This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

- Contents - Hide Contents - Home - Section 9

Previous Next

8000 8050 8100 8150 8200 8250 8300 8350 8400 8450 8500 8550 8600 8650 8700 8750 8800 8850 8900 8950

8850 - 8875 -



top of page bottom of page up down


Message: 8875 - Contents - Hide Contents

Date: Sat, 27 Dec 2003 03:03:25

Subject: Re: 5-limit, 12-note Fokker blocks

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:


> (The scales can be obtained by letting i range from 1 to > 12.)
I forgot to put in that you must round the exponents to integers; if you don't you'll merely get 12-equal.
> (16/15)^i (2048/2025)^round(-5i/12) (81/80)^round(-2i/12) > No Scala name > > > (16/15)^i (81/80)^rounnd(8i/12) (648/625)^round(-5i/12) > No Scala name > > > (16/15)^i (128/125)^round(-8i/12) (648/625)^round(3i/12) > No Scala name, a mode of what I've called the thirds scale > > > (16/15)^i (2048/2025)^round(-3i/12) (128/125)^round(-2i/12) > Scala identifies it as lumma5r.scl > > ! lumma5r.scl > ! > Carl Lumma's scale, 5-limit just version, TL 19-2- > 99 ________________________________________________________________________ ________________________________________________________________________
To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
top of page bottom of page up down


Message: 8876 - Contents - Hide Contents

Date: Sun, 28 Dec 2003 01:53:47

Subject: Re: 5-limit, 12-note Fokker blocks

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> Scala has many others, including Marpurg's Monochord #1, Ramos/Ramis, > etc . . . see the Gentle Introduction to Fokker Periodicity Blocks.
"Many" is a little vague, though I see tamil_vi.scl is a mode of ramis.scl. Can you tell me what two commas give Marpurg #1 and Ramis? Do you know of any other specific examples?
top of page bottom of page up down


Message: 8877 - Contents - Hide Contents

Date: Sun, 28 Dec 2003 03:48:41

Subject: Re: 5-limit, 12-note Fokker blocks

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:

> "Many" is a little vague, though I see tamil_vi.scl is a mode of > ramis.scl. Can you tell me what two commas give Marpurg #1 and Ramis? > Do you know of any other specific examples?
I'm going to answer part of my own question. Determination of whether or not something was a Fokker block might be considered as an addition to Scala. Consider the intervals of ramis.scl. These go 135/126, 256/243, 16/15, 135/128, 16/15, 135/128, 16/15, 256/243, 135/128, 16/15, 135/128, 16/15 There are only three sizes (in the 5-limit, there can be as many as four) and none is a singleton, which we should avoid using as a basis. Pick 16/15 as the "basis", and take ratios with the other two step sizes to find the commas: (16/15)/(135/128) = 2048/2025, and (16/15)/(256/243) = 81/80. Now invert the matrix whose rows are the monzos for 16/15, 81/80, 2048/2025 and get the matrix whose columns are the standard vals h12, -h2, -h5. If we take the triples [h12(q), h2(q), h5(q)] for q in the Ramis scale, we get 1 [0, 0, 0] 135/128 [1, 0, 1] 10/9 [2, 1, 1] 32/27 [3, 1, 1] 5/4 [4, 1, 2] 4/3 [5, 1, 2] 45/32 [6, 1, 3] 3/2 [7, 1, 3] 128/81 [8, 2, 3] 5/3 [9, 2, 4] 16/9 [10, 2, 4] 15/8 [11, 2, 5] 2 [12, 2, 5] The values for h2 and h5 are nondecreasing, and for h12 they go up one at a time (the scale is h12-epimorphic.) This tells us we have a Fokker block. Doing the same kind of analysis for Marpurg Monochord 1, we get 16/15, 128/125 and 81/80, and h12, h5, and h3 as the vals to check it with. This time we get 1 [0, 0, 0] 25/24 [1, 1, 0] 9/8 [2, 1, 1] 6/5 [3, 1, 1] 5/4 [4, 2, 1] 4/3 [5, 2, 1] 45/32 [6, 3, 2] 3/2 [7, 3, 2] 25/16 [8, 4, 2] 5/3 [9, 4, 2] 9/5 [10, 4, 3] 15/8 [11, 5, 3] 2 [12, 5, 3] Once again, we have the correct monotonic and epimorphic properties, and Marpurg Monochord 1 is a Fokker block. ________________________________________________________________________ ________________________________________________________________________ To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
top of page bottom of page up down


Message: 8878 - Contents - Hide Contents

Date: Mon, 29 Dec 2003 08:26:01

Subject: Re: 5-limit, 12-note Fokker blocks (attn Manuel)

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:


> Consider the intervals of ramis.scl. These go > > 135/126, 256/243, 16/15, 135/128, 16/15, 135/128, 16/15, 256/243, > 135/128, 16/15, 135/128, 16/15 > > There are only three sizes (in the 5-limit, there can be as many as > four) and none is a singleton, which we should avoid using as a basis. > Pick 16/15 as the "basis", and take ratios with the other two step > sizes to find the commas: (16/15)/(135/128) = 2048/2025, and > (16/15)/(256/243) = 81/80. Now invert the matrix whose rows are the > monzos for 16/15, 81/80, 2048/2025 and get the matrix whose columns > are the standard vals h12, -h2, -h5. If we take the triples > [h12(q), h2(q), h5(q)] for q in the Ramis scale, we get > > 1 [0, 0, 0] > 135/128 [1, 0, 1] > 10/9 [2, 1, 1] > 32/27 [3, 1, 1] > 5/4 [4, 1, 2] > 4/3 [5, 1, 2] > 45/32 [6, 1, 3] > 3/2 [7, 1, 3] > 128/81 [8, 2, 3] > 5/3 [9, 2, 4] > 16/9 [10, 2, 4] > 15/8 [11, 2, 5] > 2 [12, 2, 5]
If now we take [i-h12(q), i/6-h2(q), 5i/12-h5(q)] we get 0 [0, 0, 0] 1 [0, 1/6, -7/12] 2 [0, -2/3, -1/6] 3 [0, -1/2, 1/4] 4 [0, -1/3, -1/3] 5 [0, -1/6, 1/12] 6 [0, 0, -1/2] 7 [0, 1/6, -1/12] 8 [0, -2/3, 1/3] 9 [0, -1/2, -1/4] 10 [0, -1/3, 1/6] 11 [0, -1/6, -5/12] The midpoint of values of the second column is (-2/3 +1/6)/2 = -1/4, and the midpoint of the third column is (-7/12 + 1/3)/2 = -1/8. Hence we can express the exponents in terms of round(-5i/12 - 1/8) and round(-i/6 - 1/4), and get ramis[i] = (16/15)^i * (2048/2025)^round(-5i/12-1/8) * (81/80)^round(-i/6-/14) We can automate this process with the following steps 1. Find the set of intervals between steps. If there are more of these than there are primes involved in the factorization of the scale degrees, return false. 2. Check if the scale is epimorphic. If not, return false. 3. Pick a "base" element from the scale steps, by any means (first step, smallest step, most common step, least Tenney height, etc.) Now take ratios of the basis element with the other scale steps, finding (n-1) other ratios (where n is the number of primes involved) which together with the basis element gives a matrix of row vectors whose determinant is +-1 (unimodular matrix.) If this is impossible, return false. 4. Invert the above unimodular matrix, and obtain vals from the columns of the inverted matrix. 5. For each of these vals aside from the first, take the maximum and minimum values of v(2)*i/N, where N is the number of scale steps and i goes from 0 to N-1, and find the average of the maximum and minimum (the midpoint value.) 6. Using the midpoint value as an offset, calculate a scale by the above method. If it gives the correct result, return true and the data defining the means of computing the scale (v(2) for the various vals, corresponding commas, and offsets. Just giving the formula would be good.) Can anyone improve on this?
top of page bottom of page up down


Message: 8881 - Contents - Hide Contents

Date: Tue, 30 Dec 2003 17:25:17

Subject: Re: Meantone reduced blocks

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

>> The >> meatone reduction therefore is >> >> 7i - 12(round(i/3) + round(i/4+1/8)) >
> Please express this meantone scale in conventional letter-name-and- > accidental notation.
Here is the scale in terms of meantone fifths: [0, 7, -10, -3, 4, -1, -6, 1, -4, 3, -2, -7] Using an "f" for my flat symbol, here it is in sharps and flats: [C, C#, Eff, Ef, E, F, Gf, G, Af, A, Bf, Cf] I'm not sure yet how uncommon this sort of thing is, but probably not very common. There is only one possible {128/125, 648/625} scale up to transposition, and one question is what other comma pairs give us something differing from Meantone[12] on reduction.
top of page bottom of page up down


Message: 8883 - Contents - Hide Contents

Date: Tue, 30 Dec 2003 21:55:14

Subject: The Six Syndia Scales

From: Gene Ward Smith

These are the six possible Fokker blocks, up to transpositional 
equivalence, which can be obtained from the 81/80 (SYNtonic)
and 2048/2045 (DIAschismic) commas. Since midpoints between numbers 
of the form i/12 are numbers of the form n/24, I took all offsets
n/24 for n ranging from -12 to 12 to obtain these, though in fact 
taking only odd n should suffice. The first two are self-dual, or 
whatever the word is (and if there isn't one, there should be) for a 
scale transpositionally equivalent to its inverse. Then 3 and 4, 5 
and 6 are inversionally related pairs. When a name already existed in 
the Scala archives, I used that form of the scale, otherwise I just 
picked one of the 12 which looked nice. All of these on reduction by 
meantone lead to Meantone[12].

! syndia1.scl
First 81/80 2048/2025 Fokker block = ramis.scl
12
!
135/128
10/9
32/27
5/4
4/3
45/32
3/2
128/81
5/3
16/9
15/8
2

! syndia2.scl
Second 81/80 2048/2025 Fokker block
12
!
16/15
256/225
6/5
32/25
4/3
64/45
3/2
8/5
128/75
9/5
256/135
2

! syndia3.scl
Third 81/80 2048/2025 Fokker block
12
!
135/128
9/8
1215/1024
5/4
675/512
45/32
3/2
405/256
27/16
225/128
15/8
2

! syndia4.scl
Fourth 81/80 2048/2025 Fokker block
12
!
135/128
9/8
6/5
5/4
4/3
45/32
3/2
8/5
27/16
16/9
15/8
2

! syndia5.scl
Fifth 81/80 2048/2025 Fokker block = pipedum_12.scl
12
!
135/128
9/8
75/64
5/4
4/3
45/32
3/2
405/256
5/3
16/9
15/8
2

! syndia6.scl
Sixth 81/80 2048/2025 Fokker block
12
!
135/128
9/8
6/5
5/4
4/3
45/32
3/2
405/256
27/16
16/9
15/8
2


top of page bottom of page up down


Message: 8884 - Contents - Hide Contents

Date: Tue, 30 Dec 2003 02:08:04

Subject: Meantone reduced blocks

From: Gene Ward Smith

It would be nice to classify 12-note, 5-limit Fokker blocks at least
up to meantone reduction. While pondering that, I thought I'd see how
an example which does not reduce to Meantone[12] worked out.

The "thirds" scale, the genus derived from 6/5 and 5/4, can be
analyzed as a Fokker block using the method I've given as 

thirds[i] = (25/24)^i (128/125)^round(i/3) (648/625)^round(i/4+1/8)

In meantone, 25/24 maps to 7, and 128/125 and 648/625 to -12. The
meatone reduction therefore is

7i - 12(round(i/3) + round(i/4+1/8))

I don't see a better way offhand of writing this function. Graphing it
gives a result which looks somewhat erratic.


top of page bottom of page up down


Message: 8885 - Contents - Hide Contents

Date: Tue, 30 Dec 2003 02:09:26

Subject: Re: 5-limit, 12-note Fokker blocks (attn Manuel)

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:

> We can automate this process with the following steps > > 1. Find the set of intervals between steps. If there are more of these > than there are primes involved in the factorization of the scale > degrees, return false.
Sorry, I forgot my own analysis. Skip this step.
> 2. Check if the scale is epimorphic. If not, return false. > > 3. Pick a "base" element from the scale steps, by any means (first > step, smallest step, most common step, least Tenney height, etc.) Now > take ratios of the basis element with the other scale steps, finding > (n-1) other ratios (where n is the number of primes involved) which > together with the basis element gives a matrix of row vectors whose > determinant is +-1 (unimodular matrix.) If this is impossible, return > false. > > 4. Invert the above unimodular matrix, and obtain vals from the > columns of the inverted matrix. > > 5. For each of these vals aside from the first, take the maximum and > minimum values of v(2)*i/N, where N is the number of scale steps and i > goes from 0 to N-1, and find the average of the maximum and minimum > (the midpoint value.) > > 6. Using the midpoint value as an offset, calculate a scale by the > above method. If it gives the correct result, return true and the data > defining the means of computing the scale (v(2) for the various vals, > corresponding commas, and offsets. Just giving the formula would be good.) > > Can anyone improve on this?
top of page bottom of page up down


Message: 8886 - Contents - Hide Contents

Date: Tue, 30 Dec 2003 08:56:52

Subject: Re: Meantone reduced blocks

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> It would be nice to classify 12-note, 5-limit Fokker blocks at least > up to meantone reduction. While pondering that, I thought I'd see how > an example which does not reduce to Meantone[12] worked out. > > The "thirds" scale, the genus derived from 6/5 and 5/4, can be > analyzed as a Fokker block using the method I've given as > > thirds[i] = (25/24)^i (128/125)^round(i/3) (648/625)^round(i/4+1/8) > > In meantone, 25/24 maps to 7, and 128/125 and 648/625 to -12. > > The > meatone reduction therefore is > > 7i - 12(round(i/3) + round(i/4+1/8))
Please express this meantone scale in conventional letter-name-and- accidental notation.
top of page bottom of page up down


Message: 8887 - Contents - Hide Contents

Date: Tue, 30 Dec 2003 16:54:18

Subject: Re: 5-limit, 12-note Fokker blocks (attn Manuel)

From: Manuel Op de Coul

I've paid attention. It will be a useful addition, so I've put it
on the todo list.

Manuel




________________________________________________________________________
________________________________________________________________________




------------------------------------------------------------------------
Yahoo! Groups Links

To visit your group on the web, go to:
 Yahoo groups: /tuning-math/ * [with cont.] 

To unsubscribe from this group, send an email to:
 tuning-math-unsubscribe@xxxxxxxxxxx.xxx

Your use of Yahoo! Groups is subject to:
 Yahoo! Terms of Service * [with cont.]  (Wayb.)


top of page bottom of page up down


Message: 8888 - Contents - Hide Contents

Date: Wed, 31 Dec 2003 00:44:18

Subject: The Four Syndie Scaless

From: Gene Ward Smith

The name this time comes from SYNtonic and DIEesis, or 81/80 and 
128/125. These, unsurprisingly, turn out to be already known; what is 
new is that there are only four of them. This time I did things 
systematically and made the meantone reduction run from -3 to 8. The 
first two are inversions of each other, and 3 and 4 are self-dual or 
inversionally similar.

! syndie1.scl
First Syndie scale ~ sauveur_ji.scl
12
!
135/128
9/8
6/5
5/4
27/20
45/32
3/2
25/16
27/16
9/5
15/8
2

! syndie2.scl
Second Syndie scale = fogliano1.scl
12
!
25/24
10/9
6/5
5/4
4/3
25/18
3/2
25/16
5/3
16/9
15/8
2

! syndie3.scl
Third Syndie scale ~ duodene.scl = efg33355.scl
12
!
25/24
10/9
32/27
5/4
4/3
25/18
40/27
25/16
5/3
16/9
50/27
2

! syndie.scl
Fourth Syndie scale = marpurg1.scl
12
!
25/24
9/8
6/5
5/4
4/3
45/32
3/2
25/16
5/3
9/5
15/8
2


top of page bottom of page up down


Message: 8889 - Contents - Hide Contents

Date: Wed, 31 Dec 2003 22:52:19

Subject: Re: The Two Diadie Scales

From: Carl Lumma

>The two scales using the DIAschisma and the DIEsis of 128/125 are >both known, and this seems like a Carl Lumma speciality. They don't >reduce to Meantone[12], but 22-et, pajara or orwell seem more to the >point. Reduction by 22-et or pajara leads to Pajara[12], but >reduction by orwell leads to two interesting new scales. Or at least >one is new, reducing the first diadie scale gives us something quite >close to lumma.scl, which Carl presented back in 1999.
I did a non-thorough by-hand search for 12-tone 5- and 7-limit 'Fokker blocks' (before I knew the term, and before the subject had been explored by the list -- I certainly wasn't checking for epimorphism or monotonicity). Some of this was done before I joined the list, on paper with the rectangular lattices I'd learned about from Doty's JI Primer. -Carl ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
top of page bottom of page up down


Message: 8890 - Contents - Hide Contents

Date: Wed, 31 Dec 2003 02:15:38

Subject: The Three Majsyn Scales

From: Gene Ward Smith

These are the three possible Fokker blocks arising from the MAJor 
diesis of 648 and the SYNtonic comma of 81/80. The first two are 
inversional pairs, the third is self-dual.

! majsyn1.scl
First Majsyn 648/625 81/80 scale
12
!
250/243
10/9
6/5
100/81
4/3
25/18
40/27
125/81
5/3
16/9
50/27
2

! majsyn2.scl
Second Majsyn 648/625 81/80 scale
12
!
25/24
10/9
6/5
5/4
4/3
25/18
3/2
125/81
5/3
9/5
50/27
2

! majsyn3.scl
Third Majsyn 648/625 81/80 scale ~ duodene_skew.scl
12
!
25/24
9/8
6/5
5/4
27/20
25/18
3/2
25/16
5/3
9/5
15/8
2


top of page bottom of page up down


Message: 8891 - Contents - Hide Contents

Date: Wed, 31 Dec 2003 08:31:08

Subject: The Twelve Ragisyn Scales

From: Gene Ward Smith

The interval 6561/6250 is shy of a minor semitone of 21/20 by a 
ragisma, so we may consider it a RAGIismic minor semitone; together
with the SYNtonic comma of 81/80 we have the name "ragisyn" for this 
type of Fokker block. Below I list all twelve of them; the first two
are self-dual, the rest are inversively related in successive pairs.
As before, they all reduce to -3 to 8 under the meantone val.


! ragisyn1.scl
Ragasyn 6561/6250 81/80 scale
12
!
250/243
10/9
6/5
100/81
4/3
25/18
3/2
125/81
5/3
9/5
50/27
2

! ragisyn2.scl
Ragisyn 6561/6250 81/80 scale
12
!
250/243
10/9
6/5
100/81
4/3
1000/729
40/27
10000/6561
400/243
16/9
4000/2187
2

! ragisyn3.scl
Ragasyn 6561/6250 81/80 scale
12
!
25/24
9/8
243/200
5/4
27/20
25/18
3/2
125/81
5/3
9/5
50/27
2

! ragisyn4.scl
Ragisyn 6561/6250 81/80 scale
12
!
250/243
10/9
6/5
100/81
4/3
1000/729
40/27
10000/6561
400/243
16/9
50/27
2

! ragisyn5.scl
Ragisyn 6561/6250 81/80 scale
12
!
250/243
10/9
6/5
100/81
4/3
1000/729
40/27
125/81
5/3
9/5
50/27
2

! ragisyn6.scl
Ragisyn 6561/6250 81/80 scale
12
!
250/243
10/9
6/5
5/4
27/20
25/18
3/2
125/81
5/3
9/5
50/27
2

! ragisyn7.scl
Ragisyn 6561/6250 81/80 scale
12
!
250/243
10/9
6/5
100/81
4/3
1000/729
40/27
10000/6561
400/243
9/5
50/27
2

! ragisyn8.scl
Ragasyn 6561/6250 81/80 scale
12
!
250/243
9/8
243/200
5/4
27/20
25/18
3/2
125/81
5/3
9/5
50/27
2

! ragisyn9.scl
Ragisyn 6561/6250 81/80 scale
12
!
250/243
10/9
6/5
100/81
4/3
1000/729
3/2
125/81
5/3
9/5
50/27
2

! ragisyn10.scl
Ragisyn 6561/6250 81/80 scale
12
!
250/243
10/9
6/5
100/81
27/20
25/18
3/2
125/81
5/3
9/5
50/27
2

! ragisyn11.scl
Ragisyn 6561/6250 81/80 scale
12
!
250/243
10/9
6/5
100/81
4/3
1000/729
40/27
10000/6561
5/3
9/5
50/27
2

! ragisyn12.scl
Ragisyn 6561/6250 81/80 scale
12
!
250/243
10/9
243/200
5/4
27/20
25/18
3/2
125/81
5/3
9/5
50/27
2




________________________________________________________________________
________________________________________________________________________




------------------------------------------------------------------------
Yahoo! Groups Links

To visit your group on the web, go to:
 Yahoo groups: /tuning-math/ * [with cont.] 

To unsubscribe from this group, send an email to:
 tuning-math-unsubscribe@xxxxxxxxxxx.xxx

Your use of Yahoo! Groups is subject to:
 Yahoo! Terms of Service * [with cont.]  (Wayb.)


top of page bottom of page up down


Message: 8892 - Contents - Hide Contents

Date: Thu, 01 Jan 2004 05:26:56

Subject: The Two Diadie Scales

From: Gene Ward Smith

Let me start out by saying that 81/80 with either the schisma or the 
pythagorean commas (or those two taken together) give us the 12-note 
Pythagorean scale, and that this completes the classification for 
scales using 81/80, unless you want to go past 0.75 in epimericity.

The two scales using the DIAschisma and the DIEsis of 128/125 are 
both known, and this seems like a Carl Lumma speciality. They don't 
reduce to Meantone[12], but 22-et, pajara or orwell seem more to the 
point. Reduction by 22-et or pajara leads to Pajara[12], but 
reduction by orwell leads to two interesting new scales. Or at least 
one is new, reducing the first diadie scale gives us something quite 
close to lumma.scl, which Carl presented back in 1999.

! diadie1.scl
First Diadie 2048/2025 128/125 scale = lumma5r.scl
12
!
16/15
9/8
75/64
5/4
4/3
45/32
3/2
8/5
5/3
225/128
15/8
2

! diadie2.scl
Second Diadie 2048/2025 128/125 scale ~ pipedum_12a.scl
12
!
16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
128/75
225/128
15/8
2

! diadieorw1.scl
84-et version of diadie1.scl (similar to lumma.scl)
12
!
114.285714
200.000000
271.428571
385.714286
500.000000
585.714286
700.000000
814.285714
885.714286
971.428571
1085.714286
1200.000000

! diadieorw2.scl
84-et version of diadie2.scl
12
!
114.285714
200.000000
314.285714
385.714286
500.000000
585.714286
700.000000
814.285714
928.571429
971.428571
1085.714286
1200.000000


top of page bottom of page up down


Message: 8893 - Contents - Hide Contents

Date: Thu, 01 Jan 2004 20:50:53

Subject: 225/224-planar equal temperaments

From: Gene Ward Smith

This is a kind of generic tuning to apply to large enough (and 12
seems to be large enough) 5-limit JI scales, since they always seem to
have these relations. I ran a badness filter on the rms 225/224-planar
temperaments, treating it just like a 5-limit JI computation but using
the adjusted values for 3 and 5, and got the following list of
temperaments, where the first column is the et (to 10000), the second
the 225/224-planar val, the third column where the standard val maps
225/224 (if this is 0, the standard val and the 225/224 val are the
same in the 7-limit) and the fourth the badness measure, set to be
less than 1. From 72 and 175 we can conclude that 7-limit miracle is a
good 225/224-planar tuning, but 156 and 228 give duodecimal (mapping
[<12 19 28 34|, <0 0 -1 -2|]) with rms error 1.5 cents vs 1.64 for
miracle. Beyond that, 403, 559 and 631 work; 559 gives us two vals,
one an excellent 5-limit JI val <559 886 1298| and the other an
excellent 225/224-planar val <559 885 1297|. Since there is nothing
especially magical about the rms 225/224 tuning versus, eg, minimax
the results beyond this point are pretty meaningless, I suspect. A
kind of adaptive tuning, where the first 559 val was used for purely 5
limit chords and the second, with major thirds and fifths shifted down
a schisma, for chords involving 7 would be possible.


1 [1, 2, 2] 0 .736966
2 [2, 3, 5] 0 .743974
3 [3, 5, 7] 1 .433823
4 [4, 6, 9] -1 .665419
5 [5, 8, 12] 1 .892828
7 [7, 11, 16] -1 .637629
12 [12, 19, 28] 0 .548311
15 [15, 24, 35] 1 .977303
19 [19, 30, 44] 0 .360557
31 [31, 49, 72] 0 .857806
53 [53, 84, 123] 0 .666832
72 [72, 114, 167] 0 .522038
84 [84, 133, 195] 0 .989711
103 [103, 163, 239] 0 .939217
156 [156, 247, 362] 0 .720534
175 [175, 277, 406] 0 .741466
228 [228, 361, 529] 0 .536746
403 [403, 638, 935] 4 .411588
559 [559, 885, 1297] 4 .946951
631 [631, 999, 1464] 4 .635276
1034 [1034, 1637, 2399] 7 .896596
1593 [1593, 2522, 3696] 11 .551130
1996 [1996, 3160, 4631] 15 .989239
2224 [2224, 3521, 5160] 14 .632155
2627 [2627, 4159, 6095] 18 .805402
3817 [3817, 6043, 8856] 25 .870642
4220 [4220, 6681, 9791] 29 .567603
6444 [6444, 10202, 14951] 41 .504831
8037 [8037, 12724, 18647] 50 .945557


top of page bottom of page up down


Message: 8894 - Contents - Hide Contents

Date: Thu, 01 Jan 2004 05:39:28

Subject: Re: The Two Diadie Scales

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

I made a triad circle (around the circle of fifths) for the two 
diadieorw scales, but neglected to include them. Here they are.

Diadieorw1 triad circle

-5 [385.714286, 314.285714, 700.0000000]
-4 [385.714286, 271.4285714, 657.1428571]
-3 [428.5714286, 271.4285714, 700.0000000]
-2 [428.5714286, 300.0000000, 728.5714286]
-1 [385.7142854, 314.2857143, 700.0000000]
0 [385.714286, 314.2857143, 700.0000000]
1 [385.714286, 314.2857143, 700.0000000]
2 [385.714286, 300.0000000, 685.7142857]
3 [428.5714286, 271.4285714, 700.0000000]
4 [428.5714286, 271.4285717, 700.0000000]
5 [385.714286, 314.2857143, 700.0000000]
6 [385.714286, 342.8571429, 728.5714286]

I count five excellent 84-et major triads, one weird flat one, 
and three supermajor triads.

Diadieorw2 triad circle

-5 [385.714286, 314.285714, 700.000000]
-4 [385.714286, 314.2857143, 700.0000000]
-3 [385.714286, 271.4285714, 657.1428571]
-2 [428.5714286, 300.0000000, 728.5714286]
-1 [428.5714283, 271.4285714, 700.0000000]
0 [385.714286, 314.2857143, 700.0000000]
1 [385.714286, 314.2857143, 700.0000000]
2 [385.714286, 342.8571429, 728.5714286]
3 [385.714286, 271.4285714, 657.1428571]
4 [428.5714286, 271.4285717, 700.0000000]
5 [428.5714286, 271.4285714, 700.0000000]
6 [385.714286, 342.8571429, 728.5714286]

Four 84-et major triads, three supermajor triads.


top of page bottom of page up down


Message: 8895 - Contents - Hide Contents

Date: Thu, 01 Jan 2004 20:54:55

Subject: Re: The Two Diadie Scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> Let me start out by saying that 81/80 with either the schisma or the > pythagorean commas (or those two taken together) give us the 12- note > Pythagorean scale, and that this completes the classification for > scales using 81/80, unless you want to go past 0.75 in epimericity. > > The two scales using the DIAschisma and the DIEsis of 128/125 are > both known, and this seems like a Carl Lumma speciality. They don't > reduce to Meantone[12], but 22-et, pajara or orwell seem more to the > point. Reduction by 22-et or pajara leads to Pajara[12],
Gene -- you keep saying Pajara but don't you mean Diaschismic?
top of page bottom of page up down


Message: 8896 - Contents - Hide Contents

Date: Thu, 01 Jan 2004 20:59:35

Subject: Re: The Two Diadie Scales

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

>> The two scales using the DIAschisma and the DIEsis of 128/125 are >> both known, and this seems like a Carl Lumma speciality. They don't >> reduce to Meantone[12], but 22-et, pajara or orwell seem more to > the
>> point. Reduction by 22-et or pajara leads to Pajara[12], >
> Gene -- you keep saying Pajara but don't you mean Diaschismic?
I'm assuming that in 22-equal, it is more correctly called Pajara.
top of page bottom of page up down


Message: 8897 - Contents - Hide Contents

Date: Thu, 01 Jan 2004 21:04:02

Subject: The Four Tertiadie Scales

From: Gene Ward Smith

If we consider 262144/253125 to be a third of a tone or a tertiatone, 
these may be named TERTIAtone DIEesis scales.


! tertiadie1.scl
First Tertiadie 262144/253125 128/125 scale
12
!
16/15
256/225
75/64
5/4
4/3
64/45
375/256
25/16
5/3
2048/1125
15/8
2

! tertiadie2.scl
Second Tertiadie 262144/253125 128/125 scale
12
!
16/15
1125/1024
75/64
5/4
4/3
5625/4096
3/2
8/5
128/75
225/128
15/8
2

! tertiadie3.scl
Third Tertiadie 262144/253125 128/125 scale
12
!
16/15
1125/1024
75/64
5/4
4/3
45/32
3/2
8/5
128/75
225/128
15/8
2

! tertiadie4.scl
Fourth Tertiadie 262144/253125 128/125 scale
12
!
2048/1875
32768/28125
6/5
32/25
512/375
8192/5625
3/2
8/5
128/75
2048/1125
15/8
2


top of page bottom of page up down


Message: 8898 - Contents - Hide Contents

Date: Thu, 01 Jan 2004 21:20:33

Subject: Re: The Two Diadie Scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >
>>> The two scales using the DIAschisma and the DIEsis of 128/125 are >>> both known, and this seems like a Carl Lumma speciality. They > don't
>>> reduce to Meantone[12], but 22-et, pajara or orwell seem more to >> the
>>> point. Reduction by 22-et or pajara leads to Pajara[12], >>
>> Gene -- you keep saying Pajara but don't you mean Diaschismic? >
> I'm assuming that in 22-equal, it is more correctly called Pajara.
No, Pajara is simply the 7-limit extension of Diaschsimic that you do get in 22-equal (and pretty much in no other ET): GX Networks * [with cont.] (Wayb.) As long as you're talking 5-limit though, there's no reason to bring Pajara into it.
top of page bottom of page up down


Message: 8899 - Contents - Hide Contents

Date: Thu, 01 Jan 2004 21:56:21

Subject: Re: The Two Diadie Scales

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> No, Pajara is simply the 7-limit extension of Diaschsimic that you do > get in 22-equal (and pretty much in no other ET): > > GX Networks * [with cont.] (Wayb.) > > As long as you're talking 5-limit though, there's no reason to bring > Pajara into it.
We are tuning Diaschismic so that the 7-limit works well, whether we want to acknowledge that fact or not. It's a Pajara tuning of Diaschismic, in other words, and so correctly called Pajara. Your method gives two precisely equal scales, one of which is called Diaschismic[12] and the other Pajara[12].
top of page bottom of page up

Previous Next

8000 8050 8100 8150 8200 8250 8300 8350 8400 8450 8500 8550 8600 8650 8700 8750 8800 8850 8900 8950

8850 - 8875 -

top of page