Meta-version of Miracle temperament

This came to me after reading this brillant article of Graham mentionned by Carl :
http://x31eq.com/paradigm.html
And the explanations it gives about George Secor's system.

It inspired me a fractal form of Miracle's octave division, based on the identities :
octave/generator = neutral third/small interval (= fifth/2s = neutral 7th/3s) = neutral sixth/(generator less small interval) = (etc., there are certainly more of those),
= 2 * 7^(1/2) + 5 = 10.2915026221292

what I call here "small interval" or "s" has 33.9895 c.
the semitone generator 116.601 c.
septimal tone 233.2 c.
neutral third 349.803 c .
the fifth 699.606 c.
major third 383.79266 c.
neutral fourth 549.0157 c.
smaller semitone 83.61154 c.
in an octave of 1200 c.

This is like what the "Golden scale" is to the meantone, a general frame that allows an infinity of
variations. You will find of course among them the octave of 72 steps / 7 steps as generator, as well as less accurate 31/3, 41/4 etc., but also more complex 103/10, 247/24, etc.
All are found in the recurrent series of x^2 = 10x + 3 :

1 10 103 1060
3 31 319
4 41 422
7 72 741
17 175 1801
24 247 2542

etc., that converge rapidly towards the miraculous ratio.

- - - - - - -
Jacques

Transposed to a symmetrical version (S,L,S,L,... and symmetrical on degree 0) this gives us a "Blackjack" tuning of:

0:......1/1
1:.........33.990 cents
2:.......116.601 cents
3:.......150.591 cents
4:.......233.202 cents
5:.......267.192 cents
6:.......349.803 cents
7:........383.793 cents
8:........466.404 cents
9:........500.394 cents
10:.......583.005 cents
11:........616.995 cents
12:........699.606 cents
13:........733.596 cents
14:........816.207 cents
15:.......850.197 cents
16:.......932.808 cents
17:.......966.798 cents
18:.......1049.409 cents
19:........1083.399 cents
20:........1166.010 cents
21:........1200.000 cents

The "7-limit" is clearly excellent, but the approximation to 11/8, which is otherwise also excellent, seems to be disjointed from the keys with 7-limit intervals. More interesting to me in this tuning are things like 0-6-14 and 0-8-15.

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> This came to me after reading this brillant article of Graham
> mentionned by Carl :
> http://x31eq.com/paradigm.html
> And the explanations it gives about George Secor's system.
>
> It inspired me a fractal form of Miracle's octave division, based on
> the identities :
> octave/generator = neutral third/small interval (= fifth/2s = neutral
> 7th/3s) = neutral sixth/(generator less small interval) = (etc.,
> there are certainly more of those),
> = 2 * 7^(1/2) + 5 = 10.2915026221292
>
> what I call here "small interval" or "s" has 33.9895 c.
> the semitone generator 116.601 c.
> septimal tone 233.2 c.
> neutral third 349.803 c .
> the fifth 699.606 c.
> major third 383.79266 c.
> neutral fourth 549.0157 c.
> smaller semitone 83.61154 c.
> in an octave of 1200 c.
>
> This is like what the "Golden scale" is to the meantone, a general
> frame that allows an infinity of
> variations. You will find of course among them the octave of 72
> steps / 7 steps as generator, as well as less accurate 31/3, 41/4
> etc., but also more complex 103/10, 247/24, etc.
> All are found in the recurrent series of x^2 = 10x + 3 :
>
> 1 10 103 1060
> 3 31 319
> 4 41 422
> 7 72 741
> 17 175 1801
> 24 247 2542
>
> etc., that converge rapidly towards the miraculous ratio.
>
> - - - - - - -
> Jacques
>

Hi Cameron !

First, I am not at all expert of the Miracle temperament ! What I did is just a "photosonic disk" of the logarithmic scale in a circle (= a horagram), with improved image periodicity through this fractal transformation.
All what I know is that it doesn't changes the logic of the structure, and it shows how to find all possible edo subsets versions of Miracle temperaments.
What I found is that the generator can be seen as an octave at a different scale, just like in an "abyssal image" (don't know if this translates well the french "image en abime", an image that contains a picture of the whole image).
In fact, in its pure fractal form, it's impossible to say if "the octave" is 1200 cents, or 116.6o105 c. or even 10.2915026221292 * 1200 = 12 349.803 cents !
But let's say the octave is 1200 cents, that speaks better (it can also be extended for optimisation without problem).

The main basic division to retain of the octave here is :
neutral third = 349.803 c.
fifth = 699.606 c.
neutral sixth (= octave less neutral third) = 850.197 c.
octave = 1200 c.
And these have the same proportions as 33.9895 c., 67.979 c., 82.6115 c., and 116.601 c., that form the basic division of the generator, 116.601 c.

The 11/8 approximation has to be one "double-fifth" wholetone + one neutral third :
= 199.2125 + 349.803 = 549.0155 c.
It is attained by 4 generators + one fraction of a generator, equal to the proportion of the neutral sixth to the octave :
(850.197 / 1200) *116.601 = 82.6115 <error of 1 c. in my first message>
So "11/8" = 4* 116.6o1 c. + 82.6115 c. = 549.0155 c.

If we express the intervals in 72-edo (which is also probably Miracle's most common use ?), the generator is
72 / 10.2915026221292 = roughly 7 steps, divided 2 + 2 + 1 + 2 (or steps 2-4-5-7)
The 11/8 will then be approximated by 4 * 7 + 5 = 33 steps.
The Blackjack subset goes 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2, or steps
2-7-9-14-16-21-23-28-30-35-37-42-44-49-51-56-58-63-65-70-72
It has no 33 steps from 1/1, it has 33 steps only in other places : between 2 and 35, 9 and 42, etc.
The blackjack subset has no regular whole tone (199.2125 c.), nor the "6/5" approximation neither (315.8135 c.).
31 and 41 subsets have all those.

Just tried your triads suggestions. Both sound strange, but may be I need more of the context !!
It's quite limited to test such a temperament with a 12 keys keyboard of course, but still it's open.
On my first test what striked me were the 7/6, I found them terrific ! Nothing new, I found these are quasi-perfectly just.
Then one scale I was happy with that came out is that kind of weird "septimal Shur" :

1/1
150.59050
315.81350
466.40400
699.60600
816.20734
966.79800
2/1

allmost all in the BlackJack subset, except 315.813 - but with 349.803 instead it's not uninteresting.
- - - - - - -
Jacques

Cameron wrote :

> Transposed to a symmetrical version (S,L,S,L,... and symmetrical on > degree 0) this gives us a "Blackjack" tuning of:
>
> 0:......1/1
> 1:.........33.990 cents
> 2:.......116.601 cents
> 3:.......150.591 cents
> 4:.......233.202 cents
> 5:.......267.192 cents
> 6:.......349.803 cents
> 7:........383.793 cents
> 8:........466.404 cents
> 9:........500.394 cents
> 10:.......583.005 cents
> 11:........616.995 cents
> 12:........699.606 cents
> 13:........733.596 cents
> 14:........816.207 cents
> 15:.......850.197 cents
> 16:.......932.808 cents
> 17:.......966.798 cents
> 18:.......1049.409 cents
> 19:........1083.399 cents
> 20:........1166.010 cents
> 21:........1200.000 cents
>
> The "7-limit" is clearly excellent, but the approximation to 11/8, > which is otherwise also excellent, seems to be disjointed from the > keys with 7-limit intervals. More interesting to me in this tuning > are things like 0-6-14 and 0-8-15.

> --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
>> > This came to me after reading this brillant article of Graham
>> > mentionned by Carl :
>> > http://x31eq.com/paradigm.html
>> > And the explanations it gives about George Secor's system.
>> >
>> > It inspired me a fractal form of Miracle's octave division, >> based on
>> > the identities :
>> > octave/generator = neutral third/small interval (= fifth/2s = >> neutral
>> > 7th/3s) = neutral sixth/(generator less small interval) = (etc.,
>> > there are certainly more of those),
>> > = 2 * 7^(1/2) + 5 = 10.2915026221292
>> >
>> > what I call here "small interval" or "s" has 33.9895 c.
>> > the semitone generator 116.601 c.
>> > septimal tone 233.2 c.
>> > neutral third 349.803 c .
>> > the fifth 699.606 c.
>> > major third 383.79266 c.
>> > neutral fourth 549.0157 c.
>> > smaller semitone 82.61154 c.
>> > in an octave of 1200 c.
>> >
>> > This is like what the "Golden scale" is to the meantone, a general
>> > frame that allows an infinity of
>> > variations. You will find of course among them the octave of 72
>> > steps / 7 steps as generator, as well as less accurate 31/3, 41/4
>> > etc., but also more complex 103/10, 247/24, etc.
>> > All are found in the recurrent series of x^2 = 10x + 3 :
>> >
>> > 1 10 103 1060
>> > 3 31 319
>> > 4 41 422
>> > 7 72 741
>> > 17 175 1801
>> > 24 247 2542
>> >
>> > etc., that converge rapidly towards the miraculous ratio.
>> >
>> > - - - - - - -
>> > Jacques

Hi Jacques,

I'm certainly not an expert on the Miracle temperament either, though there are those here who are. But, I have a number of observations from my own experiences.

In Blackjack 11/8 shows up in hexads rather than in a JI harmonic scale kind of way, but not as often as 7/5 appears as an "11th". So it is not a "regular" mapping in that way, or as regards to the minor thirds, which are mostly near-perfect 7/6s, and eight 6/5s mixed in. And consequently the 5/3s are scattered... basically, I see the Blackjack "scale" as an ingenious way of packing of as many near-perfect low-limit JI intervals as possible into a small space.

There is a huge amount to discuss here, it'll take some time (I'm "multitasking" at the moment:-)) For example I have a spectral image clearly illustrating some very interesting things, present in you "Shur" scale as well as well as many other Phi tunings- and Blackjack- which you might dig.

Looking forward to hearing your new photosonic disc!

By the way, in English and Slovene, and I believe many other languages, the expression for the larger-within-smaller... is mise en abîme, directly from the French. Or recursion in more scientific or "scientific" contexts. "Turtles all the way down" is another related expression, from the probably bogus, but funny, anectode of the Indian sage explaining cosmology. The two-mirrors version is my argument for the idea that the logical fallacy of infinite regress may not be a fallacy after all.

-Cameron Bobro

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Hi Cameron !
>
> First, I am not at all expert of the Miracle temperament ! What I did
> is just a "photosonic disk" of the logarithmic scale in a circle (= a
> horagram), with improved image periodicity through this fractal
> transformation.
> All what I know is that it doesn't changes the logic of the
> structure, and it shows how to find all possible edo subsets versions
> of Miracle temperaments.
> What I found is that the generator can be seen as an octave at a
> different scale, just like in an "abyssal image" (don't know if this
> translates well the french "image en abime", an image that contains a
> picture of the whole image).
> In fact, in its pure fractal form, it's impossible to say if "the
> octave" is 1200 cents, or 116.6o105 c. or even 10.2915026221292 *
> 1200 = 12 349.803 cents !
> But let's say the octave is 1200 cents, that speaks better (it can
> also be extended for optimisation without problem).
>
> The main basic division to retain of the octave here is :
> neutral third = 349.803 c.
> fifth = 699.606 c.
> neutral sixth (= octave less neutral third) = 850.197 c.
> octave = 1200 c.
> And these have the same proportions as 33.9895 c., 67.979 c., 82.6115
> c., and 116.601 c., that form the basic division of the generator,
> 116.601 c.
>
> The 11/8 approximation has to be one "double-fifth" wholetone + one
> neutral third :
> = 199.2125 + 349.803 = 549.0155 c.
> It is attained by 4 generators + one fraction of a generator, equal
> to the proportion of the neutral sixth to the octave :
> (850.197 / 1200) *116.601 = 82.6115 <error of 1 c. in my first
> message>
> So "11/8" = 4* 116.6o1 c. + 82.6115 c. = 549.0155 c.
>
> If we express the intervals in 72-edo (which is also probably
> Miracle's most common use ?), the generator is
> 72 / 10.2915026221292 = roughly 7 steps, divided 2 + 2 + 1 + 2
> (or steps 2-4-5-7)
> The 11/8 will then be approximated by 4 * 7 + 5 = 33 steps.
> The Blackjack subset goes 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2,
> or steps
> 2-7-9-14-16-21-23-28-30-35-37-42-44-49-51-56-58-63-65-70-72
> It has no 33 steps from 1/1, it has 33 steps only in other places :
> between 2 and 35, 9 and 42, etc.
> The blackjack subset has no regular whole tone (199.2125 c.), nor the
> "6/5" approximation neither (315.8135 c.).
> 31 and 41 subsets have all those.
>
> Just tried your triads suggestions. Both sound strange, but may be I
> need more of the context !!
> It's quite limited to test such a temperament with a 12 keys keyboard
> of course, but still it's open.
> On my first test what striked me were the 7/6, I found them
> terrific ! Nothing new, I found these are quasi-perfectly just.
> Then one scale I was happy with that came out is that kind of weird
> "septimal Shur" :
>
> 1/1
> 150.59050
> 315.81350
> 466.40400
> 699.60600
> 816.20734
> 966.79800
> 2/1
>
> allmost all in the BlackJack subset, except 315.813 - but with
> 349.803 instead it's not uninteresting.
> - - - - - - -
> Jacques
>
>
>
> Cameron wrote :
>
> > Transposed to a symmetrical version (S,L,S,L,... and symmetrical on
> > degree 0) this gives us a "Blackjack" tuning of:
> >
> > 0:......1/1
> > 1:.........33.990 cents
> > 2:.......116.601 cents
> > 3:.......150.591 cents
> > 4:.......233.202 cents
> > 5:.......267.192 cents
> > 6:.......349.803 cents
> > 7:........383.793 cents
> > 8:........466.404 cents
> > 9:........500.394 cents
> > 10:.......583.005 cents
> > 11:........616.995 cents
> > 12:........699.606 cents
> > 13:........733.596 cents
> > 14:........816.207 cents
> > 15:.......850.197 cents
> > 16:.......932.808 cents
> > 17:.......966.798 cents
> > 18:.......1049.409 cents
> > 19:........1083.399 cents
> > 20:........1166.010 cents
> > 21:........1200.000 cents
> >
> > The "7-limit" is clearly excellent, but the approximation to 11/8,
> > which is otherwise also excellent, seems to be disjointed from the
> > keys with 7-limit intervals. More interesting to me in this tuning
> > are things like 0-6-14 and 0-8-15.
>
>
>
> > --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@> wrote:
> >
> >> > This came to me after reading this brillant article of Graham
> >> > mentionned by Carl :
> >> > http://x31eq.com/paradigm.html
> >> > And the explanations it gives about George Secor's system.
> >> >
> >> > It inspired me a fractal form of Miracle's octave division,
> >> based on
> >> > the identities :
> >> > octave/generator = neutral third/small interval (= fifth/2s =
> >> neutral
> >> > 7th/3s) = neutral sixth/(generator less small interval) = (etc.,
> >> > there are certainly more of those),
> >> > = 2 * 7^(1/2) + 5 = 10.2915026221292
> >> >
> >> > what I call here "small interval" or "s" has 33.9895 c.
> >> > the semitone generator 116.601 c.
> >> > septimal tone 233.2 c.
> >> > neutral third 349.803 c .
> >> > the fifth 699.606 c.
> >> > major third 383.79266 c.
> >> > neutral fourth 549.0157 c.
> >> > smaller semitone 82.61154 c.
> >> > in an octave of 1200 c.
> >> >
> >> > This is like what the "Golden scale" is to the meantone, a general
> >> > frame that allows an infinity of
> >> > variations. You will find of course among them the octave of 72
> >> > steps / 7 steps as generator, as well as less accurate 31/3, 41/4
> >> > etc., but also more complex 103/10, 247/24, etc.
> >> > All are found in the recurrent series of x^2 = 10x + 3 :
> >> >
> >> > 1 10 103 1060
> >> > 3 31 319
> >> > 4 41 422
> >> > 7 72 741
> >> > 17 175 1801
> >> > 24 247 2542
> >> >
> >> > etc., that converge rapidly towards the miraculous ratio.
> >> >
> >> > - - - - - - -
> >> > Jacques
>

Cameron wrote :

> Hi Jacques,
>
> I'm certainly not an expert on the Miracle temperament either,
> though there are those here who are. But, I have a number of
> observations from my own experiences.
> In Blackjack 11/8 shows up in hexads rather than in a JI harmonic
> scale kind of way, but not as often as 7/5 appears as an "11th". So
> it is not a "regular" mapping in that way, ...

Hi Cameron,

May be we need some light from temperaments specialists here.
In Miracle has 11/8 a regular mapping, or not ? Graham's article
indicates it is :
[< 1, 1, 3, 3, 2],
< 0, 6, -7, -2, 15]>
What I am guessing is that Miracle originally was perhaps made to temper 225/224, 2401/2400 and other 7-limit commas,
but produces indirectly a very correct 11 factor as well, which can
therefore be added to the "main factors" 3, 5, and 7.
It does so because of the proximity of 11/8 to 48/35, which is the 7-
limit path used to arrive to it.
11/8 is higher than 48/35 by 385/384, so the question is : does
Miracle tempers also 385/384 ?
I would think that from any given tempering of 2, 3, 5, 7 we can
define a potential tempering of 11 that dissolves 385/384.
So the tempering of 11/8 should be a property of Miracle.

(Cameron) :
> ..., or as regards to the minor thirds, which are mostly near-
> perfect 7/6s, and eight 6/5s mixed in. And consequently the 5/3s
> are scattered... basically, I see the Blackjack "scale" as an
> ingenious way of packing of as many near-perfect low-limit JI
> intervals as possible into a small space.

(Jacques) :
If we retrace the intervals covered by a cycle of the generator
(asymmetrically), we have :
0… 1/1
1… 16/15
2… 8/7
3… 128/105 (... or 11/9)
4… 21/16 (or 64/49)
5… 7/5 (or 45/32)
6… 3/2
7… 8/5
8… 12/7
9… 64/35 (... or 11/6)
10… 63/32 (or 96/49)
11… 21/20 (or 135/128)
12… 9/8
13… 6/5
14… 9/7 (or 32/25)
15… 48/35 (... or 11/8)
16… 189/128 (or 72/49)
17… 63/40 (or 11/7)
18… 27/16
19… 9/5
20… 27/14
21… 36/35 (or 33/32)
...

It shows that a long way before 11/8 appears, 11/9 is the first 11-
limit interval to show up, and 11/8 only appears after the issue of
9/8 -
(the order of resolution of the first harmonics or their products up
to 32 is : (2) - 15 - 7 - 21 - 3 - 5 - 9 - 25 - 11 - 27)
7/5 arrives much before 11/8, and 7/6 arrives sooner than 6/5, this
confirms your remarks with Blackjack.
Blackjack (20 times the generator, 21 tones) is certainly a nice
subset, but is limited by a lack of "9/8" intervals (for example it's
missing "9/8" over 1/1 in the symmetrical version) and in
consequence, also a lack of "6/5" and "11/8".
The 31 tones version of Miracle is both more complete and more
balanced. But every DE of Miracle (...10, 11, 21, 31, 41, 72) has its
interest and for the harmonic relations it weaves, it is a fantastic
and very open temperament (it's a unconditionnal JI addict that
speaks ! ;-).

(Cameron):
> There is a huge amount to discuss here, it'll take some time (I'm
> "multitasking" at the moment:-)) For example I have a spectral
> image clearly illustrating some very interesting things, present in
> you "Shur" scale as well as well as many other Phi tunings- and
> Blackjack- which you might dig.
> Looking forward to hearing your new photosonic disc!

(Jacques):
Great ! I'll send you the Meta-Miracle fractal waveform disk as soon
as I transfer it from my photosonic computer, and scan it in a
light .jpg version,
and later on, some sounds from it.
Of course these sounds will not relate to a Miracle scale, but to the
10.2915026221292 ratio, just like a logarithmic Kornerup/Golden scale
used as a waveform would sound as any fractal Phi disk using the
1.618 ratio.
However, because Miracle signs a strong 72/7 ratio, this is a 9/7 at
the third octave and belongs by coïncidence to the JI version of the
temperament.

(Cameron):
> By the way, in English and Slovene, and I believe many other
> languages, the expression for the larger-within-smaller... is mise
> en abîme, directly from the French. Or recursion in more scientific
> or "scientific" contexts. "Turtles all the way down" is another
> related expression, from the probably bogus, but funny, anectode of
> the Indian sage explaining cosmology.
> The two-mirrors version is my argument for the idea that the
> logical fallacy of infinite regress may not be a fallacy after all.
>
> -Cameron Bobro

(Jacques):
I appreciate these semantic notes - I like "Turtles all the way
down", I can imagine some kind of Escher illustration, weaving North-Amerindian turtles...
But your double mirrors metaphor of an infinite "mise en abîme" is an
impressive vision. I have to think about it, for my next photosonic
experiments with mirrors ;-)

One more thing, the "basic division" I suggested in my previous post
was only refering in fact to the 41 tones version of the temperament.
To be more complete, it should go all the way to the 72 tones : then
it reveals that the octavial scale that would serve as proportional
reference, wether in symmetrical or asymmetrical developpment, is a 7
tones Mohajira (s L s L s L s), that belongs to all subsets starting
from Blackjack (3 6 9 12 15 18 21) -
- interesting crossing of musical worlds... that shows another very
special relation of Miracle to its semi-fifth "11/9" interval.

(Meta-Miracle 7 tones Mohajira, symmetrical form, in cents) :
1/1
150.59056
349.8031
500.3937
699.6063
850.19685
1049.4094
2/1

Which scaled at the generator level gives (2nd column indicating
minimal DE subset required) :
1/1 1
14.6325 72
33.9895 21
48.6220 72
67.9790 41
82.6115 31
101.9685 72
116.6010 10
(to be repeated over and under 1/1 alternatively to complete each
correspondant subsets - higher subsets including lower ones of course)

OK, let us not be lazy, you gave the Blackjack, here is the 31
version (s = 33.9895 cents, L = 48.6220 cents) :

0… 1/1
1… 33.9895
2… 82.6115
3… 116.601
4… 150.5906
5… 199.2126
6… 233.202
7… 267.1916
8… 315.8136
9… 349.8031
10… 383.7927
11… 432.4147
12… 466.4042
13… 500.3937
14… 549.0157
15… 583.0052
16… 616.9948
17… 650.9843
18… 699.6063
19… 733.5958
20… 767.5853
21… 816.2073
22… 850.1969
23… 884.1864
24… 932.8084
25… 966.7979
26… 1000.7874
27… 1049.4094
28… 1083.399
29… 1117.3885
30… 1166.0105
31… 1200.000

(41, 72, and 741 tones versions only on request !)
- - - - - - -
Jacques

> --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
> >
> > Hi Cameron !
> >
> > First, I am not at all expert of the Miracle temperament ! What I
> did
> > is just a "photosonic disk" of the logarithmic scale in a circle
> (= a
> > horagram), with improved image periodicity through this fractal
> > transformation.
> > All what I know is that it doesn't changes the logic of the
> > structure, and it shows how to find all possible edo subsets
> versions
> > of Miracle temperaments.
> > What I found is that the generator can be seen as an octave at a
> > different scale, just like in an "abyssal image" (don't know if this
> > translates well the french "image en abime", an image that
> contains a
> > picture of the whole image).
> > In fact, in its pure fractal form, it's impossible to say if "the
> > octave" is 1200 cents, or 116.6o105 c. or even 10.2915026221292 *
> > 1200 = 12 349.803 cents !
> > But let's say the octave is 1200 cents, that speaks better (it can
> > also be extended for optimisation without problem).
> >
> > The main basic division to retain of the octave here is :
> > neutral third = 349.803 c.
> > fifth = 699.606 c.
> > neutral sixth (= octave less neutral third) = 850.197 c.
> > octave = 1200 c.
> > And these have the same proportions as 33.9895 c., 67.979 c.,
> 82.6115
> > c., and 116.601 c., that form the basic division of the generator,
> > 116.601 c.
> >
> > The 11/8 approximation has to be one "double-fifth" wholetone + one
> > neutral third :
> > = 199.2125 + 349.803 = 549.0155 c.
> > It is attained by 4 generators + one fraction of a generator, equal
> > to the proportion of the neutral sixth to the octave :
> > (850.197 / 1200) *116.601 = 82.6115 <error of 1 c. in my first
> > message>
> > So "11/8" = 4* 116.6o1 c. + 82.6115 c. = 549.0155 c.
> >
> > If we express the intervals in 72-edo (which is also probably
> > Miracle's most common use ?), the generator is
> > 72 / 10.2915026221292 = roughly 7 steps, divided 2 + 2 + 1 + 2
> > (or steps 2-4-5-7)
> > The 11/8 will then be approximated by 4 * 7 + 5 = 33 steps.
> > The Blackjack subset goes 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2,
> > or steps
> > 2-7-9-14-16-21-23-28-30-35-37-42-44-49-51-56-58-63-65-70-72
> > It has no 33 steps from 1/1, it has 33 steps only in other places :
> > between 2 and 35, 9 and 42, etc.
> > The blackjack subset has no regular whole tone (199.2125 c.), nor> the
> > "6/5" approximation neither (315.8135 c.).
> > 31 and 41 subsets have all those.
> >
> > Just tried your triads suggestions. Both sound strange, but may be I
> > need more of the context !!
> > It's quite limited to test such a temperament with a 12 keys
> keyboard
> > of course, but still it's open.
> > On my first test what striked me were the 7/6, I found them
> > terrific ! Nothing new, I found these are quasi-perfectly just.
> > Then one scale I was happy with that came out is that kind of weird
> > "septimal Shur" :
> >
> > 1/1
> > 150.59050
> > 315.81350
> > 466.40400
> > 699.60600
> > 816.20734
> > 966.79800
> > 2/1
> >
> > allmost all in the BlackJack subset, except 315.813 - but with
> > 349.803 instead it's not uninteresting.
> > - - - - - - -
> > Jacques
> >
> >
> >
> > Cameron wrote :
> >
> > > Transposed to a symmetrical version (S,L,S,L,... and
> symmetrical on
> > > degree 0) this gives us a "Blackjack" tuning of:
> > >
> > > 0:......1/1
> > > 1:.........33.990 cents
> > > 2:.......116.601 cents
> > > 3:.......150.591 cents
> > > 4:.......233.202 cents
> > > 5:.......267.192 cents
> > > 6:.......349.803 cents
> > > 7:........383.793 cents
> > > 8:........466.404 cents
> > > 9:........500.394 cents
> > > 10:.......583.005 cents
> > > 11:........616.995 cents
> > > 12:........699.606 cents
> > > 13:........733.596 cents
> > > 14:........816.207 cents
> > > 15:.......850.197 cents
> > > 16:.......932.808 cents
> > > 17:.......966.798 cents
> > > 18:.......1049.409 cents
> > > 19:........1083.399 cents
> > > 20:........1166.010 cents
> > > 21:........1200.000 cents
> > >
> > > The "7-limit" is clearly excellent, but the approximation to 11/8,
> > > which is otherwise also excellent, seems to be disjointed from the
> > > keys with 7-limit intervals. More interesting to me in this tuning
> > > are things like 0-6-14 and 0-8-15.
> >
> >
> >
> > > --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@> wrote:
> > >
> > >> > This came to me after reading this brillant article of Graham
> > >> > mentionned by Carl :
> > >> > http://x31eq.com/paradigm.html
> > >> > And the explanations it gives about George Secor's system.
> > >> >
> > >> > It inspired me a fractal form of Miracle's octave division,
> > >> based on
> > >> > the identities :
> > >> > octave/generator = neutral third/small interval (= fifth/2s =
> > >> neutral 7th/3s) = neutral sixth/(generator less small
> interval) = (etc.,
> > >> > there are certainly more of those),
> > >> > = 2 * 7^(1/2) + 5 = 10.2915026221292
> > >> >
> > >> > what I call here "small interval" or "s" has 33.9895 c.
> > >> > the semitone generator 116.601 c.
> > >> > septimal tone 233.2 c.
> > >> > neutral third 349.803 c .
> > >> > the fifth 699.606 c.
> > >> > major third 383.79266 c.
> > >> > neutral fourth 549.0157 c.
> > >> > smaller semitone 82.61154 c.
> > >> > in an octave of 1200 c.
> > >> >
> > >> > This is like what the "Golden scale" is to the meantone, a
> general
> > >> > frame that allows an infinity of variations.
> > >> > You will find of course among them the octave of 72
> > >> > steps / 7 steps as generator, as well as less accurate 31/3,
> 41/4
> > >> > etc., but also more complex 103/10, 247/24, etc.
> > >> > All are found in the recurrent series of x^2 = 10x + 3 :
> > >> >
> > >> > 1 10 103 1060
> > >> > 3 31 319
> > >> > 4 41 422
> > >> > 7 72 741
> > >> > 17 175 1801
> > >> > 24 247 2542
> > >> >
> > >> > etc., that converge rapidly towards the miraculous ratio.
> > >> >
> > >> > - - - - - - -
> > >> > Jacques

To Jacques:

Try this:
http://x31eq.com/temper/vectors.html
and enter "385/384, 225/224, 2401/2400, 81/80" into the "Unison vectors" field. This gives you four temperaments, three of which are 2D and the fourth one is equal.

Petr

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Cameron wrote :
>
> > Hi Jacques,
> >
> > I'm certainly not an expert on the Miracle temperament either,
> > though there are those here who are. But, I have a number of
> > observations from my own experiences.
> > In Blackjack 11/8 shows up in hexads rather than in a JI harmonic
> > scale kind of way, but not as often as 7/5 appears as an "11th". So
> > it is not a "regular" mapping in that way, ...
>
> Hi Cameron,
>
> May be we need some light from temperaments specialists here.
> In Miracle has 11/8 a regular mapping, or not ? Graham's article
> indicates it is :
> [< 1, 1, 3, 3, 2],
> < 0, 6, -7, -2, 15]>
> What I am guessing is that Miracle originally was perhaps made to
> temper 225/224, 2401/2400 and other 7-limit commas,
> but produces indirectly a very correct 11 factor as well, which can
> therefore be added to the "main factors" 3, 5, and 7.
> It does so because of the proximity of 11/8 to 48/35, which is the 7-
> limit path used to arrive to it.
> 11/8 is higher than 48/35 by 385/384, so the question is : does
> Miracle tempers also 385/384 ?
> I would think that from any given tempering of 2, 3, 5, 7 we can
> define a potential tempering of 11 that dissolves 385/384.
> So the tempering of 11/8 should be a property of Miracle.

Hi Jacques,

Yes, the Miracle temperament was recognized as 11-limit from the very start. I've documented both its original discovery (1974) and its rediscovery (2001) in this paper (2006, Xenharmonikon 18):
http://www.anaphoria.com/SecorMiracle.pdf

--George

Jacques Dudon wrote:

> May be we need some light from temperaments specialists here.
> In Miracle has 11/8 a regular mapping, or not ? Graham's article > indicates it is :
> [< 1, 1, 3, 3, 2],
> < 0, 6, -7, -2, 15]>
> What I am guessing is that Miracle originally was perhaps made to temper > 225/224, 2401/2400 and other 7-limit commas,
> but produces indirectly a very correct 11 factor as well, which can > therefore be added to the "main factors" 3, 5, and 7.
> It does so because of the proximity of 11/8 to 48/35, which is the > 7-limit path used to arrive to it.
> 11/8 is higher than 48/35 by 385/384, so the question is : does Miracle > tempers also 385/384 ?

Yes, and you can check this by looking at the mapping. You can represent 385/384 as [-7, -1, 1, 1, 1> (2^-7 * 3^-1 * 5^1 * 7^1 * 11^1). If you multiply each element of <1, 1, 3, 3, 2] by the corresponding element of [-7, -1, 1, 1, 1> and add the results, you get 0; you get the same result with <0, 6, -7, -2, 15]. In other words, the tempered equivalent of 385/384 is [0, 0> : 0 periods + 0 generators. So, no matter what values you use for the size of the generators, the result is 0: the interval is tempered out.

This is also, in general, how you can find the tempered equivalent of any rational interval. Take 7/4 for instance: represent it as [-2, 0, 0, 1, 0>, and do the multiplication (you can do this in a spreadsheet as a matrix multiplication), which gives you a result of [1, -2>. So in miracle temperament, 7/4 is one octave up, 2 secors down.

|1 1 3 3 2| |-2| | 1|
|0 6 -7 -2 15| x | 0| = |-2|
| 0|
| 1|
| 0|

Waaoh ! Thanks Petr (and Graham - ) !,
I did not know such tools were existing.

Is there a reason why do you suggest to enter 81/80 here (It's not tempered by Miracle, isn'it ?)
But it works also with 1029/1024 instead.
What I do not understand is what means "1/9" in the result (1/9, 116,7 c.),
and what's the meaning, following the commas I entered, of the 3 other "unison vectors"
243:242
1029:1024
1375:1372

And can anyone explain me what's the difference between "linear temperament" and "regular temperament" ?
- - - - - - -
Jacques

Petr wrote :

> To Jacques:
>
> Try this:
> http://x31eq.com/temper/vectors.html
> and enter "385/384, 225/224, 2401/2400, 81/80" into the "Unison > vectors" field. This gives you four temperaments, three of which > are 2D and the fourth one is equal.
>
> Petr

Jacques wrote:

> Is there a reason why do you suggest to enter 81/80 here
> (It's not tempered by Miracle, isn'it ?)

To show how different some temperaments can be even though they use similar commas. While 11-limit miracle tempers out 385/384, 225/224, and 2401/2400 (or 1029/1024), 11-limit meantone tempers out 385/384, 225/224, and 81/80. So there are actually two intervals in common here which are tempered out but the resulting temperaments don't sound like the same at all.

> What I do not understand is what means "1/9" in the result (1/9, 116,7 c.),

I'm not sure myself, perhaps an approximation to something (Graham?).

> and what's the meaning, following the commas I entered, of the 3 other "unison vectors"

These are some other intervals that are tempered out but I'm not sure how the program finds them.

> And can anyone explain me what's the difference
> between "linear temperament" and "regular temperament" ?

If I'm not mistaken, a regular temperament can be any 2-dimmensional temperament (including the 2D version of Bohlen-Pierce which uses a period of 3/1 and a generator of ~440 cents), while a linear temperament is usually a regular temperament which has a period of 2/1 or close to that (i.e. things like meantone, schismatic, hanson or miracle).

Petr

Let me ask a question spurred by this Miracle / 31 EDo discussion:

So, with 31 edo it would be possible to approximate adaptive JI in a series
of approximated mean tone tunings?

Thanks,

Chris

Hi George,

I have been reading (and will explore again) this article with deep interest.
I am glad to know that the 31 and 41 tones subsets of Miracle have names (Canasta and StudLoco), and I shall use them from now on.
Also I hope you don't mind if I call the (2 * 7^(1/2) +5)^-1 octave or 116.60105 cents, the Meta-Miracle generator , a meta-secor ? ;-)
Glad to see in your graph it was bad at all.

I was impressed by your decimal keyboard. Has it been manufactured, and how many people have one ?
One detail catched my attention. You named several keys such as 17/13, 20/19, 77/51, 14/13, 13/10, 20/17 etc. from approximation of "higher-than-11" -limit ratios, certainly because being simpler and closer to the intervals generated.
I would try to understand something :
If I take for example "20/17" (23 secors), which could be mapped also as "33/28" or "88/75" in 11-limit, am I correct if I say then that Miracle also dissolves their ratios with 20/17, which are 561/560 and 375/374 ?
And same with 273/272 and 833/832 for the "17/13" approximation (as equivalent to 21/16 and 64/49) ?

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

George Secor wrote :

> Hi Jacques,
>
> Yes, the Miracle temperament was recognized as 11-limit from the > very start. I've documented both its original discovery (1974) and > its rediscovery (2001) in this paper (2006, Xenharmonikon 18):
> http://www.anaphoria.com/SecorMiracle.pdf
>
> --George

Hi Herman,

Thanks for the answer !
Except that for the demonstration I am lacking the necessary maths to understand your third line :

> If you multiply each element of <1, 1, 3, 3, 2] by the > corresponding element of
> [-7, -1, 1, 1, 1> and add the results, you get 0;

and also in the end what is meant by "multiplication" :

> Take 7/4 for instance: represent it as [-2, 0, 0, 1, 0>,
> and do the multiplication (you can do this in a spreadsheet as a
> matrix multiplication), which gives you a result of [1, -2>.
> So in miracle temperament, 7/4 is one octave up, 2 secors down.
>
> |1 1 3 3 2| |-2| | 1|
> |0 6 -7 -2 15| x | 0| = |-2|
> | 0|
> | 1|
> | 0|

But thanks for the effort, I've got the idea, and I am sure others understand.
- - - - - - -
Jacques

Herman Miller wrote :

> Jacques Dudon wrote:
>
> > May be we need some light from temperaments specialists here.
> > In Miracle has 11/8 a regular mapping, or not ? Graham's article
> > indicates it is :
> > [< 1, 1, 3, 3, 2],
> > < 0, 6, -7, -2, 15]>
> > What I am guessing is that Miracle originally was perhaps made to > temper
> > 225/224, 2401/2400 and other 7-limit commas,
> > but produces indirectly a very correct 11 factor as well, which can
> > therefore be added to the "main factors" 3, 5, and 7.
> > It does so because of the proximity of 11/8 to 48/35, which is the
> > 7-limit path used to arrive to it.
> > 11/8 is higher than 48/35 by 385/384, so the question is : does > Miracle
> > tempers also 385/384 ?
>
> Yes, and you can check this by looking at the mapping. You can > represent
> 385/384 as [-7, -1, 1, 1, 1> (2^-7 * 3^-1 * 5^1 * 7^1 * 11^1). If you
> multiply each element of <1, 1, 3, 3, 2] by the corresponding > element of
> [-7, -1, 1, 1, 1> and add the results, you get 0; you get the same
> result with <0, 6, -7, -2, 15]. In other words, the tempered > equivalent
> of 385/384 is [0, 0> : 0 periods + 0 generators. So, no matter what
> values you use for the size of the generators, the result is 0: the
> interval is tempered out.
>
> This is also, in general, how you can find the tempered equivalent of
> any rational interval. Take 7/4 for instance: represent it as [-2, > 0, 0,
> 1, 0>, and do the multiplication (you can do this in a spreadsheet > as a
> matrix multiplication), which gives you a result of [1, -2>. So in
> miracle temperament, 7/4 is one octave up, 2 secors down.
>
> |1 1 3 3 2| |-2| | 1|
> |0 6 -7 -2 15| x | 0| = |-2|
> | 0|
> | 1|
> | 0|
>

Chris wrote:
> So, with 31 edo it would be possible to approximate adaptive JI in a series of approximated mean tone tunings?
I'm not sure what you mean.
Petr

Hi Petr,

It seems to me that one of the draw backs of adaptive JI is the complexity.

From what I gather in this conversation 31 EDO can come close to
aproximating a number of mean tone tunings - and mean tone tunings all
imitate JI but in a limited fashion

So....

A draw back of mean tone tunings is that modulation to far keys are messy or
unusable. BUT if one changed the mean tone tuning on the fly like adaptive
JI then modulations far keys could be possible.

My question is then, in this context, can 31 EDO give a workable but less
complex, means to implement a good approximation to adaptive JI by using a
series of mean tone tuning approximations?

I hope that is clearer.

Best regards,

Chris

On Thu, Feb 11, 2010 at 11:36 AM, Petr Parízek <p.parizek@...> wrote:

>
>
> Chris wrote:
> > So, with 31 edo it would be possible to approximate adaptive JI in a
> series of approximated mean tone tunings?
> I'm not sure what you mean.
> Petr
>
>
>

Jacques wrote:
> Also I hope you don't mind if I call the (2 * 7^(1/2) +5)^-1 octave
> or 116.60105 cents, the Meta-Miracle generator , a meta-secor ? ;-)
#1. May I ask how you got this number?
#2. Didn't George himself suggest an optimal generator to be the 19th root of 18/5?
Petr

Chris wrote:

> From what I gather in this conversation 31 EDO can come close to aproximating
> a number of mean tone tunings - and mean tone tunings all imitate JI
> but in a limited fashion

Depends on what you mean by "a number of meantones". In my oppinion, 31-EDO can be used as one particular version of meantone which has slightly wider fifths than quarter-comma meantone.

> A draw back of mean tone tunings is that modulation to far keys are messy or unusable.
Of course, if you have only 12 tones in an octave, then modulations are pretty limited in meantone. The usual historical choice was 2 flats to 3 sharps (i.e. from Eb to G#). But in some cases, there were variants like 1 flat to 4 sharps, or 3 flats to 2 sharps, or no flats and 5 sharps. And in other cases, there were instruments with split keys, which made it possible to have 17 or 19 tones in an octave. And in other cases, there were two 12-tone manuals with each manual managing a different part of the chain of fifths, making it possible to have as much as 24 tones in the scale (although I'm not sure how easy or difficult it was to play the instruments).

> BUT if one changed the mean tone tuning on the fly like adaptive JI
> then modulations far keys could be possible.

I'm not sure if each of us isn't incidentally meaning something else when saying "adaptive JI". I would rather say that if you could somehow alternate between tuning D# and Eb even with a conventional 12-tone keyboard, you could play more distant modulations than with only D# or only Eb.

> My question is then, in this context, can 31 EDO give a workable but less complex,
> means to implement a good approximation to adaptive JI by using
> a series of mean tone tuning approximations?

#1. Please, what's meant by "a series of meantone tuning approximations"?

#2. I think I should say a few more words about adaptive JI to make sure both of us mean the same thing. For example, let's say we want to play the harmonic progression of "C major, A minor, D minor, G major, C major" in pure 5-limit JI (i.e. major thirds of 5/4 and minor thirds of 6/5) and we want both the first and last chord to have the same pitch, although the latter C major would originally (i.e. in strict JI) be a syntonic comma lower if the common tones of consecutive chords are preserved. Actually, we can do this in two ways. One possibility is to raise one pitch by a syntonic comma in one of the chords . For example, you can use one D in the D minor chord and a slightly higher D in the G major chord. So in fact, by doing this, we're raising the pitch of G major and C major by a syntonic comma, when compared to the "original" JI version. But there's one more possibility. Instead of making one great pitch shift from one chord to another by a full comma, we can do four progressive quarter-comma pitch shifts within the entire chord progression. This is what adaptive JI is about. If I'm not mistaken, its greatest promoter was the Italian Nicola Vicentino (in the later half of the 16th century) who devised an adaptive tuning by stacking two meantone chains tuned ~5.5 cents apart. Details here:
http://sonic-arts.org/td/schulter/vicentino.htm

Petr

Hi Chris & Petr,

Excuse the interjection, maybe I can help.

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
> It seems to me that one of the draw backs of adaptive JI is the
> complexity.
>
> From what I gather in this conversation 31 EDO can come close
> to aproximating a number of mean tone tunings - and mean tone
> tunings all imitate JI but in a limited fashion

31-ET is what happens if you extend the chain of historical
quarter-comma meantone fifths out to 31 fifths (as noted in
the 16th century by Vicentino and 17th century by Huygens).

The whole point of adaptive JI is to distribute a comma so
it's less noticeable. So if the score we're performing assumes
that the syntonic comma vanishes, we need steps smaller than
a syntonic comma. A step of 31 is ~ 39 cents, which is larger
than the syntonic comma, so no, 31 can't be used for that kind
of adaptive JI. This is why adaptive JI has the drawback of
complexity - there's no getting around that you need lots
of notes.

> A draw back of mean tone tunings is that modulation to far keys
> are messy or unusable.

That's only true for the historical 12-tone subsets of
meantone. If you're using 31-ET, you can modulate all you
want.

-Carl

Thanks Carl,

I think understand what you are saying.

I had the impression that 31 edo had some unique advantages here. Such as 19
EDO being usable for performing 12 edo music.
So, by saying I can modulate all I want in 31 ET you are saying -
modulations within the context of 31 ET.
Just like one can modulate in the context of any edo according to its own
rules [probably similar to 12 edo rules] (in theory)

(Petr, I'm still very new to tuning though I'm trying => I need to study
your response a bit.)

Thanks,

Chris

On Thu, Feb 11, 2010 at 1:36 PM, Carl Lumma <carl@...> wrote:

>
>
> Hi Chris & Petr,
>
> Excuse the interjection, maybe I can help.
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
> > It seems to me that one of the draw backs of adaptive JI is the
> > complexity.
> >
> > From what I gather in this conversation 31 EDO can come close
> > to aproximating a number of mean tone tunings - and mean tone
> > tunings all imitate JI but in a limited fashion
>
> 31-ET is what happens if you extend the chain of historical
> quarter-comma meantone fifths out to 31 fifths (as noted in
> the 16th century by Vicentino and 17th century by Huygens).
>
> The whole point of adaptive JI is to distribute a comma so
> it's less noticeable. So if the score we're performing assumes
> that the syntonic comma vanishes, we need steps smaller than
> a syntonic comma. A step of 31 is ~ 39 cents, which is larger
> than the syntonic comma, so no, 31 can't be used for that kind
> of adaptive JI. This is why adaptive JI has the drawback of
> complexity - there's no getting around that you need lots
> of notes.
>
>
> > A draw back of mean tone tunings is that modulation to far keys
> > are messy or unusable.
>
> That's only true for the historical 12-tone subsets of
> meantone. If you're using 31-ET, you can modulate all you
> want.
>
> -Carl
>
>
>

Carl>"The whole point of adaptive JI is to distribute a comma so it's less noticeable. So if the score we're performing assumes
that the syntonic comma vanishes, we need steps smaller than a syntonic comma."

Novice question...does the tempering out in adaptive JI involve use an area near one note between two notes IE B and B+ yet somehow also involve putting the tempered note in slightly different places leaning a bit toward the just interval it is supposed to emulate? It seems to me like a contradiction of terms on the surface as I was under the impression tempering out of the comma was designed to use one tone to represent two enharmonic notes.
Another one...what's the difference between tempering out of a comma and the concept of temperament in general?

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Thanks Carl,
>
> I think understand what you are saying.
>
> I had the impression that 31 edo had some unique advantages
> here. Such as 19 EDO being usable for performing 12 edo music.
> So, by saying I can modulate all I want in 31 ET you are
> saying - modulations within the context of 31 ET.
> Just like one can modulate in the context of any edo according
> to its own rules [probably similar to 12 edo rules] (in theory)

19 can't work for all 12-ET music. If you modulate by min3rds
or through dim7 chords you're gonna get a different result
in 19. The commonality is that they are both meantones. So
early classical music, for example, only assumes the syntonic
comma vanishes (it doesn't use dim7-based modulations) so 19'll
work. When you get to the Romantic period, bets are off.

The same is true for 31. It's a meantone, but again it doesn't
dispose of the diminished comma. The general rule for this, by
the way, is that the ET will be divisible by 4.

But any ET lets you modulate anything playable in it to any
note of itself (but you knew that). The problem with modulating
in historical meantone comes from the fact that fifths of the
size that give you pure 3rds require 31 tones/oct to equalize,
but people only had keyboards with 12 notes/oct.

-Carl

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Carl>"The whole point of adaptive JI is to distribute a comma so
> it's less noticeable. So if the score we're performing assumes
> that the syntonic comma vanishes, we need steps smaller than a
> syntonic comma."
>
> Novice question...does the tempering out in adaptive JI
> involve use an area near one note between two notes IE B and B+
> yet somehow also involve putting the tempered note in slightly
> different places leaning a bit toward the just interval it is
> supposed to emulate?

Hi Michael. I'm afraid I can't parse this question.
Normally I'd ask you to clarify, but in the past, I've learned
that doing so can lead to long threads that don't go anywhere.
So, without meaning any disrespect, I'm writing to encourage
you to think about this. I used to think it was just something
between the two of us, but now I see it's happening with others.
I don't know the cause, but I have two pieces of advice:
1. After you're done writing your post, but before sending it,
edit it down to half its original length.
2. When asking questions, use examples. I've suggested this
before. Above you gave an example of B and B+. But you should
flesh this out more if possible. In fact, when asking questions
about adaptive JI, you should probably try adaptively tuning a
short fragment of music on paper and pen. Then your questions
will come naturally, and answers will produce practical results.

By the way, there's no reason to be a novice. You've been on
these lists a while now, and have obviously spent considerable
time at it. I know adaptive JI hasn't been your focus, but
certainly there are plenty of introductory materials on the
subject, such as this page

http://tonalsoft.com/enc/a/adaptive-ji.aspx

or this one

http://www.hermode.com

> It seems to me like a contradiction of terms on the surface as
> I was under the impression tempering out of the comma was
> designed to use one tone to represent two enharmonic notes.

That's correct.

> Another one...what's the difference between tempering out
> of a comma and the concept of temperament in general?

Again, hard to answer something about vague things like the
"concept of temperament". The concept is to increase the number
of consonances in a collection of tones. That's done by
tempering out commas. This is very clearly explained in
Paul Erlich's paper, The Forms of Tonality

http://lumma.org/tuning/erlich/erlich-tFoT.pdf

For the record, I would usually expect replies to this post
to contain specific questions about the materials cited.

-Carl

Carl said>1. After you're done writing your post, but before sending it,
edit it down to half its original length.

Responded to
> Novice question...does the tempering out in adaptive JI involve use an area near one note between two notes IE B and B+
> yet somehow also involve putting the tempered note in slightly different places leaning a bit toward the just interval it is
> supposed to emulate?

Carl, that was in fact shortened/edited; how perfect does my effort here have to be?
List etiquette: does anyone else in fact agree with Carl the above question is indeed phrased in a way that's too long?!

"While retuning motions and/or drifts in strict JI renditions are one or
more commas in magnitude, adaptive JI reduces these to fractions of a
comma by spreading the motion among all the chord changes in a
progression."-http://tonalsoft.com/enc/a/adaptive- ji.aspx
Fair enough, your link provided a good answer.

"4) Adaptive JI: Temper the melodic occurrences of some of the consonant
intervals by a fraction of a syntonic comma but keep the harmonies pure.
(Disadvantage –- two slightly different pitches are needed for each
scale note.)" -http://lumma.org/tuning/erlich/erlich-tFoT.pdf

This, from your Erlich link of http://lumma.org/tuning/erlich/erlich-tFoT.pdf, explains further how melodic tones "drift" within a very close distance to the target "purified" intervals (with one of the two notes it drifts between creating the pure interval).

>"For the record, I would usually expect replies to this post to contain specific questions about the materials cited."
It did branch, but it branched off on a statement you made about how temperament works. Nonetheless, I changed the post topic so it is not a "branch" on the topic.

-Michael

Well, it's not that clear, anyway. Could you have come up with a numerical example illustrating your point? I believe he suggested at least doing that, if not using it in the post. Plus it would have been easier to answer!

-----Original Message-----
From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of Michael
Sent: Thursday, February 11, 2010 3:55 PM
To: tuning@yahoogroups.com
Subject: [tuning] Drift in adaptive JI and how adaptive JI handles temperament

Carl said>1. After you're done writing your post, but before sending it, edit it down to half its original length.

Responded to
> Novice question...does the tempering out in adaptive JI involve use an
> area near one note between two notes IE B and B+ yet somehow also
> involve putting the tempered note in slightly different places leaning a bit toward the just interval it is supposed to emulate?

Carl, that was in fact shortened/edited; how perfect does my effort here have to be?
List etiquette: does anyone else in fact agree with Carl the above question is indeed phrased in a way that's too long?!

"While retuning motions and/or drifts in strict JI renditions are one or more commas in magnitude, adaptive JI reduces these to fractions of a comma by spreading the motion among all the chord changes in a progression."-http://tonalsoft.com/enc/a/adaptive- ji.aspx <http://tonalsoft.com/enc/a/adaptive-ji.aspx>
Fair enough, your link provided a good answer.

"4) Adaptive JI: Temper the melodic occurrences of some of the consonant intervals by a fraction of a syntonic comma but keep the harmonies pure.
(Disadvantage –- two slightly different pitches are needed for each scale note.)" -http://lumma.org/tuning/erlich/erlich-tFoT.pdf <http://lumma.org/tuning/erlich/erlich-tFoT.pdf>

This, from your Erlich link of http://lumma.org/tuning/erlich/erlich-tFoT.pdf <http://lumma.org/tuning/erlich/erlich-tFoT.pdf> , explains further how melodic tones "drift" within a very close distance to the target "purified" intervals (with one of the two notes it drifts between creating the pure interval).

>"For the record, I would usually expect replies to this post to contain specific questions about the materials cited."
It did branch, but it branched off on a statement you made about how temperament works. Nonetheless, I changed the post topic so it is not a "branch" on the topic.

-Michael

possible re-write, for parsing.

Does the 'tempering out' in Adaptive JI involve using an area near one note, between two notes--i.e. B and B+--but also involve putting the tempered note in slightly different places, approximating the Just interval it is supposed to come close to?

But, having parsed it, I still don't understand it, partly because it's not clear, and partly because I don't have the technical knowledge to answer.

On Feb 11, 2010, at 4:59 PM, Cornell III, Howard M wrote:

> > Novice question...does the tempering out in adaptive JI involve > use an
> > area near one note between two notes IE B and B+ yet somehow also
> > involve putting the tempered note in slightly different places > leaning a bit toward the just interval it is supposed to emulate?

>"Well, it's not that clear, anyway. Could you have come up with a numerical example illustrating your point?"
Ok,
Would tones in a melody played in adaptive JI over a chord where 5/4 provides just harmony and where a 6/5 provides just harmony drift as such: 5/4 -> 99/80 -> 97/80 -> 6/5 with the 99/80 "leaning" closely to the 5/4 and 97/80 "leaning" closely to the 6/5? Or would it drift more like 5/4 -> 98/80 -> 6/5, where the tempered switching melodic tone would be dead in the middle of the just tones. I agree with giving a numerical example for clarity, but still can't find out a way to explain my question in a shorter manner.

Caleb>"Does the 'tempering out' in Adaptive JI involve using an area near one
note, between two notes--i.e. B and B+--but also involve putting the
tempered note in slightly different places, approximating the Just
interval it is supposed to come close to?"
Odd you say you don't understand it, because that's essentially what I'm asking. :-)

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

> George Secor wrote :
>
> > Hi Jacques,
> >
> > Yes, the Miracle temperament was recognized as 11-limit from the
> > very start. I've documented both its original discovery (1974) and
> > its rediscovery (2001) in this paper (2006, Xenharmonikon 18):
> > http://www.anaphoria.com/SecorMiracle.pdf
> >
> > --George
>
> Hi George,
>
> I have been reading (and will explore again) this article with deep interest.
> I am glad to know that the 31 and 41 tones subsets of Miracle have
> names (Canasta and StudLoco), and I shall use them from now on.
> Also I hope you don't mind if I call the (2 * 7^(1/2) +5)^-1 octave
> or 116.60105 cents, the Meta-Miracle generator , a meta-secor ? ;-)

If you see some particular advantage in using that interval as a generator and need a convenient name, then you may call it whatever you like. After all, I had no part in the naming of the temperament (Miracle), its MOS scales (Blackjack, etc.), or its generator (the "secor" -- the term can be used for any generator in the range of ~116.6 to ~116.7 cents, or roughly midway between 15:16 and 15:14).

> Glad to see in your graph it was bad at all.

I think you meant to say that it was not bad at all, and I agree. The graph indicates that primes 3, 5, 7, and 11 have almost the same error, so that the interval between any two of these (5:6, 5:7, 6:7, 6:11, etc.) will have a very small error. Another desirable consequence is that the remaining tempered 11-limit consonances (2:3, 4:7, 7:9, etc.) will have beat rates approximating simple proportions (or nearly synchronous beating) when they occur together in a chord, e.g., 4:5, 4:7, 5:9, and 7:9 in a 4:5:7:9 tetrad.

> I was impressed by your decimal keyboard. Has it been manufactured,
> and how many people have one ?

No, it has not been manufactured, so you (or anyone else reading this) may still have the distinction of owning the very first one. :-)

> One detail catched my attention. You named several keys such as
> 17/13, 20/19, 77/51, 14/13, 13/10, 20/17 etc. from approximation of
> "higher-than-11" -limit ratios, certainly because being simpler and
> closer to the intervals generated.
> I would try to understand something :
> If I take for example "20/17" (23 secors), which could be mapped also
> as "33/28" or "88/75" in 11-limit, am I correct if I say then that
> Miracle also dissolves their ratios with 20/17, which are 561/560 and
> 375/374 ?
> And same with 273/272 and 833/832 for the "17/13" approximation (as
> equivalent to 21/16 and 64/49) ?

There is a scale of generators to the left that line up with the cross (or +) markings in the center of each key, so you can see how many generators (or secors) distant each ratio is from the origin, 1/1. Since we are assuming octave equivalence, all powers of 2 are 0G (zero generators). Each prime then maps to the following number of generators:
3/2 or 3: +6G
5/4 or 5: -7G
7/4 or 7: -2G
11/8 or 11: +15G
13/8 or 13: -34G
17/16 or 17: -30G
19/16 or 19: -18G

If you factor any ratio, such as 20/17 into 2*2*5/17, you can then add the number of generators corresponding to each factor (subtracting for factors in the denominator), 0+0-7+30=23G to get the number of secor generators for that ratio. You can then test 33/28 (+6+15-0+2=23G) and 88/75 (+15-6+7+7=23G) to demonstrate that they are equivalent in 19-limit Miracle. The mapping for prime 13 involves a choice between +34G and -38G, and I chose the former (you may disagree).

--George

Jacques Dudon wrote:
> Hi Herman,
> > Thanks for the answer !
> Except that for the demonstration I am lacking the necessary maths to > understand your third line :
> >> If you multiply each element of <1, 1, 3, 3, 2] by the corresponding >> element of
>> [-7, -1, 1, 1, 1> and add the results, you get 0;

Just that (1 * -7) + (1 * -1) + (3 * 1) + (3 * 1) + (2 * 1) = 0.

> and also in the end what is meant by "multiplication" :
> >> Take 7/4 for instance: represent it as [-2, 0, 0, 1, 0>,
>> and do the multiplication (you can do this in a spreadsheet as a
>> matrix multiplication), which gives you a result of [1, -2>.
>> So in miracle temperament, 7/4 is one octave up, 2 secors down.
>>
>> |1 1 3 3 2| |-2| | 1|
>> |0 6 -7 -2 15| x | 0| = |-2|
>> | 0|
>> | 1|
>> | 0|

The column layout unfortunately got messed up in this example. Yahoo is unreliable in that respect. In any case, it's supposed to be a representation of a 2x5 matrix multiplied by a 5x1 matrix, but there isn't a way of representing long square brackets in an email. Matrix multiplication is one of the basic operations in linear algebra, but the important thing for tuning purposes is that spreadsheets such as OpenOffice.Org Calc and Excel have built-in functions for doing the calculation, which is convenient if you want to work with temperaments (it saves the trouble of doing all the multiplications and additions by hand).

So for this particular example, you might enter the numbers 1, 1, 3, 3, 2 in cells A1 to E1, and the numbers 0, 6, -7, -2, 15 in the cells A2 to E2. The rectangle of cells from A1 to E2 is a 2x5 matrix that represents a typical generator mapping of miracle temperament. Then, you could enter the numbers -2, 0, 0, 1, 0 in the cells A4 to A8 (it doesn't matter exactly where, as long as they're next to each other in the same column). This is a column vector that represents the exponents of the prime factors of the ratio 7/4 (2^-2 * 7^1).

How you specify a matrix multiplication differs from one spreadsheet to another, but in OpenOffice.org Calc you can select two vertically adjacent cells for the result (say, G1 and G2), type

=MMULT(A1:E2;A4:A8)

into the input line (without pressing Enter), and then hold Shift+Ctrl down while pressing Enter. The result tells you how many of each generator to use if you want the tempered equivalent of the ratio 7/4. Once you've set this up, you can use the same spreadsheet for any ratio and any 7-limit rank 2 temperament (and this same general method works for any regular temperament).

Petr wrote :

> Jacques wrote:
>
> > What I do not understand is what means "1/9" in the result (1/9, > 116,7 c.),
>
> I'm not sure myself, perhaps an approximation to something (Graham?).

From other requests it looks like a rough approximation of the generator/period ratio.

> > and what's the meaning, following the commas I entered, of the 3 > other "unison vectors"
>
> These are some other intervals that are tempered out but I'm not > sure how the program finds them.

Also I tried Graham's "linear temperament from two ETs" tool, by entering :
first ET 31, second ET 103, odd limit 19
here's what I get:

13/134, 116.6 cent generator...

basis:
(1.0, 0.097165178105073524)

mapping by period and generator:
[(1, 0), (1, 6), (3, -7), (3, -2), (2, 15), (7, -34), (7, -30), (6, -18)]

mapping by steps:
[(103, 31), (163, 49), (239, 72), (289, 87), (356, 107), (381, 115), (421, 127), (438, 132)]

highest interval width: 49
complexity measure: 49 (72 for smallest MOS)
I entered 31 because of Canasta, and 103 because of the series 1 10 103 1060 - I wanted to test a "Meta-Miracle" edo different from 72 edo...
I think it answers my questions to George about how Miracle tempers 13 and 17, and even 19)

> > And can anyone explain me what's the difference
> > between "linear temperament" and "regular temperament" ?
>
> If I'm not mistaken, a regular temperament can be any 2-> dimmensional temperament (including the 2D version of Bohlen-Pierce > which uses a period of 3/1 and a generator of ~440 cents), while a > linear temperament is usually a regular temperament which has a > period of 2/1 or close to that (i.e. things like meantone, > schismatic, hanson or miracle).
>
> Petr

Got it,
thanks !
- - - - - - -
Jacques

Petr wrote :

> Jacques wrote:
> > Also I hope you don't mind if I call the (2 * 7^(1/2) +5)^-1 octave
> > or 116.60105 cents, the Meta-Miracle generator , a meta-secor ? ;-)
>
> (Petr #1) : May I ask how you got this number?

Good question, except that it will not be easy to answer in a few lines ! Just ask if something is not clear. There is also more to it in the exchanges I had with Cameron.

Two square root of 7 + 5, or 10.2915026221292, approximatively 247/24, is the octave/generator ratio (in interval sizes) that divides endlessy the octave and generator and other generated intervals exactly by the same size proportions.
Just like in the Kornerup/Golden scale the (15 - square root of 5) / 22 used as a fifth/octave proportion generates 5, 7, 12, 19, 31, 50... subsets of a meantone where all intervals are divided endlessly by new ones in the golden mean proportion, so does this octave/generator ratio for the Miracle temperament.
In the Golden scale, a cycle of a fifth (or a fourth) provides divisions of the minor 9th, minor 6th, fourth, minor third, tone, semitone, lagu, diesis, and ad libitum always in the same (square root of 5 - 1) / 2 = 0.618 proportion.
In the Meta-Miracle temperament it's even simpler : the reiteration of the "meta"-secor divides the period, and the generator themselves, in the same size proportions.

More generally, if I try to define what I mean here by a "Meta-temperament", it is : a regular temperament whose generator's cycle divides endlessly the intervals of the scale in the same size proportions
(it would perhaps need more lines to be expressed in a more mathematical way).
I have done several meta-temperaments that way and I should be able to resolve meta-versions of about any regular temperament just as well, I can try on request if anyone is interested.

How I found this precise number is just as I said in my first post (February 1st), by looking at Graham's description of the Backjack tuning.
If I interpret a Blackjack (or a Canasta) horagram (logarithmic domain) as a fractal waveform (frequency domain), I find that there is a Mohajira heptatonic scale (sLsLsLs), subset of Blackjack, that zoomed in the generator's dimension can divide the Secor the very same way.
If I express the meta-secor size as 1, and x is the octave, you can verify that x^2 = 12000 + 349.8031 cents = ten octaves plus 3 secors, so x^2 = 10x + 3, and square root of 7 + 5 is the solution.

Just to give a picture of what sort of cooking I do to resolve a meta-temperament, imagine a scale (frequency domain) collecting the temperament's DE (or MOS) such as, for Miracle :
!
1/1
10/1
21/1
31/1
41/1
72/1

And suppose I want to find a timber that will have optimal inharmonics for such a scale, "a la Sethares"...
I don't know how would Sethares do, but that's what my fractal waveform does, by repeating an arc that will generate these frequencies with best possible semi-periodicity.
This creates a "fractal waveform" whose transpositions would be best suited to play this scale (that is, the 1 : 10 : 21 : 31 : 41 : 72 scale, not the Miracle scale), and its eventual octavial variations.
Interpretation of this fractal waveform as an horagram defines the Meta-temperament, which resumes here to this precise octave/"meta"-secor ratio.

> (Petr #2) : Didn't George himself suggest an optimal generator to > be the 19th root of 18/5?
> Petr

18/5, attained by 19 secors it is certainly a very valuable approximation, and George Secor has certainly good reasons for it,
but there are many others as you would very well know :
7th root of 8/5 -> 116.2409 cents
8th root of 12/7 -> 116.6411 cents
12th root of 9/4 -> 116.9925 cents
15th root of 11/4 -> 116.7545 cents
(19th root of 18/5 -> 116.7156 cents)
(24/247 of an octave -> 116.5992 cents, "meta"-secor = 116.60105 cents)
Others will use different tempering criterias and weights to define "optimal" generators (TOP, etc.).
I do not claim myself any particular acoustic quality to the "meta"-generator I suggested, not more than the golden scale could claim to be a better meantone than others, or I would be the first one to prove the contrary.
But no other generator will have the fractal properties of a meta-generator and its main interest is in the infinity of valid series its algorithm generates, converging towards a specific H(n) / H(n-1) proportion, and that may be considered as some kind of "gravity center" between all possible proportions, nothing more and nothing less.
Or may be yes, just one thing more : their horagrams, used as photosonic disks, will make better sounds...
- - - - - -
Jacques

George Secor wrote :

> > (Jacques) :
> > Also I hope you don't mind if I call the (2 * 7^(1/2) +5)^-1 octave
> > or 116.60105 cents, the Meta-Miracle generator , a meta-secor ? ;-)
>
> If you see some particular advantage in using that interval as a > generator and need a convenient name, then you may call it whatever > you like. After all, I had no part in the naming of the temperament > (Miracle), its MOS scales (Blackjack, etc.), or its generator (the > "secor" -- the term can be used for any generator in the range of > ~116.6 to ~116.7 cents, or roughly midway between 15:16 and 15:14).

I appreciate your confidence ! You may see my answer to Petr if you wish to make yourself an idea of what I suggest as a "meta-version of Miracle temperament". Of course if anyone comes up with a better "meta-secor", it can gain the title ;-)

> > Glad to see in your graph it was bad at all.
>
> I think you meant to say that it was not bad at all, and I agree.

Yes, the "not" dropped away, I was not that modest. Coming from you I can't imagine better compliment, if you find this generator applies for Miracle's identity.

> The graph indicates that primes 3, 5, 7, and 11 have almost the > same error, so that the interval between any two of these (5:6, > 5:7, 6:7, 6:11, etc.) will have a very small error. Another > desirable consequence is that the remaining tempered 11-limit > consonances (2:3, 4:7, 7:9, etc.) will have beat rates > approximating simple proportions (or nearly synchronous beating) > when they occur together in a chord, e.g., 4:5, 4:7, 5:9, and 7:9 > in a 4:5:7:9 tetrad.

It does not seem impossible and I do think too that these would be among the best possible solutions.
Based on the differential coherence of Miracle's 7/6 intervals, I found this recurrent sequence is working as well :
8 - 4s^8 = s^2 (where s = the secor ratio with 1/1)
That means that any "7/6" has its 1st order difference tone in tune with the "4/3" that is 2 secors higher.
Its solution is s = 1.069670002459 ~116,59894558 cents. = close to the 24/247 of an octave approximation of the "meta"-secor, I haven't made it on purpose !

> > I was impressed by your decimal keyboard. Has it been manufactured,
> > and how many people have one ?
>
> No, it has not been manufactured, so you (or anyone else reading > this) may still have the distinction of owning the very first one. :-)

I would love to own one ! (not necessarily to be the first one :-) Once I make a fortune by manufacturing my own photosonic disk, I will think about it ! :-D
What about virtual Miracle decimal keyboards, as a beginning, are some of them available somewhere ?

> > One detail catched my attention. You named several keys such as
> > 17/13, 20/19, 77/51, 14/13, 13/10, 20/17 etc. from approximation of
> > "higher-than-11" -limit ratios, certainly because being simpler and
> > closer to the intervals generated.
> > I would try to understand something :
> > If I take for example "20/17" (23 secors), which could be mapped > also
> > as "33/28" or "88/75" in 11-limit, am I correct if I say then that
> > Miracle also dissolves their ratios with 20/17, which are 561/560 > and
> > 375/374 ?
> > And same with 273/272 and 833/832 for the "17/13" approximation (as
> > equivalent to 21/16 and 64/49) ?
>
> There is a scale of generators to the left that line up with the > cross (or +) markings in the center of each key, so you can see how > many generators (or secors) distant each ratio is from the origin, > 1/1. Since we are assuming octave equivalence, all powers of 2 are > 0G (zero generators). Each prime then maps to the following number > of generators:
> 3/2 or 3: +6G
> 5/4 or 5: -7G
> 7/4 or 7: -2G
> 11/8 or 11: +15G
> 13/8 or 13: -34G
> 17/16 or 17: -30G
> 19/16 or 19: -18G
>
> If you factor any ratio, such as 20/17 into 2*2*5/17, you can then > add the number of generators corresponding to each factor > (subtracting for factors in the denominator), 0+0-7+30=23G to get > the number of secor generators for that ratio. You can then test > 33/28 (+6+15-0+2=23G) and 88/75 (+15-6+7+7=23G) to demonstrate that > they are equivalent in 19-limit Miracle. The mapping for prime 13 > involves a choice between +34G and -38G, and I chose the former > (you may disagree).
>
> --George

That's what I thought, and it's open enough. You may see in my second last answer to Petr that Graham's tools of course totally agree with you :

116.6 cent generator
...
mapping by period and generator :
[(1, 0), (1, 6), (3, -7), (3, -2), (2, 15), (7, -34), (7, -30), (6, -18)]

So I presume that 10 secors, (63/32 on the decimal keyboard), would then mean altogether :
128/65
63/32
112/57
55/28
108/55
51/26
100/51
49/25
96/49

- - - - - - -
Jacques

Herman Miller wrote:
...
The result tells you how many of each
generator to use if you want the tempered equivalent of the ratio 7/4.
Once you've set this up, you can use the same spreadsheet for any ratio
and any 7-limit rank 2 temperament (and this same general method works
for any regular temperament).

That's very kind of you to explain in detail this method, and the principle of "matrix multiplication".
I love maths and I regret I did quit too soon university to play in a group, when I was young...
Thanks !
- - - - - - -
Jacques

On 11 February 2010 17:49, Petr Parízek <p.parizek@...> wrote:
>
> Jacques wrote:
>
> > Is there a reason why do you suggest to enter 81/80 here
> > (It's not tempered by Miracle, isn'it ?)
>
> To show how different some temperaments can be even though
> they use similar commas. While 11-limit miracle tempers out
> 385/384, 225/224, and 2401/2400 (or 1029/1024), 11-limit
> meantone tempers out 385/384, 225/224, and 81/80. So there
> are actually two intervals in common here which are tempered
> out but the resulting temperaments don’t sound like the same at all.

It's not at all surprising that two temperament classes requiring 3
defining commas can be written with two commas in common. I
don't know if it always works but there are tricks you can do with
exterior algebra to find the intersection.

> > What I do not understand is what means "1/9" in the result (1/9, 116,7 c.),
>
> I’m not sure myself, perhaps an approximation to something (Graham?).

The generator is 1 step on a 9 note octave scale. 116.7 cents would give
10.3 steps to the octave, so you can see that 9 note scale is far from
equally tempered.

> > and what's the meaning, following the commas I entered, of the 3 other "unison vectors"
>
> These are some other intervals that are tempered out but I’m not sure
> how the program finds them.

Not very well because the results aren't always very good. If your
example looks like mine, it's an example of this. The unison vectors
it calculated are more complex than the ones you started with.

>
> > And can anyone explain me what's the difference
> > between "linear temperament" and "regular temperament" ?

A regular temperament gives the same size to every tempered
version of a given ratio. A linear temperament is a rank 2
regular temperament, that is it has two defining intervals. It
may also be required to have no equal divisions of the octave.
Which is to say the octave is one of the generators, and all notes
lie on a chain of the other generator when octave reduced.

I'd prefer "linear temperament" and "rank 2 regular temperament"
to be synonymous, but the orthodox definition is as above. You
can think of a rank 2 temperament as a line joining two equal
temperaments.

Incidentally, I may have used this miracle tuning. I haven't worked
it out. There are two phi-based tunings either side of the optimum.
Something like

3 + 4*phi
-----------------
31 + 41*phi

and

4 + 3*phi
-----------------
41 + 31*phi

Graham

Graham wrote :

> > > (Jacques) : What I do not understand is what means "1/9" in the
> result (1/9, 116,7 c.),
> >
> > (Petr) : I´m not sure myself, perhaps an approximation to
> something (Graham?).
>
> (Graham) : The generator is 1 step on a 9 note octave scale. 116.7
> cents would give
> 10.3 steps to the octave, so you can see that 9 note scale is far from
> equally tempered.

But should it not rather give "1/10" ?
I found it also gives "13/134" on several various Miracle series,
which is better but surprising.

> .../...
> You can think of a rank 2 temperament as a line joining two equal
> temperaments.

Interesting concept, and that's what your temperament finder
achieves, but I don't understand what you mean exactly by a "line
joining two equal
temperaments", unless it's an image as you woud say "the shortest
path"...

> Incidentally, I may have used this miracle tuning. I haven't worked
> it out.
> There are two phi-based tunings either side of the optimum.
> Something like
>
> 3 + 4*phi
> -----------------
> 31 + 41*phi
>
> and
>
> 4 + 3*phi
> -----------------
> 41 + 31*phi

Hahaha ! These look like Golden scale-inspired constructions...
The second fraction is precisely based on the Mohajira structure ( 4s
+ 3L / 41s + 31L ), the first one not.
(unless by Phi you mean 0.618, then it would be the other way around)
For any ""phi"" ratio between 1 and 2 at least, both will give
fractions close enough to the octave/meta-secor, and eventually start
correct "meta-Miracle" series.
For ""phi"" = 1, both fractions equal 7/72 (meta-Miracle 7
72 741 series)
For ""phi"" = 2, the second one suggests 10/103, (that belongs to
the 1 10 103 1060 series, see the example used with your
temperament finder I gave to Petr),
and the first one 113/11 (11 113 1163 series).
I don't deny the possible use of Phi to make other musical scales
than the Golden scale, but not all of them will be what I call "meta-
temperaments".
1.618 cannot be the solution for all musical L/s ratios, and besides
in meta-temperaments we can frequently find alterning L/s sets,
patterns phase shifts, not to mention L + m + s or more steps, etc.
The second structure here is a meta-temperament for the L/s ratio of
the meta-Miracle basic 7 tones Mohajira (sLsLsLs), that is :
199.2126 c. / 150.59056 c. = (square root of 7) /2 = 1.3228756555323 :
41 + 31(L/s) / 4 + 3(L/s) = 82.0091453215013 / 7.9686269665969 = 2*
(square root of 7) + 5 = octave/meta-secor.

- - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Graham wrote :
> > You can think of a rank 2 temperament as a line joining two equal
> > temperaments.
>
> Interesting concept, and that's what your temperament finder
> achieves, but I don't understand what you mean exactly by a "line
> joining two equal temperaments", unless it's an image as you woud
> say "the shortest path"...

Hi Jacques & Graham,

Please excuse the interjection, but I thought a picture might help:

http://lumma.org/stuff/2DScaleTree2.gif

Do you see any lines here, Jacques?

-Carl

On 14 February 2010 15:26, Jacques Dudon <fotosonix@...> wrote:

> But should it not rather give "1/10" ?
> I found it also gives "13/134" on several various Miracle series, which is better but surprising.

It could be better. But there is a 9 note MOS as well and the
algorithm happens to find it.

> You can think of a rank 2 temperament as a line joining two equal
> temperaments.
>
> Interesting concept, and that's what your temperament finder achieves, but I don't understand what you mean exactly by a "line joining two equal
> temperaments", unless it's an image as you woud say "the shortest path"...

I have a strict geometry in which this is true, and Carl's given you a
projection of it. If you plot the tuning of each interval, you'll see
that all tunings of a regular temperament class lie on a straight
line, including the equal temperaments.

> I don't deny the possible use of Phi to make other musical scales than the Golden scale, but not all of them will be what I call "meta-temperaments".
> 1.618 cannot be the solution for all musical L/s ratios, and besides in meta-temperaments we can frequently find alterning L/s sets, patterns phase shifts, not to mention L + m + s  or more steps, etc.

Maybe, but there is a general pattern, and a paper (by Dave Keenan?)
explaining it. For any two generators of equal temperaments n1/d1 and
n2/d2 you can get a generator

n1 + phi*n2
----------------
d1 * phi*d2

that bisects them in a maximally complex way.

Graham

Carl wrote :

> Hi Jacques & Graham,
>
> Please excuse the interjection, but I thought a picture might help:
>
> http://lumma.org/stuff/2DScaleTree2.gif
>
> Do you see any lines here, Jacques?
>
> -Carl

Very well thanks Carl !

I can see Syntonic, Hanson, Srutal, Magic and others.
I am gessing where is Miracle but it is harder to find. What's the reason for more croudy lines than others ?
Does Paul explains somewhere how he figured this kind of "Shri Yantra" of the temperaments ?
Also what's the meaning of the blue lines, and the centre defined by their intersection ?
And of the size of the numbers (it would make a good eyes test !! ;-) ?

- - - - - - -
Jacques

Graham wrote :

> > (Jacques) :I don't deny the possible use of Phi to make other > musical scales than the Golden scale, but not all of them will be > what I call "meta-temperaments".
> > 1.618 cannot be the solution for all musical L/s ratios, and > besides in meta-temperaments we can frequently find alterning L/s > sets, patterns phase shifts, not to mention L + m + s or more > steps, etc.
>
> Maybe, but there is a general pattern, and a paper (by Dave Keenan?)
> explaining it. For any two generators of equal temperaments n1/d1 and
> n2/d2 you can get a generator
>
> n1 + phi*n2
> ----------------
> d1 * phi*d2

you probably meant :

n1 + phi*n2
----------------
d1 + phi*d2

> that bisects them in a maximally complex way.
>
> Graham

Does "maximally" refers to one precise definition of "complex" ?
That would be true then only if the "complexity" of Phi could be measured as "superior" to other fractals.
From another point of view I would say that Phi's recurrent complexity is the most "simple" of the whole universe.
.
- - - - - - -
Jacques

Hi Jacques!

> > http://lumma.org/stuff/2DScaleTree2.gif
> >
> > Do you see any lines here, Jacques?
>
> Very well thanks Carl !
>
> I can see Syntonic, Hanson, Srutal, Magic and others.
> I am gessing where is Miracle but it is harder to find. What's
> the reason for more croudy lines than others?

Not sure what you mean...

> Does Paul explains somewhere how he figured this kind of "Shri
> Yantra" of the temperaments?

Herman Miller was actually the first to publish a graph
like this (though on a square plot instead of the hexagonal
plot used here). Paul, Gene, and others developed it further.
Of course the 1-D scale tree was a creation of Erv Wilson.
Which in math is known as the Stern-Brocot tree (Wilson may
have independently discovered it, I don't know).

> Also what's the meaning of the blue lines, and the centre
> defined by their intersection?

The center is just intonation. Sorry, I had forgotten the
axes are not labeled in this version. This should make it
much more clear:
http://lumma.org/stuff/2DScaleTree1.gif

> And of the size of the numbers (it would make a good eyes test!!

The size of the numbers is inversely proportional to their size.
:)

-Carl

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Cameron wrote :
>
> > Hi Jacques,
> >
> > I'm certainly not an expert on the Miracle temperament either,
> > though there are those here who are. But, I have a number of
> > observations from my own experiences.
> > In Blackjack 11/8 shows up in hexads rather than in a JI harmonic
> > scale kind of way, but not as often as 7/5 appears as an "11th". So
> > it is not a "regular" mapping in that way, ...
>
> Hi Cameron,
>
> May be we need some light from temperaments specialists here.
> In Miracle has 11/8 a regular mapping, or not ? Graham's article
> indicates it is :
> [< 1, 1, 3, 3, 2],
> < 0, 6, -7, -2, 15]>
> What I am guessing is that Miracle originally was perhaps made to
> temper 225/224, 2401/2400 and other 7-limit commas,
> but produces indirectly a very correct 11 factor as well, which can
> therefore be added to the "main factors" 3, 5, and 7.
> It does so because of the proximity of 11/8 to 48/35, which is the 7-
> limit path used to arrive to it.
> 11/8 is higher than 48/35 by 385/384, so the question is : does
> Miracle tempers also 385/384 ?

Sorry, I wasn't clear. By "in that way", I meant that 11/8 won't
show up in each »fingering« in the Blackjack tuning (because Blackjack is a subset).

Specifically, I mentioned this because of the coincident
discussion here on the list about set theory and regular
temperament: in actual practice, a regular temperament is
usually going to appear in a subset such as Blackjack. But
applying set theory to a Blackjack tuning would create sets
in which 56/55 would be tempered out on paper, but wouldn't
be in the actual tuning. That is, you'd have "the same"
interval as far as number, but sometimes the actual
sounding interval would be an 11/8, and usually it would be a 7/5.

Obviously, to some degree or other this discrepancy
between number and actual tone is going to happen
in any tuning without the symmetries to prevent it,
of course.

> Blackjack (20 times the generator, 21 tones) is certainly a nice
> subset, but is limited by a lack of "9/8" intervals (for example >it's
> missing "9/8" over 1/1 in the symmetrical version) and in
> consequence, also a lack of "6/5" and "11/8".
> The 31 tones version of Miracle is both more complete and more
> balanced. But every DE of Miracle (...10, 11, 21, 31, 41, 72) has >its
> interest and for the harmonic relations it weaves, it is a >fantastic
> and very open temperament (it's a unconditionnal JI addict that
> speaks ! ;-).

Yes the JI approximations are really excellent. I'm very much
on the record here as being against considering intervals more than
fewcents off as JI „approximations". This is because I find that there is
a small (and asymmetrical!) zone around JI intervals, that is, those
intervals with audible coincidence of partials, (and slight mistuning
within this zone can even be preferable to „perfection"), but getting „near" this
zone, not in it, can be one of the most „out of tune" things of all to do.

It is „good", but not „excellent" approximations which can make for a
particular queasy and spongey sound, perhaps because they are close
enough to clearly indicate what they are „supposed" to be, but not
close enough to „be", or to beat pleasantly against, what they
allegedly „are". It is a very complex thing- and very, very context sensitive- but in general, I'd say that either exact/excellent or „way off" intervals are more useful than „good" intervals. „Way off" intervals can establish their own identity, for other reasons of both spectral and usage natures.

More later, trying to do many things at once here....

-Cameron Bobro

On 15 February 2010 16:27, Jacques Dudon <fotosonix@...> wrote:

> you probably meant :
> n1 + phi*n2
> ----------------
> d1 + phi*d2

That's it!

>> that bisects them in a maximally complex way.

> Does "maximally" refers to one precise definition of "complex" ?
> That would be true then only if the "complexity" of Phi could be measured as "superior" to other fractals.

It's the sense in which phi is the hardest number to approximate as a
rational fraction. Every pair of numbers from the Fibonacci sequence
gets you closer, but nothing is very close. That formula gives you
the most irrational number between the pair of rationals you seed it
with (or one such as there are two different ways of choosing the
rationals).

In practical terms, it's the harderst generator to approximate with
equal temperaments in the given range. So it's useful as a way of
choosing an unequal temperament.

The same numbers have also been suggested as maximally dissonant
frequency ratios, or special ratios which aren't so dissonant after
all for mysterious reasons.

> From another point of view I would say that Phi's recurrent complexity is the most "simple" of the whole universe.

Maybe these numbers share that property. I don't know.

Graham

Carl wrote :

> > (Jacques) : I can see Syntonic, Hanson, Srutal, Magic and others.
> > I am gessing where is Miracle but it is harder to find.
> > What's the reason for more croudy lines than others?
>
> Not sure what you mean...

Of course, because it was not english ! I meant "crowdy" (in quantity of numbers along the lines)...
But I think you gave me one of the reasons after : because the size of the numbers is inversely proportional to their size.
Therefore such lines as Miracle are not so present, because of higher numbers (and also 10 was oustside the first plot).

> Herman Miller was actually the first to publish a graph
> like this (though on a square plot instead of the hexagonal
> plot used here). Paul, Gene, and others developed it further.
> Of course the 1-D scale tree was a creation of Erv Wilson.
> Which in math is known as the Stern-Brocot tree (Wilson may
> have independently discovered it, I don't know).

Compliments to Herman for the creation of this amazing 2-D representation, and to others to develop it.
I suppose the "Syntonic line" was the first one to be discovered ? (I found it by myself as well years ago)
And I wondered what was the next one that got discovered :
Magic may be because of the 19-edo property ? or Hanson for the same reason ?
And what was the first triangle of three lines ever plotted ?

> > (Jacques) : Also what's the meaning of the blue lines, and the > centre
> > defined by their intersection?
>
> The center is just intonation. Sorry, I had forgotten the
> axes are not labeled in this version. This should make it
> much more clear:

http://lumma.org/stuff/2DScaleTree1.gif

The indications concerning the 5-limit consonances are very useful indeed.
I wondered why some edos are missing :
In the syntonic line between 17 and 5, 22 is missing, may be it got only forgotten.
But what makes me curious is for 24-edo and 36-edo, since these are ET commonly in use.
I presume it is because 24-edo and 36-edo would not differ from 12-edo in the 5-limit respect ?
Another question, would there be a another 2-D figure for factors 3 and 7, etc. ?

> > (Jacques) : And of the size of the numbers (it would make a good > eyes test!! ;-) ?
>
> The size of the numbers is inversely proportional to their size. :)

That's an interesting choice, but I would suggest another one : all numbers could have the same length.
Two-digit would be two times smaller than one-digit, and three-digit three times smaller. The crossroads would be better figured.

- - - - - - -
Jacques

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

> Compliments to Herman for the creation of this amazing 2-D
> representation, and to others to develop it. I suppose the
> "Syntonic line" was the first one to be discovered ? (I
> found it by myself as well years ago) And I wondered what
> was the next one that got discovered: Magic may be because
> of the 19-edo property? or Hanson for the same reason?

We already knew about all of those temperaments, so their
lines were spotted together. By the way, here is a version
of the plot with the lines labeled:
http://lumma.org/stuff/DeviationsOfETsBelow100.gif

> In the syntonic line between 17 and 5, 22 is missing, may be
> it got only forgotten.

22 wouldn't be on the syntonic line...

> But what makes me curious is for 24-edo and 36-edo, since
> these are ET commonly in use.
> I presume it is because 24-edo and 36-edo would not differ
> from 12-edo in the 5-limit respect?

Yes, I think that's correct.

> Another question, would there be a another 2-D figure for
> factors 3 and 7, etc.?

That would be interesting. As well as a 3-D figure for
3 5 & 7.

-Carl

Graham wrote :

> >> that bisects them in a maximally complex way.
>
> > (Jacques) : Does "maximally" here refers to one precise > definition of "complex" ?
> > That would be true then only if the "complexity" of Phi could be > measured as "superior" to other fractals.
>
> It's the sense in which phi is the hardest number to approximate as > a rational fraction.

I don't think so. Pi, and an infinity of others would certainly be harder. Phi is not at all hard to approximate.
What characterises Phi compared to other numbers is, on the contrary, that it will be approximated by a maximum of ratios.

> Every pair of numbers from the Fibonacci sequence gets you closer, > but nothing is very close.

Not really. In terms of convergence, Phi series are strongly convergent.
Powers of Phi as well rapidly converge towards whole numbers.

> That formula gives you the most irrational number between the pair > of rationals you seed it
> with (or one such as there are two different ways of choosing the > rationals).

I don't know what's the "most irrational number" means.

> In practical terms, it's the harderst generator to approximate with
> equal temperaments in the given range.

I'm not sure of what you mean by "in the given range", but I don't think so.
The Golden scale for example, has a generator that can be approximated by a maximum of different edos.
Generalize its formula for any generator is easy and always possible, but it would not always be the most pertinent. And not for Miracle anyway.
But Phi would work for example for Pajara. Has anyone tried 4 + Phi as a period/generator (106.7989267 c.) for Pajara ?
That's a Meta-temperament and its 22-subset for example would be increasingly well approximated in 34, 56, 90, 146, 236, or also 78, 124, 202-edos.

> So it's useful as a way of choosing an unequal temperament.

When appropriate, but this would be true of other fractals, it may be for this reason that they can propose at once all best possible edo subsets.

> The same numbers have also been suggested as maximally dissonant
> frequency ratios, or special ratios which aren't so dissonant after
> all for mysterious reasons.

Exactly, this is one of Phi's paradoxes. Is it the most consonant, or the most dissonant number ?
I have done many photosonic disks with Phi fractal waveforms and scales and I played them a lot. And still I can never say before I play them if I will go to Hell, or to Heavens.
At least in architecture it makes the best acoustics.

> > From another point of view I would say that Phi's recurrent > complexity is the most "simple" of the whole universe.
>
> Maybe these numbers share that property. I don't know.

Other fractals will also, each one in a different way.

- - - - - - -
Jacques

On 16 February 2010 20:57, Jacques Dudon <fotosonix@...> wrote:

> I don't think so. Pi, and an infinity of others would certainly be harder. Phi is not at all hard to approximate.
> What characterises Phi compared to other numbers is, on the contrary, that it will be approximated by a maximum of ratios.

No finite number is absolutely hard to approximate. You take the
continued fraction. Phi takes the longest to converge. It has a
maximum number of approximations, yes, but none of them are very good.
Pi is easier. There are approximations that do very well for their
size. 22/7 is the famous one. To three decimal places, it's 3.143
where the target is 3.142. The equivalent for phi is 21/8 = 1.615
which is three places out, right?

In terms of MOS scales, it means you'll eventually reach one that's
almost equally spaced if you start with a random generator. But phi
means each pair of step sizes are unequally spaced by the same degree.
That's the property that's targeted by the formula I gave.

> Every pair of numbers from the Fibonacci sequence gets you closer, but nothing is very close.

> I don't know what's the "most irrational number" means.

The same property as above.

>> In practical terms, it's the harderst generator to approximate with
>> equal temperaments in the given range.
>
> I'm not sure of what you mean by "in the given range", but I don't think so.

Between n1/d1 and n2/d2. It's the same property phi has from
continued fractions, or the Stern-Brocot tree. You alternate taking
the left or right branch. That gives the maximum number of
convergents, but none of them are very strong.

Graham

Graham wrote :

> No finite number is absolutely hard to approximate. You take the
> continued fraction. Phi takes the longest to converge. It has a
> maximum number of approximations, yes, but none of them are very good.
> Pi is easier. There are approximations that do very well for their
> size. 22/7 is the famous one. To three decimal places, it's 3.143
> where the target is 3.142. The equivalent for phi is 21/8 = 1.615
> which is three places out, right?

That's perfectly right. For some reason I made myself a wrong idea about the strength of convergence of Phi, at least compared to some rational approximations of Pi, that are very good (and 355/113 is even better).
But whether *NONE* of Phi 's approximations are very good, that's relative. I compared Phi's approximations with different irrationals of the same range, and I found the game was very much equal for ANY number...
Here is one example :
Between Phi and 1.622154674978, with d<1000 and for same range of whole numbers, Phi gives better approximations 5 times, and the other 3 times.
So the statement that phi is the hardest number to approximate as a rational fraction isn't that true. It's only harder to find its best approximations, because none of its offsets will be irrregularly better or worse. That's why, out of any context, a ratio of 1.618 has such a mysterious meaning.
Also, I am not so sure now after all that Phi has MORE "better" approximations than any other irrational ! Because the better approximations it has at a given range of n/d are only its ratios between Fibonaccis numbers...

> In terms of MOS scales, it means you'll eventually reach one that's
> almost equally spaced if you start with a random generator. But phi
> means each pair of step sizes are unequally spaced by the same degree.
> That's the property that's targeted by the formula I gave.

Of course. My remark was just that Phi is not the only number having this property of, I would say, "endless even distribution". And as I said, for different L/s ratios and other reasons proper to each tuning, other noble numbers will be more pertinent - such as 2 sqrt of 7 + 5 for my resolution of a Miracle meta-temperament that produces different L/s = 1.3228756555 ; or another one in front of my eyes that will have recurrently a L/s of 1.15470053838, with a generator of (3 - sqrt of 3) /4 octave or 380.384757729362 cents, if it reminds you something...

- - - - - - -
Jacques

Carl wrote :

>> > (Jacques): In the syntonic line between 17 and 5, 22 is missing, >> may be
>> > it got only forgotten.
>
>
> 22 wouldn't be on the syntonic line...

Fifth of 13 steps, major third of 8 steps...
instead of 7 in Magic-22's version.
Not too far from 9/7, a bit more yang I admit, but that's what happens with 4 fifths -

- - - - - - -
Jacques

The physical reality of Phi, especially placed into a harmonic series, works out to have very distinct features: it is the "anti-octave" in a yin-yang sense, in that it is unique in returning itself as a difference tone (a true mirror image), in comparison/contrast to 2:1 returning 1; the harmonic partials take the "biggest forever (Cantorian)" to converge (as far as I know), yet at the same time the a:b:c proportions of a golden cut, where b lies between a and c, are maximally simple: a:c to a:b has the identical proportion as a:b does to b:c (only the geometric mean of a:b=b:c is more simple).

As far as I know, it is actually also correct to think of Phi as
having the "most" rational approximations, or better said, "requiring" the most rational approximations. But this is
based on the idea of maximum irrationality (requiring the largest infinity of integers to "reach"), which I understand to be true of Phi, but might very well not be true.

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Graham wrote :
>
> > No finite number is absolutely hard to approximate. You take the
> > continued fraction. Phi takes the longest to converge. It has a
> > maximum number of approximations, yes, but none of them are very good.
> > Pi is easier. There are approximations that do very well for their
> > size. 22/7 is the famous one. To three decimal places, it's 3.143
> > where the target is 3.142. The equivalent for phi is 21/8 = 1.615
> > which is three places out, right?
>
>
> That's perfectly right. For some reason I made myself a wrong idea
> about the strength of convergence of Phi, at least compared to some
> rational approximations of Pi, that are very good (and 355/113 is
> even better).
> But whether *NONE* of Phi 's approximations are very good, that's
> relative. I compared Phi's approximations with different irrationals
> of the same range, and I found the game was very much equal for ANY
> number...
> Here is one example :
> Between Phi and 1.622154674978, with d<1000 and for same range of
> whole numbers, Phi gives better approximations 5 times, and the other
> 3 times.
> So the statement that phi is the hardest number to approximate as a
> rational fraction isn't that true. It's only harder to find its best
> approximations, because none of its offsets will be irrregularly
> better or worse. That's why, out of any context, a ratio of 1.618 has
> such a mysterious meaning.
> Also, I am not so sure now after all that Phi has MORE "better"
> approximations than any other irrational ! Because the better
> approximations it has at a given range of n/d are only its ratios
> between Fibonaccis numbers...
>
> > In terms of MOS scales, it means you'll eventually reach one that's
> > almost equally spaced if you start with a random generator. But phi
> > means each pair of step sizes are unequally spaced by the same degree.
> > That's the property that's targeted by the formula I gave.
>
>
> Of course. My remark was just that Phi is not the only number having
> this property of, I would say, "endless even distribution". And as I
> said, for different L/s ratios and other reasons proper to each
> tuning, other noble numbers will be more pertinent - such as 2 sqrt
> of 7 + 5 for my resolution of a Miracle meta-temperament that
> produces different L/s = 1.3228756555 ; or another one in front of my
> eyes that will have recurrently a L/s of 1.15470053838, with a
> generator of (3 - sqrt of 3) /4 octave or 380.384757729362 cents, if
> it reminds you something...
>
> - - - - - - -
> Jacques
>

In the "regular temperament paradigm" developed at this list over many years as far as I know, 22 wouldn't fall on that line because it does not "temper out" the syntonic comma. The syntonic comma remains there, in that view, between the intervals of 22-equal which are very close to 5/4 and 9/7. That is, there remains a comma between
four-fifths-modulo-2 and 5/4, just as there does between four 3/2s and 5/4.

They (regular mapping) regard 22 as a "pajara" temperament, which means in actual practice splitting the difference between 7/4 and fa-of-fa (16:9), but not splitting the difference between equivalents of Pythagorean ditones and Just M3s.

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Carl wrote :
>
> >> > (Jacques): In the syntonic line between 17 and 5, 22 is missing,
> >> may be
> >> > it got only forgotten.
> >
> >
> > 22 wouldn't be on the syntonic line...
>
>
> Fifth of 13 steps, major third of 8 steps...
> instead of 7 in Magic-22's version.
> Not too far from 9/7, a bit more yang I admit, but that's what
> happens with 4 fifths -
>
> - - - - - - -
> Jacques
>

Hi Jacques.

22-EDO doesn't meet the condition that the nearest interval to 5/1 is 4 times the size of the nearest to 3/2 and therefore doesn't temper out the syntonic comma. If you use a fifth of 13/22 octave, you get superpyth rather than meantone. While meantone maps 4:5:6 to C-E-G, superpyth maps 4:5:6 to C-D#-G. The step of D#-E (which would normally work simply as a minor second) then represents the syntonic comma and has a quartertone-like size. So you can then do some sort of "comma shift imitation" by playing something like C-G-C-D#, G#-G#-C-D#, C#-F-G#-C#, G-G-A#-D, C-G-C-D#.

Petr

Petr wrote :

> Hi Jacques.
> 22-EDO doesn't meet the condition that the nearest interval to 5/1 > is 4 times the size of the nearest to 3/2 and therefore doesn't > temper out the syntonic comma. If you use a fifth of 13/22 octave, > you get superpyth rather than meantone. While meantone maps 4:5:6 > to C-E-G, superpyth maps 4:5:6 to C-D#-G. The step of D#-E (which > would normally work simply as a minor second) then represents the > syntonic comma and has a quartertone-like size. So you can then do > some sort of "comma shift imitation" by playing something like C-G-> C-D#, G#-G#-C-D#, C#-F-G#-C#, G-G-A#-D, C-G-C-D#.
> Petr

Cameron wrote :

> In the "regular temperament paradigm" developed at this list over > many years as far as I know, 22 wouldn't fall on that line because > it does not "temper out" the syntonic comma. The syntonic comma > remains there, in that view, between the intervals of 22-equal > which are very close to 5/4 and 9/7. That is, there remains a comma > between
> four-fifths-modulo-2 and 5/4, just as there does between four 3/2s > and 5/4.
> They (regular mapping) regard 22 as a "pajara" temperament, which > means in actual practice splitting the difference between 7/4 and > fa-of-fa (16:9), but not splitting the difference between > equivalents of Pythagorean ditones and Just M3s.

Hi Petr, and Cameron,

Thanks for your both precisions concerning 22-edo, which converge and make sense.
I have to hear the "comma shift imitation" suggested by Petr.
It makes sense that 22-edo would be considered mainly as a Pajara temperament, along with the "two other side possibilities" I mentionned.

Nevertheless, my comment was about the "syntonic line" that appears in a figure by Paul Erlich (and according to Carl, originally published by Herman Miller) :

http://lumma.org/stuff/DeviationsOfETsBelow100.gif

It shows an alignment of edos between 7 and 12 on one side (7 - 26 - 19 - 50 - 31 -43 - 12 among others), and its natural continuation on the same (geometrical) line between 12 and 5, for what you call Petr "superpyths" (or what I call also "extended fifths") and where Paul's graph included 17-edo and 5-edo.
It's not what was mainly interesting me in it, but this remark here was : if this line (in the geometrical and logical sense) includes 17 and 5 , it must also pass by 22.
(BTW 5 itself is coherent with the Golden scale series 5 7 12 19 31 50 etc. well present here and where we can see the perspective "point de fuite" of this Golden series resized numbers, like if Phi was "swallowing" all these numbers...! ) -

Then the objection that 22's best approximation of 5/4 is not 8 but 7 steps would apply also to 17-edo, present in the figure, where four fifths of 10 steps lead to a 6 steps large major third, while a better approximation of 5/4 again would be 5 steps.
But never mind that, the more interesting point here is that I was not refering to a "meantone line" but to a "syntonic line", as two different things in my mind, and therefore there is a purely semantic question here I am asking :

Should "syntonic fifths" be synonymous of "meantone fifths", or can they englobe extended fifths passed 700 cents ?

Some first importance searchers and luthiers under the term "syntonic" englobe fifths between 686 and 720 cents, that include 7, 53, 17, 22, and 5-edos :

http://homepages.cae.wisc.edu/~sethares/software/TFSdocs/aboutdynamictonality.html

My feeling is that 7-edo and 5-edo are the "yin and yang" extreme attractors of one global "fifth" entity, and in case "syntonic" was not a correct name for the line, another one would be welcomed.
- - - - - - -
Jacques

Jacques Dudon wrote:
> Hi Petr, and Cameron,
> > Thanks for your both precisions concerning 22-edo, which converge and > make sense.
> I have to hear the "comma shift imitation" suggested by Petr.
> It makes sense that 22-edo would be considered mainly as a Pajara > temperament, along with the "two other side possibilities" I mentionned.
> > Nevertheless, my comment was about the "syntonic line" that appears in a > figure by Paul Erlich (and according to Carl, originally published by > Herman Miller) :
> > http://lumma.org/stuff/DeviationsOfETsBelow100.gif

I did a chart of this sort, but not as detailed and much cruder (I believe it was a chart from a Microsoft Works spreadsheet). It mainly showed the middle part of this chart from 12 to 19 on the left, 15 to 22 on the right (the line from 12 to 15 was horizontal), although I think a few of the outer ETs were there also (particularly the "diminished" line from 12 to 28, and possibly also the line from 21 to 35).

> It shows an alignment of edos between 7 and 12 on one side (7 - 26 - 19 > - 50 - 31 -43 - 12 among others), and its natural continuation on the > same (geometrical) line between 12 and 5, for what you call Petr > "superpyths" (or what I call also "extended fifths") and where Paul's > graph included 17-edo and 5-edo.
> It's not what was mainly interesting me in it, but this remark here was > : if this line (in the geometrical and logical sense) includes 17 and 5 > , it must also pass by 22.

The chart shows the closest approximation to JI for the ET's that are 5-limit consistent, or the three closest approximations for inconsistent ETs. 17 happens to be inconsistent, so you can see that it's shown in two places on the chart (the third place would be off the bottom of the chart). 22 is between 5 and the 17 on the right side of the chart (on a line that would be labeled "superpyth" if the chart had a line there, which also goes through 27 and 49). You could also put 22 between 5 and the other 17, but that wouldn't be a typical mapping of 22; it would have the same size fifth but the major third approximation would be way off.

> (BTW 5 itself is coherent with the Golden scale series 5 7 12 19 31 50 > etc. well present here and where we can see the perspective "point de > fuite" of this Golden series resized numbers, like if Phi was > "swallowing" all these numbers...! ) -
> > Then the objection that 22's best approximation of 5/4 is not 8 but 7 > steps would apply also to 17-edo, present in the figure, where four > fifths of 10 steps lead to a 6 steps large major third, while a better > approximation of 5/4 again would be 5 steps.

The closest approximation to 5/4 in 17-edo is almost half a step flat (.47 of a step). 22 has a much closer approximation of 5/4, only 0.08 steps flat.

> But never mind that, the more interesting point here is that I was not > refering to a "meantone line" but to a "syntonic line", as two different > things in my mind, and therefore there is a purely semantic question > here I am asking :
> > Should "syntonic fifths" be synonymous of "meantone fifths", or can they > englobe extended fifths passed 700 cents ?

I think that's more a matter of the historical nature of the words "syntonic" and "meantone". The line itself doesn't stop at 700 cents, but historically points on the line beyond 700 cents haven't generally been included in the range of "meantone" tunings. "Syntonic" on the other hand refers to the 81/80 comma, so I think it's perfectly fair to use that name for any tuning that tempers it out, but in tuning-math land, it's more common just to use the label "meantone" for the whole range.

> Some first importance searchers and luthiers under the term "syntonic" > englobe fifths between 686 and 720 cents, that include 7, 53, 17, 22, > and 5-edos :
> > http://homepages.cae.wisc.edu/~sethares/software/TFSdocs/aboutdynamictonality.html > > > My feeling is that 7-edo and 5-edo are the "yin and yang" extreme > attractors of one global "fifth" entity, and in case "syntonic" was not > a correct name for the line, another one would be welcomed.
> - - - - - - -
> Jacques

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> I did a chart of this sort, but not as detailed and much
> cruder (I believe it was a chart from a Microsoft Works
> spreadsheet). It mainly showed the middle part of this chart
> from 12 to 19 on the left, 15 to 22 on the right (the line
> from 12 to 15 was horizontal), although I think a few of the
> outer ETs were there also (particularly the "diminished" line
> from 12 to 28, and possibly also the line from 21 to 35).

I took the liberty of placing it in your folder here, for
posterity in case you no longer have it (feel free to delete).

-C.

Herman wrote :

http://lumma.org/stuff/DeviationsOfETsBelow100.gif

> I did a chart of this sort, but not as detailed and much cruder (I
> believe it was a chart from a Microsoft Works spreadsheet). It mainly
> showed the middle part of this chart from 12 to 19 on the left, 15 > to 22
> on the right (the line from 12 to 15 was horizontal), although I > think a
> few of the outer ETs were there also (particularly the "diminished" > line
> from 12 to 28, and possibly also the line from 21 to 35).

/tuning/files/HermanMiller/ET-Scales.png

So you answer to one of my previous questions here :
the first combination of lines had many triangles !
and a cool one between Meantone, Magic, Pajara...

> The chart shows the closest approximation to JI for the ET's that are
> 5-limit consistent, or the three closest approximations for > inconsistent
> ETs. 17 happens to be inconsistent, so you can see that it's shown in
> two places on the chart (the third place would be off the bottom of > the
> chart). 22 is between 5 and the 17 on the right side of the chart > (on a
> line that would be labeled "superpyth" if the chart had a line there,
> which also goes through 27 and 49). You could also put 22 between 5 > and
> the other 17, but that wouldn't be a typical mapping of 22; it would
> have the same size fifth but the major third approximation would be > way off.

Thanks Herman for these precisions.
At the level of my knowledge of temperaments, this is helpful.
So if I summarize your answer to my question and if I understood well,
The line from 7 to 5, passing by 19, 31, 12, and possibly 17 would be called Meantone (or Syntonic),
but is not considered as relevant passed 12 or 17 in direction of 5 for a "syntonic" temperament.
But you find 17 and 22 in consistent places on the "Superpyth" line, that also passes by 5 (and which, I suppose, tempers 64/63 instead of 81/80).
So, in a continuous extension of a fifth between 4/7 to 3/5 of an octave, what's happening ?
"Useful" temperaments in practice will shift from Meantone to Superpyth at the level of 7oo c. ? or am I simplifying ?

Now I count (at least) 6 lines passing by 22 ! (in fact I did not open the right chart before) -
And the next thing that puzzles me now is how at the level of these crossroads like 22 or 34, 19, 12, 7 etc., how these differenciate. It's something I have been working on from a complete other direction, and that inspires me many meta-temperaments ! ;-)
- - - - - - -
Jacques

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> >> I did a chart of this sort, but not as detailed and much
>> cruder (I believe it was a chart from a Microsoft Works
>> spreadsheet). It mainly showed the middle part of this chart
>> from 12 to 19 on the left, 15 to 22 on the right (the line
>> from 12 to 15 was horizontal), although I think a few of the
>> outer ETs were there also (particularly the "diminished" line
>> from 12 to 28, and possibly also the line from 21 to 35).
> > I took the liberty of placing it in your folder here, for
> posterity in case you no longer have it (feel free to delete).
> > -C.

That's it! I found the original message in tuning-math where I described it.

/tuning-math/message/390

Jacques Dudon wrote:

> Thanks Herman for these precisions.
> At the level of my knowledge of temperaments, this is helpful.
> So if I summarize your answer to my question and if I understood well,
> The line from 7 to 5, passing by 19, 31, 12, and possibly 17 would be > called Meantone (or Syntonic),
> but is not considered as relevant passed 12 or 17 in direction of 5 for > a "syntonic" temperament.
> But you find 17 and 22 in consistent places on the "Superpyth" line, > that also passes by 5 (and which, I suppose, tempers 64/63 instead of > 81/80).

Superpyth as a 7-limit temperament does temper out 64/63, although this chart is 2D so it only shows the 5-limit temperaments (you'd need a 3D chart to see all the 7-limit temperaments). As a 5-limit temperament superpyth tempers out 20480/19683.

> So, in a continuous extension of a fifth between 4/7 to 3/5 of an > octave, what's happening ?
> "Useful" temperaments in practice will shift from Meantone to Superpyth > at the level of 7oo c. ? or am I simplifying ?

From 700 cents to around 703.4 cents (12ET - 29ET) a more "useful" mapping is schismatic, which tempers out 32805/32768.

> Now I count (at least) 6 lines passing by 22 ! (in fact I did not open > the right chart before) -
> And the next thing that puzzles me now is how at the level of these > crossroads like 22 or 34, 19, 12, 7 etc., how these differenciate. It's > something I have been working on from a complete other direction, and > that inspires me many meta-temperaments ! ;-)
> - - - - - - -
> Jacques

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> Maybe, but there is a general pattern, and a paper (by Dave Keenan?)
> explaining it. For any two generators of equal temperaments n1/d1 and
> n2/d2 you can get a generator
>
> n1 + phi*n2
> ----------------
> d1 + phi*d2
>
> that bisects them in a maximally complex way.

Sorry I don't check the list very often these days. My photovoltaics teaching job and electric Miata/MX-5 project are keeping me busy.
http://www.evalbum.com/2507

You would be thinking of this paper by Margo Schulter and myself:
http://dkeenan.com/Music/NobleMediant.txt

Regards
-- Dave Keenan

Good to see some of you guys coming back! It's a brand new tuning list now.

-Mike

On Tue, Apr 6, 2010 at 8:20 PM, dkeenanuqnetau <d.keenan@bigpond.net.au>wrote:

>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Graham Breed
> <gbreed@...> wrote:
> > Maybe, but there is a general pattern, and a paper (by Dave Keenan?)
> > explaining it. For any two generators of equal temperaments n1/d1 and
> > n2/d2 you can get a generator
> >
> > n1 + phi*n2
> > ----------------
> > d1 + phi*d2
> >
> > that bisects them in a maximally complex way.
>
> Sorry I don't check the list very often these days. My photovoltaics
> teaching job and electric Miata/MX-5 project are keeping me busy.
> http://www.evalbum.com/2507
>
> You would be thinking of this paper by Margo Schulter and myself:
> http://dkeenan.com/Music/NobleMediant.txt
>
> Regards
> -- Dave Keenan
>
>
>