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

Re: [tuning] Some rather unfamiliar 5-limit temperaments

Expand Messages
  • Petr Pařízek
    ... Okay, now the remaining question is if this temperament is „musically fruitful“ enough to be worth finding a name for. But I’m not sure if I can
    Message 1 of 29 , Mar 1, 2009
    • 0 Attachment
      

      Herman wrote:

      > it would be just to the right of porcupine, passing through 7, 18,

      > 25,
      and 32.

      Okay, now the remaining question is if this temperament is „musically fruitful“ enough to be worth finding a name for. But I’m not sure if I can comment on this because there were other people who have, unlike me, found something of use in temperaments like father or bug.

      > I had an example on my old Comcast web page that used gorgo temperament.

      > I chose it as much for its melodic properties as anything else, but it
      > also illustrates some of the potential for 7-limit
      harmony.

      Well, you seem to be more successful than I was. I think you would also be better at choosing the „right“ amount of tempering for dicot (another temperament that I was never satisfied with). Maybe my problem there was that I find the two intervals of 6/5 and 5/4 to be of such a different nature that I simply don’t know how to „effectively“ treat a tuning where they turn into one thing. Perhaps the main purpose is to use neutral thirds? Another possibility would be to use a generator equal to the cube root of either 15/8 (which would make 3/2 and 5/4 mistuned by the same amount)or 9/5 (then this would go for 3/2 and 6/5). But this would mean a clear preference for major thirds over minor thirds or vice versa. Okay, the cbrt(15/8) choice sounds acceptable to me, maybe I’ll try it one day too - and hear what it sounds like.

      Petr

       

       

    • Petr Parízek
      ... Exactly. This is what I ve been thinking about for some time. IIRC, the first time I used some sort of mathematical models for comma pumps in actual
      Message 2 of 29 , Mar 1, 2009
      • 0 Attachment

        Carl wrote:

        > Not only is there a wealth of existing music which

        > assumes the
        disappearance of the syntonic comma (and thus can't
        > be adequately
        performed in strict JI), there is a like quantity
        > of music that assumes
        the disappearance of other commas, which
        > has yet to be
        written.

        Exactly. This is what I’ve been thinking about for some time. IIRC, the first time I used some sort of „mathematical models“ for comma pumps in actual music was in my „Run Down The Whistle 3“, where I calculated the version which used 9 chords in the 16-tone semisixthscale (which eventually turned out to be the least number of chords and the least number of tones in the scale with which this was possible). And last week, I did something similar for porcupine when I made „Among Other Things“. Details here: http://groups.yahoo.com/group/tuning/message/81696

        Petr

         

         

      • Marcel de Velde
        ... I must disagree with this. All music can be performed in strict JI. Just await the Lasso example I ll post later today and I ll think even you will be
        Message 3 of 29 , Mar 1, 2009
        • 0 Attachment
          It's one point that's getting lost in this whole JI vs. meantone
          thread. Not only is there a wealth of existing music which
          assumes the disappearance of the syntonic comma (and thus can't
          be adequately performed in strict JI), there is a like quantity
          of music that assumes the disappearance of other commas, which
          has yet to be written.

          I must disagree with this. All music can be performed in strict JI.
          Just await the Lasso example I'll post later today and I'll think even you will be convinced :)

          Marcel
        • Petr Pařízek
          ... will be convinced :) Have you heard the comma pump demo which I posted yesterday? If you have, then I wonder what you re still trying to convince us about.
          Message 4 of 29 , Mar 1, 2009
          • 0 Attachment
            Marcel wrote:

            > I must disagree with this. All music can be performed in strict JI.
            > Just await the Lasso example I'll post later today and I'll think even you
            will be convinced :)

            Have you heard the comma pump demo which I posted yesterday? If you have,
            then I wonder what you're still trying to convince us about.

            Petr
          • caleb morgan
            I looked for this, but I didn t see it? Can you post the link? I d like to hear it. ... I looked for this, but I didn t see it? Can you post the link? I d
            Message 5 of 29 , Mar 1, 2009
            • 0 Attachment
              I looked for this, but I didn't see it?

              Can you post the link?  I'd like to hear it.


              On Mar 1, 2009, at 9:22 AM, Petr Pařízek wrote:

              Marcel wrote:

              I must disagree with this. All music can be performed in strict JI.
              Just await the Lasso example I'll post later today and I'll think even you
              will be convinced :)

              Have you heard the comma pump demo which I posted yesterday? If you have,
              then I wonder what you're still trying to convince us about.

              Petr




              ------------------------------------

              You can configure your subscription by sending an empty email to one
              of these addresses (from the address at which you receive the list):
               tuning-subscribe@yahoogroups.com - join the tuning group.
               tuning-unsubscribe@yahoogroups.com - leave the group.
               tuning-nomail@yahoogroups.com - turn off mail from the group.
               tuning-digest@yahoogroups.com - set group to send daily digests.
               tuning-normal@yahoogroups.com - set group to send individual emails.
               tuning-help@yahoogroups.com - receive general help information.
              Yahoo! Groups Links

              <*> To visit your group on the web, go to:
                 http://groups.yahoo.com/group/tuning/

              <*> Your email settings:
                 Individual Email | Traditional

              <*> To change settings online go to:
                 http://groups.yahoo.com/group/tuning/join
                 (Yahoo! ID required)

              <*> To change settings via email:
                 mailto:tuning-digest@yahoogroups.com
                 mailto:tuning-fullfeatured@yahoogroups.com

              <*> To unsubscribe from this group, send an email to:
                 tuning-unsubscribe@yahoogroups.com

              <*> Your use of Yahoo! Groups is subject to:
                 http://docs.yahoo.com/info/terms/


            • Marcel de Velde
              ... Yes I have. And nothing proves it can t be played in pure JI. Not saying I m going to attempt the interpretation but you can t say it can t be played in
              Message 6 of 29 , Mar 1, 2009
              • 0 Attachment
                Have you heard the comma pump demo which I posted yesterday? If you have,
                then I wonder what you're still trying to convince us about.

                Yes I have.
                And nothing proves it can't be played in pure JI.
                Not saying I'm going to attempt the interpretation but you can't say it can't be played in pure JI either.

                Marcel
              • Petr Pařízek
                ... It should still be there: www.sendspace.com/file/s5t9bz Petr
                Message 7 of 29 , Mar 1, 2009
                • 0 Attachment

                  Caleb wrote:

                  > I looked for this, but I didn't see it?

                  > Can you post the link?  I'd
                  like to hear it.

                  It should still be there: www.sendspace.com/file/s5t9bz

                  Petr

                   

                   

                • Carl Lumma
                  ... I keep about 1/3 of the files you post in a folder on my hard drive with your name on it, and these two are in it. Can you show your work or explain more
                  Message 8 of 29 , Mar 1, 2009
                  • 0 Attachment
                    > IIRC, the first time I used some sort of "mathematical models"
                    > for comma pumps in actual music was in my "Run Down The Whistle 3",
                    > where I calculated the version which used 9 chords in the 16-tone
                    > semisixthscale (which eventually turned out to be the least number
                    > of chords and the least number of tones in the scale with which
                    > this was possible). And last week, I did something similar for
                    > porcupine when I made "Among Other Things". Details here:
                    > http://groups.yahoo.com/group/tuning/message/81696
                    >
                    > Petr

                    I keep about 1/3 of the files you post in a folder on my hard drive
                    with your name on it, and these two are in it. Can you show your
                    work or explain more about your method?

                    [Actually, I have Run Down the Whistle ... I don't know which
                    version.]

                    -Carl
                  • Herman Miller
                    ... Well, it doesn t rank very favorably on the badness scale, but that really only measures two properties that are easily quantifiable: complexity and
                    Message 9 of 29 , Mar 1, 2009
                    • 0 Attachment
                      Petr Pařízek wrote:
                      > 
                      >
                      > Herman wrote:
                      >
                      >> it would be just to the right of porcupine, passing through 7, 18,
                      >> 25, and 32.
                      >
                      > Okay, now the remaining question is if this temperament is „musically
                      > fruitful“ enough to be worth finding a name for. But I’m not sure if I
                      > can comment on this because there were other people who have, unlike me,
                      > found something of use in temperaments like father or bug.

                      Well, it doesn't rank very favorably on the "badness" scale, but that
                      really only measures two properties that are easily quantifiable:
                      complexity and error. Many other temperaments of the same or lower
                      complexity are more accurate. But there are other considerations. It has
                      a 7-note MOS, which is usually a good sign that it might have one or
                      more musically useful modes. In this case, it has two modes with a tonic
                      chord that has a perfect fifth (using A for the large step and B for the
                      smaller step): A A B A B A B and A B A A B A B. For any other consonant
                      fifths you have to use notes outside the basic MOS. One thing that's
                      apparent is how little difference there is between major and minor. The
                      1/4 keyboard in Erv Wilson's generalized keyboard classification system
                      (http://www.anaphoria.com/key.PDF) is a good match for this temperament.
                      One interesting thing is that it splits the major third into two equal
                      steps without being a meantone.

                      >> I had an example on my old Comcast web page that used gorgo temperament.
                      >> I chose it as much for its melodic properties as anything else, but it
                      >> also illustrates some of the potential for 7-limit harmony.
                      >
                      > Well, you seem to be more successful than I was. I think you would also
                      > be better at choosing the „right“ amount of tempering for dicot (another
                      > temperament that I was never satisfied with). Maybe my problem there was
                      > that I find the two intervals of 6/5 and 5/4 to be of such a different
                      > nature that I simply don’t know how to „effectively“ treat a tuning
                      > where they turn into one thing. Perhaps the main purpose is to use
                      > neutral thirds? Another possibility would be to use a generator equal to
                      > the cube root of either 15/8 (which would make 3/2 and 5/4 mistuned by
                      > the same amount)or 9/5 (then this would go for 3/2 and 6/5). But this
                      > would mean a clear preference for major thirds over minor thirds or vice
                      > versa. Okay, the cbrt(15/8) choice sounds acceptable to me, maybe I’ll
                      > try it one day too - and hear what it sounds like.
                      >
                      > Petr

                      One way to deal with something like dicot temperament, to create a
                      distinction similar to major vs. minor, is to change the spacing of
                      chords. I.e. rather than CEG - ACE, do something like CGE - CAE. Also,
                      dicot is one of those temperaments that sounds better with slightly
                      stretched octaves.
                    • Herman Miller
                      ... That s a pretty bold claim. If that s true, you should have no trouble with this simple comma pump: http://www.io.com/~hmiller/music/ex/dim12.mid This
                      Message 10 of 29 , Mar 1, 2009
                      • 0 Attachment
                        Marcel de Velde wrote:
                        > It's one point that's getting lost in this whole JI vs. meantone
                        > thread. Not only is there a wealth of existing music which
                        > assumes the disappearance of the syntonic comma (and thus can't
                        > be adequately performed in strict JI), there is a like quantity
                        > of music that assumes the disappearance of other commas, which
                        > has yet to be written.
                        >
                        >
                        > I must disagree with this. All music can be performed in strict JI.
                        > Just await the Lasso example I'll post later today and I'll think even
                        > you will be convinced :)

                        That's a pretty bold claim. If that's true, you should have no trouble
                        with this simple comma pump:

                        http://www.io.com/~hmiller/music/ex/dim12.mid

                        This one's a little trickier: the goal is to keep it in the key of Eb.
                        In 12-ET it rises one semitone each time around.

                        http://www.io.com/~hmiller/midi/porcupine-12.mid

                        For an idea of what it should sound like, check the 22-ET version.

                        http://www.io.com/~hmiller/midi/porcupine-22.mid

                        Well, you did say *all* music! The first example tempers out 648/625,
                        which is a pretty modern thing to do compared with 81/80 or 128/125, but
                        can be found in early 20th century music. The second example tempers out
                        250/243, which defines what we call "porcupine" temperament. I've used
                        it, Paul Erlich has used it, and most recently Petr Pařízek. (I wouldn't
                        be surprised to find someone has used it earlier.)
                      • Marcel de Velde
                        ... I consider it the other way around. It s a pretty bold claim to me to say certain music can t be played in JI. I consider JI to be the underlying structure
                        Message 11 of 29 , Mar 1, 2009
                        • 0 Attachment

                          That's a pretty bold claim. If that's true, you should have no trouble
                          with this simple comma pump:

                          I consider it the other way around.
                          It's a pretty bold claim to me to say certain music can't be played in JI.
                          I consider JI to be the underlying structure of all music.
                          I don't see any proof to why it is not so.
                          It does not mean I can solve all comma pumps without any trouble.
                          I did just solve the Lasso piece.
                          This can be seen as strong evidence for pure JI.
                           

                          http://www.io.com/~hmiller/music/ex/dim12.mid

                          This one's a little trickier: the goal is to keep it in the key of Eb.
                          In 12-ET it rises one semitone each time around.

                          http://www.io.com/~hmiller/midi/porcupine-12.mid

                          For an idea of what it should sound like, check the 22-ET version.

                          http://www.io.com/~hmiller/midi/porcupine-22.mid

                          Well, you did say *all* music! The first example tempers out 648/625,
                          which is a pretty modern thing to do compared with 81/80 or 128/125, but
                          can be found in early 20th century music. The second example tempers out
                          250/243, which defines what we call "porcupine" temperament. I've used
                          it, Paul Erlich has used it, and most recently Petr Pařízek. (I wouldn't
                          be surprised to find someone has used it earlier.)

                          I do like the examples though and will have a go at them later :)

                          Marcel
                        • Carl Lumma
                          ... No JI performance would have remotely the same effect. Meanwhile, Marcel, you re so full of it that everybody by now wishes you d either tone it down or
                          Message 12 of 29 , Mar 1, 2009
                          • 0 Attachment
                            --- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
                            >
                            > >
                            > > Have you heard the comma pump demo which I posted yesterday?
                            > > If you have, then I wonder what you're still trying to convince
                            > > us about.
                            >
                            > Yes I have.
                            > And nothing proves it can't be played in pure JI.
                            > Not saying I'm going to attempt the interpretation but you can't
                            > say it can't be played in pure JI either.
                            >
                            > Marcel

                            No JI performance would have remotely the same effect.
                            Meanwhile, Marcel, you're so full of it that everybody by
                            now wishes you'd either tone it down or take your ball and
                            go home. You clearly haven't done the homework required
                            to back up the statements you make, since every time someone
                            asks you to produce a demonstration you start out by saying
                            you've 1. never tried that before 2. aren't good at reading
                            music 3. made a mistake when you fixed the mistake you made
                            fixing the mistake you made the day before when you fixed the
                            mistake in your original mp3 or 4. the piece of music you're
                            being asked to deal with is an aberration.

                            Just cut it out, dude. Your position that strict JI can
                            be the only answer is dogmatic, and is not based on a
                            consideration of evidence or reasoning.

                            -Carl
                          • Petr Parízek
                            ... If you downloaded it in May 2008, then it s the one I was speaking about. The overall length should be about 3:57. Well, I don t know if the text of
                            Message 13 of 29 , Mar 1, 2009
                            • 0 Attachment

                              Carl wrote:

                              > I keep about 1/3 of the files you post in a folder on my hard drive

                              > with
                              your name on it, and these two are in it. Can you show your
                              > work or
                              explain more about your method?
                              > [Actually, I have Run Down the Whistle
                              ... I don't know which
                              > version.]

                              If you downloaded it in May 2008, then it’s the one I was speaking about. The overall length should be about 3:57.

                              Well, I don’t know if the text of message # 81696 makes sense to you, but generally, if I want to find any of the „model“ comma pumps, I také the particular generator mapping as a starting point and then find the lowest and the highest difference between these (let’s call them DL and DH for the sake of this explanation). For example, in the case of 81/80, the 2,3,5 prime generator mapping is „0, 1, 4“, which means that DL is 1 (i.e. the difference between the mapping of 2 and the mapping of 3) and DH is 4 (between mappings of 2 and 5). To pick a different temperament, for 16875/16384, the generator mapping is „0, -4, 3“ and therefore DL is 3 (between mappings of 2 and 5) and DH is 7 (between mappings for 3 and 5). So if we are staying with 5-limit temperaments, then I start with a major triad and keep adding or subtracting DL from the generator numbers (let’s call them A for the lowest, B for the middle, and C for the highest number). If I’m adding, then I change C to a value lower than A in order the highest difference between the numbers was still DH. If I’m subtracting, I change A to a value higher than C (actually, only if this doesn’t get me out of the minimum generator range). I do this a few times until I eventually end up with the same numbers with which I started. So for the case of meantone, for example, I’ll start with „0_1_4“ and let’s say I’ll decide to subtract DL (which is 1 here). But instead of using the actual result of „-1_0_3“, I’ll change the -1 to 4, which means that „0_3_4“ will be the generator mapping for the second chord. Then I’ll subtract 1 again, which makes „-1_2_3“ (this time I don’t make any other changes as I want the overall generator range used so far to be at its minimum, which is 5; if I changed the -1 to 6, then this would be 6). Then I subtract 1 again and change the resulting -2 to 5, so that I get „1_2_5“. And when I subtract 1 once again, I get „0_1_4“, which is what I started with.

                              In the message # 81696, I was trying to explain similar things with porcupine.

                              ... Phew, ... I think I’m going to save this; you see, this is the first time ever that I’ve „publicly“ described my method for mathematically „modelling“ comma pumps.

                              Petr

                               

                               

                            • Charles Lucy
                              Maybe I ve lost the plot on this piece and its JI, but out of curiosity I downloaded dim12 and then tried LucyTuning it using 3b2s, which seemed to be the most
                              Message 14 of 29 , Mar 1, 2009
                              • 0 Attachment
                                Maybe I've lost the plot on this piece and its JI, but out of curiosity I downloaded dim12 and then tried LucyTuning it using 3b2s, which seemed to be the most appropriate tuning to my ears..

                                I put the result as I found it into this folder:



                                On 1 Mar 2009, at 19:45, Marcel de Velde wrote:



                                That's a pretty bold claim. If that's true, you should have no trouble
                                with this simple comma pump:

                                I consider it the other way around.
                                It's a pretty bold claim to me to say certain music can't be played in JI.
                                I consider JI to be the underlying structure of all music.
                                I don't see any proof to why it is not so.
                                It does not mean I can solve all comma pumps without any trouble.
                                I did just solve the Lasso piece.
                                This can be seen as strong evidence for pure JI.
                                 

                                http://www.io. com/~hmiller/ music/ex/ dim12.mid

                                This one's a little trickier: the goal is to keep it in the key of Eb.
                                In 12-ET it rises one semitone each time around.

                                http://www.io. com/~hmiller/ midi/porcupine- 12.mid

                                For an idea of what it should sound like, check the 22-ET version.

                                http://www.io. com/~hmiller/ midi/porcupine- 22.mid

                                Well, you did say *all* music! The first example tempers out 648/625,
                                which is a pretty modern thing to do compared with 81/80 or 128/125, but
                                can be found in early 20th century music. The second example tempers out
                                250/243, which defines what we call "porcupine" temperament. I've used
                                it, Paul Erlich has used it, and most recently Petr Pařízek. (I wouldn't
                                be surprised to find someone has used it earlier.)

                                I do like the examples though and will have a go at them later :)

                                Marcel


                                Charles Lucy

                                - Promoting global harmony through LucyTuning -

                                for information on LucyTuning go to:

                                For LucyTuned Lullabies go to:



                              • Marcel de Velde
                                ... Yes it is based on much of both. Marcel
                                Message 15 of 29 , Mar 1, 2009
                                • 0 Attachment
                                  Your position that strict JI can
                                  be the only answer is dogmatic, and is not based on a
                                  consideration of evidence or reasoning.

                                  Yes it is based on much of both.

                                  Marcel
                                • Carl Lumma
                                  ... Yep, May 21st. ... Not really, sorry. ... Whoa... I have no idea why you would do any of that! ... Your secret is safe with me, I m afraid. :( -Carl
                                  Message 16 of 29 , Mar 1, 2009
                                  • 0 Attachment
                                    --- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
                                    >
                                    > If you downloaded it in May 2008, then it's the one I was
                                    > speaking about. The overall length should be about 3:57.

                                    Yep, May 21st.

                                    > Well, I don't know if the text of message # 81696 makes sense
                                    > to you,

                                    Not really, sorry.

                                    > For example, in the case of 81/80, the 2,3,5 prime generator
                                    > mapping is "0, 1, 4", which means that DL is 1 (i.e. the
                                    > difference between the mapping of 2 and the mapping of 3) and
                                    > DH is 4 (between mappings of 2 and 5). To pick a different
                                    > temperament, for 16875/16384, the generator mapping is
                                    > "0, -4, 3" and therefore DL is 3 (between mappings of 2 and 5)
                                    > and DH is 7 (between mappings for 3 and 5). So if we are
                                    > staying with 5-limit temperaments, then I start with a major
                                    > triad and keep adding or subtracting DL from the generator
                                    > numbers (let's call them A for the lowest, B for the middle,
                                    > and C for the highest number). If I'm adding, then I change
                                    > C to a value lower than A in order the highest difference
                                    > between the numbers was still DH. If I'm subtracting, I
                                    > change A to a value higher than C (actually, only if this
                                    > doesn't get me out of the minimum generator range). I do
                                    > this a few times until I eventually end up with the same
                                    > numbers with which I started. So for the case of meantone,
                                    > for example, I'll start with "0_1_4" and let's say I'll
                                    > decide to subtract DL (which is 1 here). But instead of
                                    > using the actual result of "-1_0_3", I'll change the -1 to 4,
                                    > which means that "0_3_4" will be the generator mapping for
                                    > the second chord. Then I'll subtract 1 again, which makes
                                    > "-1_2_3" (this time I don't make any other changes as I want
                                    > the overall generator range used so far to be at its minimum,
                                    > which is 5; if I changed the -1 to 6, then this would be 6).
                                    > Then I subtract 1 again and change the resulting -2 to 5,
                                    > so that I get "1_2_5". And when I subtract 1 once again,
                                    > I get "0_1_4", which is what I started with.

                                    Whoa... I have no idea why you would do any of that!

                                    > ... Phew, ... I think I'm going to save this; you see, this
                                    > is the first time ever that I've "publicly" described my
                                    > method for mathematically "modelling" comma pumps.

                                    Your secret is safe with me, I'm afraid. :(

                                    -Carl
                                  • Petr Parízek
                                    ... Simply said, i m trying to convert my musical knowledge into numbers in order I could do similar things in other temperaments than meantone. To show the
                                    Message 17 of 29 , Mar 2, 2009
                                    • 0 Attachment

                                      Carl wrote:

                                      > Whoa... I have no idea why you would do any of that!

                                      Simply said, i’m trying to convert my musical knowledge into numbers in order I could do similar things in other temperaments than meantone. To show the meantone example at its extreme, I could happily start with major triads a fifth apart (like C, F, Bb, Eb, Ab). And I can immediately see that not only is G contained in C major but also in Eb major. And not only is C contained in C major but also in Ab major. But if I leave the progression of major triads as it is, then the highest tone in the chain of fifths will be E (i.e. 4 generators away from C) and the lowest tone in the chain will be Ab (-4 generators), which occupies a total range of 8 generators. If I change one of the generator numbers, I can stay in a smaller generator range without actually disrupting the effective descending by fifths in any way and eventually I’ll get to the same chord as the first one. So if, instead of playing C major and F major, I play C major and A minor, then the fact that E-G go down a fifth to A-C is still there but the total generator range used is still 4, not 5 as in the case of C major and F major. When I then go down one more fifth, I get D minor. I could play D major instead of D minor but this would change the total generator range to 6 so I’ll leave the D minor there to keep the total generator range at 5, which means that all three tones go down a fifth. When I go down another fifth then, I get G minor; but this time there’s no problem in changing Bb to B so I’ll use G major instead, which means that D-A goes down a fifth to G-D. And finally I play C major again, making all the tones go a fifth down once more. -- When I convert this chord progression to generator numbers, I get the thing I described in my previous message. And this eventually allows me to do similar things for other comma pumps, no matter what comma I choose to temper out and no matter what primes the comma contains.

                                      Petr

                                       

                                       

                                    • Carl Lumma
                                      ... Sorry to be dense, but maybe you could start by saying what your goal is. To find the minimum span of generators needed to perform a pump for comma X in
                                      Message 18 of 29 , Mar 2, 2009
                                      • 0 Attachment
                                        --- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
                                        >
                                        > Carl wrote:
                                        >
                                        > > Whoa... I have no idea why you would do any of that!
                                        >
                                        > Simply said, i'm trying to convert my musical knowledge into
                                        > numbers in order I could do similar things in other temperaments
                                        > than meantone. [snip]

                                        Sorry to be dense, but maybe you could start by saying what
                                        your goal is. To find the minimum span of generators needed
                                        to perform a pump for comma X in temperament Y? And why
                                        is this desirable? etc.

                                        -Carl
                                      • Petr Parízek
                                        ... Well, my first question was how to find a comma pump for any particular temperament with as few chords as possible. And my first idea for answering that
                                        Message 19 of 29 , Mar 2, 2009
                                        • 0 Attachment

                                          Carl wrote:

                                          > To find the minimum span of generators needed

                                          > to perform a pump for
                                          comma X in temperament Y? And why
                                          > is this desirable?
                                          etc.

                                          Well, my first question was how to find a comma pump for any particular temperament with as few chords as possible. And my first idea for answering that question was to start simply by repeatedly shifting major triads up or down by the same interval (like fifths in meantone) until I got to a chord which had one pitch equal to any one in the initial triad. But this didn’t sound nice to me because after, let’s say, going down a fifth for three times, I then had to go down by a minor third (or up a major sixth) to get to the initial chord (i.e. one step in the negative direction repeated three times and then three steps in the positive direction), which made the last two chords sound a bit „tonally distant“. Because I didn’t find this to be one of the best ways to do it, it soon occured to me that I also needed to minimize the generator range, which eventually gave me the possibility to basically keep going down by fifths without having to do things like going three fifths up at a time then. And because a major triad in meantone spans 4 generators, then I can do this task, if I’m subtracting, simply by changing the lowest generator number to a value which is 4 higher than the middle value (then becoming the lowest). And voila, when I keep doing this, I’ll eventually get to the same chord as the initial one. And this also confirms that, in meantone, I need at least 7 tones in the scale to be able to realize the chord progression.

                                          Petr

                                           

                                           

                                        Your message has been successfully submitted and would be delivered to recipients shortly.