Yahoo Tuning Groups Ultimate Backup tuning-math Temperaments and series in Scala

I've added two more categories of rank-2 temperaments to Scala's list in the file lintemp.par.

One is the Wilson type of equal beating temperaments where 5/4 and 3/2 have an equal (or integer proportional) beating rate, called Meta-<name> so to list them in Scala type:
LINEAR/ALL Meta-*
I also included a few 3/1 period based ones with 5/3 and 7/3 equal beating.

And the other one the "golden" versions which have the property that the ratio between large and small step size is equal to the golden ratio phi. To list them in Scala type:
LINEAR/ALL Golden

For some more information, enter the command TIP 146.

There's also a new datafile called series.par, which is used in the dialog
File:New:Recurrent series. You can type a name in the coefficients combobox and this will allow you to select a predefined series. Or take it from the dropdown list. Most of them came from Jacques Dudon, who no doubt has invented hundreds more of them but these are all I could find.
Here you'll find the same names as the Meta temperaments, which means the root
of this polynomial is the generator of that temperament. You can check this, for example select: 1 1 1/2 (Zinith, Meta-Pelog), select Recurrent series under Kind and click Find Polynomial Root. The root is then shown in the main window.
In order to create a recurrent series from this you additionally have to enter initial values under Parameters. More info in the help for POLYNOMIAL/SERIES.
Below is the current contents of series.par.

Manuel

! List of linear recurrent series to select coefficients from in File:New:Recurrent series.
!
! Invented mostly by Jacques Dudon, Gene Ward Smith and Ervin Wilson.
! PV stands for Pisot-Vijayaraghavan number, the first 10 polynomials where they're roots of are included.
! Format: coefficients (name)
! For example 1 2 3 means: x[n] := 2x[n-1] + 3x[n-2]
! To get the characteristic polynomial give the first coefficient a minus sign.
! For example 1 2 3 means: -x^2 + 2x + 3 (= 0)
!
! 2nd order:
1 1/2 1 (Dagbok)
1 1 1/2 (Zinith, Meta-Pelog)
1 1 1/4 (Aksaka)
1 1 1 (Fibonacci)
1 -1 3 (Brindi)
1 2 -1/4 (Zira'at)
1 2 1 (Pell)
1 3 -1 (Phidiama)
1 7 -8 (Phantax)
1 -7 10 (Macram)
1 8 -8 (Isrep)
1 8 -9 (Circé)
1 -8 12 (RétriXringco)
1 -9 16 (Petigrain)
1 -10 17 (Shebag)
1 11 -12 (RetriZalith)
1 -12 20 (Nueh)
1 15 -16 (Nadir)
1 30 -40 (Salamandre2)
1 -50 77 (Rwadlab)
2 -9 14 (Mohajisept)
3 1 9 (Salmad)
3 2 3 (Adibis)
3 3 1 (Amlak)
3 32 -32 (Mezyd1)
4 -1 7 (Aiguille)
4 3 3 (Popeye)
4 15 -12 (Guéry)
4 -17 24 (Meanotaur2)
4 49 -49 (Dlotkotier2)
5 1 5 (Thaeyink)
5 5 1 (Pancuazisrepl)
6 6 1 (Baka)
6 12 -1 (Tiersept)
7 1 7 (Minimak)
7 -2 14 (Sceptre1)
8 1 16 (Simdek)
8 4 9 (Squarto)
8 8 1 (Ishku)
8 8 3 (Poussah)
8 9 1 (Zalarith)
9 -1 16 (Sceptre2)
9 -2 23 (Cazdep)
9 8 4 (Terzbirat)
9 9 5 (Salamandre1)
9 -24 56 (Auruter)
10 20 -9 (Decime)
11 2 14 (Mohaban)
11 11 3 (Mohajatis)
11 21 1 (Diese11)
11 22 -10 (Brindet)
12 12 1 (Persane)
13 1 20 (Sceptre3)
16 16 1 (Dotz3)
16 20 -1 (Chandrak)
18 18 1 (Limma1)
19 36 2 (Treizenhuit)
32 32 9 (Zenarith)
32 37 -2 (Dlotkotier4)
64 15 120 (Pierdangle)
64 64 7 (Dlotkotier5)
!
! 3rd order:
1 0 1/4 2 (Mounos)
1 0 1/2 1 (Meta-Bug)
1 0 1 1/4 (Olzal)
1 0 1 1 (Meta-Slendro, PV-1)
1 0 1 3 (Dhaivati)
1 0 -1 5 (Pendragon)
1 0 -1 6 (Arijaom)
1 0 2 1/4 (Gulu-nem)
1 0 -2 5 (Preska)
1 0 4 -1 (Shakira)
1 0 8 -8 (Iph)
1 0 -8 8 (Fong)
1 0 -8 13 (Airos)
1 0 14 -14 (Totem2)
1 0 16 -16 (Noura)
1 0 -56 72 (Courette)
1 3/5 0 2 (Comptine, Meta-3/11-comma meantone)
1 3/4 0 5/3 (Mezzo)
1 1 0 1/3 (Mohabis)
1 1 0 1 (Meru II, PV-4)
1 1 1 1 (Awj)
1 -1 0 4 (Mrejdaïrim)
1 -2 0 8 (Natté)
1 -3 0 10 (Chimere)
1 3 0 -3 (Nawahbis)
1 8 -7 -1 (Ekipenta2)
2 0 -1 6 (Kepat)
2 0 -4 7 (Totem3)
2 0 -7 11 (Jade)
2 1 0 3 (Eair, Meta-5/18-comma meantone)
2 4 0 -1 (Arounh)
2 -4 0 11 (Adocleme)
2 5 0 -2 (Eronyme)
3 0 1 4 (Granlagun)
3 0 -2 9 (Meta-Bohlen)
3 0 -4 16 (Ray)
3 0 -6 19 (Radieubiz)
3 1 0 8 (Tapek)
4 0 1 5 (Eq-Bala)
4 0 -35 60 (Salamandre4)
5 0 6 3 (Poussami)
5 0 10 -2 (Brendwib)
6 0 2 5 (Dotz2)
6 18 -18 7 (Anhawyn)
7 0 1 7 (Gedeon7)
8 0 -1 16 (Wusta-Zactral)
8 0 7 28 (Sabra1)
8 0 16 -7 (Totem11)
9 0 16 -1 (Louven)
10 0 10 1 (Gustave2)
11 22 0 -13 (Kappa)
12 0 7 48 (Sabra3)
12 10 0 7 (Mohajaran)
13 16 0 8 (Soria)
15 10 0 18 (Cybozem)
18 0 7 78 (Sabra8)
20 20 0 1 (Gontrand1)
24 22 0 11 (Mohajicom)
27 0 -64 186 (Coquine)
!
! 4th order:
1 0 0 1/2 1 (Mezyd2)
1 0 0 1/2 4 (Rwato)
1 0 0 1 2 (Samak, Meta-Mavila)
1 0 0 2 1/2 (Melkis)
1 0 0 2 2 (Meta-Meantone)
1 0 0 2 4 (Carthage)
1 0 0 3 -1 (Tulpe)
1 0 0 4 1 (Dhaivata)
1 0 0 4 -1 (Diatess)
1 0 0 4 2 (Semiestel)
1 0 0 6 -4 (Meenshku)
1 0 0 6 -5 (Nestor)
1 0 0 10 -10 (Nung-Phan-Bâ)
1 0 0 16 -16 (Scylla)
1 0 1 0 1 (Raph, Meta-Sidi)
1 0 1 0 3 (Moera)
1 0 1 -10 20 (Pathos)
1 0 -2 3 5 (Hummy)
1 1 0 0 1/3 (Looz)
1 1 0 0 1 (Amaz, PV-2)
1 1 0 0 3 (Dhadou)
1 1 0 0 4 (Pansept, Meta-Slendro 2)
1 -2 0 -3 12 (Apatoïd)
1 3 0 0 -5 (Sireine)
1 -3 0 0 15 (Lucie)
1 4 0 0 -12 (Innocente)
1 4 -8 8 -2 (Houboriz)
1 -4 0 0 6 (Dotz1)
1 -4 0 0 9 (Lagune)
1 8 0 0 -22 (Wustaf)
2 0 0 2 3 (Irina)
2 0 0 2 7 (Tara, Meta-Woolhouse)
2 0 0 -2 13 (Traverse)
2 0 3 0 1 (Minsys)
2 0 -9 0 30 (Esterelle1)
2 1 0 0 4 (Keller, Skisni)
2 -1 0 8 -2 (Tibrepyt)
3 0 0 1 5 (Climene2)
4 0 0 1 4 (Dutz2)
4 0 0 1 8 (Buzurg)
4 0 0 -1 8 (Septimacle)
4 0 0 -1 14 (Quiretsim)
4 1 0 0 16 (Sapaan)
4 3 0 0 10 (Deolia)
5 0 0 1 5 (Damlas)
5 10 0 0 -8 (Susak)
5 -10 0 0 56 (Saptamak)
8 -6 0 0 5 (Michemine, Meta-Zarlino)
10 12 0 0 1 (Coureuse)
!
! 5th order:
1 0 0 0 1 1 (Meru 9)
1 0 0 0 -2 4 (Minos)
1 0 0 0 -4 8 (Aswafée, Meta-Magic)
1 0 0 0 -5 15 (Evelyne)
1 0 0 1/2 0 1 (Dlotkotier3, Meta-Porcupine, Meta-Hedgehog)
1 0 0 32 0 -64 (Cascade)
1 0 1 0 0 1 (Ifbis, Meru 7)
1 0 1 1 1 1 (PV-6)
1 0 -1 0 0 4 (Meta-Sirius)
1 0 -2 0 0 4 (Meta-Progression)
1 0 -2 0 0 32 (Topaz)
1 0 -16 0 0 32 (Mohawj)
1 1 0 0 0 1/8 (Leila)
1 1 0 0 0 1/4 (Slendra)
1 1 0 0 0 4 (Generousse)
1 1 0 1 0 1 (PV-9)
1 1 1 -1 0 1 (PV-3)
1 1 0 0 0 1/2 (Mohajira)
1 -1 0 4 -1 1 (Algebraic Sensi)
1 2 -2 2 -2 2 (Radix)
1 3 0 0 0 -4 (Awjig)
1 8 0 0 0 -32 (Gazelle)
2 0 0 0 -1 16 (Evan)
2 1 0 0 0 4 (Meta-Smate)
2 3 -1 0 0 1 (Borygma)
3 0 0 2 0 9 (Meta-Vega)
4 0 0 0 -1 32 (Sensible)
4 0 0 0 9 9 (Meta-Canopus)
4 0 1 0 0 4 (Limma2)
5 0 0 0 -4 20 (Septimage)
5 0 9 0 0 -4 (Macbeth)
5 2 0 0 0 40 (Tirzbirat)
6 0 0 0 4 13 (Magixsyx)
6 0 0 0 -4 23 (Sixmagic)
8 0 -1 0 0 64 (Mysterique)
8 -1 0 0 0 64 (Vali)
8 5 0 0 0 12 (Deolimage)
16 0 0 0 1 3 (Ellis equal beating meantone)
!
! 6th order:
1 0 0 0 0 1 2 (Meta-Sixix)
1 0 0 0 0 1/2 4 (Calice)
1 0 0 0 0 -9 36 (Meta-Arcturus)
1 0 0 0 1 0 3 (Radevati)
1 0 0 0 1 0 16 (Rachel)
1 0 0 0 9 0 1 (Bluette)
1 0 0 1 0 0 8 (Rhakub)
1 0 0 -1 0 0 8 (Bukhar)
1 0 1 1 0 -1 1 (Narquise)
1 1 0 0 0 0 1/2 (Lagunag)
1 1 0 0 0 0 2 (Radix)
1 1 1 0 -1 0 1 (PV-5)
1 2 0 0 0 0 -2 (Gretel)
1 2 -1 0 1 -1 1 (PV-8)
1 -2 0 0 0 0 8 (Carmine, Meta-Hanson, Meta-Tritikleismic)
3 2 0 0 0 0 4 (GWSmith)
3 4 0 0 0 0 -1 (Climene1)
4 0 0 0 0 7 1 (Gatetone)
4 0 0 0 4 0 1 (Limma3)
5 2 0 0 0 0 10 (GWSmith2)
9 -2 0 0 0 0 32 (Cremona)
15 0 8 0 0 0 12 (Secor59)
24 0 0 0 0 -15 52 (Secor26)
!
! 7th order:
1 0 0 0 0 0 1/2 4 (Gorgone)
1 0 0 0 0 0 1 2 (Meta-Subklei)
1 0 0 0 0 3 0 1 (Calista)
1 0 0 0 0 8 0 -1 (Albatros)
1 0 0 0 -1 0 0 4 (Meta-Laconic)
1 0 0 1/2 0 0 0 1 (Meta-Negri)
1 0 0 1 0 0 0 12 (Bhatiyar)
1 0 0 -2 0 0 0 4 (Meta-12&35)
1 0 0 16 0 0 0 -64 (Shrirag)
1 0 1 0 0 0 0 1 (Meru 13)
1 0 1 0 0 0 0 2 (Meta-Roman)
1 1 0 0 0 0 0 1 (Meru 12)
1 1 1 0 0 -1 0 1 (PV-7)
3 0 0 0 0 0 3 1 (Gedeon2)
4 0 0 3 0 0 0 9 (Meta-Toliman)
4 8 0 0 0 0 0 -5 (Gontrand2)
7 0 0 0 0 0 -6 28 (Arbas4)
!
! 8th order:
1 0 0 0 0 0 0 1 2 (Bala)
1 0 0 0 0 0 0 -8 16 (Meta-Würschmidt)
1 0 0 0 0 0 1 0 2 (Meta-Superpelog)
1 0 0 0 0 0 -2 0 8 (Meta-Mohajira)
1 0 0 0 0 0 3 0 -1 (Rabrindanath)
1 0 0 0 0 1/2 0 0 1 (Meta-Ripple)
1 0 0 0 0 2 0 0 8 (Meta-Progress)
1 0 1/2 0 0 0 0 0 2 (Meta-Godzilla)
1 1 0 0 0 0 0 0 1 (Buzurg 8, Meru 18)
1 1 1 0 0 0 -1 0 1 (PV-10)
4 4 0 0 0 0 0 0 7 (Carene)
14 0 0 0 0 0 7 0 16 (Secor53)
!
! 9th order:
1 0 0 0 0 0 0 0 2 1 (Tripletaille)
1 0 0 0 0 0 0 0 -1 40 (Navan)
1 0 0 0 0 0 0 0 4 8 (Meta-Helmholtz)
1 0 0 0 0 0 -8 0 0 64 (Pandore)
1 0 0 0 0 1 0 0 0 32 (Meancaline)
1 0 0 0 0 -1 0 0 0 4 (Meta-Tetracot)
1 0 0 0 1/2 0 0 0 0 1 (Meta-Passion)
1 0 0 0 -2 0 0 0 0 4 (Meta-Valentine)
1 0 1 0 0 0 0 0 0 4 (Pierre)
1 0 -1 0 0 0 0 0 0 16 (Meta-Sensi)
1 1/2 0 0 0 0 0 0 0 8 (Meta-Superpythagorean)
1 4 0 0 0 0 0 0 0 -64 (Helmut1, Meta-Helmholtz)
1 -1 0 0 0 0 0 0 0 64 (Helmut2)
3 2 0 0 0 0 0 0 0 64 (Helmut4)
4 1 0 0 0 0 0 0 0 128 (Helmut3)
!
! 10th order:
1 0 0 0 0 0 0 0 0 1 1 (Mani)
1 0 0 0 0 0 0 0 0 2 4 (Meta-Superkleismic)
1 0 0 0 0 0 0 0 0 -4 64 (Decam)
1 0 0 0 0 0 0 0 0 8 44 (Saptendre)
1 0 0 0 0 0 0 0 0 16 32 (Sapnat)
1 0 0 0 0 0 0 0 0 32 8 (Saprad)
1 0 0 0 0 0 0 0 0 64 -40 (Saplante)
1 0 0 0 0 0 0 0 -4 0 8 (Meta-Spell)
1 0 0 0 0 0 0 0 24 0 4 (Inflexcible)
1 0 0 0 0 0 0 -2 0 0 8 (Meta-Orwell)
1 0 0 0 0 0 0 4 0 0 16 (Meta-Triton)
1 0 0 0 0 0 1/2 0 0 0 1 (Meta-Nautilus)
1 0 0 0 0 0 -8 0 0 0 96 (Heptapand)
1 0 0 1 0 0 0 0 0 0 16 (Meta-Kangaroo)
1 0 1 0 0 0 0 0 0 0 4 (Meta-Hocus)
1 0 1 0 0 0 0 0 0 0 32 (Aigle)
1 -2 0 0 0 0 0 0 0 0 16 (Meta-Myna)
2 0 1 0 0 0 0 0 0 0 4 (Meta-Hemismate)
5 1 0 0 0 0 0 0 0 0 25 (Myna)
5 -1 0 0 0 0 0 0 0 0 35 (Myna2)
!
! 11th order:
1 0 0 0 0 0 0 0 0 0 -1 88 (Everest)
1 0 0 0 0 0 0 0 0 0 -16 32 (Meta-Magus)
1 0 0 0 0 0 0 0 -4 0 0 8 (Totem10)
1 0 0 0 0 0 0 1/2 0 0 0 1 (Meta-Slurpee)
1 0 0 0 0 -1 0 0 0 0 0 4 (Meta-Secund)
1 0 0 0 0 -2 0 0 0 0 0 4 (Meta-Sycamore)
1 0 -1 0 0 0 0 0 0 0 0 16 (Meta-Beatles)
!
! 12th order:
1 0 0 0 0 0 0 0 0 0 8 0 8 (Meta-Clyde)
1 0 0 0 0 0 0 0 0 -2 0 0 8 (Meta-Cynder)
1 0 0 0 0 0 0 -16 0 0 0 0 128 (Meta-Thuja)
1 0 0 1 0 0 0 0 0 0 0 0 32 (Meta-Liese)
1 0 -2 0 0 0 0 0 0 0 0 0 8 (Meta-Hemikleismic)
!
! 13th order:
1 0 0 0 0 0 0 0 0 0 0 0 2 192 (Batteuse)
1 0 0 0 0 0 0 0 0 0 0 1 0 1 (Dutz1)
1 0 0 0 0 0 0 0 0 0 0 1 0 2 (Meta-Nusecond)
1 0 0 0 0 0 0 0 0 2 0 0 0 2 (Meta-Kumonga)
1 0 0 0 0 0 0 0 1/2 0 0 0 0 1 (Meta-Unicorn)
1 0 0 0 0 0 -1 0 0 0 0 0 0 4 (Meta-Ampersand)
1 0 0 0 0 1 0 0 0 0 0 0 0 8 (Meta-Amity)
1 0 0 1 0 0 0 0 0 0 0 0 0 32 (Meta-Mabila)
1 0 1 0 0 0 0 0 0 0 0 0 0 8 (Meta-Ditonic)
1 0 1/2 0 0 0 0 0 0 0 0 0 0 4 (Meta-Immunity)
!
! 14th order:
1 0 0 0 0 0 0 0 0 0 0 0 0 4 8 (Meta-Parakleismic)
1 0 0 0 0 0 0 0 0 0 0 0 0 16 32 (Meta-Quasisuper)
1 0 0 0 0 0 0 0 0 0 0 4 0 0 4 (Meta-Quasitemp)
1 0 0 0 0 0 0 0 0 8 0 0 0 0 64 (Meta-Fervor)
1 0 0 0 0 0 0 0 -8 0 0 0 0 0 128 (Meta-Wilsec)
!
! 15th order:
1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 (Balasept)
1 0 0 0 0 0 0 0 0 0 0 0 -64 0 0 256 (Meta-Trismegistus)
1 0 0 0 0 0 0 0 0 1/2 0 0 0 0 0 1 (Meta-Ceratitid)
1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 32 (Meta-Ammonite)
!
! 16th order:
1 0 0 0 0 0 0 0 0 0 0 0 0 0 -8 0 16 (Meta-Hemiwur)
1 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 128 (Meta-Tritonic)
1 0 0 0 0 0 0 1/2 0 0 0 0 0 0 0 0 1 (Meta-Escapade)
1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 32 (Meta-Squares)
1 0 0 1/2 0 0 0 0 0 0 0 0 0 0 0 0 2 (Meta-Twothirdtonic)
!
! 17th order:
1 0 0 0 0 0 0 0 0 0 0 0 0 0 -4 0 0 16 (Meta-Rodan)
1 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 128 (Meta-Cypress)
1 0 0 0 0 0 0 0 0 0 0 -16 0 0 0 0 0 256 (Meta-Gravity)
1 0 0 0 0 0 0 0 0 0 0 -32 0 0 0 0 0 256 (Meta-Semisept)
1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 8 (Meta-Septimin)
1 0 1/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 (Meta-Hemithirds)

Temperaments and series in Scala

🔗manuel.op.de.coul@eon.com

🔗Herman Miller <hmiller@prismnet.com>

🔗manuel.op.de.coul@eon.com