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2001: A MOS Odyssey

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  • ligonj@northstate.net
    2001 A MOS Odyssey Inspired by the many and various fifth generator posts of late, I would like to interject some of the treasures found among the Prime
    Message 1 of 28 , Jan 19 8:01 PM
      2001 A MOS Odyssey

      Inspired by the many and various "fifth generator" posts of late, I
      would like to interject some of the treasures found among the Prime
      Series, with regard to various widths of prime ratio fifths, which
      generate scales capable of optimizing chosen harmonic intervals. In
      these scales, thirds of 3-17 Limit are optimized:

      Scales 1-10, constructed from Prime Series Ratios chains of fifths
      (seen in order of generator widths):

      #1

      Generator Approximated Ratios
      709/479 678.912
      0
      110.208
      157.824
      268.032 7/6 266.871
      315.648 6/5 315.641
      473.472
      631.296
      678.912
      789.12
      836.736
      994.56
      1152.384
      1200

      #2

      Generator Approximated Ratios
      877/587 695.060
      0
      65.417
      190.119
      255.536
      380.238
      445.656 22/17 446.363
      570.358
      695.06
      760.477
      885.179
      950.596
      1075.298
      1200

      #3

      Generator Approximated Ratios
      1087/727 696.390
      0
      74.727
      192.779
      267.506 7/6 266.871
      385.558 5/4 386.314
      460.286
      578.338
      696.39
      771.117
      889.169
      963.896
      1081.948
      1200

      #4

      Generator Approximated Ratios
      809/541 696.613
      0
      76.293
      193.227
      269.52 7/6 266.871
      386.453 5/4 386.314
      462.747
      579.68
      696.613
      772.907
      889.84
      966.133
      1083.067
      1200

      #5

      Generator Approximated Ratios
      271/181 698.764
      0
      91.346
      197.528
      288.874 13/11 289.210
      395.055
      486.402
      592.583
      698.764
      790.11
      896.291
      987.638
      1093.819
      1200

      #6

      Generator Approximated Ratios
      461/307 703.834
      0
      126.836
      207.667
      334.503 17/14 336.130
      415.335 14/11 417.508
      542.171
      623.002
      703.834
      830.67
      911.501
      1038.337
      1119.169
      1200

      #7 (for Margo Schulter)

      Generator Approximated Ratios
      359/239 704.368
      0
      130.575
      208.736
      339.311
      417.471 14/11 417.508
      548.047
      626.207
      704.368
      834.943
      913.104
      1043.679
      1121.839
      1200

      #8

      Generator Approximated Ratios
      269/179 705.176
      0
      136.231
      210.352
      346.583 11/9 347.408
      420.704 14/11 417.508
      556.935
      631.055
      705.176
      841.407
      915.528
      1051.759
      1125.88
      1200

      #9

      Generator Approximated Ratios
      191/127 706.493
      0
      145.451
      212.986
      358.437 16/13 359.472
      425.972
      571.423
      638.958
      706.493
      851.944
      919.479
      1064.93
      1132.465
      1200

      #10

      Generator Approximated Ratios
      631/419 708.828
      0
      161.794
      217.655
      379.449
      435.311 9/7 435.084
      597.105
      652.966
      708.828
      870.622
      926.483
      1088.277
      1144.139
      1200



      Feel free to try these out if you enjoy MOS. They are quite lovely
      sounding actually.



      Thanks,

      Jacky Ligon
    • D.Stearns
      Hi Jacky, The first one wouldn t be MOS as it s a three-stepsize 12-note set of 5 small 3 medium and 4 large stepsizes. For a twelve note set to be MOS with a
      Message 2 of 28 , Jan 20 12:03 AM
        Hi Jacky,

        The first one wouldn't be MOS as it's a three-stepsize 12-note set of
        5 small 3 medium and 4 large stepsizes.

        For a twelve note set to be MOS with a fifth generator the fifth must
        be greater than 1:2^(4/7), less than 1:2^(3/5) and not equal to
        1:2^(7/12).

        The ones that fall between 4/7ths and 7/12ths of an octave will have 5
        small and 7 large stepsizes. The ones that fall between 7/12ths and
        3/5ths of an octave will have 7 small and 5 large steps.

        --Dan Stearns
      • Robert Walker
        Hi Jacky, Here is a little piece using the MIDI acoustic guitar and Cor Anglais patches in your hypermos large prime scale 10.
        Message 3 of 28 , Jan 20 6:20 PM
          Hi Jacky,

          Here is a little piece using the MIDI acoustic guitar and Cor Anglais patches in your
          hypermos large prime scale 10.

          http://members.nbci.com/tune_smithy/2001_a_MOS_Odyssy_scale10.mid

          thanks

          Robert
        • ligonj@northstate.net
          ... Anglais patches in your ... Thanks so much for this Robert! It played wonderfully on my system, and is a very lovely piece. What a powerful mood! I ve been
          Message 4 of 28 , Jan 20 7:10 PM
            --- In tuning@egroups.com, "Robert Walker" <robert_walker@r...> wrote:
            > Hi Jacky,
            >
            > Here is a little piece using the MIDI acoustic guitar and Cor
            Anglais patches in your
            > hypermos large prime scale 10.
            >
            > http://members.nbci.com/tune_smithy/2001_a_MOS_Odyssy_scale10.mid
            >
            > thanks
            >
            > Robert


            Thanks so much for this Robert! It played wonderfully on my system,
            and is a very lovely piece. What a powerful mood!

            I've been told that you are a good melodist, and now I understand why.
            Very cool!


            Jacky Ligon

            P.S. I'm playing this in CakeWalk 9, and I'm not sure if it's playing
            at your intended tempo. What was the intended bpm, so I can be sure?
          • Robert Walker
            Hi Jacky, ... Thanks. It was very easy to write in your lovely scale. Just came like that, straight away. ... crotchet = 80 (smallest division heard = quaver).
            Message 5 of 28 , Jan 21 9:18 AM
              Hi Jacky,

              > Thanks so much for this Robert! It played wonderfully on my system,
              > and is a very lovely piece. What a powerful mood!

              > I've been told that you are a good melodist, and now I understand why.
              > Very cool!

              Thanks.

              It was very easy to write in your lovely scale. Just came like that,
              straight away.

              > P.S. I'm playing this in CakeWalk 9, and I'm not sure if it's playing
              > at your intended tempo. What was the intended bpm, so I can be sure?

              crotchet = 80 (smallest division heard = quaver).

              Robert
            • Paul H. Erlich
              Hi Jacky! Though I m sure you re having fun with your single-chain-of-fifths MOSs with various rational fifths, inspired perhaps by
              Message 6 of 28 , Jan 22 12:35 PM
                Hi Jacky!

                Though I'm sure you're having fun with your single-chain-of-fifths MOSs with
                various rational fifths, inspired perhaps by

                http://www.uq.net.au/~zzdkeena/Music/1ChainOfFifthsTunings.htm,

                I would urge you not to ignore the possibilities of double-chain-of-fifths
                tunings and scales as in:

                http://www.uq.net.au/~zzdkeena/Music/2ChainOfFifthsTunings.htm

                You could approximate the half-octave with 99/70, or 1393/985, or a
                convergent of intermediate complexity . . .

                Also, in the scales you posted, it might be valuable to state how many times
                each near-just interval occurs -- for example, in a 12-tone scale with
                631/419 fifths, 9:7 occurs 8 times.
              • Robert Walker
                Hi Jacky, I ve done a gif of the score for http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10.mid as
                Message 7 of 28 , Jan 22 12:40 PM
                  Hi Jacky,

                  I've done a gif of the score for

                  http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10.mid

                  as
                  http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10_low_res.gif
                  http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10_p1.gif
                  http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10_p2.gif

                  Also, here is the nwc (NoteWorthy Composer) file:
                  http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10.nwc

                  The nwc file is meant to be played through score retuning software,
                  such as FTS, hooked to play button of NWC, e.g. via Hubi's loopback cable.

                  The lyric line for the Cor Anglais shows the scale to use.

                  Idea is, C is 1/1, C# is 161.794 cents, and so on.

                  I did it in the "diatonic mode" in your scale, with a few accidentals, which
                  happens to be a nice mode to play in, but isn't anything like a major scale:

                  I.e. degrees 0 2 4 5 7 9 11 12:

                  1/1 217.655 cents 435.311 cents 597.105 cents 708.828 cents 926.483 cents 1144.14 cents
                  2/1

                  9/7 = 435.084 cents

                  = neutral third, augmented fourth, quarter tone leading note.

                  I've labelled the clefs as "Acoustic guitar patch" and "Cor anglais patch",
                  as I don't expect them to be playable on the actual instruments particularly.

                  The Cor anglais patch goes well below the range of Cor Anglais (F - c'').

                  Robert
                • ligonj@northstate.net
                  ... MOSs with ... Paul, I did go and read this when you mentioned it recently, which did help the focus of my paper. It is actually lots of fun indeed!!! : )
                  Message 8 of 28 , Jan 22 1:16 PM
                    --- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
                    > Hi Jacky!
                    >
                    > Though I'm sure you're having fun with your single-chain-of-fifths
                    MOSs with
                    > various rational fifths, inspired perhaps by
                    >
                    > http://www.uq.net.au/~zzdkeena/Music/1ChainOfFifthsTunings.htm,

                    Paul,

                    I did go and read this when you mentioned it recently, which did help
                    the focus of my paper. It is actually lots of fun indeed!!! : )

                    >
                    > I would urge you not to ignore the possibilities of double-chain-of-
                    fifths
                    > tunings and scales as in:
                    >
                    > http://www.uq.net.au/~zzdkeena/Music/2ChainOfFifthsTunings.htm
                    >
                    > You could approximate the half-octave with 99/70, or 1393/985, or a
                    > convergent of intermediate complexity . . .

                    Thanks for this! Got it coming up now.

                    >
                    > Also, in the scales you posted, it might be valuable to state how
                    many times
                    > each near-just interval occurs -- for example, in a 12-tone scale
                    with
                    > 631/419 fifths, 9:7 occurs 8 times.

                    This is a great idea Paul! I've got the better part of "part 2" ready
                    to post, and I'll definitely take this into serious consideration.
                    Unison Vectors anyone?


                    Jacky Ligon

                    P.S. What are 12 tone scales with less than the 5s7L and 7L5s called,
                    which have only two step sizes and are generated by an interval
                    around 9/7? Ran into these whilst exploring. I know they aren't MOS,
                    but are interesting all the same. Any terms will help. I found some
                    of your old MOS posts today, but haven't absorbed yet.
                  • Paul H. Erlich
                    Jacky wrote, ... Less than the 5s7L and 7l5s? What does that mean? ... Any scale with a single generator and only two step sizes is an MOS. Perhaps you had
                    Message 9 of 28 , Jan 22 1:41 PM
                      Jacky wrote,

                      >P.S. What are 12 tone scales with less than the 5s7L and 7L5s called,

                      Less than the 5s7L and 7l5s? What does that mean?

                      >which have only two step sizes and are generated by an interval
                      >around 9/7? Ran into these whilst exploring. I know they aren't MOS,

                      Any scale with a single generator and only two step sizes is an MOS. Perhaps
                      you had something else in mind?
                    • ligonj@northstate.net
                      ... http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10.m id ... http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10_l
                      Message 10 of 28 , Jan 22 2:54 PM
                        --- In tuning@egroups.com, "Robert Walker" <robert_walker@r...> wrote:
                        > Hi Jacky,
                        >
                        > I've done a gif of the score for
                        >
                        >
                        http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10.m
                        id
                        >
                        > as
                        >
                        http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10_l
                        ow_res.gif
                        >
                        http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10_p
                        1.gif
                        >
                        http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10_p
                        2.gif
                        >
                        > Also, here is the nwc (NoteWorthy Composer) file:
                        >
                        http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10.n
                        wc

                        Robert!

                        This is fantastic! Thanks so much! The quality of the gifs is great.
                        You must have saved them at higher than 72ppi.

                        >
                        > I did it in the "diatonic mode" in your scale, with a few
                        accidentals, which
                        > happens to be a nice mode to play in, but isn't anything like a
                        major scale:
                        >
                        > I.e. degrees 0 2 4 5 7 9 11 12:
                        >
                        > 1/1 217.655 cents 435.311 cents 597.105 cents 708.828 cents 926.483
                        cents 1144.14 cents
                        > 2/1
                        >
                        > 9/7 = 435.084 cents
                        >
                        > = neutral third, augmented fourth, quarter tone leading note.
                        >

                        It's beautiful what you did with it. Seems that you found "the
                        pattern" quickly. There's an innate pattern in every scale, that is
                        only revealed by playing the scale. You tapped the essense.

                        I wonder how diffcult it might be to create a MOS animation, which
                        would be fed with a data table of hundreds fourth generators ranging
                        between 2/5 and 3/7 octave, whilst showing graphically, how as the
                        size of the generator grows or shrinks, how this affects the size of
                        scale degrees? Perhaps even treating the 2/5 to 3/7 as a 360 degree
                        rotation. It would be an interesting visualization, as one could see
                        how the roles of the scale degrees changes relative to the generator
                        size. I can see this in my mind!

                        Erv Wilson says "it passes through the looking glass". So cool!

                        Thanks,

                        Jacky Ligon
                      • ligonj@northstate.net
                        ... called, ... Oops! Meant to say 5s7L and 7s5L. ... MOS, This is in error. Disregard. ... MOS. Perhaps ... Thanks for the clarification. I can t seem to find
                        Message 11 of 28 , Jan 22 3:02 PM
                          --- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
                          > Jacky wrote,
                          >
                          > >P.S. What are 12 tone scales with less than the 5s7L and 7L5s
                          called,
                          >
                          > Less than the 5s7L and 7l5s? What does that mean?

                          Oops! Meant to say 5s7L and 7s5L.

                          >
                          > >which have only two step sizes and are generated by an interval
                          > >around 9/7? Ran into these whilst exploring. I know they aren't
                          MOS,

                          This is in error. Disregard.

                          >
                          > Any scale with a single generator and only two step sizes is an
                          MOS. Perhaps
                          > you had something else in mind?

                          Thanks for the clarification. I can't seem to find the scale right
                          now. Was playing with this during having a raging fever this weekend.
                          Onward!

                          Jacky
                        • D.Stearns
                          Paul H. Erlich wrote, With all the exploratory
                          Message 12 of 28 , Jan 22 5:49 PM
                            Paul H. Erlich wrote,

                            <<Any scale with a single generator and only two step sizes is an MOS.
                            Perhaps you had something else in mind?>>

                            With all the exploratory posts involving three-term series and indexes
                            I thought I'd repost this bit from the other list as when I say
                            "indexing", two-term series are really what it's all about, as they
                            are very well defined.

                            When I started using indexing it was because I wanted to answer the
                            following question: given any arbitrary scale constructed of "a"
                            amount of small steps and "b" amount of large steps, what range can an
                            interval safely occupy so as to result in scale of two stepsizes and
                            Myhill's property... ?

                            Jacky seems to be fond of twelve note sets (or at least tends to give
                            his examples as such), so I'll use those as an example.

                            These would be the possible twelve note two-stepsize indexes.

                            [1,11]
                            [2,10]
                            [3,9]
                            [4,8]
                            [5,7]
                            [6,6]
                            [7,5]
                            [8,4]
                            [9,3]
                            [10,2]
                            [11,1]

                            Here's the procedure:

                            1) Reduce "a" and "b" by their GCD.

                            2) Scale the periodicity accordingly. So for example I'll assume "P" =
                            1:2 (as is usually the case, but "P" can be anything you'd like in
                            this process, its complete fluidity is a built in feature), and I'll
                            use the first reducible index from the above twelve note examples;
                            [2,10]. This reduces to [1,5]. Now if the index is reducible than the
                            periodicity is as well, so "P" must be logarithmically scaled by the
                            GCD, which in this case was 2. So P = 1:2 becomes P = 1:2^(1/2).

                            3) Convert the index, i.e., [a,b], into adjacent fractions. (Adjacent
                            fractions are two fractions that differ by 1 when cross-multiplied.)

                            4) These adjacent fractions can then be used to seed a Stern-Brocot
                            Tree:

                            <http://206.4.57.253/editorial/knot/SB_tree.html>

                            Now x, y, and x+y (as depicted at the above link) define the borders
                            of what range an interval can safely occupy so as to result in scale
                            of two stepsizes and Myhill's property.

                            Lets look at the first twelve note index in the above example; [1,11].
                            Following the procedure I just outlined would give the following x, y,
                            and x+y:

                            0/1 1/11
                            1/12

                            Now any interval > 1:2^(0/1) and < 1:2^(1/12) results in a [1,11]
                            scale. And any interval > 1:2^(1/12) and < 1:2^(1/11) results in the
                            opposite scale index of [11,1].

                            Of course the closer you are to any of the borders the more trivial
                            and less meaningful the 'answer' is. In the Apical weighting scheme I
                            use, I set the 'default' equal scale as x+2y (as depicted at the above
                            Stern-Brocot Tree link).

                            This weighting is the mean of the Golden and Silver constant, and it
                            also sets up a useful condition when moving to theoretical n-term
                            scales. Because x+2y is essentially a tempered version of thefollowing
                            scaling condition:

                            Let stepsizes be small to large alphabetized variables so that
                            two-stepsize = [a,b], three-stepsize = [a,b,c] and so forth and so on
                            where the alphabetized variables are any whole numbers.

                            Then assign each variable an uppercase fixed size:

                            A = (LOG(2)-LOG(1))*(1200/LOG(2))
                            B = (LOG(3)-LOG(1))*(1200/LOG(2))
                            C = (LOG(4)-LOG(1))*(1200/LOG(2))

                            etc.

                            Let this be a fixed interval template. This allows for a convenient
                            and aurally agreeable distribution of stepsizes amongst any [a,b,...]
                            index of:

                            a/b = log(3)/log(2)
                            c/b = log (4)/log(3)
                            d/c = log(5)/log(4)

                            etc.

                            Next you scale a given [a,b,...] index by a given periodicity (P).
                            This will give us a percentage (X) to scale the fixed A,B,... interval
                            template with.

                            So for a two-stepsize index you'd have:

                            (a*A)+(b*B)/P = X

                            For a three-stepsize index:

                            (a*A)+(b*B)+(c*C)/P = X

                            And so forth and so on...

                            Now A/X, B/X,... will give a given [a,b,...] index corresponding
                            stepsizes scaled to P.

                            So as I said before, x+2y in a two-term index is essentially a
                            tempered version of this. It would seem to follow suit that this would
                            be the case in other n-term scales as well... though no known parallel
                            condition to adjacent fractions exists that I'm aware of here, so
                            putting all the pieces together has proved very difficult. But for the
                            two-term indexes everything works exceptionally well, and may provide
                            a different inroad to the world of MOS scales.

                            --Dan Stearns
                          • D.Stearns
                            I wrote,
                            Message 13 of 28 , Jan 22 7:10 PM
                              I wrote,

                              <<it also sets up a useful condition when moving to theoretical n-term
                              scales. Because x+2y is essentially a tempered version of the
                              following scaling condition:>>

                              That's incorrect -- 2x+3y would be the tempered version of that
                              scaling condition. And though useful for n-term extrapolations, this
                              is not related to the mean of the Golden and Silver constant in the
                              manner I was pointing at, so please scratch that end part.

                              --Dan Stearns
                            • ligonj@northstate.net
                              ... indexes ... I want to thank Dan for his valuable post here. I think once I have totally grasped a working knowledge of the indexes, the code of MOS will be
                              Message 14 of 28 , Jan 23 9:07 AM
                                --- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
                                >
                                > With all the exploratory posts involving three-term series and
                                indexes
                                > I thought I'd repost this bit from the other list as when I say
                                > "indexing", two-term series are really what it's all about, as they
                                > are very well defined.
                                >

                                I want to thank Dan for his valuable post here. I think once I have
                                totally grasped a working knowledge of the indexes, the code of MOS
                                will be fully revealed.


                                For a friend who is working through these MOS posts, I'd like to also
                                include one of my cross posts from the New JI list:


                                MOS scales (Moments of Symmetry), can be generally "generated" by
                                stacking chains of intervals, then reducing them to fit within the
                                2/1. In this case it was with chains of exotic high prime ratio
                                fifths.

                                Many times when theorists seek to use fifths such as some of the ones
                                in my post, and intend to optimize the thirds in this manner, they
                                will sometimes choose more mathematical means, such as "weighting"
                                intervals with constants (an irrational approach). Please note that I
                                actively use both.

                                The extremely interesting thing for me, is that we can also
                                use "rational" means to achieve exactly the same thing. The Prime
                                Series Ratios provides us many such valuable things, and stands as a
                                mostly unexplored area of JI/RI.


                                Here's how #6 is generated:
                                #6

                                Generator Approximated Ratios
                                461/307 703.834
                                0
                                126.836
                                207.667
                                334.503 17/14 336.130
                                415.335 14/11 417.508
                                542.171
                                623.002
                                703.834
                                830.67
                                911.501
                                1038.337
                                1119.169
                                1200

                                If we take the ratio 461/307 @ 703.834 cents, and we
                                repeatedly "chain" it 11 times, we get the below cents values:

                                0
                                703.834
                                1407.667
                                2111.501
                                2815.335
                                3519.169
                                4223.002
                                4926.836
                                5630.67
                                6334.503
                                7038.337
                                7742.171


                                Then when we reduce this within the 2/1, we have:

                                0
                                703.834
                                207.667
                                911.501
                                415.335
                                1119.169
                                623.002
                                126.836
                                830.67
                                334.503
                                1038.337
                                542.171


                                Next we sort and add the 2/1:

                                0
                                126.836
                                207.667
                                334.503
                                415.335
                                542.171
                                623.002
                                703.834
                                830.67
                                911.501
                                1038.337
                                1119.169
                                1200


                                This is one of my favorites of this set, for the fact that it gives
                                you good 11 and 17 thirds:

                                334.503 17/14 336.130
                                415.335 14/11 417.508

                                A wonderful property!


                                Thanks,

                                Jacky Ligon
                              • ligonj@northstate.net
                                ... Dan, It s interesting how the weighted fifth also creates some interesting harmonic alignments! Thanks for this second post. I was studying your last one
                                Message 15 of 28 , Jan 23 6:40 PM
                                  --- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:

                                  Dan,

                                  It's interesting how the weighted fifth also creates some interesting
                                  harmonic alignments!

                                  Thanks for this second post. I was studying your last one this
                                  evening, and this will help me to grasp it better. So each line/scale
                                  is generated by the next weighted adjacent fraction?

                                  Thanks,

                                  Jacky Ligon
                                • D.Stearns
                                  Jacky Ligon wrote,
                                  Message 16 of 28 , Jan 23 6:45 PM
                                    Jacky Ligon wrote,

                                    <<Many times when theorists seek to use fifths such as some of the
                                    ones in my post, and intend to optimize the thirds in this manner,
                                    they will sometimes choose more mathematical means, such as
                                    "weighting" intervals with constants (an irrational approach).>>

                                    I used the weighted fifth tuning of Kornerup's as a model for
                                    generalizing two-term indexes.

                                    When a two-term index is converted into an adjacent fraction and the
                                    adjacent fractions are taken as a Fibonacci series, you can see the
                                    generators (the numerators) working their way towards a Phi-weighting.
                                    So there's a built-in logic, or internal consistency in this... and
                                    using this framework it's easy to see that the 3rd term of a Fibonacci
                                    series is the "m" and the 4th term is the "n" in an m-out-of-n set,
                                    and that the 1st and 2nd terms are the "small" and "large" steps
                                    respectively.

                                    Here's the single generator, Phi-weighted, twelve note two-term sets:

                                    [1,11]

                                    0/1, 1/11, 1/12, 2/23, ...

                                    0 103 207 310 413 516 620 723 826 930 1033 1136 1200
                                    0 103 207 310 413 516 620 723 826 930 1033 1097 1200
                                    0 103 207 310 413 516 620 723 826 930 993 1097 1200
                                    0 103 207 310 413 516 620 723 826 890 993 1097 1200
                                    0 103 207 310 413 516 620 723 787 890 993 1097 1200
                                    0 103 207 310 413 516 620 684 787 890 993 1097 1200
                                    0 103 207 310 413 516 580 684 787 890 993 1097 1200
                                    0 103 207 310 413 477 580 684 787 890 993 1097 1200
                                    0 103 207 310 374 477 580 684 787 890 993 1097 1200
                                    0 103 207 270 374 477 580 684 787 890 993 1097 1200
                                    0 103 167 270 374 477 580 684 787 890 993 1097 1200
                                    0 64 167 270 374 477 580 684 787 890 993 1097 1200


                                    [2,8]

                                    0/1, 1/5, 1/6, 2/11, ...

                                    P = 1:2^(1/2)

                                    0 107 214 320 427 534 600 707 814 920 1027 1134 1200
                                    0 107 214 320 427 493 600 707 814 920 1027 1093 1200
                                    0 107 214 320 386 493 600 707 814 920 986 1093 1200
                                    0 107 214 280 386 493 600 707 814 880 986 1093 1200
                                    0 107 173 280 386 493 600 707 773 880 986 1093 1200
                                    0 66 173 280 386 493 600 666 773 880 986 1093 1200


                                    [3,9]

                                    0/1, 1/3, 1/4, 2/7, ...

                                    P = 1:2^(1/3)

                                    0 111 221 332 400 511 621 732 800 911 1021 1132 1200
                                    0 111 221 289 400 511 621 689 800 911 1021 1089 1200
                                    0 111 179 289 400 511 579 689 800 911 979 1089 1200
                                    0 68 179 289 400 468 579 689 800 868 979 1089 1200


                                    [4,8]

                                    0/1, 1/2, 1/3, 2/5, ...

                                    P = 1:2^(1/4)

                                    0 115 229 300 415 529 600 715 829 900 1015 1129 1200
                                    0 115 185 300 415 485 600 715 785 900 1015 1085 1200
                                    0 71 185 300 371 485 600 671 785 900 971 1085 1200


                                    [5,7]

                                    3/5, 4/7, 7/12, 11/19, ...

                                    0 74 192 266 385 458 577 696 770 889 962 1081 1200
                                    0 119 192 311 385 504 623 696 815 889 1008 1126 1200
                                    0 74 192 266 385 504 577 696 770 889 1008 1081 1200
                                    0 119 192 311 430 504 623 696 815 934 1008 1126 1200
                                    0 74 192 311 385 504 577 696 815 889 1008 1081 1200
                                    0 119 238 311 430 504 623 742 815 934 1008 1126 1200
                                    0 119 192 311 385 504 623 696 815 889 1008 1081 1200
                                    0 74 192 266 385 504 577 696 770 889 962 1081 1200
                                    0 119 192 311 430 504 623 696 815 889 1008 1126 1200
                                    0 74 192 311 385 504 577 696 770 889 1008 1081 1200
                                    0 119 238 311 430 504 623 696 815 934 1008 1126 1200
                                    0 119 192 311 385 504 577 696 815 889 1008 1081 1200


                                    [6,6]

                                    P = 1:2^(1/6)

                                    0/1, 1/1, 1/2, 2/3, ...

                                    0 124 200 324 400 524 600 724 800 924 1000 1124 1200
                                    0 76 200 276 400 476 600 676 800 876 1000 1076 1200


                                    [7,5]

                                    4/7, 3/5, 7/12, 10/17, ...

                                    0 129 208 337 416 545 625 704 833 912 1041 1120 1200
                                    0 80 208 288 416 496 575 704 784 912 992 1071 1200
                                    0 129 208 337 416 496 625 704 833 912 992 1120 1200
                                    0 80 208 288 367 496 575 704 784 863 992 1071 1200
                                    0 129 208 288 416 496 625 704 784 912 992 1120 1200
                                    0 80 159 288 367 496 575 655 784 863 992 1071 1200
                                    0 80 208 288 416 496 575 704 784 912 992 1120 1200
                                    0 129 208 337 416 496 625 704 833 912 1041 1120 1200
                                    0 80 208 288 367 496 575 704 784 912 992 1071 1200
                                    0 129 208 288 416 496 625 704 833 912 992 1120 1200
                                    0 80 159 288 367 496 575 704 784 863 992 1071 1200
                                    0 80 208 288 416 496 625 704 784 912 992 1120 1200


                                    [8,4]

                                    1/2, 0/1, 1/3, 1/4, ...

                                    P = 1:2^(1/4)

                                    0 83 166 300 383 466 600 683 766 900 983 1066 1200
                                    0 83 217 300 383 517 600 683 817 900 983 1117 1200
                                    0 134 217 300 434 517 600 734 817 900 1034 1117 1200


                                    [9,3]

                                    1/3, 0/1, 1/4, 1/5, ...

                                    P = 1:2^(1/3)

                                    0 87 173 260 400 487 573 660 800 887 973 1060 1200
                                    0 87 173 313 400 487 573 713 800 887 973 1113 1200
                                    0 87 227 313 400 487 627 713 800 887 1027 1113 1200
                                    0 140 227 313 400 540 627 713 800 940 1027 1113 1200


                                    [10,2]

                                    1/5, 0/1, 1/6, 1/7, ...

                                    P = 1:2^(1/2)

                                    0 91 181 272 363 453 600 691 781 872 963 1053 1200
                                    0 91 181 272 363 509 600 691 781 872 963 1109 1200
                                    0 91 181 272 419 509 600 691 781 872 1019 1109 1200
                                    0 91 181 328 419 509 600 691 781 928 1019 1109 1200
                                    0 91 237 328 419 509 600 691 837 928 1019 1109 1200
                                    0 147 237 328 419 509 600 747 837 928 1019 1109 1200


                                    [11,1]

                                    1/11, 0/1, 1/12, 1/13, ...

                                    0 95 190 285 380 476 571 666 761 856 951 1046 1200
                                    0 95 190 285 380 476 571 666 761 856 951 1105 1200
                                    0 95 190 285 380 476 571 666 761 856 1010 1105 1200
                                    0 95 190 285 380 476 571 666 761 915 1010 1105 1200
                                    0 95 190 285 380 476 571 666 820 915 1010 1105 1200
                                    0 95 190 285 380 476 571 724 820 915 1010 1105 1200
                                    0 95 190 285 380 476 629 724 820 915 1010 1105 1200
                                    0 95 190 285 380 534 629 724 820 915 1010 1105 1200
                                    0 95 190 285 439 534 629 724 820 915 1010 1105 1200
                                    0 95 190 344 439 534 629 724 820 915 1010 1105 1200
                                    0 95 249 344 439 534 629 724 820 915 1010 1105 1200
                                    0 154 249 344 439 534 629 724 820 915 1010 1105 1200

                                    --Dan Stearns
                                  • ligonj@northstate.net
                                    ... Dan, Should not this [2,8] index reduce by its GDC and result in [1,4]? Forgive me if I ve missed something here. Jacky Ligon
                                    Message 17 of 28 , Jan 24 5:38 AM
                                      --- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
                                      >
                                      > [2,8]
                                      >
                                      > 0/1, 1/5, 1/6, 2/11, ...
                                      >
                                      > P = 1:2^(1/2)
                                      >
                                      > 0 107 214 320 427 534 600 707 814 920 1027 1134 1200
                                      > 0 107 214 320 427 493 600 707 814 920 1027 1093 1200
                                      > 0 107 214 320 386 493 600 707 814 920 986 1093 1200
                                      > 0 107 214 280 386 493 600 707 814 880 986 1093 1200
                                      > 0 107 173 280 386 493 600 707 773 880 986 1093 1200
                                      > 0 66 173 280 386 493 600 666 773 880 986 1093 1200
                                      >

                                      Dan,

                                      Should not this [2,8] index reduce by its GDC and result in [1,4]?

                                      Forgive me if I've missed something here.


                                      Jacky Ligon
                                    • ligonj@northstate.net
                                      ... Dan, I just happened to notice that the last scale here contains what was know in ancient times as The Evil Fifth . This evil generator of 2086812/1419857
                                      Message 18 of 28 , Jan 24 6:00 AM
                                        --- In tuning@egroups.com, ligonj@n... wrote:
                                        > --- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
                                        > >
                                        > > [2,8]
                                        > >
                                        > > 0/1, 1/5, 1/6, 2/11, ...
                                        > >
                                        > > P = 1:2^(1/2)
                                        > >
                                        > > 0 107 214 320 427 534 600 707 814 920 1027 1134 1200
                                        > > 0 107 214 320 427 493 600 707 814 920 1027 1093 1200
                                        > > 0 107 214 320 386 493 600 707 814 920 986 1093 1200
                                        > > 0 107 214 280 386 493 600 707 814 880 986 1093 1200
                                        > > 0 107 173 280 386 493 600 707 773 880 986 1093 1200
                                        > > 0 66 173 280 386 493 600 666 773 880 986 1093 1200
                                        > >

                                        Dan,

                                        I just happened to notice that the last scale here contains what was
                                        know in ancient times as "The Evil Fifth". This evil generator of
                                        2086812/1419857 @ 666.666 will yield a 9s1L 9 Tone MOS. I've been too
                                        afraid to compose with this though, since "Evil" is not my bag.
                                        Perhaps I'll pull it out around Halloween.

                                        666.666 The Evil 5th
                                        0: 1/1 0.000 unison, perfect prime
                                        1: 133.332 cents 133.332
                                        2: 266.664 cents 266.664
                                        3: 399.996 cents 399.996
                                        4: 533.328 cents 533.328
                                        5: 2086812/1419857 666.666
                                        6: 799.998 cents 799.998
                                        7: 933.330 cents 933.330
                                        8: 1066.662 cents 1066.662
                                        9: 1199.994 cents 1199.994


                                        0: 133.332
                                        1: 133.332
                                        2: 133.332
                                        3: 133.332
                                        4: 133.332
                                        5: 133.338
                                        6: 133.332
                                        7: 133.332
                                        8: 133.332
                                        9: 133.332

                                        Number of notes : 9
                                        Smallest interval : 133.332 cents
                                        Average interval (divided octave) : 133.333 cents
                                        Average / Smallest interval : 1.000005
                                        Largest interval of one step : 133.338 cents
                                        Largest / Average interval : 1.000040
                                        Largest / Smallest interval : 1.000045
                                        Least squares average interval : 133.3327 cents
                                        Median interval of one step : 133.332 cents
                                        Most common interval of one step : 133.332 cents, amount: 8
                                        Interval standard deviation : 0.0019 cents
                                        Interval skew : 0.0000 cents
                                        Scale is strictly proper
                                        Scale has Myhill's property
                                        generators: 1 of 133.3320 cents and 8 of 1066.6620 cents
                                        Scale is maximal even
                                        Scale is a mode of a 9-tone equal temperament
                                        Scale contains two identical pentachords
                                        Scale is a Constant Structure
                                        Number of different intervals : 16 = 2.00000 / class
                                        Smallest interval difference : 0.006 cents
                                        Most common intervals : 133.332 cents & inv., amount: 8
                                        Number of recognisable fifths : 0
                                        Scale is a chain of 4 triads 0.0 533.328 1066.662 cents
                                        Most common triad is 0.0 133.332 266.664 cents, amount: 9
                                        Rothenberg stability : 1.000000 = 1
                                        Lumma stability : 0.999960
                                        Rothenberg efficiency : 0.740741 redundancy :
                                        0.259259
                                        Prime limit : 17 (not all pitches rational)
                                        Limited transpositions :
                                        1 2 3 4 5 6 7 8
                                        Inversional symmetry on degrees :
                                        0 1 2 3 4 5 6 7 8
                                        Inversional symmetry on intervals :
                                        0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9

                                        }: )

                                        Jacky Ligon
                                      • ligonj@northstate.net
                                        Dan, What if P is irrational? How would one deal with an irrational non-octave P ? Perhaps a slightly stretched of compressed octave as P ? I m uncertain
                                        Message 19 of 28 , Jan 24 6:27 AM
                                          Dan,

                                          What if "P" is irrational?

                                          How would one deal with an irrational non-octave "P"?

                                          Perhaps a slightly stretched of compressed octave as "P"?

                                          I'm uncertain how to apply your formula to this possibility.



                                          Thanks,

                                          Jacky
                                        • ligonj@northstate.net
                                          ... weighting. ... Fibonacci ... Dan, Now I am truly lost in space! I thought the [5,7] index should be 2/5, 3/7, 5/12. Right-Wrong? I can follow your logic
                                          Message 20 of 28 , Jan 24 7:41 AM
                                            --- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
                                            >
                                            > When a two-term index is converted into an adjacent fraction and the
                                            > adjacent fractions are taken as a Fibonacci series, you can see the
                                            > generators (the numerators) working their way towards a Phi-
                                            weighting.
                                            > So there's a built-in logic, or internal consistency in this... and
                                            > using this framework it's easy to see that the 3rd term of a
                                            Fibonacci
                                            > series is the "m" and the 4th term is the "n" in an m-out-of-n set,
                                            > and that the 1st and 2nd terms are the "small" and "large" steps
                                            > respectively.
                                            >
                                            > Here's the single generator, Phi-weighted, twelve note two-term
                                            sets:
                                            >
                                            > [2,10]
                                            >
                                            > 0/1, 1/5, 1/6, 2/11, ...
                                            >
                                            >
                                            > [5,7]
                                            >
                                            > 3/5, 4/7, 7/12, 11/19, ...
                                            >
                                            > 0 74 192 266 385 458 577 696 770 889 962 1081 1200
                                            > 0 119 192 311 385 504 623 696 815 889 1008 1126 1200
                                            > 0 74 192 266 385 504 577 696 770 889 1008 1081 1200
                                            > 0 119 192 311 430 504 623 696 815 934 1008 1126 1200
                                            > 0 74 192 311 385 504 577 696 815 889 1008 1081 1200
                                            > 0 119 238 311 430 504 623 742 815 934 1008 1126 1200
                                            > 0 119 192 311 385 504 623 696 815 889 1008 1081 1200
                                            > 0 74 192 266 385 504 577 696 770 889 962 1081 1200
                                            > 0 119 192 311 430 504 623 696 815 889 1008 1126 1200
                                            > 0 74 192 311 385 504 577 696 770 889 1008 1081 1200
                                            > 0 119 238 311 430 504 623 696 815 934 1008 1126 1200
                                            > 0 119 192 311 385 504 577 696 815 889 1008 1081 1200
                                            >

                                            Dan,

                                            Now I am truly lost in space! I thought the [5,7] index should be
                                            2/5, 3/7, 5/12.

                                            Right-Wrong?

                                            I can follow your logic well up to the point of converting the
                                            indexes for [5,7] and [7,5]. Could you please explain in slow motion
                                            again how to convert these indexes into adjacent fractions? Even
                                            though I think I know the correct answers to be:

                                            [5,7] 2/5, 3/7, 5/12
                                            [7,5] 4/7, 3/5, 7/12

                                            I still think I'm getting confused by something here.


                                            Thanks for your continued assistance,

                                            Jacky Ligon


                                            I can feel that Dan.
                                          • D.Stearns
                                            Jacky Ligon wrote, As they re all supposed to be twelve note indexes, that should ve
                                            Message 21 of 28 , Jan 24 8:58 AM
                                              Jacky Ligon wrote,

                                              <<Should not this [2,8] index reduce by its GDC and result in [1,4]?>>

                                              As they're all supposed to be twelve note indexes, that should've read
                                              [2,10] and not [2,8]. Just a typo, the rest of the example is correct.
                                              Thanks for catching it.

                                              --Dan Stearns
                                            • D.Stearns
                                              Jacky Ligon wrote, As they re both adjacent fractions it doesn t really matter. One would just be the
                                              Message 22 of 28 , Jan 24 3:07 PM
                                                Jacky Ligon wrote,

                                                <<I thought the [5,7] index should be 2/5, 3/7,>>

                                                As they're both adjacent fractions it doesn't really matter. One would
                                                just be the compliment of the other.

                                                A generator (or adjacent fraction) here always has a analogous '5th'
                                                and '4th', as in a circle (or spiral/whatever) of fifths. I think this
                                                (circle of fifths/fourths) is a handy mindset to adopt when looking at
                                                this so long as one remembers that this is just a mental reference
                                                point, and that the generators are whatever they are... so in other
                                                words, they may or may not be fifths and fourths but the whole
                                                generalization is constructed in a like manner.

                                                --Dan Stearns
                                              • D.Stearns
                                                Jacky Ligon wrote,
                                                Message 23 of 28 , Jan 24 3:31 PM
                                                  Jacky Ligon wrote,

                                                  <<What if "P" is irrational? How would one deal with an irrational
                                                  non-octave "P"? Perhaps a slightly stretched of compressed octave as
                                                  "P"?>>

                                                  Yes, the value of "P" is completely fluid in the algorithm so that you
                                                  can make it anything you'd like.

                                                  So

                                                  X = P/((a+W*b))*(A+W*B)

                                                  where

                                                  "P" = a given periodicity

                                                  "W" = a given weight

                                                  "A"/"a", "B"/"b" = the two adjacent fractions of a given [a,b] index

                                                  and "X" = the resulting weighted generator

                                                  Now say you wanted an arbitrary scale of 3 small steps and 5 large
                                                  steps where your periodicity is 1225¢ and your large step is exactly
                                                  two of your small steps...

                                                  1225/((3+2*5))*(1+2*2) = X

                                                  So your generator, "X", would be ~471.154¢. And the resulting rotation
                                                  of scales would be:

                                                  0 188 377 471 660 848 942 1131 1225
                                                  0 188 283 471 660 754 942 1037 1225
                                                  0 94 283 471 565 754 848 1037 1225
                                                  0 188 377 471 660 754 942 1131 1225
                                                  0 188 283 471 565 754 942 1037 1225
                                                  0 94 283 377 565 754 848 1037 1225
                                                  0 188 283 471 660 754 942 1131 1225
                                                  0 94 283 471 565 754 942 1037 1225

                                                  Let me know if that helps.

                                                  thanks,

                                                  --Dan Stearns
                                                • ligonj@northstate.net
                                                  ... 9s1L ... you d ... scale. ... This is so interesting! I didn t even recognize that it was 9ET. I did notice the approximated 7/6 was a member of the Evil
                                                  Message 24 of 28 , Jan 24 4:42 PM
                                                    --- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
                                                    > Jacky Ligon wrote,
                                                    >
                                                    > <<This evil generator of 2086812/1419857 @ 666.666 will yield a
                                                    9s1L
                                                    > 9 Tone MOS.>>
                                                    >
                                                    > Hi Jacky,
                                                    >
                                                    > If you used the 9th multiple of 2086812/1419857 for your octave
                                                    you'd
                                                    > have an [8,1] scale, and if you used the 2/1 you'd have a [7,2]
                                                    scale.
                                                    > But in either case you have nine equal no matter how you look at it,
                                                    > and the difference in stepsizes is completely trivial in any ears on
                                                    > fashion.

                                                    This is so interesting! I didn't even recognize that it was 9ET. I
                                                    did notice the approximated 7/6 was a member of the "Evil Scale".

                                                    >
                                                    >
                                                    > <<I've been too afraid to compose with this though, since "Evil" is
                                                    > not my bag. Perhaps I'll pull it out around Halloween.>>
                                                    >
                                                    > Now, now... 9-tET is a "nice" tuning: See no evil, hear no evil!

                                                    He, he! Very funny!

                                                    >
                                                    > A while back Enrique Ubieta and "bimodalism" came up. I did a lot of
                                                    > microtonal experimentation with his "bimodal triad" (a 1-b3-3-5 four
                                                    > note chord that combines the minor and major triad). And one of the
                                                    > things I determined was that 9 equal, or preferably some optimal
                                                    > tweaking thereof, was most likely the theoretical best tuning to
                                                    > accomplish Ubieta's bimodal theory; which would harmonize all scale
                                                    > degrees to the bimodal triad.

                                                    Would it be ok to show an example of one of his scales? BTW, I've
                                                    always been hugely interested in bi-tonal chords, which were an
                                                    important part of my past 12 Tone keyboard technique.

                                                    A funny story: I went to audition for a "Prog-Rock" band back in the
                                                    90s. My main interest in them was they played in other meters than 4.
                                                    After the practice, the band leader (a fabulously talented guitarist)
                                                    and I had a little jam session when everyone else was gone. He ask
                                                    for me to begin an improvisation based on my ideas (we'd focused on
                                                    his entirely until this point), so I improvised with one hand in a
                                                    certain tonality, then the other transposed down by a tritone, and a
                                                    different third. He went "Wait a minute! What was that?!!!" He got me
                                                    to play the pattern slowly - and then he totally nailed the mode!
                                                    Then it was on! It was great to have someone relate to this - and in
                                                    Prog Rock this should be no problem. It was great fun, and I think
                                                    our private jam was the best thing of the evening. They liked what I
                                                    played, but I didn't accept the gig because of stylistic differences,
                                                    and they were too far away to be practical.

                                                    Of course Ornette had taken all this to levels inconceived of. He
                                                    *IS* the multi-tonal man. Why did Jamaladeen Tacuma just pop into my
                                                    mind?

                                                    >
                                                    > And speaking of Enrique, he sent me three of his bimodal
                                                    compositions
                                                    > just the other day, two guitar pieces and his "Sui Generis Suite"
                                                    for
                                                    > harpsichord. "Bimodal" (one of the guitar pieces) seems to me like
                                                    it
                                                    > really should be a repertory type piece, analogous to say Brouwer's
                                                    > guitar pieces... it's a very fine piece indeed.

                                                    Well (no, not a Zappa "Well"), I got to tell you that I love Brouwer,
                                                    and this is a great recommendation. Is this a commercial release? Any
                                                    chance to hear this?

                                                    Well maybe Evil ain't so bad after all!


                                                    : )

                                                    Jacky Ligon
                                                  • D.Stearns
                                                    Jacky Ligon wrote, Hi Jacky, If you used the 9th multiple of
                                                    Message 25 of 28 , Jan 24 6:38 PM
                                                      Jacky Ligon wrote,

                                                      <<This evil generator of 2086812/1419857 @ 666.666 will yield a 9s1L
                                                      9 Tone MOS.>>

                                                      Hi Jacky,

                                                      If you used the 9th multiple of 2086812/1419857 for your octave you'd
                                                      have an [8,1] scale, and if you used the 2/1 you'd have a [7,2] scale.
                                                      But in either case you have nine equal no matter how you look at it,
                                                      and the difference in stepsizes is completely trivial in any ears on
                                                      fashion.


                                                      <<I've been too afraid to compose with this though, since "Evil" is
                                                      not my bag. Perhaps I'll pull it out around Halloween.>>

                                                      Now, now... 9-tET is a "nice" tuning: See no evil, hear no evil!

                                                      A while back Enrique Ubieta and "bimodalism" came up. I did a lot of
                                                      microtonal experimentation with his "bimodal triad" (a 1-b3-3-5 four
                                                      note chord that combines the minor and major triad). And one of the
                                                      things I determined was that 9 equal, or preferably some optimal
                                                      tweaking thereof, was most likely the theoretical best tuning to
                                                      accomplish Ubieta's bimodal theory; which would harmonize all scale
                                                      degrees to the bimodal triad.

                                                      And speaking of Enrique, he sent me three of his bimodal compositions
                                                      just the other day, two guitar pieces and his "Sui Generis Suite" for
                                                      harpsichord. "Bimodal" (one of the guitar pieces) seems to me like it
                                                      really should be a repertory type piece, analogous to say Brouwer's
                                                      guitar pieces... it's a very fine piece indeed.

                                                      --Dan Stearns
                                                    • D.Stearns
                                                      Here s some links:
                                                      Message 26 of 28 , Jan 25 12:05 AM
                                                        <<Would it be ok to show an example of one of his scales>>

                                                        Here's some links:

                                                        <http://www.ubieta.com/bio.htm>
                                                        <http://www.ubieta.com/bimodalism/BimodalHarmony.htm>

                                                        It is my impression that Ubieta himself operates completely within the
                                                        usual twelve-tone equal tempered way of going about things... and when
                                                        he brings up the harmonic series (etc.) and any sort of teleology he
                                                        appears to only mean it in a purely conjectural manner; "the truth
                                                        from the facts inclines us to think so" is what he once wrote me...

                                                        The scales that I last posted about this were 14 and 23-tET temperings
                                                        of an 18:21:23 identity where if you were to lattice out the 18:21:23
                                                        scale in these temperaments you'd have a situation somewhat analogous
                                                        to the diatonic scale where the comma -- a 7889/7776 in the following
                                                        14-tET example -- is absorbed by the temperament and an additional
                                                        consonant chord results. In this case an additional "bimodal chord".

                                                        These peculiar diatonic-like extrapolations came about by taking the
                                                        first two terms of the bimodal chord -- i.e., the minor and major
                                                        thirds -- as an identity.

                                                        I then came up with a little template to generate 9-tone scales where
                                                        each scale degree is connected to a centralized tonic:

                                                        b-a
                                                        |
                                                        |
                                                        |
                                                        |
                                                        |
                                                        ai | b
                                                        \ | /
                                                        \ | /
                                                        \ | /
                                                        \ | /
                                                        \|/
                                                        (a+b)i----------t---------a+b
                                                        /|\
                                                        / | \
                                                        / | \
                                                        / | \
                                                        / | \
                                                        bi | a
                                                        |
                                                        |
                                                        |
                                                        |
                                                        |
                                                        (b-a)i

                                                        where

                                                        "a" and "b" simply represent a minor and major third respectively
                                                        (which are generally culled from an identity in the form of t:a:b)

                                                        "i" indicates an inversion, or 'inverted'

                                                        and "t" is any given pitch or tonic

                                                        So with that in mind, here's the scale that results when t:a:b =
                                                        18:21:23

                                                        23/21
                                                        /|\
                                                        / | \
                                                        / | \
                                                        / | \
                                                        / | \
                                                        12/7----+---23/18
                                                        /|\ | /|\
                                                        / | \ | / | \
                                                        / | \ | / | \
                                                        / | \ | / | \
                                                        / | \|/ | \
                                                        216/161--+----1/1----+--161/108
                                                        \ | /|\ | /
                                                        \ | / | \ | /
                                                        \ | / | \ | /
                                                        \ | / | \ | /
                                                        \|/ | \|/
                                                        36/23---+----7/6
                                                        \ | /
                                                        \ | /
                                                        \ | /
                                                        \ | /
                                                        \|/
                                                        42/23

                                                        And here's the 14-tET example with the additional consonant chord:

                                                        8-----------2
                                                        /|\ /|\
                                                        / | \ / | \
                                                        / | \ / | \
                                                        / | \ / | \
                                                        / | \ / | \
                                                        3-----+----11-----+-----5
                                                        \ | /|\ | /|\
                                                        \ | / | \ | / | \
                                                        \ | / | \ | / | \
                                                        \ | / | \ | / | \
                                                        \|/ | \|/ | \
                                                        6-----+-----0-----+-----8
                                                        \ | /|\ | /|\
                                                        \ | / | \ | / | \
                                                        \ | / | \ | / | \
                                                        \ | / | \ | / | \
                                                        \|/ | \|/ | \
                                                        9-----+-----3-----+----11
                                                        \ | / \ | /
                                                        \ | / \ | /
                                                        \ | / \ | /
                                                        \ | / \ | /
                                                        \|/ \|/
                                                        12-----------6

                                                        In 14-tET this is a proper scale. A proper scale has instances of
                                                        shared intervals amongst interval classes, and in this case the shared
                                                        (or "ambiguous") intervals are the augmented and diminished 3rds and
                                                        4ths, the 5ths and
                                                        6ths, and the 7ths and 8ths:

                                                        0 171 257 429 514 686 771 943 1029 1200
                                                        0 86 257 343 514 600 771 857 1029 1200
                                                        0 171 257 429 514 686 771 943 1114 1200
                                                        0 86 257 343 514 600 771 943 1029 1200
                                                        0 171 257 429 514 686 857 943 1114 1200
                                                        0 86 257 343 514 686 771 943 1029 1200
                                                        0 171 257 429 600 686 857 943 1114 1200
                                                        0 86 257 429 514 686 771 943 1029 1200
                                                        0 171 343 429 600 686 857 943 1114 1200

                                                        23-tET, like 14, is consistent (that is if you even allow that the
                                                        full bimodal chord can be rightfully seen in such a light here), and
                                                        thereby allows all instances of the identity's cross-set or interval
                                                        matrix to always be the best rounded (LOG(N)-LOG(D))*(T/LOG(2))
                                                        representations (were "N" and "D" are the numerator and denominator of
                                                        the relevant consonant ratios, and "T" is the temperament).

                                                        However, unlike 14-tET, 23-tET is strictly proper, and this allows for
                                                        uniquely articulated representations of all interval classes. 23-tET
                                                        recognizes the diaschisma like 285768/279841 while still hiding the
                                                        syntonic comma like 7889/7776:

                                                        I. 0 157 261 417 522 678 783 939 1043 1200
                                                        II. 0 104 261 365 522 626 783 887 1043 1200
                                                        III. 0 157 261 417 522 678 783 939 1096 1200
                                                        IV. 0 104 261 365 522 626 783 939 1043 1200
                                                        V. 0 157 261 417 522 678 835 939 1096 1200
                                                        VI. 0 104 261 365 522 678 783 939 1043 1200
                                                        VII. 0 157 261 417 574 678 835 939 1096 1200
                                                        VIII. 0 104 261 417 522 678 783 939 1043 1200
                                                        IX. 0 157 313 417 574 678 835 939 1096 1200
                                                        X. 0 157 261 417 522 678 783 939 1043 1200

                                                        Anyway, these were some of the types of ideas that I was looking at
                                                        and experimenting with. But as I said in the previous post I
                                                        eventually decided that a simple 9 equal, or preferably some optimally
                                                        tweaked variations thereof, would seem to be the scale that is most
                                                        theoretically in step with Ubieta's bimodalism.

                                                        --Dan Stearns
                                                      • MONZ@JUNO.COM
                                                        ... http://groups.yahoo.com/group/tuning/message/17939 ... Who put out that incredible blues album called Live Evil ? Was it Howlin Wolf? -monz
                                                        Message 27 of 28 , Jan 26 2:55 AM
                                                          --- In tuning@y..., ligonj@n... wrote:

                                                          http://groups.yahoo.com/group/tuning/message/17939

                                                          > Well maybe Evil ain't so bad after all!


                                                          Who put out that incredible blues album called "Live Evil"?
                                                          Was it Howlin' Wolf?



                                                          -monz
                                                        • David Beardsley
                                                          ... Miles Davis. Although there were a couple of live Howlin Wolf albums, I don t think there were any called Live Evil. -- * D a v i d B e a r d s l e
                                                          Message 28 of 28 , Jan 26 4:14 AM
                                                            MONZ@... wrote:
                                                            >
                                                            >
                                                            > --- In tuning@y..., ligonj@n... wrote:
                                                            >
                                                            > http://groups.yahoo.com/group/tuning/message/17939
                                                            >
                                                            > > Well maybe Evil ain't so bad after all!
                                                            >
                                                            > Who put out that incredible blues album called "Live Evil"?
                                                            > Was it Howlin' Wolf?

                                                            Miles Davis.

                                                            Although there were a couple of live
                                                            Howlin Wolf albums, I don't think there were any called Live Evil.


                                                            --
                                                            * D a v i d B e a r d s l e y
                                                            * 49/32 R a d i o "all microtonal, all the time"
                                                            * http://www.virtulink.com/immp/lookhere.htm
                                                            * http://mp3.com/davidbeardsley
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