Golden ratio
  • mrcoppermrcopper
    Posts: 653
    Edit: in the middle of revising and completing so temporarily deleted example ..

    Singing to myself while waiting for piglets to hatch ...

    re-edit: started new discussion for this XVII Agnus Dei: see "sight-tuning example"

    William Copper
  • Liam
    Posts: 5,092
    The Golden Ratio is overemphasized. There are many harmonic ratios that have been actively used in Christian art and architecture, and theory at least going back to Boethius. The Golden Ratio was not as salient in the Middle Ages as it became later. U am trying to remember the book that included a significant deflating of the Golden Ratio as the supreme natural mathematical ratio.

  • CHGiffenCHGiffen
    Posts: 5,193
    The Golden ratio makes an interesting appearance in musical engraving. The scale factor is the factor by which the distance following a note of a given duration is multiplied to give the distance following a note of twice that duration, when no other notes of smaller values are scored in the same position. This is best illustrated by a single music line (only one part).

    If the scale factor is 2, a half note, say, has twice the space following it as a quarter note, and a whole note has twice the space following it as a half note (hence, whole note has four times the space following it as a quarter note).

    On the other hand, if the scale factor is φ (the Golden ratio, about 1.618), the distance following a note of a given duration is multiplied by φ to get the distance following a note of twice the value. So a whole note has φ times the space following it as a half note, and a half note φ times the space following it as a quarter note; hence a whole note has φxφ (about 2.6) times the space following it at a quarter note.

    For aesthetic reasons, a scale factor of φ is much more appealing than a scale factor of 2, and some engraving programs, including Finale, use φ as the default scale factor in engraving music. It can be changed in Finale and some people use √2 which is about 1.414, or sometimes 1.5 as the scale factor.

    The attached PDF score illustrates music engraved with scale factors 2 and φ, so that you can see the difference. The first four staves are single voice (on the middle line B), with varying scale factors φ and 2, as indicated. The remaining staves show a two-part canon (from my Ave Maria a 2), first with φ-spacing and then with 2-spacing.

    Of course, this is not an instance of the Golden ratio φ in music, per se, but rather an aesthetic appearance of φ in music engraving. There are, however, some interesting studies as to how the Golden ratio makes its appearance in music, usually in terms of the lengths of various sections of a piece (which are sometimes even more directly related to Fibonacci numbers in terms of numbers of measures), and such things as the length of some pieces of music being φ times the length to the dramatic climax of a piece, etc.
    Scale factors.pdf
    45K
    Thanked by 2chonak MarkThompson
  • mrcoppermrcopper
    Posts: 653
    Interesting graphic and explanation, thank you. I began thinking (again) about this ratio because in the Agnus Dei I wrote a few months ago, the 3 equal parts left me feeling vaguely dissatisfied with the overall structure. One of my teachers made the interesting argument that because the first time we hear something it seems to take longer than the second time we hear it, proportions where a repeat is involved are more complicated than the simple measurement of time. Don't know if he was right about that, but we have all noticed that the first time travelling somewhere unfamiliar it seems to take much longer than the same route once familiar: whether walking or driving or taking a train. Hm .. flying though, where there is no passing reference of landscape and scenery, may be different.
  • francis
    Posts: 10,824
    Hey

    Can I get the φ in Sibelius?
  • Adam WoodAdam Wood
    Posts: 6,482
    I've used the Golden Ratio in print design a few times. It can help organize visual space in a way that feels organic, but is still structured.

    However: I'm a bit dubious about its application to music. I'm not aware of the Golden Ratio appearing in any naturally occurring sonic phenomenon. So, it seems like it would become just an arbitrary organizing principle, no better (or worse) than any other.
  • francis
    Posts: 10,824
    I really like the look of music set with φ in CHG's example. Very easy to read. Right now I set music using notes spaced at 82%, and then tweek optically at the end. But that mathematical approach seems very nice.
    Thanked by 1CHGiffen
  • CHGiffenCHGiffen
    Posts: 5,193
    I'm not sure about the Sibelius terminology, but I think that Sibelius and francis(?) look at note spacing from the opposite direction that Finale does. So, if I'm right ...

    ... Instead of talking (from Finale's viewpoint) in terms of the space following, say, a half note being (the scale factor) F times the space following a quarter note (F is typically between 1 and 2), francis and Sibelius seem to refer to the space following, say, a quarter note being (a contraction factor) f times the space following a half note (so that f would typically be between 0.5 and 1). In other words Finale's scale factor F (which is expansive) is the reciprocal of Sibelius's contraction factor f ... ie. f = 1/F.

    Equal beat spacing has Finale scale factor F = 2 or the Sibelius contraction factor f = 0.5 = 50%. The Golden ratio spacing has F = φ =1.618 which corresponds to f = 1/φ = φ-1 = 0.618 = 61.8%.

    When francis uses a contraction factor f = 82%, this corresponds to a scale factor F = 1/0.82 = 1.22. This is a little too cramped for my taste. But worst of all, if F = 1 = f, then the space following a half note and the space following a quarter note are exactly the same, a most unaesthetic choice.

    Another commonly used value (which I also use), if the Golden ratio spacing seems too "loose" is F = √2 = 1.414, which corresponds to a conraction factor f = (√2)/2 = 0.707 = 70.7%. Music set with this note spacing is a little more "tight" than music set with Golden ratio spacing.

    To summarize:

    F = 2 or f = 50% corresponds to "equal beat spacing"
    F = φ = 1.618 or f = φ-1 = 61.8% corresponds to "Golden ratio spacing"
    F = √2 = 1.414 or f = (√2)/2 = 70.7% corresponds to "square root 2 spacing"
    F = 1.22 or f = 82% corresponds to "Koerber spacing"
    F = 1 or f = 100% corresponds to "all note values equal spacing"

    These go from "open" or "loose" spacing to "closed" or "tight" spacing.
  • francis
    Posts: 10,824
    Sibelius' default spacing is 100%. So perhaps it is using a different type of ratio than what Finale uses?
  • francis
    Posts: 10,824
    Perhaps if I try to simulate your φ using the same passage, I can arrive at a conclusion. Can you send me a file of your passage?
  • CHGiffenCHGiffen
    Posts: 5,193
    Sibelius' default spacing is 100%. So perhaps it is using a different type of ratio than what Finale uses?
    This must be the case then. I don't have Sibelius and haven't been able to find out what sort of spacing algorithm is used.

    I can send you the Finale 2014 file of the passage ... or export it to MusicXML and send that. Preference?
  • francis
    Posts: 10,824
    Xml works
  • mrcoppermrcopper
    Posts: 653
    The xvii Agnus Dei complete and posted; I did use the golden ratio, to make the first and second parts very nearly 1.61812 times the third part. Seems to work for me, without unduly straining the sense that it is still a chant melody. Also 23 new piglets from two litters.
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