Re: Fw: golden ratios

Golden ratio tunings have been mentioned here again recently, and it raises
a question in my mind: has there been much consideration by anyone of the
use of ETs with Fibonacci cardinality as approximations to golden ratio
tunings? Obviously, the approximations will get better as one goes higher
up the series
(and as one examines the larger pc intervals of a particular temperament),
but are the proportions of, for instance, the relatively manageable 13-equal
"close enough?"

I guess the issue here is that, since many of the golden ratio intervals
tend to fall between the cracks of the familiar harmonic intervals, perhaps
their tuning doesn't have to be that accurate? (i.e. how out-of-tune can an
out-of-tune interval be and still be *in-tune* out-of-tune?) And (segue to
another fairly recent thread) what about a "pseudo-13-tet" produced by a
12-tet fretted instrument with sliding bridge? Better or worse in terms of
the proportions?

D.S.

Dale Scott wrote,

>Golden ratio tunings have been mentioned here again recently, and it raises
>a question in my mind: has there been much consideration by anyone of the
>use of ETs with Fibonacci cardinality as approximations to golden ratio
>tunings? Obviously, the approximations will get better as one goes higher
>up the series
>(and as one examines the larger pc intervals of a particular temperament),
>but are the proportions of, for instance, the relatively manageable
13-equal
>"close enough?"

Neither of the types golden ratio tunings I've described would divide the
octave logarithmically into the golden ratio, though that's what you're
going for. The golden ratio as a frequency ratio happens to be well
approximated in 13-equal, but not via the Fibonacci sequence -- the golden
ratio is about 9/13 octave (=2^(9/13)), not 8/13 octave (=2^(8/13)).
However, golden meantone tuning divides the fourth logarithmically into the
golden ratio, and Kornerup's whole spiel was that the "Fibonacci" sequence
of ETs: 5, 7, 12, 19, 31, 50, . . . approaches golden meantone tuning.

>I guess the issue here is that, since many of the golden ratio intervals
>tend to fall between the cracks of the familiar harmonic intervals, perhaps
>their tuning doesn't have to be that accurate?

Well, the golden ratio is only 19.4 cents sharp of 8/5 and 7.4 cents flat of
13/8. I don't know if there really is a "crack" between the two ratios --
one just sort of blends into the other, and sometimes 13/8 doesn't really
figure in strongly and you just get more and more of 5/3 coming in as you
move in that direction.

>And (segue to
>another fairly recent thread) what about a "pseudo-13-tet" produced by a
>12-tet fretted instrument with sliding bridge? Better or worse in terms of
>the proportions?

Clearly better in some places, worse in others.

Thirteen-tone ET has semitones of 92.307692 "cents", thus: 0, 92.3, 184.6, 276.9, 369.2, 461.5, 553.8, 664.1, 738.5, 830.8, 923.1, 1015.4, 1107.7, 1200. The ratios corresponding most closely to these and their values in cents are: 135/128 (92.2), 10/9 (182.4), 88/75 (276.7), 9/8 (368.9), 64/49 (462.3), 11/8 (551.3), 22/15 (663.0), 49/32 (737.7), 160/99 (831.1), 75/44 (923.3), 9/5 (1017.6), 256/135 (1107.8).
Note that 18/17 at 99.0 cents or 35/33 at 101.9 cents or 17/16 at 105 cents are not perceived when hearing a 100 cent semitone, in preference to a mistuned 16/15, with lots of beats.
When Thomas started making electronic organs, they used only six tone generators, instead of twelve. Each could play either of two adjacent semitones, but not both simultaneously. Apparently they realized that most music most owners would want to play wouldn't have adjacent simultaneous semitones.
All of your thirteen tones are very close to a just intonation ratio, about as close as a tempered fifth is to 3/2 (701.955 cents), but only 10/9, 9/8, 11/8 and 9/5 are likely to be perceived as consonant, although slightly tempered. 276.9 will be perceived as a badly tuned 7/6 (266.9) if at all, 461.5 as a sharp 13/10 (454.2) if at all, or a very flat 4/3 (498.0), 664.1 is unlikely to be heard as 16/11 (648.7), 738.5 as a flat 20/13 (745.8) if at all, 830.8 as a flat 13/8 (840.5) if at all, 923.1 as a flat 12/7 (933.1) if at all and 1107.7 as a sharp 17/9 (1101.1) if at all.
You can see from the above that quartertones (50 cents apart) are not so far from some intervals involving 11 and 13. But when you consider how bad the equal-tempered semitone sounds at 11.7 cents flat from 16/15, most of the ratios likely to be perceived as consonant are about as far sout of tune, which should yield a thoroughly nasty sounding scale with little to recommend it. Even if one hear the lower number ratios, the scale would approximate: 1/1, 18/17, 7/6, 9/8, 13/10, 11/8, 16/11, 20/13, 13/8, 12/7, 9/5, 17/9, 2/1. Would you want a scale like that in just intonation? Maybe for Halloween. Better still, for that use, to temper it and have beats.

The sequence of ratios giving successively closer approximations to the "golden mean" can be generated starting from 1/1 (as x/y) by successively using x+y/x. This gives 2/1, 3/2, 5/3, 8,5, 13/8, etc. These golden mean ratios occur in nature, for example in spirals such as the arrangement of seeds in the head of a sunflower. Considered a visually pleasing ratio, it is often used in painting, sculpture, architecture and furniture design.
Music, on the other hand, is a one dimensional art form, with changes in intensity over time. We do not entirely perceive it that way because of the limitations of our senses. We perceive rhythm as separate from melody, melody as only a fragment of harmony and harmony as perhaps related is some way to tone color, but independent of it. The same piece is often be played on different instruments, or different combinations of instruments, or even with different tempos, yet is usually still recognizable. If this one dimensionality were not so, recording technology would have to be different.
One can, of course, imagine dividing a vibrating string by these different ratios, or using them to multiply or divide a particular frequency. In terms of cents, these ratios represent, respectively: 0, 1200, 702, 884.4, 813.7, 840.5, etc., or unison, octave, perfect fifth, major sixth, minor sixth, etc. Would you like to generate a scale using a "circle" of these successive approximations until you happen to come close to a multiple of 1200 cents? The first four give three octaves: 2/1 X 3/2 X 5/3 X 8/5 = 8/1. Note the roundoff error of 0.1 cent. When using only four significant figures, this is to be expected at times. But cents, a logarithmic measure, allow us to add and subtract instead of multiplying and dividing fractions (ratios). They also make it easier to compare the sizes of intervals, such as 531441/523288 and 74/73. Dividing, you get 1.0136432 and 1.0136986, respectively. Only in the fourth significant figure (after the decimal and zero) does the difference show. Rounded off these would be 1.0136 and 1.0137. In cents, they are 23.5 and 23.6, respectively. Interestingly, 13 X 18 = 234, somewhat off from 233/144, and 13 X 29 = 377, right on for 377/233. But what you are looking for is another power of two, such as 16 or 32.
Generally, the stronger the nearby consonances (lower numbers in the ratio), the broader will be the "pull" toward hearing that interval instead of some higher number ratio as an alternate consonance. The more consonant ratio will be heard as out of tune, with beats. In the case of these ratios, consider them versus each other as well as versus others in the same range of cents. Ratios in which the numbers can be factored vary from the expectation, sounding more consonant, for example 16/9, which is 4 X 4/ 3 X 3.
The problem here is that although you appear to be dealing with linear measure, as the eye perceives it and in terms of string lengths, you are actually dealing with a logarithmic phenomenon in terms of hearing. From 100 to 200 cycles per second of vibration (Hertz) sounds like what we call an octave. But the next higher octave is not 300, but 400 Hz. This is the source of your difficulty in applying the idea of the golden mean to music.
Also note that as we more closely approach the golden mean, rather than arriving as some previously undiscovered wonderful consonance, we ar moving further and further away from consonance and in the direction of dissonance.

>>> Dale Scott <adelscott@mail.utexas.edu> 09/14 4:45 PM >>>
From: Dale Scott <adelscott@mail.utexas.edu>

Golden ratio tunings have been mentioned here again recently, and it raises
a question in my mind: has there been much consideration by anyone of the
use of ETs with Fibonacci cardinality as approximations to golden ratio
tunings? Obviously, the approximations will get better as one goes higher
up the series
(and as one examines the larger pc intervals of a particular temperament),
but are the proportions of, for instance, the relatively manageable 13-equal
"close enough?"

I guess the issue here is that, since many of the golden ratio intervals
tend to fall between the cracks of the familiar harmonic intervals, perhaps
their tuning doesn't have to be that accurate? (i.e. how out-of-tune can an
out-of-tune interval be and still be *in-tune* out-of-tune?) And (segue to
another fairly recent thread) what about a "pseudo-13-tet" produced by a
12-tet fretted instrument with sliding bridge? Better or worse in terms of
the proportions?

D.S.

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John F. Sprague wrote,

>9/8 (368.9)

That must be a typo. 9/8 is 203.9 cents.

>Note that 18/17 at 99.0 cents or 35/33 at 101.9 cents or 17/16 at 105 cents
are not perceived when >hearing a 100 cent semitone, in preference to a
mistuned 16/15, with lots of beats.

Unless the interval is in a larger harmonic context (such as a major seventh
chord), I totally disagree. All these ratios will be part of the perception,
with 16/15 weaker since it is farther away from the true interval.

>Generally, the stronger the nearby consonances (lower numbers in the
ratio), the broader will be the >"pull" toward hearing that interval instead
of some higher number ratio as an alternate consonance. >The more consonant
ratio will be heard as out of tune, with beats. In the case of these
ratios, consider >them versus each other as well as versus others in the
same range of cents. Ratios in which the >numbers can be factored vary from
the expectation, sounding more consonant, for example 16/9, >which is 4 X 4/
3 X 3.

This last sentence explains why you must think 16/15 dominates over 17/16
and 18/17 for the ET semitone. I think this is only true if the interval
forms part of a chord, like 9:12:16 in the case of 16/9. for intervals
sounding alone, I see no evidence for any exceptions to your sentence
beginning "generally . . ." In fact, my harmonic entropy theory provides a
framework for both explaining, and getting quantitative descriptions of,
this phenomenon, which was called by Harry Partch "Observation One".

>The problem here is that although you appear to be dealing with linear
measure, as the eye perceives >it and in terms of string lengths, you are
actually dealing with a logarithmic phenomenon in terms of >hearing. From
100 to 200 cycles per second of vibration (Hertz) sounds like what we call
an octave. >But the next higher octave is not 300, but 400 Hz. This is the
source of your difficulty in applying the >idea of the golden mean to music.

Kornerup's Golden Meantone and other tunings Paul Hahn has just described
apply the golden ratio logarithmically, not linearly.

>Also note that as we more closely approach the golden mean, rather than
arriving as some previously >undiscovered wonderful consonance, we ar moving
further and further away from consonance and in >the direction of
dissonance.

I think that's what its users wanted.

[John F. Sprague:]
>Also note that as we more closely approach the golden mean, rather
than arriving as some previously undiscovered wonderful consonance, we
ar moving further and further away from consonance and in the
direction of dissonance.

[Paul H. Erlich:]
>I think that's what its users wanted.

Heinz Bohlen made a provocative comment in his recent ~833� html
(<http://members.aol.com/hpbohlen/833cent.html>): "This implies that
an 833 cents chord might sound special, despite representing a gross
dissonance in traditional opinion."

I'm not so certain that this represents so much of a desire to move
"further and further away from consonance and in the direction of
dissonance," as it does a genuine curiosity in a "special"
relationship... and the feeling I get from this (though I really don't
know), is that special here is probably (ideologically) nearer to
'consonance' than it is 'dissonance...'

But I don't know what sort of actual sonic context (which is meant to
say something other than a largely subjective musical context) might
evidentially befriend his contention that "this implies that an 833
cents chord might sound special..."

Dan

Dan-

Heinz was referring to a chord of 3 or 4 notes constructed from 833-cent
intervals. This is very different from the 2-note 833-cent interval that
John Sprague and I were discussing!

-Paul

833 cents is a useful interval. In Dune for bassoon I use single melodic
golden mean, and I chose a spiral of 833 cent intervals to form a scale for a
movement of my solo bass composition Alternity.

Why not?

Johnny Reinhard
AFMM

Sorry about the typo. 368.9 cents should have been identified with 99/80.
I quite agree that with more than two tones present simultaneously, the intervals perceived can be shifted. But I wouldn't expect that several different intervals between the same two tones would be perceived at the same time, which you seem to imply but perhaps did not mean. (Are you referring to Henry Cowell's ideas about "tone clusters"?) For example, the tritone at 600 cents in 12 TET is midway between 7/5 at 582.5 cents and 10/7 at 617.5 cents.
I'm not familiar with Kornerup's tuning or Paul Hahn's messages, being a newcomer to this site. But it seems that much of the recent discussion involved an interval of about 833 cents, which I arrived near by linear approximatiion of the golden mean, using ratios.

>>> "Paul H. Erlich" <PErlich@Acadian-Asset.com> 09/21 4:05 PM >>>
From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>

John F. Sprague wrote,

>9/8 (368.9)

That must be a typo. 9/8 is 203.9 cents.

>Note that 18/17 at 99.0 cents or 35/33 at 101.9 cents or 17/16 at 105 cents
are not perceived when >hearing a 100 cent semitone, in preference to a
mistuned 16/15, with lots of beats.

Unless the interval is in a larger harmonic context (such as a major seventh
chord), I totally disagree. All these ratios will be part of the perception,
with 16/15 weaker since it is farther away from the true interval.

>Generally, the stronger the nearby consonances (lower numbers in the
ratio), the broader will be the >"pull" toward hearing that interval instead
of some higher number ratio as an alternate consonance. >The more consonant
ratio will be heard as out of tune, with beats. In the case of these
ratios, consider >them versus each other as well as versus others in the
same range of cents. Ratios in which the >numbers can be factored vary from
the expectation, sounding more consonant, for example 16/9, >which is 4 X 4/
3 X 3.

This last sentence explains why you must think 16/15 dominates over 17/16
and 18/17 for the ET semitone. I think this is only true if the interval
forms part of a chord, like 9:12:16 in the case of 16/9. for intervals
sounding alone, I see no evidence for any exceptions to your sentence
beginning "generally . . ." In fact, my harmonic entropy theory provides a
framework for both explaining, and getting quantitative descriptions of,
this phenomenon, which was called by Harry Partch "Observation One".

>The problem here is that although you appear to be dealing with linear
measure, as the eye perceives >it and in terms of string lengths, you are
actually dealing with a logarithmic phenomenon in terms of >hearing. From
100 to 200 cycles per second of vibration (Hertz) sounds like what we call
an octave. >But the next higher octave is not 300, but 400 Hz. This is the
source of your difficulty in applying the >idea of the golden mean to music.

Kornerup's Golden Meantone and other tunings Paul Hahn has just described
apply the golden ratio logarithmically, not linearly.

>Also note that as we more closely approach the golden mean, rather than
arriving as some previously >undiscovered wonderful consonance, we ar moving
further and further away from consonance and in >the direction of
dissonance.

I think that's what its users wanted.

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John F. Sprague wrote,

>But I wouldn't expect that several different intervals between the same two
tones would be perceived at >the same time, which you seem to imply but
perhaps did not mean

There have been psychoacoustical experiments where listeners hear more than
one "missing fundamental", and are therefore ascribing more than one set of
ratios to, a given interval or chord. If instead you go by beating, a 13:11
ratio may have audible beating between the 7th harmonic of the lower tone
and the 6th harmonic of the higher tone, as well as between the 6th harmonic
of the lower tone and the 5th harmonic of the higher tone.

>I'm not familiar with Kornerup's tuning or Paul Hahn's messages, being a
newcomer to this site. But it >seems that much of the recent discussion
involved an interval of about 833 cents, which I arrived near >by linear
approximatiion of the golden mean, using ratios.

Yes. But Kornerup's golden meantone involves the golden ratio
logarithmically (i.e., between the cents measurements of different
intervals). When measured in cents, the perfect fourth of this tuning is the
golden ratio times the minor third, the minor third is the golden ratio
times the major second, the major second is the golden ratio times the minor
second, etc. It also has very consonant triads.