Loading ...
Sorry, an error occurred while loading the content.

A new rational well-temperament

Expand Messages
  • Aaron Krister Johnson
    Hi, Spurred on by my recent Python code for rational approximations, and wanting for some time to develop a well-temperament with 24/19 instead of 81/64 as a
    Message 1 of 16 , Jun 1, 2006
    • 0 Attachment
      Hi,

      Spurred on by my recent Python code for rational approximations, and
      wanting for some time to develop a well-temperament with 24/19 instead
      of 81/64 as a wide-third basis, and inspired by George Secor and Gene
      Ward Smith's work in the area of rational temperament, I came up with
      the following yesterday.

      The idea is to have the backbone thirds E-G# and Ab-C be 24/19, and
      C-E is of course the octave residue of that. Other than that, I tried
      to use the smallest rational approximations I could while preserving
      traditional well-temperament qualities.

      Tune it up and play...I would love some comments, and I hope I might
      inspire others to take this work further, or improve it!

      ! johnson_ratwell.scl
      !
      a rational well-temperament with five 24/19's
      12
      !
      19/18
      103/92
      32/27
      361/288
      4/3
      38/27
      208/139
      19/12
      129/77
      16/9
      152/81
      2/1
    • Gene Ward Smith
      ... Great! This scale is epimorphic in more than one way, so it s a nice example among other talents. It s also an authentic well-temperament, with no fifth
      Message 2 of 16 , Jun 1, 2006
      • 0 Attachment
        --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:

        > Tune it up and play...I would love some comments, and I hope I might
        > inspire others to take this work further, or improve it!

        Great! This scale is epimorphic in more than one way, so it's a nice
        example among other talents. It's also an authentic well-temperament,
        with no fifth wider than 3/2. Scala tells me that this is similar to
        Herman Miller's "Arrow" temperaments, but searching did turn those up,
        so I hope Herman can explain.

        This mild well-temperament should suit nineteenth century music pretty
        well.
      • a_sparschuh
        ... C# 19/18 == (256/243)*(513/512) D 103/92 = (9/8)*(206/207) Eb 32/27 == (6/5)*(81/80) pyth. minor 3rd E 361/288= (5/4)*(361/360) F 4/3 F#
        Message 3 of 16 , Jun 1, 2006
        • 0 Attachment
          --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:
          >
          > ! johnson_ratwell.scl
          > !
          > a rational well-temperament with five 24/19's
          > 12
          > !
          C#> 19/18 == (256/243)*(513/512)
          D > 103/92 = (9/8)*(206/207)
          Eb> 32/27 == (6/5)*(81/80) pyth. minor 3rd
          E > 361/288= (5/4)*(361/360)
          F > 4/3
          F#> 38/27 == (1024/729)*(513/512)
          G > 208/139= (3/2)*(416/417)
          G#> 19/12 == (128/81)*(513/512)
          A > 129/77 = (5/3)*(387/385)=(27/16)*(688/693)both none-epomoric!
          Bb> 16/9
          b > 152/81 = (15/8)*(1216/1215)
          C'> 2/1

          As far as i'm able to see:
          All -but except yours "A"- deviate only from just-pure merely
          about an small epimoric cofactor in order to yield the tempering.

          Hence i can't understand:
          Why did you took the special "A" in a different way from its
          superparticular neighbourhood, unlike yours other 11 ratios?
          Please -be so kind to- explain me yours extraordinary choice on "A".
          Question: Why became that "A" not epimoric-deviating too?
          A.S.
        • Yahya Abdal-Aziz
          Hi all, ... With G# =Ab ? ... Great name! At first I thought, I know who Johnson is, but who is Ratwell?! ;-) ... Well, Aaron, I hope some day to
          Message 4 of 16 , Jun 1, 2006
          • 0 Attachment
            Hi all,

            On Thu Jun 1, 2006, Aaron Krister Johnson wrote:
            >
            > Hi,
            >
            > Spurred on by my recent Python code for rational approximations, and
            > wanting for some time to develop a well-temperament with 24/19 instead
            > of 81/64 as a wide-third basis, and inspired by George Secor and Gene
            > Ward Smith's work in the area of rational temperament, I came up with
            > the following yesterday.
            >
            > The idea is to have the backbone thirds E-G# and Ab-C be 24/19, and
            > C-E is of course the octave residue of that. ...

            With G# =Ab ?


            > ... Other than that, I tried
            > to use the smallest rational approximations I could while preserving
            > traditional well-temperament qualities.
            >
            > Tune it up and play...I would love some comments, and I hope I might
            > inspire others to take this work further, or improve it!
            >
            > ! johnson_ratwell.scl

            Great name! At first I thought, "I know who Johnson
            is, but who is Ratwell?!" ;-)


            > !
            > a rational well-temperament with five 24/19's
            > 12
            > !
            > 19/18
            > 103/92
            > 32/27
            > 361/288
            > 4/3
            > 38/27
            > 208/139
            > 19/12
            > 129/77
            > 16/9
            > 152/81
            > 2/1

            Well, Aaron, I hope some day to understand the
            virtues of a well-temperament well enough to
            use one. (Oh, OK, I do use 12-EDO for jazzy
            stuff, and for first audition of JI stuff.) But
            since most of my music doesn't require extensive
            key modulation, I don't expect I can be much use
            to you at present with this temperament - anything
            I wrote using it would almost certainly not exploit
            its potential particularly well.

            Still, I've never knowingly used the 19 limit, and
            it might be fun to try!

            Regards,
            Yahya

            --
            No virus found in this outgoing message.
            Checked by AVG Free Edition.
            Version: 7.1.394 / Virus Database: 268.8.0/353 - Release Date: 31/5/06
          • Keenan Pepper
            ... Of course; that s what makes it a well temperament. Unequal but closed.
            Message 5 of 16 , Jun 1, 2006
            • 0 Attachment
              On 6/1/06, Yahya Abdal-Aziz <yahya@...> wrote:
              > With G# =Ab ?

              Of course; that's what makes it a well temperament. Unequal but closed.
            • Gene Ward Smith
              ... Scala adds no fifth greater than 3/2 to the definition; otherwise, I suppose, it is extraordinaire.
              Message 6 of 16 , Jun 1, 2006
              • 0 Attachment
                --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
                >
                > On 6/1/06, Yahya Abdal-Aziz <yahya@...> wrote:
                > > With G# =Ab ?
                >
                > Of course; that's what makes it a well temperament. Unequal but closed.

                Scala adds "no fifth greater than 3/2" to the definition; otherwise, I
                suppose, it is extraordinaire.
              • Aaron Krister Johnson
                ... A . ... Hi, Well, I hadn t thought about it that way until you pointed it out.... ... My calculations indicate that we could change the A to 191/114 and
                Message 7 of 16 , Jun 1, 2006
                • 0 Attachment
                  --- In tuning@yahoogroups.com, "a_sparschuh" <a_sparschuh@...>
                  wrote:
                  >
                  > --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@>
                  wrote:
                  > >
                  > > ! johnson_ratwell.scl
                  > > !
                  > > a rational well-temperament with five 24/19's
                  > > 12
                  > > !
                  > C#> 19/18 == (256/243)*(513/512)
                  > D > 103/92 = (9/8)*(206/207)
                  > Eb> 32/27 == (6/5)*(81/80) pyth. minor 3rd
                  > E > 361/288= (5/4)*(361/360)
                  > F > 4/3
                  > F#> 38/27 == (1024/729)*(513/512)
                  > G > 208/139= (3/2)*(416/417)
                  > G#> 19/12 == (128/81)*(513/512)
                  > A > 129/77 = (5/3)*(387/385)=(27/16)*(688/693)both none-epomoric!
                  > Bb> 16/9
                  > b > 152/81 = (15/8)*(1216/1215)
                  > C'> 2/1
                  >
                  > As far as i'm able to see:
                  > All -but except yours "A"- deviate only from just-pure merely
                  > about an small epimoric cofactor in order to yield the tempering.
                  >
                  > Hence i can't understand:
                  > Why did you took the special "A" in a different way from its
                  > superparticular neighbourhood, unlike yours other 11 ratios?
                  > Please -be so kind to- explain me yours extraordinary choice on
                  "A".
                  > Question: Why became that "A" not epimoric-deviating too?
                  > A.S.

                  Hi,

                  Well, I hadn't thought about it that way until you pointed it
                  out....
                  :)
                  My calculations indicate that we could change the 'A' to 191/114 and
                  preserve that property entirely....any comments, Gene, or George?

                  It's possible for 'D' to be 19/17 or 28/25, too, but I don't like
                  the step sizes that result as much, so I traded them for higher
                  ratios.

                  -Aaron.
                • Carl Lumma
                  ... Insired by this, I came up with: ! 12_moh-ha-ha.scl ! Rational well temperament. 12 ! 19/18 323/288 19/16 323/256 171/128 361/256 551/368 19/12 323/192
                  Message 8 of 16 , Jun 1, 2006
                  • 0 Attachment
                    > Hi,
                    >
                    > Spurred on by my recent Python code for rational approximations,
                    > and wanting for some time to develop a well-temperament with
                    > 24/19 instead of 81/64 as a wide-third basis, and inspired by
                    > George Secor and Gene Ward Smith's work in the area of rational
                    > temperament, I came up with the following yesterday.

                    Insired by this, I came up with:

                    ! 12_moh-ha-ha.scl
                    !
                    Rational well temperament.
                    12
                    !
                    19/18
                    323/288
                    19/16
                    323/256
                    171/128
                    361/256
                    551/368
                    19/12
                    323/192
                    57/32
                    513/272
                    2
                    !

                    and

                    ! 12_fun.scl
                    !
                    Rational well temperament based on 577/289, 3/2, and 19/16.
                    12
                    !
                    19/18
                    18464/16473
                    19/16
                    361/288
                    1154/867
                    361/256
                    73856/49419
                    10963/6936
                    9232/5491
                    4616/2601
                    208297/110976
                    577/289
                    !

                    The first is a pure-octaves scale based on direct approximations
                    to 12-tET with 'simple' ratios. It's similar to Aaron's, but
                    swaps two of his '24/19' thirds for one '81/80' third on C#.

                    The second uses flat octaves, and is built from three
                    19/16-based 'diminished 7th' chords rooted on adjacent 3:2
                    fifths.

                    And don't forget strangeion...

                    ! 12_strangeion.scl
                    !
                    19-limit "dodekaphonic" scale.
                    12
                    !
                    17/16 !.......C#
                    19/17 !........D
                    19/16 !.......D#
                    323/256 !......E
                    8192/6137 !....F
                    361/256 !.....F#
                    6137/4096 !....G
                    512/323 !.....G#
                    32/19 !........A
                    34/19 !.......A#
                    32/17 !........B
                    2/1 !..........C
                    !
                    ! F#--G
                    ! / \ /
                    ! D---D#--E
                    ! / \ / \ /
                    ! B---C---C#
                    ! / \ / \ /
                    ! G#--A---A#
                    ! /
                    ! F
                    !
                    ! --- = 17/16
                    ! / = 19/16

                    I'd love to hear anybody's reactions to playing with these.

                    -Carl
                  • Gene Ward Smith
                    ... It s fine by me, though personally I find the 139-limit quite higher enough without going all the way to the 191 limit.
                    Message 9 of 16 , Jun 1, 2006
                    • 0 Attachment
                      --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:

                      > My calculations indicate that we could change the 'A' to 191/114 and
                      > preserve that property entirely....any comments, Gene, or George?

                      It's fine by me, though personally I find the 139-limit quite higher
                      enough without going all the way to the 191 limit.
                    • Aaron Krister Johnson
                      Cool! I ll have to check these out........ -Aaron.
                      Message 10 of 16 , Jun 2, 2006
                      • 0 Attachment
                        Cool! I'll have to check these out........

                        -Aaron.


                        --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
                        >
                        > > Hi,
                        > >
                        > > Spurred on by my recent Python code for rational approximations,
                        > > and wanting for some time to develop a well-temperament with
                        > > 24/19 instead of 81/64 as a wide-third basis, and inspired by
                        > > George Secor and Gene Ward Smith's work in the area of rational
                        > > temperament, I came up with the following yesterday.
                        >
                        > Insired by this, I came up with:
                        >
                        > ! 12_moh-ha-ha.scl
                        > !
                        > Rational well temperament.
                        > 12
                        > !
                        > 19/18
                        > 323/288
                        > 19/16
                        > 323/256
                        > 171/128
                        > 361/256
                        > 551/368
                        > 19/12
                        > 323/192
                        > 57/32
                        > 513/272
                        > 2
                        > !
                        >
                        > and
                        >
                        > ! 12_fun.scl
                        > !
                        > Rational well temperament based on 577/289, 3/2, and 19/16.
                        > 12
                        > !
                        > 19/18
                        > 18464/16473
                        > 19/16
                        > 361/288
                        > 1154/867
                        > 361/256
                        > 73856/49419
                        > 10963/6936
                        > 9232/5491
                        > 4616/2601
                        > 208297/110976
                        > 577/289
                        > !
                        >
                        > The first is a pure-octaves scale based on direct approximations
                        > to 12-tET with 'simple' ratios. It's similar to Aaron's, but
                        > swaps two of his '24/19' thirds for one '81/80' third on C#.
                        >
                        > The second uses flat octaves, and is built from three
                        > 19/16-based 'diminished 7th' chords rooted on adjacent 3:2
                        > fifths.
                        >
                        > And don't forget strangeion...
                        >
                        > ! 12_strangeion.scl
                        > !
                        > 19-limit "dodekaphonic" scale.
                        > 12
                        > !
                        > 17/16 !.......C#
                        > 19/17 !........D
                        > 19/16 !.......D#
                        > 323/256 !......E
                        > 8192/6137 !....F
                        > 361/256 !.....F#
                        > 6137/4096 !....G
                        > 512/323 !.....G#
                        > 32/19 !........A
                        > 34/19 !.......A#
                        > 32/17 !........B
                        > 2/1 !..........C
                        > !
                        > ! F#--G
                        > ! / \ /
                        > ! D---D#--E
                        > ! / \ / \ /
                        > ! B---C---C#
                        > ! / \ / \ /
                        > ! G#--A---A#
                        > ! /
                        > ! F
                        > !
                        > ! --- = 17/16
                        > ! / = 19/16
                        >
                        > I'd love to hear anybody's reactions to playing with these.
                        >
                        > -Carl
                        >
                      • Aaron Krister Johnson
                        ... and ... George? ... higher ... So does that mean you would prefer the first version? How important to you theoretically (or even sonically--although with
                        Message 11 of 16 , Jun 2, 2006
                        • 0 Attachment
                          --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
                          wrote:
                          >
                          > --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@>
                          wrote:
                          >
                          > > My calculations indicate that we could change the 'A' to 191/114
                          and
                          > > preserve that property entirely....any comments, Gene, or
                          George?
                          >
                          > It's fine by me, though personally I find the 139-limit quite
                          higher
                          > enough without going all the way to the 191 limit.

                          So does that mean you would prefer the first version? How important
                          to you theoretically (or even sonically--although with trying it, I
                          suspect it's hard to notice) would the 'A' missing a superparticular
                          co-factor be?

                          Are there any ways to improve the scale I posted that would:
                          1) satisfy superparticular co-factor fetishes?
                          2) satisfy being lower than 139-limit?
                          3) keep the fifths from C to E sounding smooth and perceptibly
                          similar in size?

                          I can't see any right now........am I missing something?

                          -Aaron.
                        • a_sparschuh
                          ... hi! ... Hence it looks i.m.o. nearer to pythagorean than to syntonic, basing mostly on: http://tonalsoft.com/enc/x/xenharmonic-bridge.aspx Eratosthenes
                          Message 12 of 16 , Jun 2, 2006
                          • 0 Attachment
                            --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
                            hi!
                            > ! 12_moh-ha-ha.scl
                            > !
                            > Rational well temperament.
                            > 12
                            > !
                            > 19/18 ! = (256/243)(513/512)
                            > 323/288!= (9/8)(323/324) = (10/9)(323/320)
                            > 19/16 ! = (32/27)(513/512)
                            > 323/256!= (81/64)(323/324) = (5/4)(323/320)
                            > 171/128!= (4/3)(513/512)
                            > 361/256!= (45/32)(361/360)
                            > 551/368!= (3/2)(551/552)
                            > 19/12 ! = (128/81)(513/512)
                            > 323/192!= (27/16)(323/324) = (5/3)(323/320)
                            > 57/32 ! = (16/9)(513/512)
                            > 513/272!= (32/17)(513/512) = (15/8)(171/170)
                            > 2
                            > !
                            Hence it looks i.m.o. nearer to pythagorean than to syntonic,
                            basing mostly on:
                            http://tonalsoft.com/enc/x/xenharmonic-bridge.aspx
                            " Eratosthenes 3==19 bridge, so it skips 5 primes in between"

                            That epimoric riddle-play makes real fun.
                            I think above defactorized superparticular decompositions
                            tell more about how the tempering of the intervals is done,
                            than merely only the original bare(scl-)ratios alone.
                            A.S.
                          • Carl Lumma
                            ... Interesting. Thanks, A.S.! -Carl
                            Message 13 of 16 , Jun 2, 2006
                            • 0 Attachment
                              > > ! 12_moh-ha-ha.scl
                              > > !
                              > > Rational well temperament.
                              > > 12
                              > > !
                              > > 19/18 ! = (256/243)(513/512)
                              > > 323/288!= (9/8)(323/324) = (10/9)(323/320)
                              > > 19/16 ! = (32/27)(513/512)
                              > > 323/256!= (81/64)(323/324) = (5/4)(323/320)
                              > > 171/128!= (4/3)(513/512)
                              > > 361/256!= (45/32)(361/360)
                              > > 551/368!= (3/2)(551/552)
                              > > 19/12 ! = (128/81)(513/512)
                              > > 323/192!= (27/16)(323/324) = (5/3)(323/320)
                              > > 57/32 ! = (16/9)(513/512)
                              > > 513/272!= (32/17)(513/512) = (15/8)(171/170)
                              > > 2
                              > > !
                              > Hence it looks i.m.o. nearer to pythagorean than to syntonic,
                              > basing mostly on:
                              > http://tonalsoft.com/enc/x/xenharmonic-bridge.aspx
                              > " Eratosthenes 3==19 bridge, so it skips 5 primes in between"
                              >
                              > That epimoric riddle-play makes real fun.
                              > I think above defactorized superparticular decompositions
                              > tell more about how the tempering of the intervals is done,
                              > than merely only the original bare(scl-)ratios alone.
                              > A.S.

                              Interesting. Thanks, A.S.!

                              -Carl
                            • Gene Ward Smith
                              ... If I were to choose, yes. How important ... No importance whatever. But keeping the prime limit low only has the effect for me that when I run the show
                              Message 14 of 16 , Jun 2, 2006
                              • 0 Attachment
                                --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:

                                > > It's fine by me, though personally I find the 139-limit quite
                                > higher
                                > > enough without going all the way to the 191 limit.
                                >
                                > So does that mean you would prefer the first version?

                                If I were to choose, yes.

                                How important
                                > to you theoretically (or even sonically--although with trying it, I
                                > suspect it's hard to notice) would the 'A' missing a superparticular
                                > co-factor be?

                                No importance whatever. But keeping the prime limit low only has the
                                effect for me that when I run the "show data" command with Scala, it
                                can keep its enthusiasm within better bounds.
                              • George D. Secor
                                ... instead ... Gene ... with ... tried ... Aaron, sorry I ve taken so long to reply. This is really intriguing in that it: 1) produces 8 simple
                                Message 15 of 16 , Jun 5, 2006
                                • 0 Attachment
                                  --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
                                  wrote:
                                  >
                                  > Hi,
                                  >
                                  > Spurred on by my recent Python code for rational approximations, and
                                  > wanting for some time to develop a well-temperament with 24/19
                                  instead
                                  > of 81/64 as a wide-third basis, and inspired by George Secor and
                                  Gene
                                  > Ward Smith's work in the area of rational temperament, I came up
                                  with
                                  > the following yesterday.
                                  >
                                  > The idea is to have the backbone thirds E-G# and Ab-C be 24/19, and
                                  > C-E is of course the octave residue of that. Other than that, I
                                  tried
                                  > to use the smallest rational approximations I could while preserving
                                  > traditional well-temperament qualities.
                                  >
                                  > Tune it up and play...I would love some comments, and I hope I might
                                  > inspire others to take this work further, or improve it!
                                  >
                                  > ! johnson_ratwell.scl
                                  > !
                                  > a rational well-temperament with five 24/19's
                                  > 12
                                  > !
                                  > 19/18
                                  > 103/92
                                  > 32/27
                                  > 361/288
                                  > 4/3
                                  > 38/27
                                  > 208/139
                                  > 19/12
                                  > 129/77
                                  > 16/9
                                  > 152/81
                                  > 2/1

                                  Aaron, sorry I've taken so long to reply.

                                  This is really intriguing in that it:
                                  1) produces 8 simple proportional-beating major triads (on all of the
                                  most dissonant ones), while
                                  2) keeping the max error for the major 3rd around 18 cents.

                                  I was able to accomplish each of these things in separate well-
                                  temperaments, but not both at once. (And as Gene noted, it's an
                                  excellent well-temperament.)

                                  Unfortunately, the major brats on C, G, D, and A are not simple, so I
                                  couldn't resist seeing if those could be improved. By changing the
                                  ratios for G, D, and A I was able to get simpler brats: 2.75 for C,
                                  2.25 for D, and 2 for A, with a leftover of ~2.491803 for G (pretty
                                  close to 2.5):

                                  ! AKJ-GDS-RWT.scl
                                  !
                                  A.K. Johnson/G. Secor proportional-beating rational well-temperament
                                  with five 24/19's
                                  12
                                  !
                                  19/18
                                  3629/3240
                                  32/27
                                  361/288
                                  4/3
                                  38/27
                                  431/288
                                  19/12
                                  2413/1440
                                  16/9
                                  152/81
                                  2/1

                                  Half of the minor brats are exactly 1, and the others are not all
                                  that bad, considering that most of those are approximations of
                                  reasonably simple brats. I tried it in Scala, and I think it sounds
                                  pretty good! And the 6 just fifths should make it reasonably easy to
                                  tune by ear.

                                  I've had a couple of days to decide whether or not I prefer this to
                                  my rationalized Ellis #2 (Secor-VRWT.scl). It's not an easy call,
                                  but I think I would have to go with the VRWT because of:
                                  1) its higher key contrast (more consonant C major triad), and
                                  2) my personal preference for slightly tempered (vs. just) fifths on
                                  the worst triads -- which is to say, I prefer to have the total error
                                  of the fifths of the worst triads distributed more or less equally,
                                  as opposed to putting all of that error on 1 or 2 of the fifths.

                                  --George
                                • a_sparschuh
                                  ... A.S.
                                  Message 16 of 16 , Jun 6, 2006
                                  • 0 Attachment
                                    --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
                                    > A.K. Johnson/G. Secor proportional-beating rational well-temperament
                                    > with five 24/19's
                                    > 12
                                    > !
                                    > 19/18 ! = = (256/243)(513/512)
                                    > 3629/3240! =(9/8)(3629/3780) = (10/9)(3629/3600)
                                    > 32/27 ! = = (6/5)(80/81)
                                    > 361/288 ! = (5/4)((361/360)
                                    > 4/3 ! = = = (11/8)(32/33)
                                    > 38/27 ! = = (7/5)(190/189)
                                    > 431/288 ! = (3/2)(431/432)
                                    > 19/12 ! = = (25/16)(76/75)
                                    > 2413/1440 !=(5/3)(2413/2400)
                                    > 16/9 ! = = =(7/4)(64/63)
                                    > 152/81 ! = =(243/128)(513/512)
                                    > 2/1
                                    A.S.
                                  Your message has been successfully submitted and would be delivered to recipients shortly.