring" in that case.
> Example: Take 9/8 and 3/2 times 9/8. the latter is NOT a mirror as it as not the same distance from 3/2 as 9/8 is.

What I said was that "mirroring" seemed to be when the same interval appeared twice. If you toss in 1, then 27/16 is the same distance ftrom 9/8 as 3/2 is from 1. That seems to be what you are talking about--is it?

> Second Example: 5/4 and 2/1 times 5/4. The latter is not a mirror because it is simply the same fraction (albeit an octave up).

4/3 * 3/2 = 2 (one of your examples) is mirroring but
5/4 * 8/5 = 2 isn't?

> BTW, when I say mirror I mean around a very limited number of very simple intervals such as 2/1 and 3/2 and occasionally 4/3.

So "mirroring" is when a lot of very simple and basic intervals appear in a scale?

> Gene>"Equal temperaments have more mirroring than anything else."
> Sure it does...assuming you use all notes in the entire temperament as a scale (which I've seen rarely if "ever" happens).

Well, you can do pretty well with rank two temperaments, with two generators, and get a lot of this stuff in scales.

> Here's an example of why "common practice theory" 12TET scales aren't half as mirror-full as you'd expect given the whole 12TET tuning set of notes. In 12TET when you use any half step in combination with whole steps to make a scale within the tuning, you cancel out a whole bunch of "mirrors". For example...the distance from E to G in 12TET is not the distance from G to B or G to A...those do not mirror around 3/2 AKA "G".

Why must they "mirror" around G? And in any case, G to G# equals F# to G, G to A equals F to G, etc etc etc.

Rather than complaining about my attempt to understand what is so far incomprehensible, why don't you simply give a definition of "mirroring" and explain how it differs from my proposed definition (the same interval appears twice in a scale.)

> You can easily say 12TET itself mirrors, say, around 3/2...for example G to F and G to A or G to E and G to A#...but at some point with doing this you run into tones that are simply out of key (IE A# is not in the key of C where G is the fifth).

Sorry, but you've now introduced a whole new idea, "key", which one might think depended on a scale being already implicitly constructed. This is a mess!

🔗Jacques Dudon <fotosonix@...>

Gene wrote :

> > (Jacques) : It seems you mix up differential tones perception > (Carl's answer) and equal beating phenomena here.
> > I thought we were talking of the equal-beating qualities of my > algorithms.
>
> You need to perceive the difference tones for equal-beating to make > a difference, don't you?

OK, now I see where is the misunderstanding. A big one. And one that has probably been spread on the Tuning List for years :
Thinking that dyad beatings and dyad difference tones are the same thing. From where comes that invention ?

Lets take a super-basic example, a 1/5 pyth comma fifth meantone of a 1.4959535062432 ratio (697.2784049 c.).

What's the beating of such a A:E dyad for A = 440 hz ?
(3 * A) - (2 * E) =
(3 * 440 hz) - (2 * 658.219542747 hz) = 3.5609145 hz
It's a beating frequency, not a difference tone.

The difference tone of this dyad is B - A =
658.219542747 hz - 440 hz = 218.219542747 hz

Is it more clear ?
- - - - - - - -
Jacques

🔗Michael <djtrancendance@...>

>"Rather than complaining about my attempt to understand what is so far
incomprehensible, why don't you simply give a definition of "mirroring"
and explain how it differs from my proposed definition (the same
interval appears twice in a scale.)"
I have several times, but I'll try to reword it better for clarity. It means using any interval (A) as input find an interval the same distance from one and only one simple rotational "center" interval R (where R is usually 2/1, 1/1, 3/2, or 4/3) on the other side of that interval R.

>"What I said was that "mirroring" seemed to be when the same interval
appeared twice. If you toss in 1, then 27/16 is the same distance ftrom
9/8 as 3/2 is from 1. That seems to be what you are talking about--is
it?"

No...because they have to be the same distance from one interval, not two. You are talking about a given distance from 3/2 and 1/1...to be mirrored the intervals have to be the same distance from either 1/1 or 3/2...but not both. Your example goes wrong in that
A) You use 9/8 AND 3/2 as rotation points...and there can only be one rotation point
B) You are putting both results on the same side of each rotation IE 3/2 > 1/1 and 27/16 > 9/8. In a true rotation result intervals are on opposite sides of the rotation point and rotating either interval around the center gives the other interval as a result.

>"4/3 * 3/2 = 2 (one of your examples) is mirroring but
5/4 * 8/5 = 2 isn't?"
Bad example on my part...basically in each of the above I gave only one half of the mirror (so it's neither right nor wrong, just incomplete)! :-(
Say for the example 4/3 * 3/2 = 2 (mirroring around 3/2)...the other half of the mirror would be 3/2 * 3/4 = 9/8. Make sense?

>"So "mirroring" is when a lot of very simple and basic intervals appear
in a scale? "
No it's when intervals that are built around revolving around basic intervals (typical rotating points are intervals like 3/2 and 1/1 or 2/1 and occasionally 4/3).

For example you can rotate the "not so simple" 27/20 fraction around 3/2 to get 3/2 * 20/27 = 60/54 = 30/27...that 30/27 is mirrored but is not simple.

>"Why must they "mirror" around G? And in any case, G to G# equals F# to
G, G to A equals F to G, etc etc etc. "
There's no "must". Mirroring around 1/1 and 2/1 or even 4/3 work fine too...it's just harder to explain the math in examples as the "mirrored" interval often ends up crossing over the octave.

Mike> You can easily say 12TET itself mirrors, say, around 3/2...for example G to F and G to A or G to E and G to A#...but at some point
with doing this you run into tones that are simply out of key (IE A# is
not in the key of C where G is the fifth).

Gene>"Sorry, but you've now introduced a whole new idea, "key", which one
might think depended on a scale being already implicitly constructed.
This is a mess!"
You made a statement that any TET (which, of course, is a type of tuning and not a type of scale) is very "mirrored".
I'm saying that simply isn't true because scales use only limited subsets of tunings. My example assumed the notes CDEFGAB in the key of C and shows, for example, that many notes from the key of C become rotated out of the key of C (IE become notes like A#,C#,D#,F#,G#) when mirrored. Thus 12TET in common practice theory is often not mirror-able in the same way to entire chromatic scale is. Same goes for things like MOS scales, which also take subsets which run into the same problem.

🔗Michael <djtrancendance@...>

Start with the scale 1/1 6/5 3/2 2/1.

2/1 over 3/2 is 4/3.
Mirror this around 2/1 (AKA 1/1) to get 4/3.
Thus 3/2 mirrors around the "center" 2/1 to produce the rotated interval 4/3.
...giving the new resultant scale
1/1 6/5 4/3 3/2 2/1.

Also 6/5 over 1/1 is 6/5.
Mirror this around 1/1 (AKA 2/1) to get 2/1 * 5/6 = 5/3.
Thus 6/5 mirrors around 2/1 to produce the rotated interval 5/3.
...giving the new resultant scale
1/1 6/5 4/3 3/2 5/3 2/1.

Now 5/3 over 4/3 = 5/4.
And (mirroring around 4/3) ...
4/3 over 5/4 = 16/15
...giving the new resultant scale

1/1 16/15 6/5 4/3 3/2 5/3 2/1.

Now 16/15 mirrors around 1/1 (AKA 2/1) to give
2/1 over 16/15 = 15/8.
...giving the new resultant scale
***********************
1/1 16/15 6/5 4/3 3/2 5/3 15/8 2/1.
***********************

Look familiar?

________________________________
From: genewardsmith <genewardsmith@sbcglobal.net>
To: tuning@yahoogroups.com
Sent: Mon, May 3, 2010 11:41:58 PM
Subject: [tuning] Re: The regular mapping paradigm strikes back

--- In tuning@yahoogroups. com, Michael <djtrancendance@ ...> wrote:
>
> Gene>"Judging my your example below you almost can (take any multiplication of ratios and call it a mirror)"