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 Forum index » DIY Hardware and Software » Developers' Corner
The Golden Ratio Tree
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Tony Deff



Joined: May 25, 2008
Posts: 51
Location: Suffolk, UK

PostPosted: Mon Sep 14, 2015 1:29 pm    Post subject:  The Golden Ratio Tree
Subject description: Surprising rôle of 555 in generating accurate digital intervals
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The set of ratios 555:588:623:660 spans a minor third in 3 semitones, with a worst-case error of just of a cent.
So, got your attention by false pretences, did I, you "555-analoguists" !
But stick around and see how you could reduce having to tune your 49 oscillators to just ten
(see next post).

This is the best accuracy that can be achieved using sensible numbers/divisors. (Note the "incrementing increment", +33, +35, +37)

The ratios 89:84, 107:101 & 196:185 were used by the older generation of electro-musicians to approximate an equal-temperament semitone.
These first two have opposing errors of 0·1 cent and so are practical by themselves within phase-locked loops over several octaves.

The latter ratio is legendary, having been referred to as the "magic" or "golden" ratio. We might define a "golden ratio" as one that is practical to implement and has an accuracy better than 0·1 cent: an example would be 37:22 (=900·026 cents, viz. a major sixth).

There are just three basic (non-inverted) Golden Ratios, viz.
196:185 (99.99 cents; a semitone)
  55:49 (199.98 cents; a whole tone)
  44:37 (299.97 cents; a minor third)  (44/37)^4 = 1.99988  (4 minor thirds to an octave)

Ratios can be found by brute-force-and-ignorance, bashing-away on a calculator.
However, these particular ratios have been eye-strikingly evident for 7 decades to anyone looking at a music frequency look-up table:
A3  = 220·000 Hz. (220 = 55×3) (= 44×5)
G3  = 195·998 Hz. (196 = 49×4)
F#3= 184·997 Hz. (185 = 37×5)

The missing semitone (G#3 = 207.652 Hz) has a fractional part approximating two-thirds, which suggests multiplying throughout by 3.
Doing this results in a very curious co-incidence:
 220.000 Hz × 3 = 660.000 = 660 
 207.652 Hz × 3 = 622.957 ≈ 623 = 7×89
 195.998 Hz × 3 = 587.993 ≈ 588 = 7×84
 184.997 Hz × 3 = 554.992 ≈ 555

So, the "Silver ratio" of 89:84 is related to the Golden Ratio of 196:185 via the Golden Ratio Tree!


Golden Ratio Tree.gif
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The Golden Ratio Tree - pick your own route!
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Golden Ratio Tree.gif



Golden Tree References.gif
 Description:
(Click pic to view): Related crystals generate Reference Notes nominally 3/4 cent flat
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Golden Tree References.gif



Last edited by Tony Deff on Sat Oct 31, 2015 3:57 am; edited 8 times in total
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Tony Deff



Joined: May 25, 2008
Posts: 51
Location: Suffolk, UK

PostPosted: Mon Sep 14, 2015 1:42 pm    Post subject:  The Silver Ratio Tree
Subject description: How to "do Long-Division" to implement 1-cent accuracy scale
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In a practical instrument, the super-accurate Golden Tree ratios are not required and the easier-to-implement ratios 321:303:286:270 have a very acceptable maximum error of ½ cent. This might be termed the Silver Ratio Tree, being based on 107:101 (×3).

There are 3 boards per octave, each board driven by a master oscillator tuned a major third apart.
(Overall accuracy is dictated by the manual tuning of these oscillators.)

Having lower divisors than the Golden Tree ratios means lower oscillator frequencies, allowing the use of the TS555 (which can oscillate at over 2MHz.). Every chord will be derived from at least two oscillators, so if each oscillator is modulated by a different phase of a phase-shifted LF vibrato, a rich effect can be obtained.

Tuning stability can be improved if only the lowest board of each octave (master boards) has a freely-tuned oscillator.
The other two boards (slave boards) are equipped with an extra divide-by-227 and a phase-locked loop.
The clock frequency of each board is fed upward to the next board, where it is divided by 227 and compared in the PLL to the divide-by-286 output.
(286:227 = 399.99 cents = a major third)

A universally-acceptable error for "professional instruments" has long-been 1 cent.
Allowing this extra tolerance, we can include an extra ratio (an extra semitone) in the group, thus:

340:321:303:286:270 spans a major third, spanning a 49-note keyboard with 10 boards. Each board's Master Oscillator is tuned a fourth apart (Fig. 2 below).

Divisors up to 512 are achievable using a 40103 (8-bit divider), using two passes or loops (mark + space). For even divisors this is a trivial case.
The most complicated case (divide-by-321) is detailed below.


Silver Tree divider implementation.gif
 Description:
Implementing the dividers: one dual D-type is required to implement the two odd divisors. Even divisors are trivial.
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Silver Tree divider implementation.gif



Top Thirds.gif
 Description:
Five identical PCBs driven by 5 separate HF oscillators span two octaves in a non phase-locked "digital free-phase".
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Top Thirds.gif



Last edited by Tony Deff on Thu Oct 01, 2015 2:22 pm; edited 5 times in total
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JovianPyx



Joined: Nov 20, 2007
Posts: 1988
Location: West Red Spot, Jupiter
Audio files: 224

PostPosted: Mon Sep 14, 2015 6:33 pm    Post subject: Reply with quote  Mark this post and the followings unread

Not sure if your interest is the analog or digital domain... but I've done a digital synth (in an FPGA) where I used a highly accurate tuning table (digital domain) of some 36 bits and then each oscillator could be modulated by a pseudo-random LFO so that the interactions of the tones of any interval are never exactly the same. Very pleasant effect IMO.
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Tony Deff



Joined: May 25, 2008
Posts: 51
Location: Suffolk, UK

PostPosted: Sat Sep 26, 2015 12:36 pm    Post subject: Reply with quote  Mark this post and the followings unread

JovianPyx wrote:
Not sure if your interest is the analog or digital domain... but I've done a digital synth (in an FPGA) ....

I guess you could categorize my interest as Music Research !

Although the above technology is somewhat dated and primitive compared to your enviable hardware, it is very noticeable that constructors on this site enjoy playing with oscillators and relatively easy-to-build hardware.
In case, at some future date, it tickles someone's fancy to construct a set of digital tone-generators, using old-fashioned chips and large soldering-irons, I merely seek here to post some information I have discovered.

Like me, many of them are Dual-In-Line Luddites, scared to go near your FPGA in case their large smoldering-stick dripped and bridged 6 pins together.

That said, I will be fascinated to delve in to learn more about your awesome system, whether it uses more gates than a dozen clever collies can herd sheep through and if it operates at blue-light frequencies, and how you control harmonic content.
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Tony Deff



Joined: May 25, 2008
Posts: 51
Location: Suffolk, UK

PostPosted: Sat Oct 31, 2015 5:12 am    Post subject: The Golden Ratio Tree
Subject description: ... used within a digital Top-Octave Generator
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Golden Ratios in Digital Top-Octave Generator

The term "TOGware" does not yet exist in the English dictionary, so is defined here to mean that set of 12 numerical divisors (whether implemented in hardware, firmware or software) used by a Top-Octave Generator. The traditional TOGware set used for 1-cent accuracy is 508, 538 .. 959 (last page of this PDF ).

In firmware/software, this set can neatly span a range of 9 bits plus fixed bit ([26 .. 504] + 512 = [538 ... 1016]) but is awkward to implement in hardware.
Apart from the antiquated and rare MM5555/5556 chip pair (nothing to do with timers!), this set is usually implemented with 4040s buried in a rat's-nest of diodes.

TOGware set 654, 693 ... 1235, despite the use of larger divisors, has no better accuracy, but does have distinct advantages.

a) It is the only set without an awkward prime number.
    The entire set can be broken-down into factors under 512 (Fig.1), allowing practical 2-chip implementation using a Digital Feedback Divider (Fig.2)

b) The set can can be shifted into the range 734, 778 ... (654×2), (693×2) (Fig.1b)
    Driven from a 6·144 MHz. or 12·288 MHz crystal, this is a Tuneless Top-Octave Razz , yet has an absolute and relative accuracy of 1 cent.

Digital Feedback Divider

The 40103, being an 8-bit counter, can only process division up to 256 without cascading.
This is a pain, requiring not only a second chip but an extra gate to achieve a limit of 65½ thousand (ridiculously high for electro-musical purposes).

To handle division by up to 512, we "go-round twice", using the traditional octave divider's first stage to combine two half-counts.
This has the advantage of generating a square-wave.
The HC4520 suggested here can source much more current into the traditional staircase-generator network than can 4000-series CMOS.

When the requirement is for an odd divisor, the first-stage "square" output is used to alter the start-count loaded into the 40103.
Not all odd numbers are created equal; they may be either two even numbers plus one, or two odd numbers plus one.
Half of all required odd divisors are simple cases, achieved by simply alternating the least significant bit of the loaded value.
Others require an invertor to allow bits to swap polarity, else (as in Fig. 2). biasing the mark-space to create easier binary.

Note: Digital feedback from a trailing-edge-triggered octave divider (such as a CD4040 and all "ripple-clock" counters) may cause an unpredictable race condition.


TOGtable.jpg
 Description:
Fig. 1 : Two arrangements of the same TOGware, using different Master Oscillator frequencies
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TOGtable.jpg



Top Octave implementation.jpg
 Description:
Fig. 2 Shaded areas in Table are pins of the 40103 that can be bridged by copper tracks to reduce wiring.
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Top Octave implementation.jpg



Digital Top Octave.pdf
 Description:
Combined version of Figs 1 and 2 to save on your computer.
(You may want to set it to black-and-white before printing)

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 Filename:  Digital Top Octave.pdf
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