Full thread (4 messages)
From: Gene Ward Smith (2006-01-15)
Subject: Apollo and magic
I was looking at the tuning of the "apollo" temperament, 100/99 and
225/224, and concluded that there isn't much difference in tuning
between it and 11-limit magic. If you take the least squares tuning
map for apollo, it shrinks 3125/3072 from 29.6 cents down to less than
two cents. Put another way, the major third for apollo is a good magic
temperament generator, so five of them will be close to 3 in any case,
and not much tuning damage results from assuming five of them make up
a tempered 3.
From: a_sparschuh (2006-01-16)
Subject: 100/99 sharp & 225/224 flat, was Re: Apollo and magic
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> I was looking at the tuning of the "apollo" temperament, 100/99 and
> 225/224....
1.start @ A4=448 Hz, tuning in 5ths, see 'CAPITAL'-note names,
downwards on the left, constituting the frequencies step by step.
2.The pure >>3rds=5/4 come in 'lower-case' note-names on the right
for comparision.
A: 7,14,28,56,112,224,448 Hz
E: 21......................42,84,168>>165 : C>e @=56/55 same as F>a
B: 63............................126>>125 : G>b
F# 189...........................378>>375 : D>f#
C# 283,566/567...................283>>280 : A>c# 140,70,35:=A*5
G# 53,106,212,424,848/849........106>>105 : E>g#
Eb 79,158/159....................316>>315 : B>eb
Bb 59,118,236/237............472,956>>945 :F#>bb
F: 11,22,44,88,176/177..352,704,1408>>1415:C#>f '1. flat 3rd
C: 33.....................66,132,264>>265 :G#>c '2. flat 3rd
G: 25,50,100/99.............,200,400>>395 :Eb>g @=80/79'5th 100/99sh.
D: 75........................150,300>>295 :Bb>d @=60/59
A: 7,14,28,56,112,224/225.........56>>55_ : F>a '5th D>A 225/224 flat
yielding an superparticular division of the PC into 7 subfactors:
3^12/2^19=
(567/566)(849/848)(159/158)(237/236)(177/176)(99/100)(225/224)
=531441/524288 -confirm by prime-factor decompsition!-
and the diesis "Sorge"-matrix distibution of 3rds:
Octave/(3rds^3)=2/(5/4)^3=2^7/5^3=
= (F/f)(A/a)(C#/c#) = (1415/1408)(56/55)(283/280) :1: F>A >C#>F
= (C/c)(E/e)(G#/g#) = (168/165)(106/105)(264/265) :2: C>E >G#>C
= (G/g)(B/b)(Eb/eb) = (80/79)*(126/125)*(316/315) :3: G>B >Eb>G
=(D/d)(F#/f#)(Bb/bb)= (60/59)*(378/375)*(956/945) :4: D>F#>Bb>D
= 128/125
for all four 3rd cycles subdivisions, in consecuting 5th coulumn order
consisting each as TEMPERED/pure 3rds ratio triple-product.
Or relative in chromatic order:
C: _ 1/1
C# 283/264
D:_ 25/11
Eb_ 79/66
E:_ 14/11
F:__ 4/3
F#_ 63/44
G:_ 50/33
G#_ 53/33
A:_ 56/33
Bb_ 59/33
B:_ 23/11
C:'_ 2/1
Attend the primes: 5,7,11,23,53,59,79 & 283 in it.
Have a lot of fun with the 225/224 & 100/99 tempered 5ths,
and all the others above epimoric sounding ratios here!
From: Gene Ward Smith (2006-01-16)
Subject: 100/99 sharp & 225/224 flat
--- In tuning-math@yahoogroups.com, "a_sparschuh" <a_sparschuh@y...>
wrote:
What is the idea behind this scale? What does 225/224 or 100/99 have
to do with it?
> Or relative in chromatic order:
> C: _ 1/1
> C# 283/264
> D:_ 25/11
> Eb_ 79/66
> E:_ 14/11
> F:__ 4/3
> F#_ 63/44
> G:_ 50/33
> G#_ 53/33
> A:_ 56/33
> Bb_ 59/33
> B:_ 23/11
> C:'_ 2/1
I don't know what you mean by chromatic order. If you put everything
in an octave, and sort it by increasing pitch, you get this:
! sparschuh1.scl
Sparchuh scale
12
!
23/22
283/264
25/22
79/66
14/11
4/3
63/44
50/33
53/33
56/33
59/33
2
> Have a lot of fun with the 225/224 & 100/99 tempered 5ths,
> and all the others above epimoric sounding ratios here!
Why would I temper this way?
From: a_sparschuh (2006-01-17)
Subject: Re: 100/99 sharp 5th: C>G & 225/224 flat 5th D>A as subparts of the PC
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> What is the idea behind this scale?
to demonstrate that 225/224 & 100/99 can well be used as 2 0f 7 steps
D>A flat(1 200 * ln(225 / 224)) / ln(2) = ~ 7.71152299... cents
C>G sharp(1 200 * ln(99 / 100)) / ln(2) = ~-17.3994836... cents
for tempering in a 12-circular tuning beside
F#>C# flat (1 200 * ln(567 / 566)) / ln(2) = ~ 3.05601853... c
C#>G# flat (1 200 * ln(849 / 848)) / ln(2) = ~ 2.04034679... c
G#>Eb flat (1 200 * ln(159 / 158)) / ln(2) = ~ 10.9226485... c
Eb>Bb flat (1 200 * ln(237 / 236)) / ln(2) = ~ 7.32023944... c
Bb>F flat (1 200 * ln(177 / 176)) / ln(2) = ~ 9.80871773... c
the sum of all that 7 steps is = 23.4...c the PC
> What does 225/224 ~7.7...c or 100/99 ~17.4...c have
> to do with it?
They are used as tempering steps in the PC subdivision.
>
Attend the error @ B!
> > Or relative in chromatic order:
that means ordered ascending seize as on the keys from left to right
> > C: _ 1/1
> > C# 283/264
> > D:_ 25/11
> > Eb_ 79/66
> > E:_ 14/11
> > F:__ 4/3
> > F#_ 63/44
> > G:_ 50/33
> > G#_ 53/33
> > A:_ 56/33
> > Bb_ 59/33
corr:B_ 21/11 instead wrong>B:23/11,because63/3=21 not=23 see 5ths
> > C:'_ 2/1
Sorry, but that tuning has nothing to do with the prime 23,
smuggeld in by faulty overtaking an reading error.
>
> I don't know what you mean by chromatic order. If you put everything
> in an octave, and sort it by increasing pitch, you get this:
after correcting above error the scale is already ordered in seize.
>
> ! sparschuh1.scl
> Sparchuh scale
> 12
> !
!!!!!> 23/22 omit that wrong line in the whole!!!!!!
> 283/264
> 25/22
> 79/66
> 14/11
> 4/3
> 63/44
> 50/33
> 53/33
> 56/33
> 59/33
21/11 !!!!reinsert here the correct value 21/11 instad formerly 23/11
> 2
because my old wrong "23/11">2 would make no sense.
Sorry my mistake in hurry.
Are there any other faults?
>
> > Have a lot of fun with the 225/224 & 100/99 tempered 5ths,
> > and all the others above epimoric sounding ratios here!
>
> Why would I temper this way?
In order to demonstrate the usefulness of 225/224 & 100/99
as constitues of an PC subdivision.