This track was written to demonstrate the harmonic possibilities of using higher harmonics freely whenever the composer wishes, without the bounds of a traditional scale, but within the bounds of major/minor/seventh harmony. It is also quite funky!

Harmonics of 11ths, 13th, 23rds etc are prominent in this composition, without being overwhelming in their unusualness. The highest prime harmonic used was the 127th harmonic in the final melody! E#''' was also used (125th harmonic)

It was written using a new notation system which has been devised to make accessing higher harmonics as intuitive as possible (it still takes a bit of getting one's head around!). Please contact me if you would like to see the score written in this notation system, or for details about how this notation system enables interesting harmonic composition outside of the bounds of the Western 12-note scale to be realised.

It qualifies for chiptune since although it is (much to my shame) 16-bit 44100Hz audio instead of the pure 8-bit, the sounds were synthesised from scratch using modified sin/saw waves, with a bit of band-limited noise thrown in for good measure. The synth took as input CSV 6xN matrices as sequence files, and processed in Octave/MATLAB to give MP3 output which you hear here.
---------
update 10th Apr 2015:

Here is a brief description of how the MP3 track was generated:

Just Intonation Notes (1/1 4/3 7/4 etc) <-> New Notation System (C F `Bb~7 etc) which is easier to read than pure numbers and where the letter corresponds (roughly) to the position in the 12ET scale
-> Excel block of chords and melody notes for visual presentation to composer (.txt)
-> Excel long (~1700 columns) of calculated note letters, frequencies, note lengths, etc
-> Excel final sequence file with 6 columns: channel, start time, frequency, note length, loudness, stereo position (.csv)
-> Octave/MATLAB code to take the 6 columns and turn them into chiptune (.mp3 or .wav)
-> Sometimes extra post processing in Audacity for final .wav

------------------------

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Sonata 1 'Cheery' - Chip Synth, Justly Intoned

This is a justly intoned arrangement of J. S. Bach's Prelude in C Major, the first Prelude of the 48 Preludes and Fugues, a body of work by Bach which demonstrated that with tempered tunings every major and minor key was usable.

As well as being one of my favourite pieces of music, I thought it would be worthwhile to demonstrate that this music sounds just as interesting (maybe more interesting!) in Just Intonation (JI), where exclusively rationally-spaced intervals are used. JI is the natural way of tuning intervals. It derived from harmonic series, which are produced by string and wind instruments.

Due to the 'wrap-around' nature of tempered tunings, some transitions (e.g. D minor to G major) are good transitions in tempered tunings, but sound 'further away' or less harmonious in JI. Also, transitions from ,D' to `D (10/9 to 9/8) and from C to `C~7 (1/1 to 63/64) are noticeably microtonal shifts. Some people might like them, some might not. Personally, I like them. The major advantage is the purity of all the triads and seventh chords, which in my opinion outweigh the 'disadvantages' of having noticeable microtonality.

As always, I've slipped in an 11th and 13th harmonic (the 11th is obvious, the 13th less so). I refuse to write any new music which does not contain both 11th and 13th harmonics.

Another interesting feature was Bach's usage of diminished chords. My favourite justly intoned diminished chord is the sequence 10:12:14:17:20:24:28:34:40 etc. Hence, some 17th harmonics have been used, wherever there is a diminished chord, in both 17/1 and 1/17 formats (and 5/17).

Hence, this track contains all the harmonics 2, 3, 5, 7, 11, 13 and 17. I will attempt to remedy the lack of 19, 23, ..., 127th harmonics in my next upload.

Enjoy!

Dave

------------------------

Bach number of this track: BWV846 or BWV 846

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Bach BWV846 Prelude in C major - Chip Synth, Justly Intoned

This is a justly intoned arrangement of Chopin's Etude in C Major (Opus 10, No. 1), the first Etude of his series of Etudes which demonstrated his new piano performance techniques.

Although Etudes are usually technical exercises, Chopin's are special since they are immensely listenable with intricate harmonies.

I have retuned Chopin's harmony to have much more than 12 notes to the scale. More to come (hopefully soon!) about the notes which have been used. Concisely, the major and minor triads and sevenths have almost everywhere been preserved, and Chopin's use of sequences leads the harmony down very interesting and long paths in the 7-limit tonal lattice. A couple of jarring transitions are where the harmony jumps up by a syntonic comma (or 7-limit comma). As always, 11th and 13th harmonics are smuggled in, in some of the diminished chords near the end.

Enjoy!

Dave

------------------------

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Chopin Etude in C Major - Op. 10, No. 1 - Chip Synth, Justly Intoned

Stevie Wonder has written a great song - You Are The Sunshine Of My Life. But those amazing harmonies are crying out for something - Just Intonation!

By using my new notation system to convert his piano scale harmonies to natural harmonics, a purer and more interesting harmonic structure is revealed.

As always, the underused 11th and 13th harmonics are hidden in this retuning (in the intro section using neutral thirds)

Enjoy! Dave

------------------------

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Stevie Wonder - You Are The Sunshine Of My Life - Chip Synth, Justly Intoned

This is a justly intoned arrangement of J. S. Bach's Prelude and Fugue in C Sharp Major, the third Prelude and Fugue of the 48 Preludes and Fugues, a body of work by Bach which demonstrated that with tempered tunings every major and minor key was usable. It was made using a custom sequencer in Excel+Octave which allowed exact frequencies to be specified.

'Comma Travel' means that local sections of the Prelude and Fugue were specified using simple justly intoned ratios, but the whole piece was stitched together by (from time-to-time) specifying commas, that is, microtonal intervals which make the different sections exactly match up. By altering the baseline frequency dynamically in this way, the whole piece of music suffers from 'comma travel' which is the accumulation of commas in the sharp or flat direction, making the music shift up and down in frequency. However, since there's not much in the way of 'comma travel' music already on the market, I thought this would sound interesting to some people!

A more sophisticated way of making the piece of music stay at the same base frequency would be to normalise the commas by dividing out the comma by a local moving average; this has also been done in two separate tracks uploaded at the same time.

Accumulation of commas, normalised or unnormalised, will be explained further in my second JI maths/music paper - pending! The theme of it is how to take any sheet music and retune it into JI using comma alterations, mainly of 5th and 7th harmonics. However, as always, in this track I've snuk in some 11th and 13th harmonics which you might be able to locate by ear or by examining the code blocks in Github (upload pending as of 9th Nov 2015).

Enjoy!

Dave

------------------------

Bach number of this track: BWV848 or BWV 848

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Bach BWV848 Prelude And Fugue in C Sharp Major - Chip Solo Synth, Justly Intoned, Comma Travel

This is a justly intoned arrangement of J. S. Bach's Fugue in C Sharp Major, the third Fugue of the 48 Preludes and Fugues, a body of work by Bach which demonstrated that with tempered tunings every major and minor key was usable. It was made using a custom sequencer in Excel+Octave which allowed exact frequencies to be specified.

'Normalised commas' mean that the piece was composed in simple Just Intonation by specifying local frequency ratios, the most important of which are major (4:5:6) and minor (10:12:15) triads locally. However, between different sections they may be out of tune by microtonal commas (e.g. 81/80) and to fix this the different sections need to be stitched together by multiplying the later section by an accumulation of commas. This means that we get 'comma travel' so the piece moves sharp or flat in frequency. To fix this second issue, the commas can be normalised by dividing out by a local moving average of the accumulated commas.

By comparing this normalised version to the 'comma travel' version, it is easy to detect this movement in pitch. The commas on this track were normalised over 64 beats (4 bars).

Accumulation of commas, normalised or unnormalised, will be explained further in my second JI maths/music paper - pending! The theme of it is how to take any sheet music and retune it into JI using comma alterations, mainly of 5th and 7th harmonics. However, as always, in this track I've snuk in some 11th and 13th harmonics which you might be able to locate by ear or by examining the code blocks in Github (upload pending as of 9th Nov 2015).

Enjoy!

Dave

------------------------

Bach number of this track: BWV848 or BWV 848

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Bach BWV848 Fugue in C Sharp Major - Chip Synth, Justly Intoned, Normalised Commas

This is a justly intoned arrangement of J. S. Bach's Prelude in C Sharp Major, the third Prelude of the 48 Preludes and Fugues, a body of work by Bach which demonstrated that with tempered tunings every major and minor key was usable. It was made using a custom sequencer in Excel+Octave which allowed exact frequencies to be specified.

'Normalised commas' mean that the piece was composed in simple Just Intonation by specifying local frequency ratios, the most important of which are major (4:5:6) and minor (10:12:15) triads locally. However, between different sections they may be out of tune by microtonal commas (e.g. 81/80) and to fix this the different sections need to be stitched together by multiplying the later section by an accumulation of commas. This means that we get 'comma travel' so the piece moves sharp or flat in frequency. To fix this second issue, the commas can be normalised by dividing out by a local moving average of the accumulated commas.

By comparing this normalised version to the 'comma travel' version, it is easy to detect this movement in pitch. The commas on this track were normalised over 48 beats (8 bars).

Accumulation of commas, normalised or unnormalised, will be explained further in my second JI maths/music paper - pending! The theme of it is how to take any sheet music and retune it into JI using comma alterations, mainly of 5th and 7th harmonics. However, as always, in this track I've snuk in some 11th and 13th harmonics which you might be able to locate by ear or by examining the code blocks in Github (upload pending as of 9th Nov 2015).

Enjoy!

Dave

------------------------

Bach number of this track: BWV848 or BWV 848

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Bach BWV848 Prelude in C Sharp Major - Chip Synth, Justly Intoned, Normalised Commas

This track is constructed from all chords using the numbers 1 to 6 which meet certain criteria. Using whole numbers to construct harmony is 'Just Intonation'. Both the otonal chord (a, b, c) and the utonal chord (1/c, 1/b, 1/a) were used.

This track is intended to help composers to familiarise themselves with the chords available in 5-limit harmony.

For example, (3, 4, 5) is a (G, C, E') major triad in the second inversion. It is otonal, and its utonal counterpart is (1/5, 1/4, 1/3) = (12, 15, 20) which is (G, B', E') minor triad in the first inversion.

The rules for including a chord (a, b, c) were:
  a < b < c
  gcd(a,b,c) = 1
  4/3 <= c/a <= 4/1
  chord is otonal, which is (abc)^(1/3) < lcm(abc)^(1/2)

The final otonal harmonies found were:
1	2	4
1	3	4
1	2	3
2	5	6
2	3	5
2	4	5
2	3	4
3	5	6
3	4	5
4	5	6

The corresponding utonal harmonies were:
1	2	4
1	3	4
1	2	3
2	5	6
2	3	5
2	4	5
2	3	4
3	5	6
3	4	5
4	5	6
(note that 1 2 4 does not have a utonal counterpart)

Each harmony was used 3 times, by multiplying each chord by 6, 8 and 9, giving versions in keys of C, G and D.

They were then arranged in:
  descending order of MaxRatio = c/a
  ascending order of bass note 'a'
  descending order of middle note 'b'.

I hope that you find this track useful! Dave.

PS apologies for the lack of 11th and 13th harmonics, however those were not part of the construction schema :/

------------------------

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Example track of 3 note harmonies in Just Intonation constructed from the numbers 1 to 6

This track is an example of using the harmonic series itself as a scale. Frequencies are accessed directly by specifying 6 7 8 9 10 multiplied by a set number (e.g. 30) to give direct frequencies such as 180 210 240 270 300 Hz.

It is in four parts of 8 bars each
- Simple harmonics (otonal)
- Complex harmonics
- Inverted simple harmonics (utonal)
- Inverted complex harmonics 

It was created using:
- Excel for sequencing
- Octave for turning CSV sequence file into WAV
- Audacity for mixing the 2 harmony tracks and 3 click tracks.
- SoundCloud for upload.

I hope that you find this track useful! Dave.

PS it is good to be using the 11th, 13th, 23rd, 29th and 31st harmonics again. I am extremely keen that musicians use them and that the public be exposed to these exotic harmonies ;)

------------------------

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Example track with noodling around the otonal (harmonic) and utonal series

Example track with noodling around the otonal and utonal harmonic series

This track is an example of a simple twelve bar backing track accompanying a solo melody written in free Just Intonation (JI).

JI allows access to all the frequencies, not just the ones on the piano keyboard or guitar frets. Often musicians when playing the blues, jazz, or guitar solos, bend the string to produce a higher pitch. In this track, there are no 'bent' notes, only precisely specified notes which are often noticeably different to the standard notes from 12-EDO.

Some devices used include:
- No scale - only addressing frequencies (in Hz) directly via whole numbers. E.g. 4, 5, 6 multiplied by 60 Hz would give 240Hz, 300Hz, 360Hz directly.
- Descending harmonic series, e.g. 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16
- What sounds like vibrato, but is actually a combination of holding and sliding pitches several times a beat
- Rapid switching of frequency several times a beat, this is a common chip tune device.
- Descending shifting triads, e.g. 61, 54, 47, 56, 49, 42, 51, 44, 37, 46, 39, 32...

Just Intonation is the future for those of us who like both maths and music! I understand that not everyone wants numbers in their music composition. However, that's fundamentally what frequencies are, thats what harmony is, a combination of whole numbers. And you get more interesting results when using the numbers for what they are, which is unique chords well outside of traditional harmony. I am particularly enjoying the sound of the prime numbers 11, 13, 23, 29, 31, 41, 43 and 61 in this composition.

May the prime numbers be with you! Enjoy - Dave

----------

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Example of the Twelve Bar Blues with crazy Solo in Just Intonation

This track is an example of a simple twelve bar backing track accompanying 3-note block harmony written in free Just Intonation (JI) with added sliding chords.

For example, one transition used is between a major triad 1st inversion (5 6 8) to a major triad root position (4 5 6), with 1 and then 2 different intermediate chords on the 1st and 2nd repetitions respectively:

1 intermediate chord on 1st repetition:
16   14   12
12   11   10
10    9     8

2 intermediate chords on 2nd repetition:
24   22   20   18
18   17   16   15
15   14   13   12

You could call these types of transition a ‘linear step’ since the frequencies of each note in the harmony undergo linear steps from the starting point to the end point. They are otonal, operating on the upper harmonics (the set of whole-number N, as opposed to utonal which operate on 1/N).

These types of transitions give access to higher prime numbers such as 11, 13, 17 in the examples above. In Just Intonation, hearing the higher prime numbers is the coolest thing of all and I fully anticipate this being as interesting and widespread a hobby as trainspotting or stamp collecting in the decades ahead. Imagine listening to all the prime numbers between 1 and 500! The sheer joy!! No doubt this type of thing will have a high frequency in the future.

If prime numbers wasn’t enough, some chords undergo continuous linear slides instead of linear steps. In the track icon, it is a screenshot of the spectrogram of this track, and a linear step looks like a ladder, but a linear slide looks like a slope of constant gradient. The great thing about a linear slide is that at any rational point during the slide, the notes are in a whole number ratio, but that ratio is changing constantly. So I guess it counts as some kind of extension of Just Intonation.

May your musical listening frequently slide into Just Intonation :) Dave

----------

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Example of the Twelve Bar Blues with 3-note harmony using linear steps and slides in Just Intonation

This track is an example of where the melody in each bar alternately takes utonal and otonal values, that is, uses alternately subharmonics (of a very high note) and harmonics (of a very low note).

The bass and backing was devised first, using a repeating series of 12 chords which were alternately utonal and otonal.

After that, the melody was chosen from the subharmonic or harmonic spectrum for that chord in each bar.

Its in three parts - a melody, a 2 part harmony, then all 3 parts superimposed.

Harmony is supposed to be a bit tricky and captivatingly strange at times! Enjoy the 11ths, 13ths and 29ths - Dave

--------------

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Example of alternating undertone and overtone melody and harmony

This track is an example of using high prime numbers to create melody and harmony, to refute the claim that only prime numbers up to 37 / 211 / some arbitrary limit are useful for making music. This piece of music has a prime limit of 2499949 and the twin primes 2499949, 2499947 can be heard in sequence! To be honest, the focus was more on using higher prime numbers than making a decent sounding bit of music. So it sounds a bit out of tune! Sorry about that if you don’t like microtones! But I do. So here it is.

There are 2 bass lines which create a blues backing of fifths and major sixths in 5-limit JI. The overall structure is a twelve bar blues with one repetition and intro/outro.

There are 3 melody lines which meander about all over the place. Each bar is split into 12 chords, and the twelve bars are grouped into 5 sections of 48-24-24-24-24 beats. Each section starts and ends with a low-limit chord, mostly 5-limit triads or 3-limit fifths and octaves. Various interpolation methods are used to generate approximate intervals which are matched up to a big long list of prime numbers. The result is that higher primes are used as intermediate chords. I’m sure you will agree that the result is impressive! No two intermediate chords are likely to be tuned exactly the same under this scheme, demonstrating the virtually infinite flexibility of the chords in Just Intonation. (Note that some authors use Rational Intonation for when the primes get bigger than say around 41.)

Original sequencing and matching to prime numbers: Excel
CSV sequence file to WAV file: custom MATLAB/Octave script
Post-processing: Audacity

So a lot of primes were used. Which ones? I wrote an Octave script to interrogate the CSV and find the higher primes (over 1601). This was the result, a list of the 336 highest primes used:
2499949, 2499947, 2351963, 2344751, 2335183, 2322667, 2307073, 2288449, 2266591, 2241527, 2212633, 2180333
2144629, 2105231, 2061847, 2014471, 1963319, 1908703, 1849921, 1787033, 1719701, 1648567, 1258973, 1210427
1164749, 1122427, 1083031, 1045903, 1011947, 980773, 951791, 925447, 902191, 880907, 862399, 846037
831679, 819827, 809833, 801641, 795601, 791159, 788317, 786887, 522521, 521393, 517399, 516839
514201, 507901, 507329, 503317, 501029, 493853, 492029, 486349, 485417, 479263, 476299, 473197
(some omitted... 4000 char limit!)

The highest two primes (2499949, 2499947) are played in bar 5 in the top melody, as notes 2 and 3. They sound very similar!

Also, here are the fractions used for the first six bars of the top melody, which have units of Hertz (e.g. direct frequencies):
432/1, 1809/4, 1647/4, 1917/4, 1809/4, 1971/4, 1971/4, 2133/4, 2241/4, 2241/4, 2403/4, 2403/4
2403/4, 2619/4, 2619/4, 2727/4, 2727/4, 2727/4, 2889/4, 2781/4, 2889/4, 2943/4, 2943/4, 2943/4
3051/4, 3429/4, 3429/4, 3537/4, 3537/4, 3699/4, 3699/4, 3699/4, 3753/4, 4023/4, 4023/4, 4077/4
4077/4, 4239/4, 4401/4, 4239/4, 4509/4, 4509/4, 4671/4, 4671/4, 4671/4, 4833/4, 4833/4, 1296/1
1296/1, 22499541/16384, 22499523/16384, 21167667/16384, 21102759/16384, 21016647/16384, 20904003/16384, 20763657/16384, 20596041/16384, 20399319/16384, 20173743/16384, 19913697/16384
19622997/16384, 19301661/16384, 18947079/16384, 18556623/16384, 18130239/16384, 17669871/16384, 17178327/16384, 16649289/16384, 16083297/16384, 15477309/16384, 14837103/16384, 864/1

The highest prime 2499949 can be found here in the 22499541/16384 note.

No doubt this artistic endeavour will be used to spur interest in using higher primes directly in music. It has been fun - enjoy! Dave

--------------

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Example of Prime Numbered Blues with JI prime limit of 2499949

This track has been created using an L-system, which is a type of fractal where large parts are comprised of smaller parts that look like reduced copies of the large parts.

As an example, a melody going up by a Just-Intonation perfect fifth (3/2 frequency ratio) could be comprised of three smaller copies going up 3/2, down 3/2, then up 3/2. This simple example would not be musically very interesting, however the idea is there.

In this particular track, the following mappings were used:

1 beat 1/1 --> 1/2 beat 4/3, then 1/8 beat 1/1, then 1/8 beat 9/8, then 1/4 beat 3/2 (inverted)
1 beat 3/2 --> 5/4, then 6/5 (both 1/2 beat)
1 beat 4/3 --> 10/9, then 6/5 (reversed)
1 beat 5/4 --> 9/8, then 10/9 (reversed)
1 beat 6/5 --> 4/3, then 10/9 (inverted)
1 beat 9/8 --> 1/1, then 9/8
1 beat 10/9 --> 1/1, then 10/9

Reversed means the smaller copy is played back in i) the reverse order and ii) the smaller copies are themselves reversed
Inverted means that the frequency shifts are inverted, e.g. f --> 1/f. So 10/9 inverted means ‘go down by 10/9’ or ‘shift by 9/10’

So an L-system melody has a set of melodies (in this case one each for 1/1, 3/2, 4/3, 5/4, 6/5, 9/8, 10/9) and each has its own set of ‘daughter’ melodies.

Each daughter has:
- an associated frequency shift
- a timescale, here they are all 1/2, 1/4 or 1/8 beat
- optional inversion and reversing operations, giving 4 possible orientations in total.

Rules include
- product of daughter frequency shifts must equal parent frequency shift
- sum of daughter timescales must equal parent timescale of 1 beat
- continue iterating until maximum timescale of (iterated) daughters is 1/2^n.

In this particular track, the three parts were:
Bass: reference freq 100 Hz, max timescale 1/32
Alto: ref freq 200 Hz, max timescale 1/64
Treble ref freq 400 Hz, max timescale 1/128

This gave a 3 part invention! Due to the fractal melody/harmony, the different parts were consonant with each other.

One extra option is pre-tuning or post-tuning.
- Pre-tuning means ‘take the note frequency to be from the start of the fractal block’
- Post-tuning means ‘take the note frequency to be from the end of the block’

In this track, post-tuning was used, which means that at the end of blocks the tuning tends to be in 1:2:4 (octaves) ratio. Pre-tuning could just as easily be used, meaning that blocks start off on octaves and change to other chords during each block.

Enjoy! Dave

--------------

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Example Just Intonation Fractal 3-part Invention

This is a justly intoned arrangement of J. S. Bach's Invention 1 in C Major, the first of his 2 Part Inventions.

It was made using a custom sequencer in Excel+Octave which allowed exact frequencies to be specified, with plenty of post-processing in Audacity.

The prime limit for this track is 2029. The highest primes used were 2029, 521, 61, 29, 19, 13 and 11. However most of the notes were on the 7-limit JI lattice. I only added higher primes to spice things up! Also I put in a whole stack of pitch bends and special effects which weren't in the original. Hopefully to improve on Bach's work! To be honest, even if its worse, I had fun :) and when musicians originally played this they would have ornamented and embellished it, in my defence. It now has features, its definitely not ruined/buggy, its now full of art and soul.

When arranging this music, a few 'commas' were used, which is where the key has travelled from say C4=1/1 to C[5]4=80/81. The commas were normalised out over 3 seconds, which means the base frequency was smoothed out so that the music was not away from 1/1 for more than around 3 seconds. If this had not been done then the ending key would have been out from the starting key by the product of all the commas encountered along the way.

If anyone is bothering to follow the development of my sequencers and general mixing ability (or is still reading) then I have just discovered Reverb in Audacity. Reverb reverb reverb! Its awesome! I am definitely going to binge and misuse that for the next few tracks. Apologies in advance. I wish I could go back in Time using a Time travelling device and put More Reverb on all my previous tracks (a friend commented they were 'too dry' and I now know what they mean). But SoundCloud has not this feature to modify an Existing Track. They are Set In Stone. So I won't and can't add reverb to my previous tracks. But I would if I could. Reverb rocks! wow

Enjoy!

Dave

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Bach number of this track: BWV772 or BWV 772

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Bach BWV772 Invention 1 in C Major using 2029-limit Just Intonation

This is a justly intoned arrangement of J. S. Bach's Invention 13 in A Minor, one of his set of 2 Part Inventions.

It was made using a custom sequencer in Excel+Octave which allowed exact frequencies to be specified, with plenty of post-processing in Audacity. Especially reverb. Lovely lovely reverb!

The prime limit for this track is 2579. The highest primes used were 2579, 2053, 1021, 179, 103, 97, 53, 41, 31, 23, 19, 13, 11 as occasional prime comma accidentals; the majority of the notes were 7-limit with additional [17] commas, e.g. to make C# in the key of C use C#[17]4 = 17/16 (where C4 is 1/1).

Pitch bends were also used. For serious work, they are helpful to shift between commas during a note, e.g. a pitch bend from G4=3/2 to G[5]4=40/27 which might match the surrounding harmonies better than either note in isolation. However, pitch bends also sound funky and strange, which is plenty motivation for me to load up Bach with pitch bends, like Bach on diisopropyltryptamine. After all, did not Baroque musicians embellish the music as they played?

Normalisation over 3 seconds was used. Locally, each bar or section is in a well defined key, however when stitching different sections together it might be necessary to use comma shifts to get them to match. A bit like General Relativity for musical tuning. Overall this piece had a pitch shift of around a semitone, however this was not audible since each local comma was divided out over approximately 3 seconds by a shift in reference frequency in the opposite direction.

Enjoy!

Dave

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Bach number of this track: BWV784 or BWV 784

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Bach BWV784 Invention 13 in A Minor using 2579-limit Just Intonation

This is a justly intoned arrangement of J. S. Bach's Prelude for Lute, BWV998.

It was made using a custom sequencer in Excel+Octave which allowed exact frequencies to be specified, with plenty of post-processing in Audacity. One again, there is such an abundance of reverb that it borders on reverbabusage. However, this overabundance of reverb turns a band-limited sawtooth wave into something like a stringed instrument, so must now be tolerated by the listener.

The prime limit for this track is 5623. (The observant among you will notice that I am ratcheting up the prime limit for each subsequent track. So secretly its free-JI!) The highest primes used were 5623, 5591, 5531, 5449, 5387, 2039, 1801, 1523, 1009, 769, 647, 643, 619, 509, 383, 313, 307, 257, 241, 229, 223, 191, 181, 179, 173, 149, 139, 127, 113, 101, 97 as well as all the primes to 79.

My original arrangement was mostly 7-limit, however I could not resist a liberal peppering of the aforementioned prime commas, making it sound Out Of Tune to the Philistine Ear. However, to those of us who have ears to hear, it is Pureity (sic) and Magnificence! Long live peppered Bach!

Unfortunately, I forgot to put any pitch bends in this one :( but I did make a special effect like the plug being pulled out at the end :) which makes up for it. Also, normalisation of a few comma shifts was carried out over 2 seconds. The General Relativity of Music strikes again, patching together locally tuned segments into a global tapestry of sound.

Enjoy!

Dave

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Bach number of this track: BWV998 or BWV 998

See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Bach BWV998 Prelude for Lute using 5623-limit Just Intonation

This is a justly intoned arrangement of the Allegro from W. A. Mozart’s Piano Sonata No. 16 in C Major (K545).

The Allegro has two main parts. Each part is repeated, giving the opportunity to try out two different tunings on each part. The first time through, sensible tunings (mostly 7-limit) are used which stick quite closely to what Mozart wrote. The second time through its (WARNING!) a bit avant-garde. I chucked a lot of prime commas in, and detuned/retuned notes in a very trigger-happy way. This will displease as many people as it pleases, and I welcome both positive feedback as well as criticism of my wantonness. Feel free to comment just below the track!

The overall tuning is 3049-limit JI, and the following primes are used: 3049, 2029, 1847, 1823, 1811, 1543, 1471, 1303, 1277, 1049, 1031, 1021, 823, 787, 761, 757, 631, 617, 503, 449, 431, 383, 317, 313, 307, 283, 271, 269, 257, 241, 239, 211, 193, 191, 181, 179, 167, 157, 127, 113, 109 and all the primes from 2 to 97. In addition, some of my audience have complained (BOO!) about introducing pitch bends into classical music (HISS!) so I have not really used pitch bends here, except for once very prominently in order to annoy them, and delight everyone else.

Same sequencer as before:
- Sheet music printed, prime commas annotated, also comma normalisation noted (4 seconds was used in the final version)
- Transcribed into Excel spreadsheet
- Saved to CSV
- Imported into custom MATLAB/Octave sequencer
- Each track sequenced using sawtooths of 3 different decay rates
- Exported as 9 WAV files (3 per track, 3 tracks)
- Imported into Audacity, processed with 3 levels of reverb
- Bit at the end where it sounds like the plug being pulled out. I like that

Enjoy!

Dave

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See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Mozart Piano Sonata 16 (K545) in nicely-tuned Just Intonation with (WARNING!) avant-garde repeats

JI demo using otonal interpolation

I'm into Just Intonation (JI), which is using harmonics (or undertones) to generate music. Over time, my profile will fill up with wonderful and melodious New Harmony, fresh from the infinite vaults of the original and best source of harmony, Just Intonation.

I’ve written some papers on the music theory of Just Intonation. You can check them out via the arXiv link below. Topics include the mathematical analysis of harmony (in Just Intonation), and a notation system to cover all of Just Intonation.

Enjoy listening!

David Ryan

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