Isle I-1 User Guide – Appendix A – Index of Available Pre-Defined Tuning Algorithms
 
 

The Isle I-1 provides a number of tuning algorithms out-of-the-box, which may be used as-is with little or no configuration, or serve as the basis of a completely customized tuning.  In addition to generating a frequency table, they also define note names and named intervals, which may be used in defining frequencies for Oscillators or any other component which uses frequency as a parameter.  The “canned” tunings are currently grouped among 25 categories, which always appear as the first 25 entries in the Tuning drop-downs available in various places on the System Settings dialogs.  (The exception is the empty Custom Tuning, used for creating a new Custom Tuning from scratch, which is always the LAST entry in the drop-downs; any user-defined tunings or tunings loaded from an external file appear between the canned tuning categories and the Custom Tuning entry.)

All tunings share the following common configuration parameters:

  • Reference Frequency
  • Reference Note Number
  • Reference Pitch Name
  • Reference Interval
  • Stretch Octave By Cents

12-Tone Even Temperament – the contemporary standard in Western music and its descendants and offshoots.  An octave is envely divided into 12 steps, each exactly the same size.  This tuning has no additional configuration, both for your convenience, and to be able to assign Western note names. 

N-Tone (Over Octave) Even Temperament – A way to divide an Octave (a 2:1 ratio) into any (integer) number of equal parts.  Use this to create 17-, 19-, 31-, 53-, or even 1200-edo tunings.  As there is no single standard for naming notes in any given edo, note names start at 0 and continue through each scale step.  The Notes Per Octave parameter controls the number of equal divisions of the octave, and can only be integer values.

N-Tone (Over Arbitrary Ratio) Even Temperament – A way to divide ANY interval (octaves, tritaves, half-tritaves, Pi, Phi, Euler’s constant, or any other positive real number) into ANY number of equal steps (again, any positive real number – not limited to integers).  If you want to divide an octave into a non-integer number of steps, use this tuning, with an interval of 2.0.  Use an irrational number of notes and build scales which do not repeat over the interval or over an octave!  Its additional configuration parameters are Interval of Equivalence and Notes Per Interval, each of which must be a positive value. 

Pythagorean – Actually not invented by Pythagoras or any of his disciples.  There is evidence that “Pythagorean” tuning was used by the Babylonians before it made its way to ancient Greece, and ancient bone flutes from China, dating back 10,000 years, are drilled in “Pythagorean” tuning.  The tuning is built using the 3rd harmonic, divided by 2 to bring it down to less than an octave apart from the reference frequency (3/2 ratio, or a perfect 5th), as a generator.  Multiply the reference frequency by this generator 6 times, then multiply the reference frequency by the reciprocal 5 times, and adjust to bring all of the pitches into the same octave – this traverses the entire circle of 5ths to build a 12-tone chromatic scale, albeit with pretty sharp 3rds compared to 12-TET.  All notes are the same distance apart, except one – the difference in step size is the Pythagorean comma.  The van Zwolle variant moves this larger step from between F#-C# to B-F#.

5-Limit Just Intonation – Various tunings built on 5-Limit integer ratios, where the largest-allowed prime factor of either the numerator or the denominator is 5.  While the 3rd harmonic is the most consonant non-octave ratio, the 5th is considered the next-most consonant, and 5-Limit tunings typically use the 5/4 ratio for a major 3rd, as well as others for other notes.  The note names for all 5-limit tunings use Johnston notation.  Variants offered:

  • Symmetric 1, Symmetric 2, and Asymmetric – While Pythagorean tuning uses a single ratio as a generator and applies it multiple times ascending, then the reciprocal multiple times descending, these tunings work with TWO ratios as generators – the Pythagorean 3/2 (perfect 5th), and 5/4, a just major 3rd.  Pythagorean is applied to a reference frequency, let’s call it C, twice ascending, then twice descending, to generate 5 notes.  These 5 notes are then multiplied by the 5/4 ratio, and again by its reciprocal, 4/5, giving a total of 15 pitches – the full set includes 2 D’s, 2 Bb’s, and F# and Gb as two distinct pitches – the difference between all of them is 21.5 cents, the syntonic comma.  These three 12-note variations differ by which of the “duplicate” notes are chosen – Symmetric 1 uses the 9/8 D, the 16/9 Bb, and the 64/45 tritone of Gb; Symmetric 2 uses the 10/9 D, the 9/5 Bb, and the 64/45 Gb; Asymmetric uses the 9/8 D, 64/45 Gb, and 9/5 Bb. 
  • Aysmmetric Extended – uses the 9/8 D, the 9/5 Bb, the 36/25 Gb tritone, AND an F# tritone of 25/18.
  • Full includes ALL of the above pitches, PLUS an alternate Gb of 45/32, for a total of 17 notes per octave.
  • Danielou 53-note and 36-note scales – the 20th century theorist Alain Danielou, in his 1967 book, “Semantique Musicale,” built a 53-note scale using only 2-, 3-, and 5-limit ratios.  A few years later, he worked with an engineer to build an instrument implementing 36 of the 53 notes. 

7-Limit Just Intonation – A set of tunings which incorporate the 7th harmonic, as well as the 2nd, 3rd, and 5th.  Characteristic of these tunings is the inclusion of the septimal minor 7th, a 7/4 ratio to the reference frequency.  Variations include:

  • 12-Tone – a 12-tone chromatic scale, with traditional Western note names, built with 3-, 5-, and 7-Limit steps.  Includes a 7/5 tritone.
  • Tonality Diamond – a Partch-style tonality diamond incorporating 7-limit ratios
  • 12-Tone, using mostly Septimal Ratios – a 12-tone chromatic scale using 7-limit ratios for all steps except the perfect 4th and perfect 5th.  Includes both the 7/5 and 10/7 tritones.
  • Three-Generator – Uses the 3/2, 5/4, and 7/4 ratios as generators – 3/2 is applied to the reference frequency 2x and its reciprocal twice, then 5/4 and its reciprocal are applied to those five frequencies, for a total of 15 notes.  7/4 and its reciprocal are then applied to all 15, for a total of 45 notes.  Note names are given in Johnston notation.
  • Centaur Scale (Kraig Grady) – a 12-note scale built on 3-, 5-, and 7-limit ratios, originally developed for ear training.  Invented by composer, theorist, and Erv Wilson archivist, Kraig Grady.  An example of what Wilson called a “Constant Structure,” a scale which produces each “functional” interval in a constant number of steps from any scale degree. 
  • Erv Wilson 19-tone scale – An Erv Wilson-defined “Constant Structure” also built on 3/2, 5/4, and 7/4 generators.  Erv Wilson was a pioneering tuning theorist in the latter 20th century, a genius who presented some really unique ideas on why human history has produced so many pentatonic, heptatonic, and 12-note scales (“Moments of Symmetry”), how to map such scales to a generalized keyboard capable of mapping more than 12 notes per octave (a hexagonal keyboard based on Bosanquet’s earlier design of recessed banks of keys, and since implemented on a few different MIDI controllers), and several unique tuning structures, many of which are also implemented on the I-1.  A really great introduction to Wilson’s ideas is Terumi Narushima’s “Microtonality and the Tuning Systems of Erv Wilson,” listed in the bibliography at the end of this appendix.
  • Danielou 53-note and 36-note scales – similar to the 5-limit versions of these scales, but swapping in 7-limit ratios where the ratios are simpler.

Genesis Scale – Composer Harry Partch’s Genesis scale, outlined in his book, “Genesis of a Music.”  This scale is built on a “tonality diamond” (his book also defines this notion) using 11-limit just intonation.  Partch built his own instruments to play his music on this scale; his scores don’t use traditional note names/staffs, only just intonation ratios and instrument-specific tablature.  In the I-1, note names can be ratios (select 1/1 in the Reference Note Name drop-down) or traditional Western names in Johnston notation.

Ptolemy’s Harmonika – 20 heptatonic scales compiled (and in some cases invented) by the ancient Greek philosopher Ptolemy, from his music theory treatise, Harmonika.  Tunings are attributed to several figures of Greek antiquity – Archytas, Aristoxenus, Eratosthenes, Didymus, and Ptolemy himself (including his Intense Diatonic, which is widely considered to be the basis of Western music).  The tunings are each expressed as just integer ratios, derived from monochord markings, and despite being from ancient Greece, none (except perhaps Ptolemy’s Intense Diatonic) are built by “Pythagorean” tuning.  The notes of each scale are mapped to sequential keys, despite each scale being 7 notes and containing wide gaps which could suggest skipping a key.  It may be helpful to map the scale to only the white keys on a Halberstadt-pattern keyboard, by copying the tuning for customization and inserting empty (0-frequency) scale degrees where appropriate.

22 Shruti – Based on the work of Indian music theorist Dr. Vidyadhar Oke, who also designed a harmonium capable of not only playing each of the individual notes, but shifting into subsets appropriate for different raga.  In this system, there is only one Shadja (root of the scale and recipient of the reference frequency), and only one Pancham (3/2 ratio); they are immovable (achala).  The other scale degrees each have two variations, 21.5 cents (a syntonic comma) apart, and are considered “moveable” (chala).  Which version of each should be used depends on which raga, and gets close to pure just intonation of the notes built off of a scale degree.  Dr. Oke outlines it much more clearly, and in more space, than I can:  https://22shruti.com

Arabic-Turkish-Persian – 3 variants of broadly-similar tunings and systems of music, all of which are built around the concept of maqam.  A maqam is built from 2 tetrachords – hundreds of maqam are composed of a much smaller set of tetrachords. 

  • The Eastern Mediterranean variant is built off of the work of theorist Sami Abu Shumays.  It starts as 24 edo, but the first 4 powers of 3/2 replace the corresponding edo-generated slots.  According to Shumays, however, the tuning is absolutely NOT based on 24 edo – in fact, there is no standard, mathematically-derived set of pitches.  Each town, and even individual ensembles, has its own tuning.  I would encourage you to not use the 24edo-derived tuning straight out of the box.  Instead, develop an ear for a particular ensemble, then copy the canned tuning to a custom tuning and adjust the individual pitches of the octave definition by ear, using the fine-tuning sliders.  Note names are given in Arabic, once with Latin letters and again in Arabic script.
  • Persian – based on the work of Persian theorist Hormoz Farhat, who similarly argues that there is not a single, standard, mathematically-derived Persian tuning; tunings are town- and ensemble-specific.   This tuning is defined in cents, according to a table defined by Farhat, with Western note names plus the symbols for Sori and Koron.  Again, the initial values are a starting point, and not a hard-and-fast immutable value.  Copy the tuning, then adjust using the fine-tuning sliders, by ear.
  • Turkish – 24 notes, including quarter tones, but defined using a subset of 53-edo.  The “quarter tones” are not precisely quarter tones, closer to third tones in either direction.  Note names are Turkish.  Turkish musical influence spread as far east as outer Mongolia, and across eastern Europe, via the Ottoman Empire. 

Chinese Shi’er Lu tuning – A Chinese system built on just intonation ratios.  Modes are constructed by choosing a pitch, then recalculating pitches from that frequency using the original ratios, and selecting one of a set of pentatonic scales, each with its own distinctive character.  Note names are given in Latin and traditional Chinese characters. 

Thai Classical – Based on 7edo, which is an oversimplification – like Arabic tunings, there is no single Thai tuning.  Tunings differ by region, town, ensemble, and individual instruments.  7edo is given as a starting point; you are encouraged to create a custom tuning and fine-tune the individual scale degrees.  Note names are given in Thai script.

Gamelan – 4 variants are defined – Pelog, Slendro, Degung, and Madenda – but again, these are just starting points, as there is no standard, mathematically-derived tuning for any of the 4.  For each of these, an authentic tuning can be obtained by copying the canned version for customization, then fine-tuning the notes by ear.

  • Pelog – 7-note scale uses reference frequency and notes 1, 2, 4, 5, 6, and 7 of 9edo as a starting point. 
  • Slendro – pentatonic scale built on 5 edo.
  • Degung and Madenda – two pentatonic scales used in Sundanese Gamelan, based on tunings measured from an actual Suling.  The notes are reference frequency, 107 cents, 318 cents (Degung only), 505 cents (Madenda only), 711 cents, and 812 cents.

Historic Just Intonations – A grab bag of some medieval/early-Renaissance just intonation tunings from organ builders and mathematicians of the time.  Pythagorean tuning was the dominant tuning, with the obvious intractable problem of the Pythagorean comma; there were many, many attempts to address this until 12-tone equal temperament became the norm.  Tunings included here were compiled in the highly popular book by J. Murray Barbour, “Tunings and Temperaments – A Historical Survey”.  They are:

  • DeCaus’ Monochord
  • Euler’s Monochord
  • Mersenne’s Spinet Tunings, No. 1 and No. 2
  • Mersenne’s Lute Tunings, No. 1 and No. 2
  • Kepler’s Monochords No. 1 and No. 2 (Kepler originally published them in Harmonices Mundi)
  • Vincenzo Galilei’s Approximation (Vincenzo was Galileo’s father)

Meantone Temperaments – Pythagorean tuning has a large comma present between notes at the back end of the circle of 5ths.  It also has pretty sharp 3rds, subjectively compared to contemporary ears and objectively compared to the 5/4 major 3rd.  As 3rds came into vogue in Europe, it became desirable to both flatten Pythagorean 3rds and make more keys usable in performance.  One solution was to distribute the comma across several 5ths.  Variants here include several historically prominent distributions, referred to as fractions of a comma.  ¼ Comma Meantone was the most widely used in practice, though others implemented in the I-1 are 1/3, 1/6, 2/7, 3/10, and 5/18.  The math is the same; why not try your own fraction?  Meantone Temperament Algorithm with Custom Temperament allows you to do just that —  specify a fraction of a comma as a floating point value between 0 and 1.

Well/Circular Temperaments – Throughout the Baroque era and into the early classical period, leading composers, most prominently Bach, demanded tunings where all keys were playable.  Further, and unequal, distributions of the comma improved thirds and made all 24 major and minor keys playable, though each key retains a unique color, as intervals are different from key to key.  It is important to note that these tunings are NOT 12-tone equal temperament.  These are the category of tunings for which Bach’s Well-Tempered Klavier were written.  A great many were devised, each with their own strengths and drawbacks, and several were historically noteworthy.  Bach himself likely preferred Neidhardt III (Grosse Stadt).  Circular tunings continued to be devised through the classical period, and even in the present day.  Those provided by the Isle I-1 include:

  • Werckmeister Correct Nos 1, 2, and 3 (commonly referred to as Werckmeister III, IV, and V)
  • Werckmeister Septenarius (Werckmeister VI)
  • Neidhardt Dorf, Kleine Stadt, and Grosse Stadt (1724)
  • Neidhardt’s Third Circle No. 1 (1732)
  • Valotti’s 1/6 Pythagorean Comma (1754)
  • Young No. 1 (1799)
  • Young No. 2 – 1/6 Comma (1800)
  • Kirnberger I (1/2 comma, 1771), II (1/2 comma, 1776) and III (1/4 comma, 1779)
  • Bendeler I – 1/3 Comma
  • Artusi’s Bonded Clavichord Tuning, No. 1
  • Schlick (a hypothetical tuning based on his writings, suggested by Barbour)
  • Ganassi’s Monochord
  • Nigel Taylor (a contemporary piano tuner and music theorist — https://www.ringing.info/nigel-taylor/index.html — this tuning is bundled, as a Scala file, with Apple’s Logic Pro)

Wendy Carlos – The composer and electronic music pioneer has devised several tunings over her career, featured prominently on her album, “Beauty in the Beast”.  Alpha, Beta, and Gamma use 9, 11, and 20 equal divisions of a 3/2 perfect 5th rather than an octave; octave equivalence does not exist in these tunings, but the major triads are maybe the most consonant harmony I’ve ever heard.  Also included are her Harmonic and Super Just scales, both built on just intonation ratios.

Bohlen-Pierce – Built by subdividing a 3/1 “tritave,” rather than an octave.  Developed independently around the same time by Heinz Bohlen, John R. Pierce, and Kees van Prooijen.  Variants include a just intonation version, and a version with 13 equal divisions of the tritave.

Linear Scale – The term “linear scale” was coined by Erv Wilson to describe scales produced by a single generator value, such as the Pythagorean 3/2 ratio.  This tuning accepts a generator value as a floating-point ratio greater than 0, an interval of equivalence (you’re not limited to octaves), and the number of applications in either direction.  Note names follow Wilson’s naming, which is scale degree followed by a dot.  The root is written as 0/<number of notes>.

Multiple-Generator Scale – a generalized way to generate tunings using more than one generator.  You define an interval of equivalence (again, not limited to octaves), and are given one generator to start; add generators by clicking the + in the list header (remove one by clicking its trash can icon).  The generator value may be defined as a ratio of two numbers separated by a slash, or in cents.  You also define the number of applications in either direction.  The generators are applied in order; the first is applied to the reference frequency, and corresponds to a linear scale.  The second is applied to every pitch in the first and creates a “rectangle,” or 2-dimensional grid of pitches.  Additional generators add multiple dimensions.  You can build a lattice by filling out the generators, then copying the tuning as a custom tuning, where you can remove undesired notes from the interval definition.  Take care when defining generator iterations – the number of notes calculated grows very quickly, and can get out of hand with even small numbers of applications over 3 generators.  Large numbers of generators end up creating scales with notes too close together to discern, while making it impractical to map much more than an, or even one complete, octave to an instrument.

Popular Moment of Symmetry Tunings – Another term coined by Erv Wilson, a “Moment of Symmetry” is when the pitches produced by a linear scale line up, such that there are only two (or one) scale step sizes present in the scale.  Using the Pythagorean 3/2 ratio, the first of these Moments of Symmetry happen at 5, 7, 12, and 19 scale steps.  Other generators produce other MOS.  Most of those provided use an octave as the interval of equivalence, though an octave is not required.  Provided are:

  • Miracle (10-note), Blackjack (21-note), Canasta (31-note), and Stud Loco (41-note) scales – based on George Secor’s suggestion of using 1/19th of an 18/5 ratio, or about 116.71 cents, as the generator.  Carrying this generator further ultimately produces a temperament virtually identical to 72edo.
  • Mavila – A 9-tone scale which attempts to temper the 135/128 major chroma instead of a 81/80 syntonic comma.  The result is an “anti-diatonic” heptatonic scale, with intervals reversed – the “anti-Ionian” scale, for instance, has scale steps of ssLsssL instead of the Ionian LLsLLLs.  According to the Xenharmonic wiki, “Because of the structure of this unique tuning, every existing piece of common practice music has, effectively, a shadow version in antidiatonic.”  The generator used is 162.62 cents.
  • Porcupine – 8- and 15- note scales built on a 166.0 cent generator, such that two generators form the 6/5 minor third, and three create a 4/3 perfect fourth.
  • Pajara – the only scale of this group defined over a non-octave interval, Pajara uses a 16/15 ratio as its generator, over an interval of the square root of 2.  Generates 10 notes.
  • Magic – uses a generator of 380.49 cents, tempering out the “magic comma” of 3125/3072, the difference between five 5/4 major 3rds and a tritave.  Several variants, applying the generator different numbers of times in each direction, are provided.
  • Blackwood – a 10-note scale whose generator tempers the 256/243 difference between 5 perfect fifths and the nearest octave of the first pitch of the fifths. 
  • Beatles – the 844.44 cent generator tempers out a couple of 7-limit intervals; 7-, 10-, and 17-note versions are provided. Why this is named “Beatles” seems to be lost to history.
  • Lucy Tuning – A meantone temperament tuning invented by Charles Lucy, inspired by John Harrison (the inventor of longitude).  Harrison wrote that, “The natural scale of music is associated with the ratio of the diameter of a circle to its circumference.”  He devised a scale such that the larger step in a Moment of Symmetry would be the 2*pi root of 2, or 2^(1/2*pi) – about 190.99 cents.  The smaller step would be half the difference between 5 larger notes and an octave, or 122.54 cents.  3 large steps plus a small, adding up to a 5th in this system, is 695.493 cents, which is then used as the generator for this scale.  19-, 25-, 31-, and 43-note versions of this scale are provided.  Read more at Charles Lucy’s website, and perhaps treat yourself to a Lucy-intoned guitar: https://www.lucytune.com/index.html

Erv Wilson Tuning Structures

The following 3 major groupings of tunings are based on the work of Erv Wilson and are built from his concept of  Combination Product Sets, somewhat similar to Partch’s tonality diamonds.  A Combination Product Set takes a number of tones/ratios passed in as inputs, and multiplies each combination to arrive at a number of pitches laid out along the vertices of various shapes.  When used with ratios, the original reference frequency becomes merely a suggestion, and is not itself present in the resultant set of pitches; the result is a sort of centerless framework built out of just intonation ratios.  Wilson was inspired in the late 1960s by the advent of space exploration and the weightlessness experienced by astronauts.  A much better, deeper explanation of all of the structures present here (and a few which are not) is given by Dr. Terumi Narushima in her book, “Microtonality and the Tuning Systems of Erv Wilson”, which serves as a great introduction to Wilson’s ideas. 

Hexany – A Hexany works on all combinations of 4 pitches, taken 2 at a time, producing a total of 6 pitches.  Any value expressed as a ratio of the reference frequency can be used as a parameter.  Several related structures are also provided as variants:  the Tetradic Diamond, the Hexany Diamond, and the Stellate Hexany.    

Eikosany – Similar to Hexany, but takes all combinations of 6 values, 3 at a time, for a total of 20 individual notes.  Two related structures are also provided as variants:  the Hexadic Diamond and Stellate Eikosany.

Hebdomekontany – The largest and most complex of the CPS structures, and also the most fun to type.  This takes all combinations of 8 ratios, 4 at a time, for a total of 70 pitches per octave. 

Bibliography

Some excellent reading on tuning:

  • Gann, Kyle.  “The Arithmetic of Listening.” Urbana, Illinois: Univeristy of Illinois Press, 2019.  A composer in his own right, student of Ben Johnston, and long-time music critic, this book is an indispensable journey of Western tuning theory from ancient Greece, through medieval and Renaissance Europe, through the ideas behind various meantone and circular temperament approaches, to 12-TET and over EDOs, and most of all, Just Intonations (5-limit, 7-limit), Patch’s Genesis Scale, and the work of Johnston, Partch, Gann himself, and LaMonte Young.  I can’t emphasize enough the importance of the foundations of tuning laid out in this book. 
  • Barbour, J. Murray. “Tuning and Temperament – A Historical Survey.”  East Lansing, Michigan: Michigan State College Press, 1951.  An indispensable compendium of Western, and occasionally other, tunings, up to 12-tone equal temperament. 
  • Narushima, Terumi.  “Microtonality and the Tuning Systems of Erv Wilson.”  Oxfordshire, UK: Routledge, 2018.  Erv Wilson devised so many unique insights into and structures of tuning and seemed to be a pretty unique personality as well.  This book performs the enormous task of piecing together an accessible introduction to Wilson’s ideas from an enormous collection of loose notes, hand-drawn and variously annotated diagrams, letters, and personal anecdotes and recollections by his students; Wilson himself left no such outline of his work. 
  • Farraj, Johnny and Shumays, Sami Abu.  “Inside Arabic Music.”  New York, New York: Oxford University Press, 2019.  A really comprehensive survey of theory and instrumentation of Arabic music.  Shumays gave an incredible lecture in 2024 titled, “The Politics of Maqam Scales and the Decolonization of Music Studies,” which you can watch here: https://www.youtube.com/watch?v=DsLLgKDfaOo, which was the tipping point in changing the entire direction I was taking in implementing tuning in the Isle I-1.  Coming from a background as a Westerner, in Western music, implementing something on a device that only really understands math, it’s easy to forget that music is something to be heard, something which should sound the way it was intended.  So much of tuning is done by ear, so much of music is done by ear.  I hope I put some of that back into synth tuning, that was able to allow something of the essence of music to peek through all of the mathematical precision the device requires. 
  • Farhat, Hormoz.  “The Dastgah Concept in Persian Music.”  Cambridge, United Kingdom: Cambridge University Press, 1990.  A thorough book on Persian music theory, and also reinforced the idea that I need to allow some flexibility in tunings, a way to do at least part of a tuning by ear.
  • Partch, Harry.  “Genesis of a Music.”  New York: Da Capo Press, 1974.
  • The Xenharmonic Wiki.  https://en.xen.wiki/w/Main_Page 
  • Tonalsoft Encyclopedia of Microtonal Music.  http://www.tonalsoft.com/enc/encyclopedia.aspx A website written and maintained by Joe Monzo, a ton of great information on Microtonal and Just Intonation theory.
  • Kepler, Johannes. “Harmonies of the World.” Kepler is better known as an astronomer, and the ideas relating planetary motion in our solar system to an ideal tuning and system of music contained within this book are not quite scientific (also, we know more about planetary motion today than Kepler was able to deduce, and have discovered three more planets than he knew about). Still, this is a fascinating read. There’s always been a sense of something beyond our grasp about music; mankind has never stopped trying to reach for it. Also, a recording of Laurie Spiegel’s Harmony of the Worlds, making use of the ratios Kepler outlined in this book, is on the Golden Record affixed to the Voyager I and II spacecraft.

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