Linear versus fractal generators

Hello to all,

I am trying to see in what shape this list of generators gets through
Yahoo.
(for printing I will load another version with full length lines in
the TL files,
here I have cut the lines in two)

To resume, this work tries to test the hypothesis that linear
temperaments can have matching useful solutions in the form of
fractal sequences.
Thanks to Graham for this list of Linear Temperaments, and to
translate it for me with the generators values in ratios, along with
their octave complements (second decimal number), which saved me
hours of work.
Thanks to Gene to remind me it would be nice to have it started.

Under each temperament headline are corresponding fractal
propositions. Firstly each one requires to be validated, if precise
enough, or not ; secondly, the polynomial proposition gives good
indications of how easy series can be developped from them, which is
one of the interesting applications.
Precision here varies a lot and I also suspect that some temperaments
might need more precision than others, so it is an affair of
evaluation by temperaments specialists now.
I often gave several solutions, either for the ratio or its octave
complement, whenever some simpler polynomials would occur in the
neighborhood, in order to propose eventually various types of sequences.
Note that the exponents of the polynomials, for ex. "x3", has to be
read "x^3", and sorry for the silly french keyboard comma point...
Just to give a rough idea of the matching precision, 0.001 (= one
thousandth) is equivalent to 1,7 cent for the smaller ratios and
decreases to 0,9 cent for ratios approaching 2/1.
A few fractals (Miracle, Meantone, Vincentino, Dicot, Magic,
Wuerschmidt, Schism) match exactly the values of the ratios given by
Graham. Worse approximations generally do not exceed 1 cent.
But precision required for very small generators could also be more
critical than for others.

...Your feedback is welcome !

LINEAR TEMPERAMENTS GENERATORS VERSUS FRACTAL RATIOS - version 1.0
Jacques Dudon 2010/08/01
List of linear temperaments generator ratios by Graham Breed

1.02244 1.95611 Slender
1,95557797934 Diese11 11x2 - 21x = 1

1.02642 1.94853 Quartonic
1,9487525048 Treizenhuit 19x2 - 36x = 2

1.04560 1.91278 Tertiaseptal
1,9128709292 Tiersept 12x - 6x2 = 1

1.04591 1.91221 Valentine
1,0457231904 Gontrand1 20x3 = 20x2 + 1

1.04601 1.91202 Valentine
1,0460842566 Gontrand2 4x7 = 8x6 - 5
1,0466805318 Gustave2 10x3 - 10x = 1
1,0471379812 Gedeon2 3x7 - 3x = 1
1,0475847635 Gedeon7 7x3 - 7 = x

1.04881 1.90692 Nautilus
1,0487548816 Damlas 5x4 - 5 = x
1,9067177515 Salmad 3x2 - 9 = x

1.05219 1.90080 Octacot
1,0522166467 Limma 4x6 - 4x2 = 1
1,0524868886 Limma 4x5 - 4 = x3
1,0527707984 Limma 18x2 - 18x = 1

1.05865 1.88920 Passion
1,0585432207 Dotz1 x4 = 6 - 4x3
1,0585659530 Dotz2 6x3 = 2x + 5
1,0590169944 Dotz3 16x2 - 16x = 1

1.05998 1.88684 Ripple
1,0596110214 Dutz1 x13 = x2 + 1
1,0605584628 Dutz2 4x4 - 4 = x

1.06969 1.86970 Miracle
(note : 100 eq-b Secors available)
1,0696885514 Secor26 24x6 = 52 - 15x
1,0696889604 Secor68 5x13 = 15 - 2x6

1.06972 1.86965 Miracle
1,0697172600 Secor59 15x6 = 8x4 + 12
1,0697202035 Secor78 7x14 = 10x6 + 3

1.06976 1.86958 Miracle
1,0697553108 Secor53 14x8 = 7x2 + 16

1.07525 1.86003 Negrisept
1,0739763463 Minimak 7x2 = x + 7
1,8608058531 Shakira x3 = 4x - 1

1.07534 1.85987 Negripent
1,0757660661 Mani x10 = x + 1
1,0761362109 Macbeth 5x5 = 9x3 - 4

1.07805 1.85520 ???
1,0773502692 Persane 12x2 - 12x = 1
1,0783777456 Noura x3 = 16x - 16
1,8546376797 Arounh 2x3 = 4x2 - 1

1.08829 1.83775 Bohpier
1,0873780254 Scylla x4 = 16x - 16
1,8392867552 Awj x3 = x2 + x + 1

1.08833 1.83767 Bohpier
1,0889086818 Leila 8x5 = 8x4 + 1

1.09340 1.82916 Nusecond
1,0933601923 Totem2 x3 = 14x - 14
1,0944759268 Totem3 2x3 = 7 - 4x

1.09597 1.82487 Hemikleismic
1,0958287524 Totem10 x11 = 8 - 4x3

1.09611 1.82464 Hystrix
1,0962912018 Totem11 8x3 = 16x - 7

1.09857 1.82055 Porcupine
1,0985076843 Dlotkotier2 4x2 = 49 (x - 1)

1.09865 1.82041 Porcupine
1,0991125988 Dlotkotier3 2x5 = x2 + 2
1,0994008537 Dlotkotier4 37x - 32x2 = 2

1.09933 1.81929 Porcupine
1,0994789404 Dlotkotier5 64x2 - 64x = 7
1,8196497813 Topaz x5 = 32 - 2x3

1.10665 1.80726 Sesquiquartififths
1,1049875621 Thaeyink 5x2 = x + 5

1.10711 1.80650 Tetracot
1,1071598717 Olzal 4x3 - 4x = 1

1.11806 1.78881 Luna
1,1169631198 Mezyd1 3x2 = 32x - 32
1,1173490366 Mezyd2 2x4 - 2 = x

1.11809 1.78876 Hemithirds
1,1177804428 Meanotaur2 4x2 = 24 - 17x

1.11851 1.78809 Hemiwuerschdt
1,1197626094 Minos x5 = 4 - 2x

1.14098 1.75287 Gorgo
1,1411796349 Tripletaille x9 = 2x + 1

1.14230 1.75085 Gamera

1.14259 1.75042 Gidorah

1.14342 1.74913 Cynder/Mothra
1,143453585 Rabrindanath x8 = 3x2 - 1

1.14353 1.74897 Cynder
1,1436839647 Mothra(GWSmith) x12 = 8 - 2x3
1,1438392083 Jade 2x3 = 11 - 7x
1,7484028813 Pansept x4 - x3 = 4

1.14468 1.74722 Guiron
1,1441939142 Septimacle 8 - 4x4 = x
1,7475938359 Bluette x6 = 9x2 + 1

1.14503 1.74668 Rodan
1,1452994122 Slendra 4x5 - 4x4 = 1
1,746170945 Innocente x4 = 4x3 - 12
1,1454972244 Baka 6x2 - 6x = 1

1.14868 1.74112 Penta
1,1485289581 Ekipenta2 x3 = 8x2 - 7x - 1

1.15661 1.72919 Semaphore
1,1557112298 Nadir x2 = 15x - 16

1.15711 1.72844 Semaphore
1,1569296692 Guéry 4x2 = 15x - 12
1,7277535004 Semiestel x4 = 4x + 2

1.16201 1.72115 Superpelog
1,1621966417 MiSpire 2x4 - 4x = 1

1.16231 1.72072 Bug
1,1621966417 MiSpire 2x4 - 4x = 1

1.16952 1.71010 Quasiorwell
1,1683010133 Ennea 4x8 - 4x7 = 1

1.16974 1.70978 Orwell
1,7097759952 Sabra8 18x3 = 7x + 78

1.16980 1.70970 Orwell
1,1697819921 Arbas4 7x7 = 28 - 6x

1.16982 1.70966 Orwell
1,7096688761 Sabra3 12x3 = 7x + 48

1.16988 1.70958 Orson
1,7094982727 Sabra1 8x3 = 7x + 28
1,1708203933 Pancuazisrepl 5x2 - 5x = 1

(temperaments missing here ? : Isrep, Niris, Archaos, Anom, Majinn,
Alsepdim)

1.19620 1.67197 Myna
1,1961765899 Myna 5x10 = 35 - x9
1,1952232829 Looz 3x4 - 3x3 = 1

1.19629 1.67184 Myna
1,1963051171 Myna 5x10 = x9 + 25
1,6716998817 Dhaivati x3 = x + 3

1.19968 1.66711 Parakleismic
1,1985770979 Climene1 3x6 = 4x5 - 1

1.19972 1.66705 Parakleismic
1,1989446264 Climene2 3x4 = x + 5

1.20058 1.66587 Keemun
1,2005112724 Lagunag 2x6 - 2x5 = 1
1,2005949010 Lagune x4 = 9 - 4x3

1.20076 1.66562 Catakleismic
1,2007754561 GWSmith 3x6 = 2x5 + 4
1,2008648917 GWSmith 5x6 = 2x5 + 10

1.20095 1.66535 Hanson
1,2009620088 Cremona 9x6 = 32 - 2x5

1.20103 1.66524 Countercata
1,2010459310 Carmine x6 = 8 - 2x5

1.20140 1.66473 Keemun
1,2013361798 Granlagun 3x3 = 4 + x
1,2016078128 Gretel x6 = 2x5 - 2
1,6632519388 Dhaivata x4 - 4x = 1

1.20431 1.66070 Superkleismic
1,2037682266 Aiguille 7 - 4x2 = x

1.20437 1.66062 Superkleismic
1,6580980674 Dhadou x4 - x3 = 3

1.21667 1.64384 Amity
1,216990566 Macram x2 = 10 - 7x
1,6435942530 Houboriz x4 = 4x3 -8x2 +8x -2
1,6429348843 Carthage x4 - 4 = 2x

1.22272 1.63569 Vicentino
1,2227181453 Mohaban 11x2 = 2x + 14

1.22293 1.63541 Mohajira
1,2228950302 Mohabis 3x3 - 3x2 = 1
1,2229454416 Mohawj x5 = 32 - 16x3

1.22298 1.63535 Mohajira
1,2229988055 Mohajatis 11x2 = 11x + 3

1.22303 1.63528 Mohajira
1,2229988055 Mohajatis 11x2 = 11x + 3

1.22306 1.63524 Dicot
1,2230631050 Mohajicom 24x3 = 22x2 + 11

1.22316 1.63511 Mohajira
1,2231109974 Mohajisept 2x2 = 14 - 9x

1.22321 1.63504 Mohajira
1,2232031492 Mohajaran 12x3 - 10x2 = 7
1,2232849566 Mohajira 2x5 - 2x4 = 1
1,6343652930 Arijaom x3 = 6 - x

1.22510 1.63252 Hemififths
1,2252704261 Awjig x5 = 3x4 - 4
1,2257102299 Nung-Phan-Bâ x4 = 10x -10
1,2268579888 Wusta-Zactral 8x3 = 16 - x
1,2268841133 Zalarith 8x2 - 9x = 1

1.22824 1.62835 Beatles
1,2279981273 RetriZalith x2 = 11x - 12
1,6284994191 Rachel x6 = x2 + 16
1,228868987 Zenarith 32x2 - 32x = 9

(temperaments missing here : Mogar, Buzurg, Nadir, Iph, Monik, etc.)
1,2325819246 Buzurg 4x4 = 8 + x

1.24436 1.60726 Muggles
1,2444537221 Gorgone 8x7 = 4x + 32

1.24549 1.60579 Magic
1,2454806199 Magixsyx 6x5 = 4x + 13

1.24570 1.60552 Magic
1,2456780612 Terzbirat 9x2 = 8x + 4

1.24595 1.60520 Magic
1,2459548236 Sixmagic 6x5 = 23 - 4x

1.24602 1.60512 Magic
1,245996795 Septimage 5x5 = 20 - 4x
1,2464431275 Deolimage 8x5 = 5x4 +12
1,2467941053 Aswafée x5 = 8 - 4x
1,6041140967 Generousse x5 = x4 + 4

1.25040 1.59949 Grendel
1,6002760053 Pathos x4 = x2 + 20 - 10x

1.25077 1.59901 Wuerschmidt
1,2507723627 Courette x3 = 72 - 56x

1.25107 1.59863 Wuerschmidt
1,2510688803 Coureuse 10x4 = 12x3 + 1

1.27894 1.56379 Squares
1,2781630728 Nestor x4 = 6x - 5
1,2785534816 Sceptre1 14 - 7x2 = 2x

1.27895 1.56378 Squares
1,278934683274 Sceptre2 16 - 9x2 = x

1.27919 1.56349 Squares
1,279405061915 Sceptre3 13x2 - 20 = x

1.27988 1.56265 Sidi
1,2807764064 Dagbok 2x2 = x + 2

1.29036 1.54995 Clyde
1,2903455786 Poussami 5x3 = 6x + 3
1,290569415 Poussah 8x2 - 8x = 3
1,550321208 Anhawyn 6x3 = 18x2 - 18x + 7

1.29165 1.54841 Sensipent
1,2915026221 RétriXringco x2 = 12 - 8x
1,5485837704 TierXring 3x2 = 4x + 1

1.29189 1.54812 Sensisept
1,291870800 Kappa 11x3 = 22x2 - 13
1,2919315501 Calista x7 = 3x2 + 1

1.29207 1.54790 Sensi
1,2919803206 Narquise x5 = x4 + x3 - x + 1

1.29231 1.54761 Sensisept
1,2923530911 Borygma 2x5 = 3x4 - x3 + 1
1,292469901 Adocleme 2x3 = 11 - 4x2
1,2927280143 Irina 2x4 = 2x + 3

1.30135 1.53686 Father
1,3009102569 Louven 9x2 = 16x - 1
1,3009759883 Brendwib 5x3 = 10x - 2
1,301511345 Brindet 11x2 = 22x - 10
1,3027756377 Brindi 3 - x2 = x

1.30801 1.52904 Semisept
1,307486101 Tulpe x4 = 3x - 1

1.31612 1.51962 Vulture
1,3145962123 Mrejdaïrim x3 = 4 -x2
1,5207972894 Petigrain x2 = 16 - 9x

1.31619 1.51954 Vulture
1,316227766 Decime 10x2 = 20x - 9

1.31831 1.51709 Mother
1,3187293047 Popeye 4x2 = 3x + 3
1,5174899136 Moera x4 = 3 + x2
1,5159802277 Pendragon x3 = 5 - x

1.32693 1.50724 Superpyth
1,3260051791 Mounos 4x3 = 8 + x

1.32709 1.50705 Superpyth
1,3269596286 Kepat 2x3 = 6 - x

1.32850 1.50545 Quasisuper
1,3282688557 Preska x3 = 5 - 2x

1.33233 1.50113 Dominant
1,3318145987 Tibrepyt 2x4 = 8x - x3 - 2
1,5010679722 Inflexcible x10 = 24x2 + 4
1,5015418889 Wustaf x4 = 8x3 - 22

1.33250 1.50094 Undecental
1,5008601377 Decam 64 - x10 = 4x
1,5012205255 Sensible 4x5 = 32 - x

1.33282 1.50057 Kwai
1,5006304969 Shrirag x7 = 16x4 - 64

1.33287 1.50052 Cassandra

1.33301 1.50036 Schismatic

1.33321 1.50014 Alt. Cassandra
1,5002256478 Batteuse x13 = 192 + 2x
1,5002443402 Navan 40 - x9 = x

1.33323 1.50011 Garibaldi
1,5000038428 Everest x11 = 88 - x

1.33349 1.49983 Pontiac
1,4998272798 Helmut4 3x9 - 64 = 2x8
1,4998540822 Aigle x10 = x8 + 32

1.33350 1.49981 Helmholtz
1,4998160148 Helmut3 4x9 = x8 + 128
1,4998031788 Hemut2 x9 = 64 - x8

1.33363 1.49967 Dominant
1,4997712670 Helmut1 x9 = 4x8 - 64

1.33364 1.49965 Schism
1,4996458667 Bhatiyar x7 = x4 + 12
1,4995971249 Rhakub x6 = x3 + 8
1,3336915408 Bukhar x6 = 8 - x3

1.33388 1.49938 Grackle
1,3339672093 Quiretsim 4x4 = 14 - x
1,4994021598 Mysterique 8x5 = 64 - x3

1.33473 1.49843 Sharptone
1,4984515839 Albatros x7 = 8x2 - 1
1,3344573453 Minsys 2x4 = 3x2 +1

1.33718 1.49569 Meantone
1,3369399461 Skisni 2x4 - 4 = x3
1,495672251 Traverse 2x4 = 13 - 2x

1.33743 1.49540 Meantone
1,3374838299 Cybozem 15x3 = 10x2 + 18
1,49542113797 Coquine 27x3 = 186 - 64x
1,4954155044 Sapnat x10 = 16x + 32

1.33755 1.49528 Meantone
1,495282599 Rwadlab x2 = 50x - 77
1,4952263885 Radieubiz 3x3 = 19 - 6x

1.33759 1.49522 Meanpop
1,3375853959 Apatoïd x4 = 12 - 3x - 2x3

1.33774 1.49506 Meantone
1,495093791 Auruter 9x2 = 56 - 24x
1,4950519243 Terabi x3 = x2 + 26x - 40

1.33776 1.49503 Meanpop
(~Golden Meantone 1,49503445)
1,3376959274 Sirk x4 = 8x3 + 12x - 36
1,494972766 Tara 2x4 = 2x + 7

(missing meantones here - hundreds of meantones available)

1.33965 1.49293 Flattone
1,3397247359 Squarto 8x2 = 4x + 9
1,4928556845 Mansax x2 = 50 - 32x
1,4930231803 Esterelle1 2x4 = 30 - 9x2

(lower meantones missing)

1.34777 1.48393 Marvo
1,3472963553 Nawahbis x3 = 3x2 - 3
1,4833147735 Nueh x2 = 20 - 12x

1.35050 1.48093 Mavila
1,3506928322 Susak 5x4 = 10x3 - 8
1,4807406984 Shebag x2 = 17 - 10x

1.35504 1.47597 Pelogic
1,4768727445 Saptamak 5x4 = 56 - 10x3
1,4753367636 Rwato 2x4 = x + 8
1,3542486889 Circé x2 = 8x - 9

(temperaments missing here ? - see question below)
1,366025404 Zinith 2x2 - 2x = 1

1.38799 1.44093 Liese
1,3874258867 Adibis 3x2 - 2x = 3
1,3880935089 Radix x6 - x5 = 2

1.38901 1.43988 Triton
1,4384471872 Phantax x2 = 7x - 8

1.39819 1.43043 Tritonic
1,3975274679 Salamandre1 9x2 = 9x + 5

1.39820 1.43041 Tritonic
1,3985294913 Salamandre2 x2 = 30x - 40

1.39995 1.42862 Neptune
1,4004100679 Salamandre4 4x3 = 60 - 35x

In the meantime, these are my personal conclusions :

1) There is generally no adequation between temperaments and
fractals, and even rather a more general "phase opposition" between
the two worlds. That is not surprising : a temperament is made to
offer the best possible harmonic solutions, while a fractal sequence
usually avoids harmonic attractors. If a recurrent sequence resolved
all first primes like all these temperaments do, we would know it
already.
Also simplest forms of polynomials are just not present here : most
of those fractals belong to relatively complex categories. Only a few
of those fractals have been resolved in geometric form, who become
more precious now because their specific timbres can be used for the
corresponding temperaments, and this gives us good motivations for
trying more.

2) However each time I've been working on fractals based on a
temperament's specific properties (what I did only for a few here), I
found dozens and dozens (if not hundreds) of very close solutions,
that would satisfy any need of precise generators for those
temperaments.

3) There are methods to find pertinent sequences for any given
temperament (that were applied with same success by Gene, according
to the few perfect examples he gave - some of which I added here).

4) My collection shows holes in the very-near harmonic ratios
(visible around 8/7 and 4/3), simply because I never looked at recurrences sequences in such places (which would result in very high
numbers series and no special waveform) ; some temperaments on the
contrary make use of quasi-harmonic intervals.

5) The list shows some specific musical zones I have been working on
in musical projects, that in direct or octave-complement form seem to
coincide with precise zones of linear temperaments :
Gontrand > Valentine
Salmad > Nautilus
Octacot > Limma
Totem > Hemikleismic/Hystrix
Dlotkotier > Porcupine
Olzal > Tetracot
Sabra > Orwell
Keemun > Lagune
Sceptre > Squares
Poussah > Clyde
TierXring > Sensipent
Decime > Vulture
Helmut > Pontiac/Helmholtz
Salamandre > Triton, etc.
The case of Orwell/Orson versus Sabra is interesting, as different
fractals correspond with a fair precision to the octave complements
of each version of the temperament.

6) More surprising, I found some holes in the linear temperament
list, at places where musical systems, even used in traditional music, have galaxies of fractal solutions : the "Buzurg" zone is one
of these, that would generate very musical scales : at least Margo
Schulter will not contradict me on that point (Margo, I have not read
your last article on Buzurg yet : will see if we find some
concordances on the subject !). The only reason I see for that
omission in the temperament list is that the Buzurg generator before
anything else conciliates 7 and 13, and will not come to harmonics 3,
5 and 11 before longer cycles, therefore it is not on a list that
privilegiates temperaments with full 3, 5, 7 solutions and after.
Here is a 23-limit mapping, showing main MOS at 10, 43 , 53, that
would be resolved by a "Buzurg" neutral third of 16 unequal steps of
53 steps in one octave (around 362 cents) :
<0, -8, 11, 6, 28, -1, -3, 24, 15]
(note that the 23rd harmonic, one of the best centered, would be reached precisely by 15 generators, suggesting a useful and efficient
target generator with 1.2324856 or 361.884956 c.)
note also that a pure harmonic 3 will be attained with a generator of
362.255625 c.
Many practical fractals would qualify as Buzurgs, such as :

7x^2 = 7x + 2 > 1,23192505471 > 361,09738964 c.
4x^7 = 16 + x > 1,2320003029 > 361,2031329 c.
4x^2 = 11 - 4x > 1,2320508076 > 361,2741 c.
x^8 = x^7 + 1 > 1,2320546314 > 361,2794748 c.
x^2 = 20 - 15x > 1,2321245983 > 361,37778668556 c.
3x^2 = 11x - 9 > 1,2324081208 > 361,77611271646 c.
4x^4 = 8 + x > 1.23258192464 > 362,020247741 c.
2x^5 = 8 - x^4 > 1,2326439928 > 362,107458 c.
8x^5 = 24 - x > 1,2326632514 > 362,1344722332 c.
2x^8 = 15 - x^7 > 1,2328237225 > 362,359833742 c.
x^2 = 28x - 33 > 1,2328546652 > 362,403288 c.

The simplest series are generated by x^8 = x^7 + 1, such as :
17 21 26 32 39 48 59 73 90 111 137 169 208 256 315 388 478...
(differentials 4 5 6 7 9 11 14 17 21 26 32 39 48 59 73 90...)
4x^4 = 8 + x converges to a generator by coincidence also
approximated by the main MOS ratio : 53/43...
One last thought : 43 + 53 = 96 steps/octave, a division I know to be
dear to Shaahin Mohajeri, features of course also a very good Buzurg
third of 13 + 16 = 29 steps.

7) Another empty zone, even if in a smaller region of the list is
lower meantones, around 7-edo such as Thai scales, for the same
reasons I suppose, because the generators provide only complex 3, 5
and 7 solutions.

8) Another fractal I consider as specially harmonic and musical,
Zinith, 2x2 - 2x = 1, of solution (sqrt3+1)/2 = 1.366025404 or
539.9811764 c. is not sollicited either, I think not only because
Zinith approximates 7, 11, 13, 15 and with less success 3 or 5, but
perhaps also because it quasi-ends in a 20-edo cicrcle, and perfect
h. 3 and 5 would only come after a reiteration of the generator
several hundred times. Should it be seen only as a non-3 & 5 tuning
then, or could it be possible to lower 719,8494 c. to a more
acceptable fifth ? (and eventually making the 512/x^20 comma more
useful ?) Then another linear temperament could be proposed as well :
<0, 8, -6, 4, 1, 6, -2, 5]
... a challenge that would have the advantage to be able to use the
colorful timbres of the Zira'at and Zinith waveforms, some of the
most emblematic fractal sounds I heard.

- - - - - - - - - -
Jacques

On 4 August 2010 19:10, Jacques Dudon <fotosonix@...> wrote:

> 1) There is generally no adequation between
> temperaments and fractals, and even rather a
> more general "phase opposition" between the
> two worlds. That is not surprising :  a
>temperament is made to offer the best possible
> harmonic solutions, while a fractal sequence
> usually avoids harmonic attractors. If a
> recurrent sequence resolved all first primes
> like all these temperaments do, we would
> know it already.

Is this anything to do with the idea that noble mediants are the most
dissonant intervals? (Even if not everybody finds them dissonant.)

Graham

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

> 1.04560 1.91278 Tertiaseptal
> 1,9128709292 Tiersept 12x - 6x2 = 1

One thing to keep in mind is that temperaments differ in accuracy and complexity, and hence have differing requirements for the accuracy of generators. Tertiaseptal, the 31&171 temperament, is a microtemperament and hence I think 2x^3 = 25x^2 - x - 24 would be a better choice. The largest root is much bigger than the generator, however.

> 1.14230 1.75085 Gamera

3x^3 - 10x^2 - 10x + 20
5x^3 - 5x^2 - 10x + 6
The second one has a generator which is the largest root in absolute value.

> 1.14468 1.74722 Guiron
12x^2 - 5x - 10

> 1.14503 1.74668 Rodan
8*x^3 + x^2 - 16*x + 5
x^3 - 2x^2 - 7x + 13

This should fill in the gap for generators of about 8/7.

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

> 1.10665 1.80726 Sesquiquartififths
> 1,1049875621 Thaeyink 5x2 = x + 5
20x^2 - 14x - 9
x^4 + 10x^2 - 7x - 6
3x^3 - 9x^2 + 12x - 10
Once again, for a temperament like this you need more tuning accuracy in a generator.

> 1.11851 1.78809 Hemiwuerschdt
> 1,1197626094 Minos x5 = 4 - 2x

x^2 - 19x + 20
4x^3 + 6x^2 + 16x - 31
10x^3 - 11x^2 - 22

Graham wrote :

> On 4 August 2010 19:10, Jacques Dudon <fotosonix@...> wrote:
>
> > 1) There is generally no adequation between
> > temperaments and fractals, and even rather a
> > more general "phase opposition" between the
> > two worlds. That is not surprising : a
> >temperament is made to offer the best possible
> > harmonic solutions, while a fractal sequence
> > usually avoids harmonic attractors. If a
> > recurrent sequence resolved all first primes
> > like all these temperaments do, we would
> > know it already.
>
> Is this anything to do with the idea that noble mediants are the most
> dissonant intervals? (Even if not everybody finds them dissonant.)
>
> Graham

Same kind of thing indeed ; noble mediants leave uncertainty between
many close intervals issued from fractal series (Phi series in Dave
Keenan's sense), is it a problem or a quality and should we call that
dissonant, these are other questions that depends on the musical
context, on one's taste for complexity and on the definitions of
dissonance. In a context where you would use a timbre based on the
relevant fractal, noble mediants could prove to be the more consonant
solutions !
Could the fact that a fractal is just not naturally meant to be a
temperament be enough to explain this non-adequation ?
The temperament maker uses methods to find ratios that get close to
rational numbers of certain harmonic limits. That's not the problem
of fractal ratios on their own : so if some do, it's only after a
severe selection between them, or methods to find the "low-limit
tempering" ones. I suppose that if you took any ratios at random not
many would generate temperaments either.
Also what is worse with some of the simplest fractals such as Phi,
Narayanan, Natté, Awj, Semq, or Amaz, is that the powers of their
ratios converge towards whole numbers... but of no harmonic limit, so
it's exactly what you don't want for a temperament...
- - - - - - - -
Jacques

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

> 6) More surprising, I found some holes in the linear temperament
> list, at places where musical systems, even used in traditional
> music, have galaxies of fractal solutions : the "Buzurg" zone is one
> of these, that would generate very musical scales : at least Margo
> Schulter will not contradict me on that point (Margo, I have not read
> your last article on Buzurg yet : will see if we find some
> concordances on the subject !). The only reason I see for that
> omission in the temperament list is that the Buzurg generator before
> anything else conciliates 7 and 13

There are a lot of subgroups possible, and so omitting 5 and 11, or 3, 5, and 11, out of primes up to 17 or 19 is not going to be one of the first things anyone looks at. Even when you do that, commas such as 2197/2176 or 2048/2023 are certainly usable but they don't leap out as things to try. The upshot is, by the time a temperament search got around to finding Buzurg, it would have been buried under a landslide of other possible temperaments of a generally similar kind.

I've noticed that 5 tends to be omitted a lot. Why is that? If there was some rationale for what might be omitted, it would cut down the search area.

> 6) More surprising, I found some holes in the linear
> temperament list, at places where musical systems, even
> used in traditional music, have galaxies of fractal
> solutions : the "Buzurg" zone is one of these, that would
> generate very musical scales : at least Margo Schulter will
> not contradict me on that point (Margo, I have not read
> your last article on Buzurg yet : will see if we find some
> concordances on the subject !).

Dear Jacques,

Please let me agree that the Buzurg zone is a wonderful place for
fractal solutions -- and also for the kinds of regular or planar
temperaments that I often use. Of course, I will be very
interested to read any comments you have on my Buzurg article,
which is proving far longer than I planned on, for better or
worse, maybe because the "Zalzalian 12" set has caused me to
focus on a variety of modes in the different rotations.

In fact, your discussion below of a Buzurg generator gave me an
idea for an experiment to show that -c tunings and regular or
planar temperaments can often lead to musically congenial,
although distinct, results, with both approaches lending
themselves to maqam/dastgah music. One of the goals of my "Ethno
Extras" is indeed to seek out such concordances.

And here I would add that a -c tuning like your Aulos/Soria seems
to me closer to the spirit of an irregular Persian tar fretting,
for example, than a regular or even planar temperament.

> The only reason I see for that omission in the temperament list
> is that the Buzurg generator before anything else conciliates 7
> and 13, and will not come to harmonics 3, 5 and 11 before
> longer cycles, therefore it is not on a list that privilegiates
> temperaments with full 3, 5, 7 solutions and after.

Curiously, the situation with the Buzurg generator may be like
that with my regular "e-based" temperament (704.607c), or my
Peppermint planar temperament based on the regular Wilson/Pepper
noble fifth temperament (704.096c) with a distance between the
12-note chains or "ribbons" of 58.680c. These systems, like your
Buzurg, support harmonics 7 and 13 -- and also here 3 and 11 --
but not 5, at least within the intended size of 24 notes. We
might indeed say that such temperaments, like your Buzurg,
neither privilege nor are privileged by the usual "regular
mapping" approach.

> Here is a 23-limit mapping, showing main MOS at 10, 43 , 53,
> that would be resolved by a "Buzurg" neutral third of 16
> unequal steps of 53 steps in one octave (around 362 cents) :
> <0, -8, 11, 6, 28, -1, -3, 24, 15] (note that the 23rd
> harmonic, one of the best centered, would be reached precisely
> by 15 generators, suggesting a useful and efficient target
> generator with 1.2324856 or 361.884956 c.) note also that a
> pure harmonic 3 will be attained with a generator of
> 362.255625c.

Your generators syggested an experiment: I used Scala to define a
regular temperament with 361.884956c as the diminished fourth
(e.g. G#-C) from eight fifths down or fourths up. Of course, this
method is very different from using this large neutral third of
Buzurg as itself the generator, and different again from
a -c system using a series including a 69/56 third (361.402c),
for example, maybe somewhat in the manner of the Ibina tuning
(or the very slightly larger 53/43 you mention below)!

The fifth is 704.76438 cents, almost identical to 63-EDO, and
actually a bit closer to the point where the apotome is equal to
precisely half of 7/6, which thus is virtually pure.

Your generator, interestingly, produces a fifth (from 8
generators down) of 704.920 cents, with 16 in a 24-note tuning.
And we get lots of hemifourths at 247.540 cents not present in my
regular temperament! And you have 8 locations with symmetrical
Buzurg tetrachords, (0-133-362-495-705-838-1066-1200). one of
which also includes a 23/16 step (56/39 or 13/9 in medieval JI
versions).

Here is a Buzurg mode in the regular 704.764c temperament with
the upper tetrachord like the lower:

! rbuzurg-buzurg8_Cup.scl
!
Buzurg pentachord plus 133-229-133 tetrachord at ~3/2
8
!
133.35066
361.88496
495.23562
628.58628
704.76438
838.11504
1066.64934
2/1

and here is Qutb al-Din al-Shirazi's Buzurg-Hijaz combination:

! rbuzurg-buzurg_hijaz_Cup.scl
!
Qutb al-Din al-Shirazi's Buzurg plus upper Hijaz (JI 12:11-7:6-22:21)
8
!
133.35066
361.88496
495.23562
628.58628
704.76438
857.12058
1123.82190
2/1

As you remark, in this temperament as in your "buzurg" mapping,
we get a ~23/16 step as a variation on the 56/39 of Safi al-Din
al-Urmawi and Qutb al-Din.

Of course, we could also explore this territory with a -c tuning
maybe something like Soria 17 + 2 in the A version with the
larger comma, where Buzurg or Buzurg-Rast modes turn up at a
couple of locations.

> Many practical fractals would qualify as Buzurgs,
> such as :

> 7x^2 = 7x + 2 > 1,23192505471 > 361,09738964 c.
> 4x^7 = 16 + x > 1,2320003029 > 361,2031329 c.
> 4x^2 = 11 - 4x > 1,2320508076 > 361,2741 c.
> x^8 = x^7 + 1 > 1,2320546314 > 361,2794748 c.
> x^2 = 20 - 15x > 1,2321245983 > 361,37778668556 c.
> 3x^2 = 11x - 9 > 1,2324081208 > 361,77611271646 c.
> 4x^4 = 8 + x > 1.23258192464 > 362,020247741 c.
> 2x^5 = 8 - x^4 > 1,2326439928 > 362,107458 c.
> 8x^5 = 24 - x > 1,2326632514 > 362,1344722332 c.
> 2x^8 = 15 - x^7 > 1,2328237225 > 362,359833742 c.
> x^2 = 28x - 33 > 1,2328546652 > 362,403288 c.

> The simplest series are generated by x^8 = x^7 + 1, such as :
> 17 21 26 32 39 48 59 73 90 111 137 169 208 256 315 388 478...
> (differentials 4 5 6 7 9 11 14 17 21 26 32 39 48 59 73 90...)
> 4x^4 = 8 + x converges to a generator by coincidence also
> approximated by the main MOS ratio : 53/43...

Indeed this series includes many of the intervals based on
harmonics 3-7-11-13 that I seek out in regular or planar
temperaments, and I think would be a "charmed attractor" to
anyone with a passion for these intervals. You may recall
explaining to me, when I proposed a sequence for "Mohajira,"
that what I was seeking was actually this "Buzurg 8" series.

</tuning/topicId_89500.html#89511>

A curious question is whether 69/56, like 53/43, might make an
interesting generator.

> One last thought : 43 + 53 = 96 steps/octave, a division I
> know to be dear to Shaahin Mohajeri, features of course also
> a very good Buzurg third of 13 + 16 = 29 steps.

Shaahin's creative use of 96-EDO, and your comment, make me take
note now that indeed this equal temperament has lots of support
for some traditional Persian modes. For example, symmetrical
Buzurg:

0 137.5 362.5 500.0 625.0 700.0 837.5 1062.5 1200

And Qutb al-Din's Buzurg-Hijaz:

0 137.5 362.5 500.0 625.0 700.0 850.0 1112.5 1200

So with your -c tunings or a mapping like buzurg, or Shaahin's
96-EDO, or my regular and planar temperaments, what we seem to
be looking for in common is "modal goodness to fit," intervals
which will form pleasant maqamat or dastgah-ha, etc. And so we
may find some notable concordances as well as differences.

With many thanks,

Margo Schulter
mschulter@...

It isn't clear what you mean by a fractal generator. i look at the archives and it wasn't clear there either. Is it the same as a recurrent sequence?

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Graham wrote :
>
> > On 4 August 2010 19:10, Jacques Dudon <fotosonix@> wrote:
> >
> > > 1) There is generally no adequation between
> > > temperaments and fractals, and even rather a
> > > more general "phase opposition" between the
> > > two worlds. That is not surprising : a
> > >temperament is made to offer the best possible
> > > harmonic solutions, while a fractal sequence
> > > usually avoids harmonic attractors. If a
> > > recurrent sequence resolved all first primes
> > > like all these temperaments do, we would
> > > know it already.
> >
> > Is this anything to do with the idea that noble mediants are the most
> > dissonant intervals? (Even if not everybody finds them dissonant.)
> >
> > Graham
>
>
> Same kind of thing indeed ; noble mediants leave uncertainty between
> many close intervals issued from fractal series (Phi series in Dave
> Keenan's sense), is it a problem or a quality and should we call that
> dissonant, these are other questions that depends on the musical
> context, on one's taste for complexity and on the definitions of
> dissonance. In a context where you would use a timbre based on the
> relevant fractal, noble mediants could prove to be the more consonant
> solutions !
> Could the fact that a fractal is just not naturally meant to be a
> temperament be enough to explain this non-adequation ?
> The temperament maker uses methods to find ratios that get close to
> rational numbers of certain harmonic limits. That's not the problem
> of fractal ratios on their own : so if some do, it's only after a
> severe selection between them, or methods to find the "low-limit
> tempering" ones. I suppose that if you took any ratios at random not
> many would generate temperaments either.
> Also what is worse with some of the simplest fractals such as Phi,
> Narayanan, Natté, Awj, Semq, or Amaz, is that the powers of their
> ratios converge towards whole numbers... but of no harmonic limit, so
> it's exactly what you don't want for a temperament...
> - - - - - - - -
> Jacques
>

Kraig wrote :

> It isn't clear what you mean by a fractal generator. i look at the
> archives and
> it wasn't clear there either. Is it the same as a recurrent sequence?

Hi Kraig,
The booklet of my CD "Lumieres audibles" that I think you have gave
this definition many years ago, but it is still correct :

" I discovered these very particular waveforms from the beginning of
my disk experimentations. Their sonorities, both complex and
transparent, are among the oddest, reminding those of inharmonic
tones or hisses. Their audition is often accompanied by very powerful
psychic effects, the ear recognizing precise textures, paradoxically
without being able to give them any determinated fundamental pitches.
This comes from the fact that graphically, and by analogy with the
fractal images, these waves are generated by laws of geometrical
developpment, putting in action the same organizing principles
whatever the scale they are being observed at. The synthesis, on a
disk, of a white noise, sets a very interesting mathematical problem,
which can't be resolved by shapes thrown in a hazardous way, neither
by other aleatory parameters, which would only produce a buzz, while
a white noise is the undifferenciated mixture of all frequencies.
Fractal waveforms are up to this date what I found the most
successful for a white noise imitation.
In this CD are explored three of these fractal waveforms, with their
related intonations : the "Clar" fractal waveform, and its first
developpments, in "Hexagrammes" (track 8) ; the "Phi" fractal
waveform, starting point of "Fleurs de lumière" (tracks 1-2-3) ; and
the "Mohajira" fractal waveform, at the basis of "Sumer" (tracks4-5-6-7)."

What I have been calling "fractals" for years and recurrent sequences
belong to the same phenomena, and may only differ from their
applications ; and the "fractal generator" is the convergent ratio of
a recurrent sequence, or (one of) the solution(s) of its polynomial
expression.
My first applications were sounds and rhythmns, and soon after, while resolving more and more fractals I started to be interested in their
scales, more particularly generated by the different series of their
recurrent sequences that I would be able to draw on photosonic disks.
I used this term by analogy with 2D-fractals or fractal images - and
I believe recurrent series are 1D-fractals. We could call them also
"linear fractals" from the aspect of their parenty with linear
temperaments, where we can using their irrational solutions as
generators, as well as in the form of their variations from different
recurrent series.
What's most fractal in this really is the algorithm - which can be
applied recurrently without end - meaning that whatever the scaling
you look at it from (or the reference time or pitch), you find the
same patterns.
This is the same phenomena that can applies to rhythmns, timbers
(that will sound the same for each fractal whatever their
developpment), patterns (well visible on photosonic disks), scales,
logarithmic structures, and linear temperaments.

- - - - - - -
Jacques

> --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
> >
> > Graham wrote :
> >
> > > On 4 August 2010 19:10, Jacques Dudon <fotosonix@> wrote:
> > >
> > > > 1) There is generally no adequation between
> > > > temperaments and fractals, and even rather a
> > > > more general "phase opposition" between the
> > > > two worlds. That is not surprising : a
> > > >temperament is made to offer the best possible
> > > > harmonic solutions, while a fractal sequence
> > > > usually avoids harmonic attractors. If a
> > > > recurrent sequence resolved all first primes
> > > > like all these temperaments do, we would
> > > > know it already.
> > >
> > > Is this anything to do with the idea that noble mediants are
> the most
> > > dissonant intervals? (Even if not everybody finds them dissonant.)
> > >
> > > Graham
> >
> >
> > Same kind of thing indeed ; noble mediants leave uncertainty between
> > many close intervals issued from fractal series (Phi series in Dave
> > Keenan's sense), is it a problem or a quality and should we call
> that
> > dissonant, these are other questions that depends on the musical
> > context, on one's taste for complexity and on the definitions of
> > dissonance. In a context where you would use a timbre based on the
> > relevant fractal, noble mediants could prove to be the more
> consonant
> > solutions !
> > Could the fact that a fractal is just not naturally meant to be a
> > temperament be enough to explain this non-adequation ?
> > The temperament maker uses methods to find ratios that get close to
> > rational numbers of certain harmonic limits. That's not the problem
> > of fractal ratios on their own : so if some do, it's only after a
> > severe selection between them, or methods to find the "low-limit
> > tempering" ones. I suppose that if you took any ratios at random not
> > many would generate temperaments either.
> > Also what is worse with some of the simplest fractals such as Phi,
> > Narayanan, Natté, Awj, Semq, or Amaz, is that the powers of their
> > ratios converge towards whole numbers... but of no harmonic
> limit, so
> > it's exactly what you don't want for a temperament...
> > - - - - - - - -
> > Jacques

Actually it is quite different with my Buzurg proposition :
<0, -8, 11, 6, 28, -1, -3, 24, 15]
since it tempers ALL the primes up to 23, mainly by tempering 169/168 and 273/272 (and of course 4096/4095).

How I understand why 5 would be tend to be omitted, if this is the case, is because 5 generates major and minor thirds, augmented fourths, sixths and sevenths - it does not leaves much space for alternatives - would that be a reason ?
- - - - - - -
Jacques

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

> (Jacques) :
> > 6) More surprising, I found some holes in the linear temperament
> > list, at places where musical systems, even used in traditional
> > music, have galaxies of fractal solutions : the "Buzurg" zone is one
> > of these, that would generate very musical scales : at least Margo
> > Schulter will not contradict me on that point (Margo, I have not read
> > your last article on Buzurg yet : will see if we find some
> > concordances on the subject !). The only reason I see for that
> > omission in the temperament list is that the Buzurg generator before
> > anything else conciliates 7 and 13
>
> There are a lot of subgroups possible, and so omitting 5 and 11, or 3, 5, and 11, out of primes up to 17 or 19 is not going to be one of the first things anyone looks at. Even when you do that, commas such as 2197/2176 or 2048/2023 are certainly usable but they don't leap out as things to try. The upshot is, by the time a temperament search got around to finding Buzurg, it would have been buried under a landslide of other possible temperaments of a generally similar kind.
>
> I've noticed that 5 tends to be omitted a lot. Why is that? If there was some rationale for what might be omitted, it would cut down the search area.

Thanks for clearing that up. Yes i have you CD and you might be happy to know it is still used as a listening example for two of the classes here at the University of Wollongong.

I was fortunate to have Walter O'Connell as my physics of music teacher in LA in 1971 where he played the class examples of a scale based on phi.It was through Walter that i met Erv Wilson.

I am curious about the notation that you and Gene are using and was curious if it in anyway resembles that found here. and /or if this makes sense to either of you.
Also i am not clear exactly where you two are taking this. I am not sure what hand Hanson might have played in this form of notation.

http://anaphoria.com/meruthree.PDF

there are also recurrent sequences listed in the wilson archives besides there

It is interesting your preference for the converged version.
My own work has given me a preference for the numerical often in the area right before it gets within minute variations of the converged.
I also have found the subharmonic versions just as good, but if one works with the converged these two are the same.

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Kraig wrote :
>
> > It isn't clear what you mean by a fractal generator. i look at the
> > archives and
> > it wasn't clear there either. Is it the same as a recurrent sequence?
>
>
>
> Hi Kraig,
> The booklet of my CD "Lumieres audibles" that I think you have gave
> this definition many years ago, but it is still correct :
>
> " I discovered these very particular waveforms from the beginning of
> my disk experimentations. Their sonorities, both complex and
> transparent, are among the oddest, reminding those of inharmonic
> tones or hisses. Their audition is often accompanied by very powerful
> psychic effects, the ear recognizing precise textures, paradoxically
> without being able to give them any determinated fundamental pitches.
> This comes from the fact that graphically, and by analogy with the
> fractal images, these waves are generated by laws of geometrical
> developpment, putting in action the same organizing principles
> whatever the scale they are being observed at. The synthesis, on a
> disk, of a white noise, sets a very interesting mathematical problem,
> which can't be resolved by shapes thrown in a hazardous way, neither
> by other aleatory parameters, which would only produce a buzz, while
> a white noise is the undifferenciated mixture of all frequencies.
> Fractal waveforms are up to this date what I found the most
> successful for a white noise imitation.
> In this CD are explored three of these fractal waveforms, with their
> related intonations : the "Clar" fractal waveform, and its first
> developpments, in "Hexagrammes" (track 8) ; the "Phi" fractal
> waveform, starting point of "Fleurs de lumière" (tracks 1-2-3) ; and
> the "Mohajira" fractal waveform, at the basis of "Sumer" (tracks
> 4-5-6-7)."
>
>
> What I have been calling "fractals" for years and recurrent sequences
> belong to the same phenomena, and may only differ from their
> applications ; and the "fractal generator" is the convergent ratio of
> a recurrent sequence, or (one of) the solution(s) of its polynomial
> expression.
> My first applications were sounds and rhythmns, and soon after, while
> resolving more and more fractals I started to be interested in their
> scales, more particularly generated by the different series of their
> recurrent sequences that I would be able to draw on photosonic disks.
> I used this term by analogy with 2D-fractals or fractal images - and
> I believe recurrent series are 1D-fractals. We could call them also
> "linear fractals" from the aspect of their parenty with linear
> temperaments, where we can using their irrational solutions as
> generators, as well as in the form of their variations from different
> recurrent series.
> What's most fractal in this really is the algorithm - which can be
> applied recurrently without end - meaning that whatever the scaling
> you look at it from (or the reference time or pitch), you find the
> same patterns.
> This is the same phenomena that can applies to rhythmns, timbers
> (that will sound the same for each fractal whatever their
> developpment), patterns (well visible on photosonic disks), scales,
> logarithmic structures, and linear temperaments.
>
> - - - - - - -
> Jacques
>
>
> > --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@> wrote:
> > >
> > > Graham wrote :
> > >
> > > > On 4 August 2010 19:10, Jacques Dudon <fotosonix@> wrote:
> > > >
> > > > > 1) There is generally no adequation between
> > > > > temperaments and fractals, and even rather a
> > > > > more general "phase opposition" between the
> > > > > two worlds. That is not surprising : a
> > > > >temperament is made to offer the best possible
> > > > > harmonic solutions, while a fractal sequence
> > > > > usually avoids harmonic attractors. If a
> > > > > recurrent sequence resolved all first primes
> > > > > like all these temperaments do, we would
> > > > > know it already.
> > > >
> > > > Is this anything to do with the idea that noble mediants are
> > the most
> > > > dissonant intervals? (Even if not everybody finds them dissonant.)
> > > >
> > > > Graham
> > >
> > >
> > > Same kind of thing indeed ; noble mediants leave uncertainty between
> > > many close intervals issued from fractal series (Phi series in Dave
> > > Keenan's sense), is it a problem or a quality and should we call
> > that
> > > dissonant, these are other questions that depends on the musical
> > > context, on one's taste for complexity and on the definitions of
> > > dissonance. In a context where you would use a timbre based on the
> > > relevant fractal, noble mediants could prove to be the more
> > consonant
> > > solutions !
> > > Could the fact that a fractal is just not naturally meant to be a
> > > temperament be enough to explain this non-adequation ?
> > > The temperament maker uses methods to find ratios that get close to
> > > rational numbers of certain harmonic limits. That's not the problem
> > > of fractal ratios on their own : so if some do, it's only after a
> > > severe selection between them, or methods to find the "low-limit
> > > tempering" ones. I suppose that if you took any ratios at random not
> > > many would generate temperaments either.
> > > Also what is worse with some of the simplest fractals such as Phi,
> > > Narayanan, Natté, Awj, Semq, or Amaz, is that the powers of their
> > > ratios converge towards whole numbers... but of no harmonic
> > limit, so
> > > it's exactly what you don't want for a temperament...
> > > - - - - - - - -
> > > Jacques
>

--- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@...> wrote:
>
> Actually it is quite different with my Buzurg proposition :
> <0, -8, 11, 6, 28, -1, -3, 24, 15]
> since it tempers ALL the primes up to 23, mainly by tempering 169/168 and 273/272 (and of course 4096/4095).

A basis for the commas is {99/98, 120/119, 169/168, 171/170, 176/175, 208/207, 231/230}. All the ratios are superparticular. This, however, is less impressive then it sounds, as there are so many 23-limit superparticular ratios, and there would even more ways of choosing seven of them which would lead a temperament not on your list if you were making one. The basic reason why no one found this particular 23-limit regular temperament is that no one has been trying to find such things; there are quite a lot of them.

Gene wrote :

> > (Jacques) :
> > Actually it is quite different with my Buzurg proposition :
> > <0, -8, 11, 6, 28, -1, -3, 24, 15]
> > since it tempers ALL the primes up to 23, mainly by tempering > 169/168 and
> 273/272 (and of course 4096/4095).
>
> A basis for the commas is {99/98, 120/119, 169/168, 171/170, > 176/175, 208/207,
> 231/230}. All the ratios are superparticular. This, however, is > less impressive
> then it sounds, as there are so many 23-limit superparticular > ratios, and there
> would even more ways of choosing seven of them which would lead a > temperament
> not on your list if you were making one. The basic reason why no > one found this
> particular 23-limit regular temperament is that no one has been > trying to find
> such things; there are quite a lot of them.

Sorry Gene but I am not a temperament specialist and I don't catch your answer !
I understand you listed the larger commas tempered by the mapping, but not what follows.

What interests me in Buzurg for one thing is that it is a traditional structure from Middle East, probably a very ancient one, that I have been using personally to compose many music, in which I could integrate not only Persian modes but also Indonesian Slendros and even Pelogs and Pygmy scales. I have been refretting guitars for such tunings and I experienced the musicality of this system.
Apart meantone, schismic, superpyth, mohajira, cynder/mothra (?) not many of our refined microtonal temperaments can pretend to modelise extra-european traditional tunings, and this one does.
What I think is that 11 and 19 are certainly more or less "extras" in here, and perhaps 23 except it appears by chance to be very well-centered as I explain in my answer to Margo, the ancient mode of Safi-Al-Din suggesting it in the form of the 56/39 ratio, differing by 897/896. But the way a generator of around 362 cents reunites factors 3, 5, 7, 13, 17, even if not unique in his kind, is sufficient for consideration, not to mention the clear -c of this ratio, that appears in its fractal solutions.
It has also to be compared with other temperaments around this generator : Beatles, Muggles, of which I am ignorant.

- - - - - - -
Jacques

Kraig wrote :

> Thanks for clearing that up. Yes i have you CD and you might be > happy to know it
> is still used as a listening example for two of the classes here at > the
> University of Wollongong.
>
> I was fortunate to have Walter O'Connell as my physics of music > teacher in LA in
> 1971 where he played the class examples of a scale based on phi.It > was through
> Walter that i met Erv Wilson.
>
> I am curious about the notation that you and Gene are using and was > curious if
> it in anyway resembles that found here. and /or if this makes sense > to either of
> you.
> Also i am not clear exactly where you two are taking this. I am not > sure what
> hand Hanson might have played in this form of notation.
>
> http://anaphoria.com/meruthree.PDF
>
> there are also recurrent sequences listed in the wilson archives > besides there
>
> It is interesting your preference for the converged version.
> My own work has given me a preference for the numerical often in > the area right
> before it gets within minute variations of the converged.
> I also have found the subharmonic versions just as good, but if one > works with
> the converged these two are the same.

Speaking for myself and as I explained on this list before,
the polynomial form of writing recurrent sequences is a shortcut for the more correct one, mathematically speaking, that uses Erv Wilson, and all mathematicians.
I never used the aH(n) = bH(n-m) +cH(n-o) + dH(n-p) etc... form myself, and I am glad I never did as I find it more complicate !

What is it about Hanson ? Which Hanson do you refer to, and perhaps also the Hanson temperament ??

I see what you mean about the convergence. Me too and I like to integrate divergences also whenever it brings musical results, such as :
64 : 75 : 88 : 104 : 128 : 192 : 512 : 2560 : ...

... and also when it gives negative frequencies and ratios ;)
- - - - - - -
Jacques

(PS : I am on the digest mode too and it works well for me)

> > (Jacques) :
> > Actually it is quite different with my Buzurg proposition :
> > <0, -8, 11, 6, 28, -1, -3, 24, 15]
> > since it tempers ALL the primes up to 23, mainly by tempering 169/168 and
> 273/272 (and of course 4096/4095).

Gene:
> A basis for the commas is {99/98, 120/119, 169/168, 171/170, 176/175, 208/207,
> 231/230}. All the ratios are superparticular. This, however, is less impressive
> then it sounds, as there are so many 23-limit superparticular ratios, and there
> would even more ways of choosing seven of them which would lead a temperament
> not on your list if you were making one. The basic reason why no one found this
> particular 23-limit regular temperament is that no one has been trying to find
> such things; there are quite a lot of them.

Jacques:
> Sorry Gene but I am not a temperament specialist and I don't catch your answer !

Gene gave a set of commas that are all tempered out by the mapping you
gave, and so uniquely define the temperament class. I can identify it
as this:

29/96

1199.625 cents period
362.309 cents generator

mapping by period and generator:
[<1, 4, -1, 1, -5, 4, 5, -3, 0],
<0, -8, 11, 6, 28, -1, -3, 24, 15]>

mapping by steps:
[<53, 84, 123, 149, 183, 196, 217, 225, 240],
<43, 68, 100, 121, 149, 159, 176, 183, 195]>

tuning map:
[1199.625, 1900.025, 2785.779, 3373.482, 4146.538, 4436.191, 4911.197,
5096.551, 5434.641> cents

scalar complexity: 3.743
RMS weighted error: 1.158 cents/octave
max weighted error: 1.658 cents/octave

http://preview.tinyurl.com/377kp66

> I understand you listed the larger commas tempered by the mapping, but not what follows.

He also said that it isn't such a standout temperament that anybody's
noticed it before. It isn't bad, but there are a lot of other things
that aren't bad either.

For example, go to

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

and put your commas in. (Please don't put Gene's in because they'll
hang the script.) Click the higher-limit buttons until you get to the
23-limit, and you'll see your temperament only comes in at number 5.
So even with those commas, it isn't the best. (Assuming I'm measuring
things correctly, which I'm obviously not, but it's not that bad
either.)

You said 11 and 19 were extras as well, which looks correct. There's
something similar that comes out better in the 19-limit:

http://preview.tinyurl.com/33l9qyb

The octave equivalent mapping is

<0 -8 11 6 -25 -1 -3 -29].

It disagrees with yours in the 11 and 19. But even in the
2.3.7.13.17.23-limit nothing that looks like your Buzurg come in my
top 20.

Graham

p.s. I think I spelled your name right this time.

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

> For example, go to
>
> http://x31eq.com/temper/uv.html
>
> and put your commas in. (Please don't put Gene's in because they'll
> hang the script.)

Why does it do that? Too many commas?

> You said 11 and 19 were extras as well, which looks correct.

Another fun game you can play with Graham's temperament finder is to look for temperaments in subgroups of a prime limit. For instance, you can put 2.3.7.13.17 in the prime limit box, and it will search omiting 5 and 11 from the list of primes. Subgroups exist in such profusion that Graham's answer machine is the thing to use; no one is likely to produce a survey list or to discuss many of these.

Graham wrote :

> > > (Jacques) :
> > > <0, -8, 11, 6, 28, -1, -3, 24, 15]
> > > it tempers ALL the primes up to 23, mainly by tempering 169/168 > and
> > > 273/272 (and of course 4096/4095).
>
> Gene gave a set of commas that are all tempered out by the mapping you
> gave, and so uniquely define the temperament class. I can identify it
> as this:
>
> 29/96

Right, I found this also but I would have placed 16/53 before.

> 1199.625 cents period
> 362.309 cents generator

and "10 & 53ET" entered in your software indicate also :
[1199.834, 362.059>

Both sound good to me.

What do I do if I want to find the ideal generator in a 1200 cents octave ? I suppose I have to keep the same octave/generator proportions ? and interpret them as 362.422257 and 362.109092 cents ?

> You said 11 and 19 were extras as well, which looks correct. There's
> something similar that comes out better in the 19-limit:
>
> http://preview.tinyurl.com/33l9qyb
>
> The octave equivalent mapping is
>
> <0 -8 11 6 -25 -1 -3 -29].

Great software ! Yes, I found this mapping too for 11 as a second option (quite evident, as 25 + 28 = 53)

> It disagrees with yours in the 11 and 19. But even in the
> 2.3.7.13.17.23-limit nothing that looks like your Buzurg come in my
> top 20.

So it may not be a exceptionnal temperament ! but it makes very good scales anyway.
It's interesting to see they comfort the higher range of my fractal solutions and rather pure fifths (attained by 362.2556 c.)
Thanks for the expertise !
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques

Gene wrote :

> (jacques) :
> > 1.04560 1.91278 Tertiaseptal
> > 1,9128709292 Tiersept 12x - 6x2 = 1
>
> One thing to keep in mind is that temperaments differ in accuracy and
> complexity, and hence have differing requirements for the accuracy of
> generators.

Right, that's what I said in my presentation.

> Tertiaseptal, the 31&171 temperament, is a microtemperament and
> hence I think 2x^3 = 25x^2 - x - 24 would be a better choice. The > largest root
> is much bigger than the generator, however.

It might make difficult series.

> > 1.14230 1.75085 Gamera
>
> 3x^3 - 10x^2 - 10x + 20
> 5x^3 - 5x^2 - 10x + 6
> The second one has a generator which is the largest root in > absolute value.
>
> > 1.14468 1.74722 Guiron
> 12x^2 - 5x - 10
>
> > 1.14503 1.74668 Rodan
> 8*x^3 + x^2 - 16*x + 5
> x^3 - 2x^2 - 7x + 13
>
> This should fill in the gap for generators of about 8/7.

Thanks !

I will just add : three terms polynomials have more value for me than four terms, unless you have special acoustic reasons for the four terms ones. Also generally, the lower the coefficient of the highest power, the better.
But I like 12x^2 = 5x + 10, for other reasons.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques

Gene wrote:

> Subgroups exist in such profusion that Graham's answer machine
> is the thing to use; no one is likely to produce a survey list
> or to discuss many of these.

This strikes me as a lame argument. It's not hard to list the
the best few temperaments for each subgroup. I've already done
rank 1 for all subgroups of the 17-limit.

http://lumma.org/temp/test3.txt

I haven't checked these yet, but they should be the logflat-best
two vals (assume the vals are TOP tuned) for each octave division
up to 100 (even when there's no octaves in the subgroup). Each
line is

(subgroup) (best.oct.div. (val) TOPdamage badness) (2ndbest...)

-Carl

I wrote:

> I've already done rank 1 for all subgroups of the 17-limit.

Subgroups with up to six elements, that is. -C.

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

> > Tertiaseptal, the 31&171 temperament, is a microtemperament and
> > hence I think 2x^3 = 25x^2 - x - 24 would be a better choice. The
> > largest root
> > is much bigger than the generator, however.
>
> It might make difficult series.

A problem I have is that I don't know what your criteria are for a good choice. For instance, Erv Wilson uses only monic polynomials of three terms only, but they can be of high degree. How important is the leading term? The degree? The number of terms? The relationship of the root being used to the other roots (eg, being largest in absolute value, etc.) Is it best if the leading term is at least a power of two, if not 1?

> I will just add : three terms polynomials have more value for me than
> four terms, unless you have special acoustic reasons for the four
> terms ones. Also generally, the lower the coefficient of the highest
> power, the better.
> But I like 12x^2 = 5x + 10, for other reasons.

Ah, I see you answer some of this. Leading term should be small, and the number of terms is best if 3, or at least not higher than 4. But what about degree, and the roots?

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Gene wrote:
>
> > Subgroups exist in such profusion that Graham's answer machine
> > is the thing to use; no one is likely to produce a survey list
> > or to discuss many of these.
>
> This strikes me as a lame argument. It's not hard to list the
> the best few temperaments for each subgroup.

For some value of "best". This barely scratches the surface of the question.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> I wrote:
>
> > I've already done rank 1 for all subgroups of the 17-limit.
>
> Subgroups with up to six elements, that is. -C.

And no, you didn't do all such subgroups.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@> wrote:
>
> > > Tertiaseptal, the 31&171 temperament, is a microtemperament and
> > > hence I think 2x^3 = 25x^2 - x - 24 would be a better choice. The
> > > largest root
> > > is much bigger than the generator, however.
> >
> > It might make difficult series.

What about x^3+x^2+19x-47?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > I wrote:
> >
> > > I've already done rank 1 for all subgroups of the 17-limit.
> >
> > Subgroups with up to six elements, that is. -C.
>
> And no, you didn't do all such subgroups.

What'd I miss? -C.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > >
> > > I wrote:
> > >
> > > > I've already done rank 1 for all subgroups of the 17-limit.
> > >
> > > Subgroups with up to six elements, that is. -C.
> >
> > And no, you didn't do all such subgroups.
>
> What'd I miss? -C.

Lots of things, since omitting primes is not the only way to define subgroups:

http://xenharmonic.wikispaces.com/Just+intonation+subgroups

Gene wrote:

> > What'd I miss? -C.
>
> Lots of things, since omitting primes is not the only way
> to define subgroups:
>
> http://xenharmonic.wikispaces.com/Just+intonation+subgroups

Yes I thought that's what you meant, but I don't understand
them fully. I fail to see how something like 2,3,7/5 is not
going to come out the same as 2,3,5,7 if Tenney weighting
is used. And if it isn't, I fail to see the psychoacoustics
justification for it.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> Yes I thought that's what you meant, but I don't understand
> them fully. I fail to see how something like 2,3,7/5 is not
> going to come out the same as 2,3,5,7 if Tenney weighting
> is used.

How do you propose to get 5 as a product of 2, 3, and 7/5?

And if it isn't, I fail to see the psychoacoustics
> justification for it.

What would be a psychoacoustic justification for anything? How can not playing certain notes need one?

Hi Jacques~
Larry Hanson worked with Erv on a fairly regular basis.
When Larry would run across something that would help Erv he would pass it on.
He also looked up specific things at the cal tech library.
The formula in question is not the aH(n) = bH(n-m) +cH(n-o) + dH(n-p) etc.
but the ones in the various forms of (1+G)^(1/2) as an example that are on page 2.
Page 16 to the end shows others ways of using this with more secondary type recurrent sequences.
I guess this is a formula that Hanson found. But i could be wrong.

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Kraig wrote :
>
> > Thanks for clearing that up. Yes i have you CD and you might be
> > happy to know it
> > is still used as a listening example for two of the classes here at
> > the
> > University of Wollongong.
> >
> > I was fortunate to have Walter O'Connell as my physics of music
> > teacher in LA in
> > 1971 where he played the class examples of a scale based on phi.It
> > was through
> > Walter that i met Erv Wilson.
> >
> > I am curious about the notation that you and Gene are using and was
> > curious if
> > it in anyway resembles that found here. and /or if this makes sense
> > to either of
> > you.
> > Also i am not clear exactly where you two are taking this. I am not
> > sure what
> > hand Hanson might have played in this form of notation.
> >
> > http://anaphoria.com/meruthree.PDF
> >
> > there are also recurrent sequences listed in the wilson archives
> > besides there
> >
> > It is interesting your preference for the converged version.
> > My own work has given me a preference for the numerical often in
> > the area right
> > before it gets within minute variations of the converged.
> > I also have found the subharmonic versions just as good, but if one
> > works with
> > the converged these two are the same.
>
>
> Speaking for myself and as I explained on this list before,
> the polynomial form of writing recurrent sequences is a shortcut for
> the more correct one, mathematically speaking, that uses Erv Wilson,
> and all mathematicians.
> I never used the aH(n) = bH(n-m) +cH(n-o) + dH(n-p) etc... form
> myself, and I am glad I never did as I find it more complicate !
>
> What is it about Hanson ? Which Hanson do you refer to, and perhaps
> also the Hanson temperament ??
>
> I see what you mean about the convergence. Me too and I like to
> integrate divergences also whenever it brings musical results, such as :
> 64 : 75 : 88 : 104 : 128 : 192 : 512 : 2560 : ...
>
> ... and also when it gives negative frequencies and ratios ;)
> - - - - - - -
> Jacques
>
>
> (PS : I am on the digest mode too and it works well for me)
>

Gene wrote :

> How important is the leading term? The degree? The
> number of terms? The relationship of the root being used to the > other roots (eg,
> being largest in absolute value, etc.) Is it best if the leading > term is at
> least a power of two, if not 1?

> (Jacques) :
> > I will just add : three terms polynomials have more value for me > than
> > four terms, unless you have special acoustic reasons for the four
> > terms ones. Also generally, the lower the coefficient of the highest
> > power, the better.
> > But I like 12x^2 = 5x + 10, for other reasons.
>
> Ah, I see you answer some of this. Leading term should be small, > and the number
> of terms is best if 3, or at least not higher than 4. But what > about degree, and
> the roots?

My personal order of criterias, based on the research of recurrent series, would be :

1) Number of terms : should generally rather be always three, as a basic condition for acoustic qualities.
2) The leading term should be small, because you have to multiply the whole series by this term to get the series going on, and this leads rapidly to very high numbers ; so it depends also on the number of notes needed.
3) Leading term = 1 is the best, but 2, 4, ... 2^n are OK whenever the octave is a period, as numbers can be reduced at least by 2^n.
4) High degree is OK for a temperament usually practiced with a high number of notes, but the lower the better.
(a high degree obliges you to create yourself the starting series, which is a very interesting game)
5) Coefficients belonging to primes or multiples of primes well approximated by the recurrent series bring acoustic qualities.
6) Equality or proportionality between 2 coefficients is also a quality.
7) About the roots, existence of a another root higher than the generator leads to divergent series and that will not serve temperaments with many notes. Other positive roots below the generator are not a problem, and negative roots of higher absolute value certainly have an influence, that I have to investigate more. But divergent series are not a problem, they just have to be examined in their opposite direction.
There are more criterias, but these are the simplest.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques

On 7 August 2010 17:32, genewardsmith <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>>
>> http://x31eq.com/temper/uv.html
>>
>> and put your commas in.  (Please don't put Gene's in because they'll
>> hang the script.)
>
> Why does it do that? Too many commas?

It's too difficult to find 20 equal temperaments, which is what it was
doing, by looking for good mappings and filtering them by the commas.
I've fixed it now so that it only looks for 5, and it's a bit slow but
it works.

Graham

On 7 August 2010 18:07, Jacques Dudon <fotosonix@...> wrote:

> What do I do if I want to find the ideal generator in a 1200 cents octave ?
> I suppose I have to keep the same octave/generator proportions ? and
> interpret them as 362.422257 and 362.109092 cents ?
>

You can stretch the whole scale so that octaves are pure. I get a result of
362.423 for this, which isn't much different to what you have by ignoring
the impure octaves, so it doesn't matter much.

> So it may not be a exceptionnal temperament ! but it makes very good scales
> anyway.
> It's interesting to see they comfort the higher range of my fractal
> solutions and rather pure fifths (attained by 362.2556 c.)
>

If they give good scales, they must be good temperaments. And if the
software doesn't recognize that, maybe it should be removed.

I've noticed that this Buzurg is very close to a 3/10 octave generator. And
that the resulting 10 note scale is unequal enough to have two good fifths.

Graham

I wrote:

> If they give good scales, they must be good temperaments.  And if the
> software doesn't recognize that, maybe it should be removed.

Improved, I meant. I noticed this right after I pressed the "Send" button.

Graham

Please don't remove your software Graham !
Actually, these Buzurg scales sound quite terrible :(
And Margo will confirm these are some of the worse persian out-of-tune bizarre things we never heard ! ;-)
- - - - -
Jak

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> I wrote:
>
> > If they give good scales, they must be good temperaments. And if
> > the software doesn't recognize that, maybe it should be removed.
>
> Improved, I meant. I noticed this right after I pressed the "Send" button.
>
>
> Graham

Gene wrote:

> > Yes I thought that's what you meant, but I don't understand
> > them fully. I fail to see how something like 2,3,7/5 is not
> > going to come out the same as 2,3,5,7 if Tenney weighting
> > is used.
>
> How do you propose to get 5 as a product of 2, 3, and 7/5?

I didn't, but you're right, for 2 3 7/5 I get

(70 (70 111 34) 0.355200957179477 208.02770958109716)
(31 (31 49 15) 1.9261689480442783 332.45809959703246)

which is different than what I get for 2 3 5 7. The extra
intervals do seem to change the error.

-Carl