Music originally written for a production of the Chekhov play 'Platonov', Helsinki 2009. Recorded live from a concert in the 'Tulkinnanvaraista' series, Helsinki, 10 Dec, 2014. Eija Kankaanranta, kantele; Kimmo Hakola, piano, Juhani Nuorvala, keyboard and electronics. 
The keyboard (Pianoteq) and the soundtrack are tuned to 13-limit JI.


Pianotarha (Music from Platonov)

Excerpt from Besaxung-Album "Go!"
Composition P. Gerschlauer

Besaxung:
Philipp Gerschlauer (Sax)
Felix Roßkopf (microtonal Keyboard)
Reza Askari (Bass)
Thomas Sauerborn (Drums)

www.gerschlauermusic.com
www.besaxung.com

Mr. P (comp. Gerschlauer) with Besaxung

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

Title:
Tetration

Dates:
20-Jul-2024 (inception), 21-Jul-2024 (finalised), 25-Aug-2024 (released)

Artists:
Music composed, performed, produced by Norman Freund
Artwork by Norman Freund

BandCamp Purchase:
https://normanfreund.bandcamp.com/track/tetration

Description:
(cut down description, SoundCloud limits, see my BandCamp or FaceBook page full details)
Inspired by a Tetration tuning question on the "Microtonal Music and Tuning Theory" FaceBook group by Steven Socha (18 July 2024).

A simple description of tetration, is to recursively exponentiate a number. For each tetration operation (or iteration) yields another level of tetration. For example the tetration of 2.3 to three levels is:

(2.3^((2.3^2.3))) = 286.234

and the tetration of 2.3 to two levels is:
2.3^2.3=6.792

So there are two components or inputs to a tetration operation, the base number (in the above example 2.3) and the number of integer levels (in the above examples (3 and 2 respectively).

So what base to choose? I felt in a Just Intonation (JI) mood (is that a thing?), so decided to use a base of 3/2. 

Now for a formal definition of the tetration process:

The tetration notation used here is that of:
T=i:R
where
T = the tetraion result (floating point number)
i = the number of tetrations or tetration level (positive integer number)
R = the base of tetration (floating point or integer number, *(1) )
: = tetration symbol (or authors may us a different symbol, convenient to use here)

In general R can be a real or complex number, here I chose to use real positive numbers.

for example
2:R = R^R
3:R = R^(R^R)
4:R = R^(R^(R^R))
5:R = R^(R^(R^(R^R)))
...

The tetration tuning ratio T, for steps i=0,1,2 ... 8 here were:
0	1/1 = 1. = 0 c
1	(3/2)^(3/2) = 1.837 = 1052 c
2	(3/2)^((3/2)^(3/2)) = 2.106 = 1290 c
3	(3/2^((3/2)^((3/2)^(3/2))) = 2.349 = 1478 c
4	(3/2)^((3/2^((3/2)^((3/2)^(3/2)))) = 2.592 = 1659 c
5	(3/2)^((3/2)^((3/2^((3/2)^((3/2)^(3/2))))) = 2.860 = 1819 c
6	(3/2)^((3/2)^((3/2)^((3/2^((3/2)^((3/2)^(3/2)))))) = 3.189 = 2008 c
7	(3/2)^((3/2)^((3/2)^((3/2)^((3/2^((3/2)^((3/2)^(3/2))))))) = 3.644 = 2239 c
8	(3/2)^((3/2)^((3/2)^((3/2)^((3/2)^((3/2^((3/2)^((3/2)^(3/2)))))))) = 4.383 = 2558c 

Above is the whole scale, which shows the tuning frame was not an octave ratio of 2/1=2.,
instead a ratio of 4.383 or 2.132 octaves, so made for some interesting improvisations.

This tuning was placed in a Ableton Live (release 12) scala file along with meta definitions to give:
! Tetration_01-1.ascl
!
Tetration_of_3 divided_by_2 for a 8_note_scale_with_an_offset_of_1
 8
!
 1052.93250 cents
 1289.57373 cents
 1478.45996 cents
 1648.89600 cents
 1819.48535 cents
 2007.90125 cents
 2238.76245 cents
 2558.12329 cents
!
! @ABL REFERENCE_PITCH 3 0 440
! @ABL NOTE_RANGE_BY_INDEX -5 0 10 7

Harmonic, melodic and rhythmic ideas were developed in tandem. The master time signatures were 4/4 and 5/4, however the recorded clips used a variety of time signatures:

Time signatures used:
	Master time signatures: 4/4 and 5/4
	Clip Time signatures:
		Percussion: 4/4, 47/32, 5/4, 17/16, 5/16
		Rhythmic melodic: 41/16, 5/4, 
		Melodic and Harmony: 4/4, 5/4

Instrument choice was simple:
* Two drum tracks using the same drum kit.
* One bell sound track
* One bass sound track (very wide range used)
* Four pad sound tracks using the same pad instrument

The primary role of the drums were to provide rhythmic interest and to provide variety of time signatures.

The bells took on the role of a cross between melody and rhythm, and in particular contrasting to the rhythm of the drums, the drum and bells not necessarily having simultaneously the same time signatures. 

The bass took on the roles of from a spanner in the works to it's own melodic content. Hey it's a wacky tuning, so a wacky bass line felt in order!

The pads took on the role to establish the harmony as it exists in this tetration tuning system, long sustain chords, drones and at times playing it's own melody.

Buy Tetration

Tetration

Strange Waves

3/2=695 cents, 2/1=1205 cents

I stole the original image from google images. Sorry original content creator.

Memore (in approximately 19 tone equal temperament)

5-ed2, thanks to Sean Archibald for the title.

Ouch My Causality (in 5 tone equal temperament)

3/2=701, 5/4=387

Made using The Viking, a synth from dynamictonality.com, and a free drum plugin.

The scale repeats at the "fifth" (aka the 3/2), and is generated by the "major third" (5/4). I shrunk the 3/2 and stretched the 5/4 slightly so that there was a bigger contrast between the large and small step in the scale.

The scale structures are Mavila-esque, and the tuning here is pretty close to 11-ed3/2

The Stupistest Freakin' Decision (in approximately Carlos-Alpha Tuning)

After an overnight computational search for perfectly balanced scales with lots of consonant intervals, I came across this beauty. This is a piece I wrote in that scale the next day.

The scale is: 0, 76.7, 261.4, 490.4, 642.3, 763.9, 967.9 1200.0 cents.

just a bunch of noodling really -- nothing too serious.

Catnip (2016)

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

From "Mikrojazz - Neue Expressionistische Musik"

Available on CD / Vinyl / Digital Download. 
A RareNoise Records Release. 

https://www.rarenoiserecords.com/gers...

"November" composed by Philipp Gerschlauer

Microtonal keyboard programmed by Philipp Gerschlauer

Philipp Gerschlauer, alto saxophone
David Fuze Fiuczynski, microtonal guitars
Jack DeJohnette, drums
Matthew Garrison, fretless bass
Giorgi Mikadze, microtonal keyboards

https://www.facebook.com/Gerschlauer/...
http://gerschlauermusic.com/home-eng....

LaMonte's Gamelan Jam (comp. Gerschlauer) Gerschlauer, Fiuczynski, DeJohnette, Garrison, Mikadze

Musically, I'm interested in the roiling interstices between the improvised and the composed, the unfamiliar and the familiar, the directed and the collaborated.

Catnip (2016)

Andrew J Milne

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