Extending Just Intonation

Discussion in 'Music Theory, Lessons & Techniques' started by bostjan, Aug 22, 2018.

  1. bostjan

    bostjan MicroMetal Contributor

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    Because so many different variations of some intervals are given by differing sources, I decided ages ago to eschew the more mathematical approach to extended just intonation (JI) that most mathematicians take (including Kyle Gann, Harry Partch, etc.) which sets JI intervals based on limiting the size of the numbers used in the ratio, and, instead, extended the above modal approach to all scales that can be spelled with a formula of 1 2 3 4 5 6 7 and accidentals bb, b, #, and natural. This limits there to being 23 intervals, then I minimized the number of step sizes (W, w, h, H, d, and A) necessary to connect them all, in order to find the JI intervals for the more dissonant intervals, like diminished thirds, and such, and then sanity checked those intervals by ear to make sure I didn't get something wrong. IMO, this approach works, but, the part that stumped me is determining the appropriate neutral intervals this way, which, I also feel, are important to understand.

    So, overall, my goal is to determine tonalities that are least dissonant:

    Unison can only be 1:1 ratio, or zero cents difference. This, in theory, should always be a perfect interval, "p1." "p" for "perfect," and "1" for "first scale interval," which is basically the root note and itself.

    The second, from modal theory, should have major and minor variants. Once you expand modal theory through equal temperament into more exotic scales, like the harmonic minor (1 2 b3 4 5 b6 7, the "Yngwie Malmsteen's favourite" scale), you will also have an augmented second interval. I've arbitrarily decided that there ought to be a diminished version of this interval as well, since I'm going to have a step that takes us from a perfect interval to a diminished interval. So, the second should come in flavours of major "M", minor "m", diminished "d", and augmented "A". So, in theory, the second should be either "d2," "m2," "M2," or "A2."

    The third follows exactly the same logic as the second.

    There is no major or minor fourth, as such, there should only be diminished "d4," perfect "p4," or augmented "A4."

    The fifth is like the fourth, the sixth is like the third, and the seventh is like the second.

    So, all the intervals, according to this system are:

    p1, d2, m2, M2, d3, A2, m3, M3, d4, A3, p4, A4, d5, p5, d6, A5, m6, M6, d7, A6, m7, M7, A7.

    There are, of course, other intervals. The neutral versions of intervals that can be major or minor are very important in Persian, Turkish, and Arabic music. The tritone is a sort of important one that maximizes the distance between two notes when the root note is ambiguous. This is a sort of clever way of maximizing dissonance. There's also a harmonic seventh that is interesting, as well as intervals that come out of the harmonic series, mathematically, or out of the natural harmonics of your strings (if they are thin and uniform enough).

    I think most of those are special cases. In fact, even though I believe the neutral intervals are very important, they really aren't a western (i.e. European music theory) idea, so you might be able to excuse their absence in a tuning temperament if you are focused on western music fundamentals. Obviously, it will get more an more difficult to formulate a universal model of world music theory without figuring out some things about specific musical cultures. I don't mean to downplay non-western music theory, I just don't know very much about those cultures (maybe just enough to know for sure that there are some things that will fit this same train of thought and some things that derail it, like the fact that Indian Classical music has multiple versions of what western music sees as the same interval, particularly the sixths).

    The JI system is meant to be an open system, with kind of "whatever" for intervals, but it makes it very difficult to delve into it or do any research. Maybe we can have subsets of the JI system?

    On the guitar, I'm more interested in using this for comparison of different fretting systems.
     
  2. bostjan

    bostjan MicroMetal Contributor

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    For anyone who is interested, here are the cents for the intervals I mentioned above:

    p1 (unison) 0 ¢
    d2 (diminished second) 41.0589 ¢
    m2 (minor second) 111.7313 ¢
    M2 (major second) 203.9100 ¢
    d3 (diminished third) 244.9689 ¢
    A2 (augmented second) 274.5824 ¢
    m3 (minor third) 315.6413 ¢
    M3 (major third) 386.3137 ¢
    d4 (diminished fourth) 427.3726 ¢
    A3 (augmented third) 478.4924 ¢
    p4 (perfect fourth) 498.0450 ¢
    A4 (augmented fourth) 590.2337 ¢
    d5 (diminished fifth) 609.7763 ¢
    p5 (perfect fifth) 701.9550 ¢
    d6 (diminished sixth) 743.0139 ¢
    A5 (augmented fifth) 772.6274 ¢
    m6 (minor sixth) 813.6863 ¢
    M6 (major sixth) 884.3587 ¢
    d7 (diminished seventh) 925.4176 ¢
    A6 (augmented sixth) 976.5374 ¢
    m7 (minor seventh) 1017.596 ¢
    M7 (major seventh) 1088.269 ¢
    A7 (augmented seventh) 1158.0410 ¢
    p8 (octave) 1200 ¢
     
  3. The Omega Cluster

    The Omega Cluster n00b

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    How did you arrive at those numbers? What ratios have you used?
     
  4. ElRay

    ElRay Mostly Harmless

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    Wow. I need to chew the mental cud on this one.

    But, it does raise one issue I’ve had with JI. It’s supposed to be based on “pure” ratios, but somebody complains that 4/5 isn’t quite right, so 7/10 is better, but than somebody else says, no you should use 13/20, etc. etc etc. until you get essentially (beyond any humanly measurable precision’s) irrational ratios like 393/492.

    JI has always seemed like folks trying to force something neat and tidy on something that isn’t.
     
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  5. bostjan

    bostjan MicroMetal Contributor

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    So, I started with well-known intervals, like major and minor thirds and perfect fourths and fifths, and then defined interval steps, then pushed the numbers through the rest of the JI scale, trying to stick with the most well-accepted values, to minimize the number of interval steps required, then calculated those.

    diminished step "d": 41.05886 cents [128/125]
    half step "h": 111.7313 cents [16/15]
    superior half step "H" (necessary for mixolydian and dorian scales): 133.2376 cents [27/25]
    inferior whole step "w": 182.4037 cents [10/9]
    superior whole step "W": 203.9100 cents [9/8]
    augmented step "A": 274.5824 cents [75/64]

    Everything else is constructed from those:

    d2 = 128/125 [d]
    m2 = 16/15 [h]
    M2 = 9/8 [W]
    A2 = 75/64 [A]

    d3 = 144/125 [Wd or Hh]
    m3 = 6/5 [Wh or Ad]
    M3 = 5/4 [Ah or Ww]
    A3 = 675/512 [WA]

    d4 = 32/25 [d3+w, m3+h, or M3+d]
    p4 = 4/3 [m3+w or M3+h]
    A4 = 45/32 [m3+A, M3+W, or A3+h]

    d5 = 64/45 [d4+w or p4+h]
    p5 = 3/2 [d4+A, p4+W, or A4+h]
    A5 = 25/16 [p5+A or A5+w]

    d6 = 192/125 [d5+H or p5+d]
    m6 = 8/5 [d5+W, p5+h, or A5+d]
    M6 = 5/3 [d5+A, p5+w, or A5+h]
    A6 = 225/128 [p5+A or A5+W]

    d7 = 128/75 [d6+w, m6+h, or M6+d]
    m7 = 9/5 [d6+A, m6+W, M6+H, or A6+d]
    M7 = 15/8 [m6+A, M6+W, or A6+h]
    A7 = 125/64 [M6+A or A6+w]

    Then, to get to the octave, you can go d7+A, m7+w, M7+h, or A7+d.

    The only thing you can't do is go from augmented to diminished or vice-versa, otherwise, it's a closed system with those intervals and those six step sizes.

    I don't know that this system really solves that, but I at least wanted there to be some logical reason behind each interval, other than just "it's a fraction and I think this is what sounds best," although I admit everything is still a little bit rooted in that. I think pretty much everyone agrees about the major and minor thirds and the perfect fourth and fifth, though, and since half of the system is based on that, and the other half is mostly made to dovetail as closely as possible with that, maybe it's at least a step in the right direction to tidy this all up.

    But yeah, 192/125, for example, doesn't sound that exciting, in terms of simplicity, but if you start from 64/45, and 128/75, which are not universally accepted, but accepted, I think, by a majority or at least a large portion of JI experts, and then choose something in-between that fits with pre-defined steps from each of those two endpoints, it at least means that you can walk through a scale with a nice roadmap.
     
  6. The Omega Cluster

    The Omega Cluster n00b

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    So you've developed a 22-tone system out of not-really-JI intervals?
     
  7. bostjan

    bostjan MicroMetal Contributor

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    23 note. Every interval is acceptable as JI.
     
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  8. The Omega Cluster

    The Omega Cluster n00b

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    I'm not sure I comprehend what you intend to achieve with that.
     
  9. bostjan

    bostjan MicroMetal Contributor

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    I just wanted a subset of JI that was a closed system where the notes in the system could be explained by something other than shrugging.
     
  10. The Omega Cluster

    The Omega Cluster n00b

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    Well you can achieve that with any number. For example the "p5" of 12-EDO (700¢) can be seen as a JI ratio of 300000/200226 to the point where it couldn't even be noticeably different (a difference of less than 0.0002¢). Even then, that's only because I'm too lazy to keep going until I reach a better approximation.
     
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  11. bostjan

    bostjan MicroMetal Contributor

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    It's literally entirely different. I guess I don't see where you are trying to take this discussion, so I'm not sure exactly how I should respond to that to get wherever you are going without coming off with a weird attitude. I'm honestly only confused.

    Maybe I'm just reiterating everything I said earlier in the thread, but the idea was to settle on the simplest JI ratios for the major and minor intervals, determine the step sizes used for those, then determine the minimum number of additional step sizes necessary to get the diminished and augmented intervals (which ended up being one for each), and generating the extended intervals through the use of the minimum number of JI step sizes. Each interval was sanity-checked by listening to ascending and descending scales. I'm not going to guarantee this process is repeatable, but I would love for someone else to try and compare. I feel somewhat confident that these intervals are the best that they can be, but the doubt in that from personal bias is why I started this thread. Also, someone else probably has done this approach before. I have done numerous searches, and found nothing, but maybe someone here can see this and say, "Oh, so-and-so did this exact thing back in 1906 or whatever and here's why no one has heard of it..."



    If I took 300000/200226 as the only ratio necessary to generate a scale, I'd fail right away, since two of these intervals stacked together is already more than an octave. The sanity check would also fail, as 3/2 is clearly a better fifth than 300000/200226. Also, the premise of taking an arbitrary approximation of the fifth from 12-EDO in itself is missing the point of the process. :scratch:


    So, every other approach to JI that I had come across was either a) done entirely by ear or b) done by setting a maximum number ("limit") of factorization, mathematically, and then defining the intervals by the simplest fractions. I don't want to detract from either of those methods, but I propose this other method because a) diminished and augmented tonalities are intrinsically dissonant, yet perfectly viable in music theory, so finding the least dissonant version of a dissonant interval solely by ear can prove difficult, and b) I do not feel that the simplest fraction in terms of the smallest prime factor and smallest number of factors involved is exactly the same way that the human brain perceives consonance; I believe that the method I used of relating an interval not only to the root note, but also to other intervals, is justified psychoacoustically, because the human brain has a memory, and will think not only of an interval in the moment, but recall the tonal context around it.
     
  12. The Omega Cluster

    The Omega Cluster n00b

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    Seems like we're both confused, then. I still don't understand your approach and what you're trying to do. Heh.
     
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  13. bostjan

    bostjan MicroMetal Contributor

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    Maybe my initial explanation wasn't clear. If this gets tedious for you, please feel no obligation to respond.

    Here's a less detailed explanation:

    We choose a set of "steps." Each step is a rational number that increases the scale degree by one, where the old and new scale degrees are of arbitrary tonality (major, minor, diminished, augmented, anything in-between).

    The tuning system will be generated such that it will contain one root note, and all possible diatonic scales, within some set of rules (we care not to mix augmented and diminished tonalities in the same scale, otherwise everything is fair game).

    So, each scale will consist of the root note, then a second, third, fourth, fifth, sixth, and seventh note, then the perfect octave to close it back up.

    The steps allowed are:

    1. The diminished step, which alters the tonality of the next scale degree flatter.
    2. The half step
    3. The whole step
    4. The augmented step, which alters the tonality of the next scale degree sharper.
    *. There will be allowed one variant of each step as necessary to obtain the best JI intervals

    The specific steps I found were:

    d - The diminished step: ratio 128/125 (41.1 ¢)
    h - The half step: ratio 16/15 (111.7 ¢)
    H - The superior half step: ratio 27/25 (133.2 ¢)
    w - The inferior whole step: ratio 10/9 (182.4 ¢)
    W - The superior whole step: ratio 9/8 (203.9 ¢)
    A - The augmented step: 75/64 (274.6 ¢)

    To make, the major scale, for example, you would start on the root note, then step up W, w, h, W, w, W, h (we know the series of whole and half steps from music theory, the variations in the whole steps is necessary to get higher scale degrees to sound the most consonant).

    To break that example down:

    Root note: 1/1 or 0 ¢
    go up one superior whole step (9/8 or 203.9 ¢): 1/1 * 9/8 = 9/8 or 0 ¢ + 203.9 ¢ = 203.9 ¢
    Major second: 9/8 or 203.9 ¢
    go up one inferior whole step (10/9 or 182.4 ¢): 9/8 * 10/9 = 5/4 or 203.9 ¢ + 182.4 ¢ = 386.3 ¢
    Major third: 5/4 or 386.3 ¢
    go up one half step (16/15 or 111.7 ¢): 5/4 * 16/15 = 4/3 or 386.3 ¢ + 111.7 ¢ = 498.0 ¢
    Perfect fourth: 4/3 or 498.0 ¢
    go up one superior whole step (9/8 or 203.9 ¢): 4/3 * 9/8 = 3/2 or 498.0 ¢ + 203.9 ¢ = 701.9 ¢ (there's a little rounding error here, just sig fig's, let's agree to call it 702.0 ¢, since we can double check against the ratios, which are expressed without rounding: 1200 ¢ = 702.0 ¢ log2 3/2)
    Perfect fifth: 3/2 or 702.0 ¢
    go up one inferior whole step (10/9 or 182.4 ¢): 3/2 * 10/9 = 5/3 or 702.0 ¢ + 182.4 ¢ = 884.4 ¢
    Major sixth: 5/3 or 884.4 ¢
    go up one superior whole step (9/8 or 203.9 ¢): 5/3 * 9/8 = 15/8 or 884.4 ¢ + 203.9 ¢ = 1088.3 ¢
    Major seventh: 15/8 or 1088.3 ¢
    go up one half step (16/15 or 111.7 ¢): 15/8 * 16/15 = 2/1 or 1088.3 ¢ + 111.7 ¢ = 1200.0 ¢

    The cents here are for reference.

    So, each scale is a set of seven operations chosen from the steps: d, (h or H), (w or W), and/or A. The seventh operation has to conclude the scale at the perfect octave.

    You can make a fully diminished scale (1 bb2 bb3 b4 b5 bb6 bb7), a scale that doesn't actually exist in 12-EDO:

    Root note: 1/1 (0 ¢)
    go up one diminished step (128/125 or 41.1 ¢)
    Diminished second: 128/125 (41.1 ¢)
    go up one superior whole step (9/8 or 203.9 ¢)
    Diminished third: 144/125 (245.0 ¢)
    go up one inferior whole step (10/9 or 182.4 ¢)
    Diminished fourth: 32/25 (427.4 ¢)
    go up one inferior whole step (10/9 or 182.4 ¢)
    Diminished fifth: 64/45 (609.8 ¢)
    go up one superior half step (27/25 or 133.2 ¢)
    Diminished sixth: 192/125 (743.0 ¢)
    go up one inferior whole step (10/9 or 182.4 ¢)
    Diminished seventh: 128/75 (925.4 ¢)
    go up one augmented step (75/64 or 274.6 ¢)
    Perfect octave: 2/1 (1200.0 ¢)

    So, as the major scale is WwhWwWh, the fully diminished scale in this system is dWwwHwA.

    This is the sort of template used to generate a list of all possible intervals in the key, including one diminished and one augmented variation for each scale degree.

    Why?

    By just making up whatever ratio for each interval, you get an arbitrarily different step size between each consecutive scale degree. When playing a melody in a single key, some scale degrees with leave a memory image in the listener's mind's ear, each with an arbitrary amount of consonance. There are bound to be some double stops within a scale that are not as consonant with respect to each other as they are with respect to the root note, but I feel that this approach emphasizes consonance back to the root note whilst still accounting somewhat for consonance between intervals within a scale (so long as a scale does not contain both augmented and diminished intervals). It also has the added benefit of an extra level of simplicity when considering performance on a fretted or keyboard instrument, since the step sizes are discrete (if you don't want to take that for granted, it could be another discussion entirely, maybe).
     
  14. bostjan

    bostjan MicroMetal Contributor

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    Mega-long double post
     
  15. The Omega Cluster

    The Omega Cluster n00b

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    I think I get it. That's an interesting idea, but difficult to apply to physical instruments like piano and guitar, even for 1 tonal centre. I can see the implications for electronic music though, that's pretty interesting!
     
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  16. ElRay

    ElRay Mostly Harmless

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    I think Bostjan is using JI as a guide, much as Buddhists can use christianity as a guide, whereas TOC May have encountered too many dogmatic JI evangelicals that the knee-jerk reaction is to be skeptical.

    It took a bit, but I’m almost tracking what Bostjan is getting at — but it’s more along the lines of having an academic understanding of tennis but never having played it. Maybe I need to play with Ixlramp’s microtonal offset tuning to get a feeling for what’s going on.

    I started looking at non-12-EDO temperaments because I could hear that the Bb a cello/violinist plays is not the same as the A# they play. Or more accurately, I could hear when they missed it. That’s why my route into microtonal music came through Lucy Tuning. When they started talking about 19-Frets-per-octave, it was mind blowing.
     
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  17. bostjan

    bostjan MicroMetal Contributor

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    Well, the impetus to start this, for me, was to come up with a better way to measure different EDO tuning temperaments. There exist digital pianos that play in JI using an AI ("AIJI?") that tracks your playing in real time. The same algorithm ought to be expandable into microtonal playing options, if anybody had the time and the need to justify doing it (which is unlikely, I know). There is also an increasing interest in fretlets now, as evident by other threads recently. Those are already utilized to make JI guitars that play in one key with open strings tuned one particular way. Since there is really little consensus as to what an augmented fifth in JI is supposed to sound like, yet it is already a sound basis in standard western music theory that relies on that interval, I figured it was time to evaluate the entire augmented and diminished tonalities as objectively as I could. I know it's not perfect, but I hope it can contribute something to the discussion that might lead to a better understanding.
     
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  18. ElRay

    ElRay Mostly Harmless

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    This is on the "eventually" list. I still want a guitar that is tempered ala Bach's "Well Tempered Clavier" and (ideally) tuned in M3rds. I haven't worked-out the fret positions yet, so I don't know how the frets will line-up (I fear not well) on adjacent strings.
     
  19. Winspear

    Winspear Tom Winspear Vendor

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    This seems pretty cool, I'll have to try it out. I've usually assembled JI scales by sweeping two tones and jotting down the intervals I like. I found I do pretty much get the regular agreed upon diatonic+chromatic low ratios, plus 10 or so 'microtonal' points of interest/variants, for a scale of similar size in the low twenties. This one looks nicely thought out. Will have to try it, particularly interested to see how it works from other roots. Scala has a nice tool to analyse that
     

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