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

Optimization criteria

Expand Messages
  • domilare
    Hello group, on 29 november 2002 Margo Schulter, on the topic Re: Unison vectors (Peppermint, 72-EDO, etc.) wrote us :
    Message 1 of 5 , Dec 1, 2002
      Hello group, on 29 november 2002 Margo Schulter, on the topic "Re:
      Unison vectors (Peppermint, 72-EDO, etc.)" wrote us :

      <For my own purposes of optimization, if I were using notes taken from
      a single chain of fifths in a regular temperament around this region,
      I might choose a generator of around 704.160 cents>
      <......>
      <In this scheme, the representations of 2-3-7-9-11-13 and 14:17:21 are
      much as in Peppermint 24, where an arbitrary "quasi-diesis" of ~58.680
      cents is used to achieve a pure 7:6.

      <Note, however, that from your 17-limit perspective, there's an
      important disadvantage in this approach: the noncontiguous chains
      don't include the excellent approximation of 4:5 as 21 fifths up
      (augmented second plus natural 12-diesis).>

      <From my own point of view, I might prefer Peppermint 24, because ...>
      <....>
      <From my practical perspective as a keyboardist,... >

      <The main advantage of a regular and contiguous temperament such as
      the
      e-based tuning (Blackwood's R=e) is that we have free
      transposibility...>

      <Also,
      ratios of 11:12:13:14 are overall more accurately represented. Each
      scheme has its own charms.

      Most appreciatively,

      Margo Schulter
      mschulter@...>

      Indeed, many of us are looking for "a better tuning" or "a better
      scale", and are taking an optimization approach, as some look for the
      better mousetrap.

      My question is: has one of us, or another friend of music, reason or
      beauty, attempted to list or sort out optimization criteria ? Has one
      also tried to optimize the search for such criteria, and what could
      be the criteria for that new optimizing process? Are there scales of
      criterion futility?

      With deep interest and cordiality

      Dominique Larré
    • wallyesterpaulrus
      ... reason or ... one ... could ... these, if i understand you correctly, have been major concerns of the tuning-math list. i refer you to that list, its
      Message 2 of 5 , Dec 1, 2002
        --- In tuning@y..., "domilare" <Dominique.Larre@w...> wrote:

        > My question is: has one of us, or another friend of music,
        reason or
        > beauty, attempted to list or sort out optimization criteria ? Has
        one
        > also tried to optimize the search for such criteria, and what
        could
        > be the criteria for that new optimizing process?

        these, if i understand you correctly, have been major concerns of
        the tuning-math list. i refer you to that list, its archives, and its
        members, for more info . . .

        > Are there scales of
        > criterion futility?

        i'm not sure how to understand that sentence, though "criterion
        futility" sounds like a good name for a james bond movie or
        something . . . :)
      • M. Schulter
        Hello, there, Dominique Larre, and thank you for raising a question in response to some recent comments of mine which is a most happy one: What is the basis
        Message 3 of 5 , Dec 1, 2002
          Hello, there, Dominique Larre, and thank you for raising a question in
          response to some recent comments of mine which is a most happy one:
          "What is the basis for optimizing a tuning, or more generally a
          musical style or outlook?"

          While a group like tuning-math might be the ideal place to discuss the
          mathematical fine points of finding a precise "ideal" size of fifth
          for a given genre of temperament, for example, I consider the question
          of _why_ one prefers a given type of tuning for a given style very
          much a characteristic topic here. Of course, this is not to suggest
          that this kind of philosophical discussion would be out of place for a
          more specialized kind of forum either, only to say that "optimization"
          is a question which invites considering "Why?" as well as "How?"

          For example, the meantone temperaments associated with the Renaissance
          music of 16th-century Europe seem nicely to fit the style: thirds with
          ratios at or near 5 (5:4, 6:5) fit with a smoothly flowing texture of
          mostly homogenous and saturated concords based on these ratios, while
          the rather wide diatonic semitones comport with the gentleness of the
          style. This isn't to say that there isn't an element of tension and
          contrast -- the suspension, especially, beautifully fulfilling this
          role -- but that the tuning system and style seem in harmony.

          Of course, musicians and mathematicians can debate the precise degree
          of meantone temperament which ideally serves a given taste, for a
          specific piece: thus Mark Lindley prefers 1/4-comma (pure major thirds
          at 4:5) for some pieces, but 2/7-comma (major and minor thirds equally
          impure) for slower-moving Venetian organ pieces, and 1/5-comma (major
          thirds a bit larger than pure) for some English keyboard music around
          1600 where the narrower diatonic semitones or other nuances are more
          "sprightly" in effect.

          A theorist might come up with some "ideal" meantone from the viewpoint
          of some mathematical function or formula -- and this has been done
          more than once -- but different people can have different preferences,
          now as in the 16th century, and listeners such as Lindley suggest that
          the "optimal" answer might be different for one piece than another.

          With this prelude on a more familiar kind of tuning system, I'll try
          to share a few aesthetic considerations for what I call a "common
          temperament" (_participatio communis_) where fifths around about two
          cents wider than pure -- about the same amount of temperament as in
          12-tone equal temperament, but in the opposite direction.

          At one level, the "common music" for which such a temperament is
          favored is largely based on a "classical" European background -- that
          is, the European tradition in the era from Perotin to Machaut, or
          about 1200-1400. From this viewpoint, the basic 12-note temperament on
          which a _participatio communis_ tuning is based can be seen as a
          somewhat accentuated form of medieval Pythagorean tuning.

          At the same time, however, the aesthetic of this _common music_ also
          focuses on categories of intervals not discussed in the medieval
          European tradition, but recognized in medieval Near Eastern traditions
          and used in certain modern Near Eastern musics also. Some of these
          type of ratios and intervals, interestingly, also arise from the logic
          of experimentation in a European-type style with the new sizes of
          intervals made available by a 21st-century tuning with wide fifths.

          One criterion is that the tuning should produce a regularly arranged
          and very attractive diatonic scale. With fifths of around 704 cents,
          for example, whole-tones are around 208 cents, about 4 cents larger
          than the Pythagorean 9:8 (~203.91 cents), while diatonic semitones are
          close to 22:21 (~80.54 cents). These semitones are rather more narrow
          and incisive than the 100-cent steps of 12-equal, or even the classic
          256:243 steps of medieval Pythagorean tuning (~90.22 cents).

          At the same time, two other sizes of melodic steps often play a
          prominent role. Chromatic semitones at around 14:13 (~128.30 cents)
          add a contrasting "color" to the music; while in a usual "common
          temperament" of 24 notes, a small semitone or "thirdtone" often a bit
          smaller than 28:27 (~62.96 cents) is often used in cadences involving
          resolutions of intervals with ratios at or near 7.

          In a 24-note system such as Peppermint 24, possibly the quintessential
          "common temperament," there are other steps closely approximating
          medieval Near Eastern ratios such as 12:11 and 13:12, often described
          as neutral seconds. Here the "variety of steps" is deemed a musical
          virtue, with a special elegance resulting when the steps closely
          approach superparticular ratios (n+1:n).

          While sonorities involving fifths and fourths mark the standard of
          stable concord and richness, as in 13th-14th century European music,
          unstable intervals such as thirds, sixths, major seconds, and minor
          sevenths are often used in "relatively blending" sonorities and
          textural passages, sometimes involving prolonged parallel motion with
          an eventual cadence to a stable sonority. The fauxbourdon of the early
          15th century provides a kind of model for some 21st-century forms of
          parallelism involving sonorities such as a tempered 7:9:12 or
          12:14:18:21.

          Regular or usual diatonic ratios for major and minor thirds are
          generally around 14:11 (~417.51 cents) and 13:11 (~289.21 cents) or
          33:28 (~284.45 cents), and these usual ratios do much to set the tone
          and color for "common music" in such a temperament.

          Augmented seconds at around 17:14 (~336.13 cents) and diminished
          fourths at around 21:17 (~365.83 cents) provide a contrasting color,
          with the "Four Convivial Ratios" (14:11, 13:11, 17:14, 21:17) serving
          both as especially esteemed ideals for the imperfect concords, and as
          friendly landmarks to describe a given fine degree of temperament. In
          Peppermint, all four ratios are within 1.5 cents of pure, a
          distinction giving it a special place on the musical map.

          The regular major thirds around 14:11, and major sixths around 56:33
          or 22:13, are often praised as having a "bright," "active," or "sunny"
          quality, and as lending a certain "buzz" to directed cadential
          progressions. They are also beloved intervals for passages in
          conventional 15th-century fauxbourdon, where the observation by
          Johannes Boen (1357) about parallel thirds and sixths before a cadence
          is often cited: these pleasing intervals are like "forerunners and
          handmaidens" of the stable concords which typically follow.

          In a 24-note system such as Peppermint, a related kind of effect but
          with a somewhat color is achieved by playing in parallel sonorities
          such as approximate 7:9:12 ratios (~0-435-933 cents in just
          intonation, and around 0-437-933 cents in Peppermint, with 7:12 pure
          but 7:9 about 2.14 cents wide, the same amount by which the fourth is
          narrow). These 7:9:12 sonorities are available on 10 of the 12 scale
          steps of the upper keyboard manual: one plays the lower note on the
          upper manual, and the notes visually a fourth and minor seventh higher
          on the lower manual.

          While in a conventional late medieval European style or related genre
          of improvisation involves a variety of intervals, and even strict
          15th-century fauxbourdon with little ornamentation involves a routine
          contrast between major and minor types of thirds of sixths
          (e.g. A3-C4-F4 vs. G3-B3-E4, with C4 showing middle C), a texture of
          parallel 7:9:12 sonorities involves _literal_ parallelism, with all
          voices moving by identically-sized steps. The effect might recall both
          an organ mixture stop and some of the textures of Debussy at the turn
          of the 20th century.

          Another Debussyian effect possible in Peppermint is the use of
          parallel sonorities with ratios near 11:14:18, with a usual 11:14
          major third and a large 7:9 major third forming an outer neutral sixth
          at 11:18. This feels to me very rich and "different," with a floating
          quality that could suggest some sonorities of Debussy involving more
          notes. In "common music" there is often a preference for relatively
          mild timbres where complex ratios will have an effect or richness or
          subtlety rather than "dissonance."

          As already mentioned, a "common temperament" such as Peppermint
          includes many intervals characteristic of medieval or modern Near
          Eastern music, and this "bicultural" aspect of style further enriches
          the music. A Near Eastern scale might be used in a primarily melodic
          context, or as a source for vertical progressions introducing some
          curious variations on "standard" medieval European harmonies and
          cadences.

          Having described some especially valued features of "common music," I
          should add that a "common temperament" or _participatio communis_ is
          only one type of tuning system that might be used for realizing this
          kind of music. For example, a 17-note system, either equal, or
          well-tempered (as in George Secor's superb 17-note well-temperament),
          or based on integer ratios, could be used to play much of the music we
          are discussing very pleasantly.

          There is also an awareness that a "common temperament" is indeed a
          _modern_ tuning, in contrast to the classic Pythagorean intonation of
          medieval Europe -- although the medieval Near East provides its own
          classic setting for many of the interval types and sizes which
          result.

          A final "optimization" feature might bear mentioning: in a tuning
          system like Peppermint, while each keyboard is arranged in a standard
          diatonic fashion, transposing or mixing notes from the two keyboards
          makes possible many small nuances and distinctions between step sizes
          and modes. Although a fixed-pitch instrument like a keyboard cannot
          fully achieve the flexibility of ensemble performers either as
          suggested by the medieval European theory of Marchettus of Padua
          (1318) or reported for Near Eastern performances of _maqaam_ music,
          the variety of step sizes and intervals at least makes it possible to
          approximate some of these distinctions.

          Here I have attempted to suggest some general aspects of
          "optimization" -- tastes and preferences as to the best sizes for
          intervals such as major and minor thirds in a given style -- along
          with considerations such as variety of step sizes. I've picked a
          rather conventionalized, although possibly not so widely familiar,
          kind of 21st-century tuning to illustrate some of these points.

          Most appreciatively,

          Margo Schulter
          mschulter@...
        • Gene Ward Smith
          ... It s easier to do a calculation than wait for 100 years of practice and testing to suggest an answer. The calculation gives an immediate and practical
          Message 4 of 5 , Dec 2, 2002
            --- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

            > A theorist might come up with some "ideal" meantone from the viewpoint
            > of some mathematical function or formula -- and this has been done
            > more than once -- but different people can have different preferences,
            > now as in the 16th century, and listeners such as Lindley suggest that
            > the "optimal" answer might be different for one piece than another.

            It's easier to do a calculation than wait for 100 years of practice and testing to suggest an answer. The calculation gives an immediate and practical answer to the question "How should I tune in this system?" Since there is more than one way of doing such a calculation, it also suggests we not take our answer as engraved in stone.
          • Joseph Pehrson <jpehrson@rcn.com>
            ... http://groups.yahoo.com/group/tuning/message/41328 ... degree of meantone temperament which ideally serves a given taste, for a specific piece: thus Mark
            Message 5 of 5 , Dec 13, 2002
              --- In tuning@yahoogroups.com, "M. Schulter" <MSCHULTER@V...> wrote:

              http://groups.yahoo.com/group/tuning/message/41328

              >
              > Of course, musicians and mathematicians can debate the precise
              degree of meantone temperament which ideally serves a given taste,
              for a specific piece: thus Mark Lindley prefers 1/4-comma (pure major
              thirds at 4:5) for some pieces, but 2/7-comma (major and minor thirds
              equally impure) for slower-moving Venetian organ pieces, and 1/5-
              comma (major thirds a bit larger than pure) for some English keyboard
              music around 1600 where the narrower diatonic semitones or other
              nuances are more "sprightly" in effect.
              >

              ***Thank you, Margo, for posting this. This is terrifically
              interesting... I guess I had a rather *hazy* idea about these
              distinctions, but not much of one. I note that all of Mark Lindley's
              books on Amazon.com are currently out of print, as has been about
              every book I've wanted to get for about the last two years... Some
              eventually come in, others don't...

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