An arrangement of Silent Night for porcupine recorder quartet. Porcupine recorders don't exist yet, but they will.
Porcupine is a temperament generated by stacking minor tones of ~163c, three of which make a perfect fourth. It does not exist in 12edo. This arrangement is in 22edo, grounded in Porcupine temperament, and makes use of 11-limit harmony available in Porcupine in 22edo.

22 Silent Porcupines

A little duet for my Xenharmonic instruments: My Axis-49 and an Irish Whistle I built and tuned to Superpyth temperament in 22edo. Superpyth temperament is like meantone but in the other direction - the fifths are wider such that tones 9:8 and 8:7 are equated, (instead of meantone's equating of 9:8 and 10:9), and two of these 'eventones' make a septimal supermajor third of 9:7, (rather than meantone's major third of 5:4). The diatonic semitones, at 55cents are much smaller than normal. 22 equal divisions of the octave supports this temperament very well. In the A section I use the 7 note superpyth diatonic and 12 note superpyth chromatic scales, retunings of the scales we know and love, and in the B section I use 22edo as an ultra-chromatic scale. The form is ABAB. The recording is a bit rough, the piece is to demonstrate the tuning, the temperament and the instruments. I hope it confuses you a little but not too much :)

Duet for Axis-49 and Superpyth Irish Whistle in 22edo.

A fun little piece written in Negri temperament, tuned to 19edo (19 equal divisions of the octave).
The idea behind Negri temperament is that 3 semitones (of 16/15) makes a major third (of 5/4), and 4 make a perfect fourth (of 4/3). I use Negri's 9 note Moment of Symmetry Scale in it's disjuct pentachordal mode Negri[9] 4|4, which in 19edo is steps of 222232222. The three step interval is the disjuct tone (9/8) between the tetrachords and the two step interval is the semitone mentioned above. So in 19edo Meantone notation the scale is C Db D# E F G Ab A# B C.
The interval from a Db to a D# and from an Ab to an A# in 19edo are the same. D# and A# are 'interspetimals' representing 7/6 and 8/7 and 7/4 and 12/7 respectively.
The piece starts with a Negri comma pump filling a bar of 7 4. Repetitions of this comma pump make the A section ( The piece is in Ternary form). The comma that is tempered out is what defines the temperament. So the Negri comma is the difference between three semitones of 16/15 and a major third of 5/4 (or between 4 semitones and a perfect fourth of 4/3), equal to 16875/16384. (The meantone comma, which you may be used to, if not aware of, is 81/80, the difference between the major and minor tones 9/8 and 10/9, hence the name 'meantone' when this difference is tempered out. The major scale is the 7 note Moment of Symmetry Scale of Meantone Temperament) A comma pump is a chord progression that leads to the tonic only if the comma is tempered out. The meantone comma pump is a I-vi-ii-V-I progression in it's simplest form. The Negri comma pump I used involves going up a major third and down a fourth four times to get to V, and then up a fourth back to I. Music is available if anyone is curious or has a 19edo guitar and what's to record it :). I will build the flute to play it eventually...

Negri Shmegri

After William Lynch's excellent example: https://soundcloud.com/pianofreak96/a-septimal-christmas-extravaganza-22-edo-christmas-piano-medley
I have retuned and reharmonised 'We Wish You A Merry Christmas' into Pajara[12], tuned to 22edo. I use the septimal major and minor tetrads as base chords, occasionally also making use of a diminished tetrad. Merry Christmas!

Merry Pajaristmas!

Superpyth temperament can be thought of as a mirror opposite of meantone temperament. In meantone temperament four fifths reduced an octave is equated to the 5-limit just major third 5/4, where 4 pure fifths of 3/2 reduced an octave give 81/64, larger than 5/4 by the meantone comma, 81/80. In quarter comma meantone each fifth is flattened by a quarter of this comma so that four of them reduced an octave give an exact 5/4 and the meantone comma is tempered out. In superpyth four fiths reduced an octave is instead taken to be equivalent to the septimal, supermajor third 9/7. Two fourth then give the harmonic seventh 7/4. In quarter-comma superpyth  each fifth is sharpened by a quarter of the archytas comma 64/63, the difference between two fourth of 4/3 and a harmonic seventh of 7/4 then tempered out and fourth fifths reduced an octave giving a pure 9/7. Quarter-comma meantone can be well approximated in 31edo (equal divisions of the octave), and quarter-comma superpyth in 22edo. Diatonic notes in Meantone are 5-limit (subgroup 2.3.5), where chromatic intervals may be septimal (7-limit), i.e. the augemented sixth is 7/4 and the diminished fourth is 9/7. Diatonic notes in superpyth are of the 2.3.7 subgroup, with major and minor thirds at 9/7 and 7/6 respectively. Chromatic intervals may be 5-limit, ie an augmented second of 5/4 and a diminished fourth of 6/5. Using a gamut of Ab-C# (traversed by fifths - the generator), the black notes host the octave species of the 5-limit enharmonic trichord, the 5-note scale: 1/1:5/4:4/3:3/2:15/8, our regular 5-limit M3, P4, P5 and M7. This study is a 5 chord riff that makes use of the 5-limit chromatic intervals of Superpyth[12] (12 notes of superpyth temperament generated by a continuous string of fifths), along with the diatonic intervals. The chords include the diatonic subminor seventh, subminor major seventh and included supermajor seventh, along with the chromatic major and minor seventh chords.

Superpyth[12] chromatic riff

12 bar blues in my new tuning system: 97.5c-ET. It's like a squished 12TET, with each semitone at 97.5c. The intonation of 5/4 - major third, 7/5 - tritone and 5/3 - major sixth (and therefore 7/4 and 7/3) is improved from 12TET, but the inverses of these intervals no longer exist and there are no octaves or fifths. The 4/3 - perfect fourth of 97.5c-ET is 10.5c flat of pure, reminiscent of Superpyth temperament, where 2 4/3's give 7/4, tempering out 64/63. 8 semitones gives an accurate 11/7 and 3 an accurate 13/11, extending the system to the 4/3.5/3.7/3.11/3.13/3 subgroup. The chords I used:
1:5/4:7/4:7/3 on C - C:E:Bb:Eb
1:5/4:7/4 on F - F:A:Eb, and
1:5/4:7/4:7/3:(11/4):13/4 on G - G:B:F:Bb:(C#):E

Blues 97.5

Strange Waves

Duet for Axis-49 and Superpyth Irish Whistle in 22edo rearranged for Irish Whistle, Cello and Piano. Recorded at Xenharmonic Praxis Summer Camp, 2015.

Trio in Superpyth Temperament for Irish Whistle, Cello and Piano

Agnus Dei from Mass in 22edo.
Recorded at Xenharmonic Praxis Summer Camp 2015.
This movement is in Pajara temperament. All notes used are from the decatonic scale, Pajara[10], a 10-note MOS scale that can function as the albitonic scale of Pajara temperament. The chord progression pumps the diaschisma, 2048/2025, by root movements of 5/4, 6/5, 5/4, 3/2, 5/4, 6/5, 5/4, 3/2. The same comma pump is repeated in the 'miserere' section, but using Pajara's major and minor tetrads the progression can be shorter due to more common tones available.

Mass in 22edo - Agnus Dei

Sanctus from Mass in 22edo.
Recorded at Xenharmonic Praxis Summer Camp 2015.
This movement is in Porcupine Temperament. Apart from one accidental, all notes used for this are from Porcupine[7], a 7-note MOS scale that can function as the albitonic scale of Porcupine temperament. The chord progression of the first two repeated phrases pumps the Porcupine comma, 250/243, by root movements of 3/2, 6/5, 6/5, 3/2, 3/2, 9/5. The hosanna section features the harmonic 11th, readily available in Porcupine, in Major add harmonic 11 chords. 

Mass in 22edo - Sanctus

A Functional Auto-harmonisation of Blacksmith[10] 1|0 (5) (the 10-note Blackwood major scale or Blackwood decatonic LsLsLsLsLs) in 15edo (2121212121) using major tetrads of 4:5:6:7, 1-4-7-9 in the scale or 0-5-9-12 in 15edo.
The chord sequence in 10-note Roman numerals is
I-IX-III-I-V-III-VII-V-IX-VII-I
The top part plays the scale ascending. The harmonisation forms a two-chord pattern repeating 5 times (one for each period of the scale). In the first chord the scale pitch is the root (1/1) of the chord and in the second it is the 5/4, which in this 10-note scale in a major fourth rather than a major third. The five functions in Blacksmith functional harmony are the 5 periods of the scale to an octave. Each major tetrad has a relative minor, whose root lies a 6/5 (minor fourth) below the major, and a leading-tone minor, whose root lies a 5/4 (major fourth) above the major. The fourth note of the minor tetrad is the same in this scale as the fourth note of the major tetrad. In just intonation however the minor tetrad would be 1/1-6/5-3/2-12/7. The minor tetrad has two notes in common with its relative or leading-tone major.

Blacksmith[10] Functional Auto-harmonisation

A Functional Auto-harmonisation of Pajara[10] 2|2 (2) (the 10-note Pajara major scale, or symmetric decatonic ssLssssLss) in 22edo (2232222322) using major tetrads of 4:5:6:7, 1-4-7-9 in the scale or 0-7-13-18 in 22edo.
The chord sequence in 10-note Roman numerals is
I-VI-VII-I-II-VI-I-II-VI-VII-I.
The sequence may be divided in half, where the second half is a repetition of the first half, transposed up the half-octave period of the scale.
The top part plays the scale ascending. The four functions in Pajara functional harmony are the tonic (T), chord I; the dominant (D), chord VII; the tonic transposed a (half-octave) period (T1/2), chord VI; and the dominant transposed a (half-octave) period (D1/2), chord II. Each major tetrad has a leading-tone minor, root lies a 5/4 (major fourth in 10-note scales) above the major, and only the dominant and dominant half-octave have relative minors, whose root lies a 6/5 (minor fourth) below the major. In just intonation the minor tetrad can be expressed as 1/1-6/5-3/2-12/7, or in 22edo also as 10:12:15:17 (where 10:12:15 is the minor triad in just intonation). The fourth note of the minor tetrad lies 1 step of 22edo below the fourth note of the major tetrad, a minor third rather than major third above the seventh (3/2). The minor tetrad has two notes in common with its leading-tone major, or relative major (though this will not always be in the scale).

Pajara[10] Functional Auto-harmonisation

Easier to hear what's going on in this one I think. A Functional Auto-harmonisation of Pajara[10] 2|2 (2) (the 10-note Pajara major scale, or symmetric decatonic ssLssssLss) in 22edo (2232222322) using major tetrads of 4:5:6:7, 1-4-7-9 in the scale or 0-7-13-18 in 22edo.
The chord sequence in 10-note Roman numerals is
I-VI-VII-I-II-VI-I-II-VI-VII-I.
The sequence may be divided in half, where the second half is a repetition of the first half, transposed up the half-octave period of the scale.
The top part plays the scale ascending. The four functions in Pajara functional harmony are the tonic (T), chord I; the dominant (D), chord VII; the tonic transposed a (half-octave) period (T1/2), chord VI; and the dominant transposed a (half-octave) period (D1/2), chord II. Each major tetrad has a leading-tone minor, root lies a 5/4 (major fourth in 10-note scales) above the major, and only the dominant and dominant half-octave have relative minors, whose root lies a 6/5 (minor fourth) below the major. In just intonation the minor tetrad can be expressed as 1/1-6/5-3/2-12/7, or in 22edo also as 10:12:15:17 (where 10:12:15 is the minor triad in just intonation). The fourth note of the minor tetrad lies 1 step of 22edo below the fourth note of the major tetrad, a minor third rather than major third above the seventh (3/2). The minor tetrad has two notes in common with its leading-tone major, or relative major (though this will not always be in the scale).

Pajara[10] Functional Auto-harmonisation 2

Initial improv in the following 10-note 5-limit JI scale:
10/9 9/8 5/4 4/3 40/27 3/2 5/3 27/16 15/8 2/1

This scale contains both the Pythagorean (3-limit) pentatonic scale:
9/8 4/3 3/2 27/16 2/1 on every second step.
And the Ptolemaic sequence, the just diatonic scale:
9/8 5/4 4/3 3/2 5/3 15/8 2/1
As well as another tetrachordal just diatonic scale:
10/9 5/4 4/3 3/2 5/3 15/8 2/1

Both tetrachordal just diatonic scales are trivalent, which means that each generic interval, i.e. 2nd (a ‘step’), 3rd, 4th, etc., comes is three specific sizes.

The entire 10-note scale may be described as LsmsLsLsms, where ‘L’ designates a large step, ‘m’ designates a medium sized step, and ‘s’ designates a small step. It has 5 large steps of 10/9, the minor tone; 2 medium sized steps of 16/15, the diatonic semitone; and 3 small steps of 81/80, the syntonic (Meantone) comma. It is not trivalent however the total number of specific intervals is less than for a trivalent scale of the same cardinality. It has 3 sizes of 2nd and 10th, 2 sizes of 3rd, 5th, 7th and 9th, and 4 sizes of 4th, 6th and 8th. It is also invariant under inversion, which means that it’s inversion:  
16/15 32/27 6/5 4/3 27/20 3/2 8/5 16/9 9/5 2/1, is one of its modes. It is an example of a class of scale I have recently discovered, which I am calling step-nested scales. SN scales can be uniquely labelled by their step sizes and multiplicities. This scale is defined by the step signature and mapping (5 2 3) (10/9 16/15 81/80). The first mode listed is mode 2, and the second mode listed is mode -2, where the modes are listed in lexicological order.

The trivalent tetrachord just diatonic scales contained within the 10-note scale are not invariant under inversion, in fact, the scales, which clearly differ, are inversion of modes of each other. The just diatonic scale with 10/9 as the first step is available after the step pattern 1212121 of the first 10-note scale in its original mode. The scale found after the same step pattern of its inversion:
16/15 6/5 4/3 3/2 8/5 9/5 2/1
is a mode of the Ptolemaic sequence. The step pattern 1212121 is the symmetric mode of the maximally even sequence for 7 in 10. It can be generated by taking 3 fourths of the scale down from the tonic, and 3 fourths up.

The two just diatonic subsets of the 10-note scale have 3 large steps of 9/8, the major tone, 3 medium steps of 10/9 and two small steps of 16/15. ‘m’ and ‘s’ correspond to ‘L’ and ‘m’ of the 10-note scale, respectively, and ‘L’ corresponds to the addition of ‘L’ and ‘s’ (10/9 x 81/80 = 9/8). 

Let’s look at the two scales: 
10/9 5/4 4/3 3/2 5/3 15/8 2/1 – mLsLmLs and its inverse:
16/15 6/5 4/3 3/2 8/5 9/5 2/1 – sLmLsLm
In these modes they are in the form of disjunct tetrachordal scales, where the same steps in the same order reach tetrachords from 1/1 to 4/3 and from 3/2 to 2/1, separated by a 9/8 in the middle.
The definition of ‘tetrachordal’ I am using requires at least one mode of a scale to be expressible in this same form; with equivalent, disjunct tetrachords.
The tetrachord of the first scale is 10/9 5/4 4/3 – mLs, and the tetrachord of the second is 16/15 6/5 4/3 – sLm.

This improvisation was made using an axis-49 loaded into Pianoteq 5, using the default mapping. Try this out!

white square

Initial improv on my Axis-49, using Pianoteq, in the 17-note line of purples scale.

The name line of purples was inspired by this video which I saw around the time I discovered / constructed the scale: https://www.youtube.com/watch?v=8gU3IA4u-J8

Titled 'Bisexual Lighting: the Rise of Pink, Purple, and Blue' the video explores an emerging trend in pink, purple and blue lighting not only in association with bisexuality, through the use of the colours on the bisexual flag and the tendency to signify bisexual characters with the lighting in movies, but with LGBT(QIA) people in general and the political left wing of YouTube. Bisexual lighting comprises colours from the line of purple, the colours that can be made through combining red with blue (with no green). Such colours do not correspond to the incandescence of any natural light, and could not be produced until relatively recently. The video comments on the use of these colours in club settings and media to provide comfort to and showcase the beauty of marginalised groups of people, mentioning the use of blue light to showcase the beauty of dark skinned people when more regularly used lighting did not do them justice, as well as the empowering use of the lighting in queer-friendly places and in association with queerness in media.

If I may attempt to connect these ideas to tuning theory, where Western common practice music since the Renaissance comprises ratios of primes 2, 3 and 5, it is often the ratios of five, particular the 'sweet' major third of 5/4 and major sixth of 5/3 are though to be particularly beautiful, and much of such music showcases this. Ratios of higher primes such as 7, 11 and 13 have been almost completely left in the shadows. As the line of purples provides light to the darkness of marginalised groups of people and showcases their beauty, this scale I hope might do the same for neglected ratios of 7, 11 and 13 (comprising ratios of 7, 11 and 13 in place of those of 5)

The scala fie for the scale, in cents:

! lineofpurples.scl
17-note 3-SIS OTC 3-chain TE Parapyth, symmetric mode
!
17
!
58.31
149.30
207.60
288.28
346.59
437.58
495.88
554.19
645.18
703.49
761.80
852.78
911.09
991.77
1050.08
1141.06
1199.37

It has a step pattern ABACABAABAABACABA, where:
A is a small step of 58.31c, representing 28/27~33/32.
B is a large step of 90.99c, representing 81/77~96/91.
C is a medium sized step of 80.68c, representing the limma, 256/243~22/21.

It is a SIS scale - Simple, Inversionally Symmetric, like white square, which is a generalisation of MOS/well-formed scales, which themselves are generalizations of the diatonic scale. It is made from the Pythagorean diatonic scale:
9/8 32/27 4/3 3/2 27/16 16/9 2/1, by adding the small step representing 28/27 and 33/32 from either end of each step of 9/8.

Like white square (previous track), this scale is also OTC, omnitetrachordal, where each mode may be represented as two equivalent divisions of 4/3, with the remaining 9/8 divided a subset of the division of 4/3. The diatonic scale is OTC.

The diatonic scale, white square and line of purples are all inversionally symmetric, which means that taking their inversion - taking the scale from the top down instead of the bottom up - leads you to a mode of the original scale.

As shown in the diagram, the scale can be conceived of as a 7 note chain of fifths from F-B, making a diatonic scale, and then two five note chains (making pentatonic scales), from Gv-Bv and from F^-A^, where 'v' and '^' take you up and down by the small step of the scale representing 28/27~33/32 respectively. 

I make use in the improv of the scale's accurate representations of 7/4, 11/8, 7/6 and 11/6, as well as the diatonic, pentatonic and tetrachordal subsets.

line of purples

Overworld Theme from Super Mario Bros. Composed by Koji Kondo and arranged for Reedefined Clarinet Quartet by Gareth Hearne (bass clarinet).

Audio (and Video) production by Stephen Le Page
http://www.youtube.com/user/stephenlepage

Performed at our lunchtime concert in Octagon Theatre, UWA, Thursday 20th September 2012.

Super Mario Bros. Theme for Clarinet Quartet

My arrangement of Bon Iver's 'Woods',
composed by Justin Vernon,
performed by Naya Chorale, under the direction of Adam Brockway at our 'Unplugged' Concert in November, 2012.

Naya Chorale - Woods by Bon Iver, Arr. Lillian Hearne

Clarinet, sax, and keyboard player; singer, composer, and theorist.

Lillian Hearne

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