Yahoo Tuning Groups Ultimate Backup tuning Retuned via Scala

I just found out Scala can do a lot more in the retuning department
than I thought it could. I've uploaded some examples of retuning using
the out-of-order scale given below; since it has not yet been
determined if I am worthy to join tuning_files I've put the files here
for now.

/tuning/files/Gene/diatonictogamelan/brmqt1.mid
/tuning/files/Gene/diatonictogamelan/brmqt1t.mid
/tuning/files/Gene/diatonictogamelan/k330.mid
/tuning/files/Gene/diatonictogamelan/k330t.mid
/tuning/files/Gene/diatonictogamelan/k519.mid
/tuning/files/Gene/diatonictogamelan/k519t.mid

Here is the "scale":

! tra.scl
!
81/80 ==> 1029/512
12
!
-1232.777790
733.111116
-499.666674
1466.222232
233.444442
2199.333348
966.555558
-266.222232
1699.666674
466.888884
2432.777790
1200.000000

🔗Carl Lumma <ekin@lumma.org>

>I just found out Scala can do a lot more in the retuning department
>than I thought it could. I've uploaded some examples of retuning using
>the out-of-order scale given below;

What does the order matter?

>! tra.scl
>!
>81/80 ==> 1029/512
> 12
>!
>-1232.777790
>733.111116
>-499.666674
>1466.222232
>233.444442
>2199.333348
>966.555558
>-266.222232
>1699.666674
>466.888884
>2432.777790
>1200.000000

Since you appear to be using off-the-shelf midi files, how are
you not goofing them up with something like this?

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> What does the order matter?

Plenty. Try the following for size:

! back.scl
!
backwards transform
12
!
1100.000000
1000.000000
900.000000
800.000000
700.000000
600.000000
500.000000
400.000000
300.000000
200.000000
100.000000
1200.000000

Since Scala does not assume scales must be given in ascending order,
it actually retunes by remapping, allowing us to do various nifty
things with little effort.

> >! tra.scl
> >!
> >81/80 ==> 1029/512
> > 12
> >!
> >-1232.777790
> >733.111116
> >-499.666674
> >1466.222232
> >233.444442
> >2199.333348
> >966.555558
> >-266.222232
> >1699.666674
> >466.888884
> >2432.777790
> >1200.000000
>
> Since you appear to be using off-the-shelf midi files, how are
> you not goofing them up with something like this?

Because that isn't any old goofed-up transformation, but one which
sends an approximate 3/2 to an appoximate 7/4 and so 81/80~1 to
1029/512~2, and hence transforms diatonic harmony into something
harmonic, rather than simply wrecking it.

🔗Carl Lumma <ekin@lumma.org>

>>What does the order matter?
>
>Plenty. Try the following for size:
>
>! back.scl

This inverts everything. That's no big deal.

>> >! tra.scl
>> >!
>> >81/80 ==> 1029/512
>> > 12
>> >!
>> >-1232.777790
>> >733.111116
>> >-499.666674
>> >1466.222232
>> >233.444442
>> >2199.333348
>> >966.555558
>> >-266.222232
>> >1699.666674
>> >466.888884
>> >2432.777790
>> >1200.000000
>>
>> Since you appear to be using off-the-shelf midi files, how are
>> you not goofing them up with something like this?
>
>Because that isn't any old goofed-up transformation, but one which
>sends an approximate 3/2 to an appoximate 7/4 and so 81/80~1 to
>1029/512~2, and hence transforms diatonic harmony into something
>harmonic, rather than simply wrecking it.

I don't get it. Why should such a transform depend on the order?

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>What does the order matter?
> >
> >Plenty. Try the following for size:
> >
> >! back.scl
>
> This inverts everything. That's no big deal.

Does not. :)

> >Because that isn't any old goofed-up transformation, but one which
> >sends an approximate 3/2 to an appoximate 7/4 and so 81/80~1 to
> >1029/512~2, and hence transforms diatonic harmony into something
> >harmonic, rather than simply wrecking it.
>
> I don't get it. Why should such a transform depend on the order?

It tells you to send note 1 to f1, note 2 to f2, etc.

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

>>>>! tra.scl
>>>>!
>>>>81/80 ==> 1029/512
>>>> 12
>>>>!
>>>>-1232.777790
>>>>733.111116
>>>>-499.666674
>>>>1466.222232
>>>>233.444442
>>>>2199.333348
>>>>966.555558
>>>>-266.222232
>>>>1699.666674
>>>>466.888884
>>>>2432.777790
>>>>1200.000000
>>>
>>>Since you appear to be using off-the-shelf midi files, how are
>>>you not goofing them up with something like this?
>>
>>Because that isn't any old goofed-up transformation, but one which
>>sends an approximate 3/2 to an appoximate 7/4 and so 81/80~1 to
>>1029/512~2, and hence transforms diatonic harmony into something
>>harmonic, rather than simply wrecking it.
>
>I don't get it. Why should such a transform depend on the order?

Gene, perhaps you could tell us a little more about this fascinating
Thing? Anybody else following out there?

Why, for example, can't the 11th degree just be 32.777 cents?

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Gene, perhaps you could tell us a little more about this fascinating
> Thing? Anybody else following out there?
>
> Why, for example, can't the 11th degree just be 32.777 cents?

It could be; or it could be 1232.777 cents, which would mean it didn't
wander far from its starting position. This was an example retuning
scale, not the only one possible.

If n is a 12-et midi note number, we can set u = n/7 mod 12 (reduced
to the range -5..6) and v = (n - 7u)/12. Then n = 12v+7u. If for v we
substitute 1200 (changing to cents) and for u we put the approximation
for 7/4 in the 1029/1024 planar temperament, we get the scale I gave.
Other values of u lead to different results, and we can also always
ajust things by octaves _ad lib_. I've worked with the following scale
transforms, all of which give interesting results. Except for the
first, where it doesn't much matter, you should change the 1/1 in
Scala to correspond to the key.

! back.scl
!
backwards transform (fifth-->fourth reduced)
12
!
1100.000000
1000.000000
900.000000
800.000000
700.000000
600.000000
500.000000
400.000000
300.000000
200.000000
100.000000
1200.000000

tra.scl
!
81/80 ==> 1029/512
12
!
-1232.777790
733.111116
-499.666674
1466.222232
233.444442
2199.333348
966.555558
-266.222232
1699.666674
466.888884
2432.777790
1200.000000

! tre.scl
!
81/80 ==> 1029/512 ==> reduction
12
!
-32.778000
733.111000
700.333000
266.222000
233.444000
999.333000
966.556000
933.778000
499.667000
466.889000
1232.778000
1200.000000

! treb.scl
!
reversed 81/80 ==> 1029/512 ==> reduction
12
!
32.778000
466.889000
499.667000
933.778000
966.556000
999.333000
233.444000
266.222000
700.333000
733.111000
1167.22200
1200.000000

! trx.scl
!
reduced 3/2->7/6 5/4->11/6 scale
12
!
1086.963920
525.214432
412.178352
1050.428864
937.392784
375.643295
262.607216
149.571136
787.821648
674.785568
113.036080
1200.000000

! trxb.scl
!
reversed reduced 3/2->7/6 5/4->11/6 scale
12
!
113.036080
674.785568
787.821648
149.571136
262.607216
375.643295
937.392784
1050.428864
412.178352
525.214432
1086.963920
1200.000000

! tr7_13.scl
!
81/80 ==> 28561/28672
12
!
-610.538616
484.215446
-126.323169
968.430892
357.892277
1452.646339
842.107723
231.569108
1326.323169
715.784554
1810.538616
1200.000000

! tr7_13r.scl
!
reduced 81/80 ==> 28561/28672
12
!
589.461384
484.215446
1073.676831
968.430892
357.892277
252.646339
842.107723
231.569108
126.323169
715.784554
610.538616
1200.000000

! tr7_13b.scl
!
reverse reduced 81/80 ==> 28561/28672
12
!
610.538616
715.784554
126.323169
231.569108
842.107723
252.646339
357.892277
968.430892
1073.676831
484.215446
589.461384
1200.00000

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> If n is a 12-et midi note number, we can set u = n/7 mod 12 (reduced
> to the range -5..6) and v = (n - 7u)/12. Then n = 12v+7u. If for v we
> substitute 1200 (changing to cents) and for u we put the approximation
> for 7/4 in the 1029/1024 planar temperament, we get the scale I gave.
> Other values of u lead to different results, and we can also always
> ajust things by octaves _ad lib_.

To explain why this works, note that meantone has the following
property: it has a generator g (say, ~3/2) such that g and g^4 belong
to the same chord. If I take an approximate 7/4 instead of 3/2 for g,
I note that (7/4)^4 (or equivalently, 7^4) reduced to the octave is
2401/2024, which is near 7/6--(2401/2024) / (7/6) = 1029/1024. If I
pick a tuning for g which is a good one for the 1029/1024 temperament,
I have g and g^4 belonging to the same chord. Hence, the mapping above
sends diatonic harmony {2,3,5} 81/80-harmony to {2,3,7} 1029/1024-harmony.

Other things belonging to the same chord, more or less, are
(13/8) and (13/8)^4~7, and 7/6 and (7/6)^4~11/6.

🔗Carl Lumma <ekin@lumma.org>

>> Gene, perhaps you could tell us a little more about this fascinating
>> Thing? Anybody else following out there?
>>
>> Why, for example, can't the 11th degree just be 32.777 cents?
>
>It could be; or it could be 1232.777 cents, which would mean it didn't
>wander far from its starting position. This was an example retuning
>scale, not the only one possible.

Aha!

>If n is a 12-et midi note number, we can set u = n/7 mod 12 (reduced
>to the range -5..6) and v = (n - 7u)/12. Then n = 12v+7u. If for v we
>substitute 1200 (changing to cents) and for u we put the approximation
>for 7/4 in the 1029/1024 planar temperament, we get the scale I gave.

If n=2, what's u?

Substitute 1200 for v in what? ie "n = 12*1200 + 7u"?

>! tra.scl
>!
>81/80 ==> 1029/512
> 12
>!
>-1232.777790
>733.111116
>-499.666674
>1466.222232
>233.444442
>2199.333348
>966.555558
>-266.222232
>1699.666674
>466.888884
>2432.777790
>1200.000000
//
>! tr7_13b.scl
>!
>reverse reduced 81/80 ==> 28561/28672
> 12
>!
>610.538616
>715.784554
>126.323169
>231.569108
>842.107723
>252.646339
>357.892277
>968.430892
>1073.676831
>484.215446
>589.461384
>1200.00000

Well, this is fascinating.

-Carl

🔗Carl Lumma <ekin@lumma.org>

>> If n is a 12-et midi note number, we can set u = n/7 mod 12 (reduced
>> to the range -5..6) and v = (n - 7u)/12. Then n = 12v+7u. If for v we
>> substitute 1200 (changing to cents) and for u we put the approximation
>> for 7/4 in the 1029/1024 planar temperament, we get the scale I gave.
>> Other values of u lead to different results, and we can also always
>> ajust things by octaves _ad lib_.
>
>To explain why this works, note that meantone has the following
>property: it has a generator g (say, ~3/2) such that g and g^4 belong
>to the same chord. If I take an approximate 7/4 instead of 3/2 for g,
>I note that (7/4)^4 (or equivalently, 7^4) reduced to the octave is
>2401/2024, which is near 7/6--(2401/2024) / (7/6) = 1029/1024. If I
>pick a tuning for g which is a good one for the 1029/1024 temperament,
>I have g and g^4 belonging to the same chord. Hence, the mapping above
>sends diatonic harmony {2,3,5} 81/80-harmony to {2,3,7}
>1029/1024-harmony.

So what are the two maps involved (fill in the blanks)?

................2..3..5..7
g(meantone)....12..1..4..?
g(1029/1024)....?..?..?..1

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

Mod 12, we have 2/7 = 2*7 = 14 = 2; then v is (2-2*7)/12 = -12/12 = -
1. Now n = 12*(-1) + 7*2 = 2.

> Substitute 1200 for v in what? ie "n = 12*1200 + 7u"?

Now we can substitute, 1200*(-1) + 7*g, where g is whatever we want
(say, an approximate 7/4.)

🔗Carl Lumma <ekin@lumma.org>

>>>If n is a 12-et midi note number, we can set u = n/7 mod 12
>>>(reduced to the range -5..6) and v = (n - 7u)/12.
>>>Then n = 12v+7u. If for v we substitute 1200 (changing to
>>>cents) and for u we put the approximation for 7/4 in the
>>>1029/1024 planar temperament, we get the scale I gave.
>>
>>If n=2, what's u?
>
>Mod 12, we have 2/7 = 2*7 = 14 = 2; then v is
>(2-2*7)/12 = -12/12 = - 1. Now n = 12*(-1) + 7*2 = 2.

K, I follow that except for 2/7 = 2*7 ... does mod allow one
to convert / to * or something?

Hrm, so...

n - an interval we want to remap, in steps of a linear temperament

g - the generator of the linear temperament in cents
(in this case, 700 cents)

u - the generator of the linear temperament in steps
(in this case, 7)

v - ?

>> Substitute 1200 for v in what? ie "n = 12*1200 + 7u"?
>
>Now we can substitute, 1200*(-1) + 7*g, where g is whatever we want
>(say, an approximate 7/4.)

You lost me. This looks vaguely like n = 12v+7u, where *12* has
been substituted with 1200, and u with g.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >Mod 12, we have 2/7 = 2*7 = 14 = 2; then v is
> >(2-2*7)/12 = -12/12 = - 1. Now n = 12*(-1) + 7*2 = 2.
>
> K, I follow that except for 2/7 = 2*7 ... does mod allow one
> to convert / to * or something?

Modulo 12, all of the invertible elements (which are the ones
relatively prime to 12) are their own inverses, so / and * are the same:

1^2 = 1, 5^2 = 25 = 1, 7^2 = 49 = 1, 11^2 = 121 = 1

> Hrm, so...
>
> n - an interval we want to remap, in steps of a linear temperament

If you are using the "12" and "7" business, you are assuming you are
remapping 12 diatonic notes.

> g - the generator of the linear temperament in cents
> (in this case, 700 cents)
>
> u - the generator of the linear temperament in steps
> (in this case, 7)
>
> v - ?

v is the number of octaves.

> >> Substitute 1200 for v in what? ie "n = 12*1200 + 7u"?

n = 12*v + 7*u.

Now, in terms of cents,

1200*v + 676.578*u gives 1/4 comma meantone

1200*v + 700*u gives 12-et

1200*v + 500*u, reduced mod 1200, gives the "back.scl" scale

1200*v + 966.55555*u gives the "tra.scl" scale

etc.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> > wrote:
> > >
> > > Hence, the mapping above
> > > sends diatonic harmony {2,3,5} 81/80-harmony to {2,3,7}
1029/1024-
> > >harmony.
> >
> > is this basically what herman miller did in mapping pachelbel's
canon
> > to blackjack tuning?
>
> I doubt it. Pachbel is 5-limit JI,

i disagree, based on having tried to perform it that way.

> and maps easily to the known
> universe without any such tricks anyway.

what do you mean by the known universe and by tricks?

i guess i just meant, is one of his blackjack renditions of
pachelbel's canon equivalent to the result of doing this?

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > > I doubt it. Pachbel is 5-limit JI,
> >
> > i disagree, based on having tried to perform it that way.
>
> Sorry, I was going by what Herman said about it.
>
> > i guess i just meant, is one of his blackjack renditions of
> > pachelbel's canon equivalent to the result of doing this?
>
> You'd need to ask him, but it seems unlikely.

why is it unlikely? doesn't he reinterpret the {2,3,5} dimension as
{2,3,7}, effecting exactly the mapping you were talking about?

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > why is it unlikely? doesn't he reinterpret the {2,3,5} dimension
as
> > {2,3,7}, effecting exactly the mapping you were talking about?
>
> No, that doesn't do it. There is a map from 5 to 7 limit which
works,
> but even that may not give the same results, since it involves first
> lifting a diatonic piece to 5-limit JI.

whoops, i wrote that wrong . . . anyway, check out the 7-limit JI
utonal (1/1:7/6:7/4) version -- close to what you're doing, yes?

did you actually read far enough down herman's page to see what he's
doing?:

"
Blackjack harmonies are represented here in decimal notation.
Although Blackjack doesn't have enough fifths in a row for a
traditional diatonic scale, it is rich in other harmonic structures.
For the Blackjack retunings, I've tried a number of different ways to
remap the harmony of the canon in a way that fits into the Blackjack
lattice. Two of these tunings exploit the 7-limit harmonies that
Blackjack is known for; there is an otonal version based on the
harmonic series, approximating 4:5:7, and a utonal version
approximating 1/(7:5:4). There is also a strange beating version,
based on a discordant 1/1 : 11/9 : 32/21 triad, and a version that
shows off the very small steps available in the Blackjack tuning,
based on a 7:8:10 triad. The structure of the remapped scales works
like this:

otonal utonal beating microstep
5> 3> 1> 7 5 3 7 3 9> 7 2 7>
2 0 8> 6> 2 0 8> 6> 4 0 6> 2> 5 0 5> 0>

original otonal utonal beating microstep
D 0 0 0 0
E 6> 6> 2> 0>
F# 3> 5 3 2
G 2 2 4 5
A 8> 8> 6> 5>
B 5> 7 7 7
C# 1> 3 9> 7>

Notice what this does to the melody!
Original: F# E D C# B A B C# | D C# B A G F# G E
otonal 3> 6> 0 1> 5> 8> 5> 1> | 0 1> 5> 8> 2 3> 2 6>
utonal 5 6> 0 3 7 8> 7 3 | 0 3 7 8> 2 5 2 6>
beating 3 2> 0 9> 7 6> 7 9> | 0 9> 7 6> 4 3 4 2>
microstep 2 0> 0 7> 7 5> 7 7> | 0 7> 7 5> 5 2 5 0>
"

🔗Carl Lumma <ekin@lumma.org>

>> >Mod 12, we have 2/7 = 2*7 = 14 = 2; then v is
>> >(2-2*7)/12 = -12/12 = - 1. Now n = 12*(-1) + 7*2 = 2.
>>
>> K, I follow that except for 2/7 = 2*7 ... does mod allow one
>> to convert / to * or something?
>
>Modulo 12, all of the invertible elements (which are the ones
>relatively prime to 12) are their own inverses, so / and * are
>the same:
>
>1^2 = 1, 5^2 = 25 = 1, 7^2 = 49 = 1, 11^2 = 121 = 1

Cool; thanks.

>> Hrm, so...
>>
>> n - an interval we want to remap, in steps of a linear temperament
>
>If you are using the "12" and "7" business, you are assuming you are
>remapping 12 diatonic notes.

Yeah, I'm hoping for a generalized version by the end of this.

>Now, in terms of cents,
>
>1200*v + 676.578*u gives 1/4 comma meantone
>
>1200*v + 700*u gives 12-et
>
>1200*v + 500*u, reduced mod 1200, gives the "back.scl" scale
>
>1200*v + 966.55555*u gives the "tra.scl" scale
>
>etc.

Ah, I see what you're doing. This should be published (here, at
least).

-Carl

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Except for the
> >first, where it doesn't much matter, you should change the 1/1 in
> >Scala to correspond to the key.
>
> What command have you been using to do that?
>
> Based on a file in C, these seem more destructive than the Brahms
> examples would indicate, treb being the least so.

I was talking about "back.scl", where setting the 1/1 changes the
voice leading and melodic line, but not the octave-equivalent
harmony. Setting the 1/1 to a whole tone below the key seems to work
pretty well. Of course you can always try all twelve choices and see
which one you prefer.

🔗Carl Lumma <ekin@lumma.org>

>>>Except for the first, where it doesn't much matter, you should
>>>change the 1/1 in Scala to correspond to the key.
>>
>> What command have you been using to do that?
>>
>> Based on a file in C, these seem more destructive than the Brahms
>> examples would indicate, treb being the least so.
>
>I was talking about "back.scl", where setting the 1/1 changes the
>voice leading and melodic line, but not the octave-equivalent
>harmony.

Huh, I wouldn't think it would change anything, and in fact that's
what I thought you were saying: the key "doesn't much matter" for
back.scl. Anyway...

>Setting the 1/1 to a whole tone below the key seems to work
>pretty well. Of course you can always try all twelve choices
>and see which one you prefer.

Sure. What command are you using?

-Carl