Name: Hypercube Version III (for Chamber Ensemble: Strings, Pianos, and drumsets) by pthoege
Duration: 7 min 55 s
Description: This is a composite of 12 tracks: 4 strings, 4 pianos, and 4 drumsets all in four different tempos and 4 different tunings. All instruments are playing the original Hypercube score! No added notes.

I like the different tempos and the different tunings but I would prefer it without the drums

Intriguing moment. I'm enjoying those chords building a repetitive tension.

Title:
Crackle

Dates:
28-Apr-2024 (inception), 01-May-2024 (finalised), 28-May-2025 (released)

Artists:
Music composed, performed, produced by Norman Freund
Artwork by Norman Freund
Photography by Zachary Freund
Astrophotography by John Holgate

BandCamp Purchase:
https://normanfreund.bandcamp.com/track/crackle
YouTube Video:
https://youtu.be/WeRj2nls07A

Description:

Midnight Run Through The Port Hills
	
Valley run,
Moon light night,
Tree shadows sweep by.

Forest behind,
Sweeping bends ahead.

Tension builds,
Revs increase,
Hill climb begins.
City lights flicker below.

Sky lights appear,
Ascending light beam,
Hatch silhouette,
Galaxy engulfs,
Silence,
Darkness.


Behind the scenes music notes:

The main motivation behind Crackle is the Just Intonation (JI) tuning used, taking all the harmonics from octaves 0 to 5, octave reduce, remove duplicates and re-order in ascending order. This resulted in a 29 frequency ratios (not including in the 2/1 octave). These 30 frequency ratios were then transposed by JI perfect fifths (3/2 ratio) to expand the scale in violin style ready for a four string violin. When applied to violin, this means there are 30 notes per string spanning one octave.

The scale outline (** full details see my social media and BandCamp **):
Scale Index(i), [string number], JI Equation, Cent
0[1], 1/1, 0c
1[1], 33/32, 53c
2[1], 17/16, 105c
3[1], 9/8, 204c
4[1], 37/32, 251c
5[1], 19/16, 298c
6[1], 39/32, 342c
7[1], 5/4, 386c
8[1], 41/32, 429c
9[1], 21/16, 471c
10[1], 43/32, 512c
11[1], 11/8, 551c
12[1], 23/16, 628c
13[1], 47/32, 666c
14[1], 3/2, 702c
15[1], 49/32, 738c
16[1], 25/16, 773c
17[1], 51/32, 807c
18[1], 13/8, 841c
19[1], 53/32, 874c
20[1], 27/16, 906c
21[1], 55/32, 938c
22[1], 7/4, 969c
23[1], 57/32, 999c
24[1], 29/16, 1030c
25[1], 59/32, 1059c
26[1], 15/8, 1088c
27[1], 61/32, 1117c
28[1], 31/16, 1145c
29[1], 2/1, 1200c
30[2], 3/2, 702c
31[2], 99/64, 755c
...
59[2], 3/1, 1902c
60[3], 9/4, 1404c
61[3], 297/128, 1457c
...
89[3], 9/2, 2604c
90[4], 27/8, 2106c
91[4], 891/256, 2159c
...
118[4], 837/128, 3251c
119[4], 27/4, 3306c


It was then time to start playing around with the scale. The reference note used for this scale was not as per a concert pitch violin, rather concert pitch middle C, then some differing octave transpositions for the various string tracks.

The string sections were played on a 64 note midi controller, the Ableton Push2, so only a subset of the full 120 notes could be played at once. Played "organ style", as chords, well beyond double stop capabilities of a single physical string instrument. These tracks provided the basic harmony and melodic concepts.

Next the Soma Pipe (hardware synthesiser, here accepting the human voice as input) was played to provide improvised lead and percussive parts. The lead parts used various Pipe algorithms (Filtera, Harcho, MadDelay and Synth), the tuning was not necessarily strictly as per the table above, but rather improvised live tuning, where I as the performer, used my ears and emotions to guide the tuning choices. Some of the Pipe parts used a digital amplitude modulation device (Max for Live, custom programmed, the NOF_AENV Release 06). This amplitude modulation tool, used two simultaneous sine wave modulations with JI frequency ratios.

A wave table synthesiser in Ableton Live was used for the string section and for one additional instrument.

** See BandCamp and Social Media links for video production notes. **

Artist Links:
Norman Freund; Social media: https://www.facebook.com/norman.freund.545; YouTube: https://youtube.com/@normanfreund; Music: https://soundcloud.com/norman-freund, https://normanfreund.bandcamp.com

John Holgate; Social media: https://www.facebook.com/blundering.fool.3; YouTube: https://www.youtube.com/@quedecree; Music: johnholgate.bandcamp.com

Zachary Freund, Social Media: https://www.facebook.com/zac.freund.9, Instagram: https://www.instagram.com/zac_freund

Crackle

There is a slender stream in my output, but extending back five decades now, of music based on scales repeating, not at the octave, but at the fifth (or fourth).  In particular, while the effect here is always "locally pentatonic," as the piece progresses it grows increasing pantonal.  By the end all the pitch classes get equal treatment.  

By a "hypercanon," I mean a canon whose theme is itself a canon, and perhaps, and so on.  

Here the  ultimate theme and patterns of canonic repetitions have been chosen very carefully so that music has an overall constant density.  Get the math wrong and the music will just grow muddy or thin out prematurely.

A pulse unifies the rhythmic surface, but it beats so fast, on the order of 50 clicks per second, that the audible rhythm will not imply it even the pulse arguably still informs the rhythm.  So just as the tonality here doesn't count as either diatonic or traditionally atonal, the rhythmic profile is neither entirely "free" nor based on a perceptible count.

Another quite different interpretation of the underlying structure is forthcoming.  Watch this space.

Sometimes the New Place to Go Sits In Between the Places You Know, for piano

A fact of nature: one and the same thing may admit both the simplest of specifications, yet also exhibit rich behavior verging on the quasi-random.  

If a 3-note chord has pitches sitting x, y and z semitones below C7, that chord occurs here at time x^3 + y^3 + z^3.

That's the whole score of the work (modulo a simple orchestration strategy).

So all the rhythmic complexity, and all the counterpoint and voice leading simply emerge from the implications of that short sentence. 

I thus invite you to listen for the structure, the generating intention.
But I invite you, equally, to listen for the phenomena, the temporal extension.  

To my ears what unfolds has something subjective, even expressive about it, even though I know it's all implicit already in that simple formula.

The Treachery of Simple Arithmetic, for 10 instruments

When living things gain the ability to extract information from their environment and react to it, they can have a survival advantage.   So did our senses evolved;e.g. an animal might "hear that" a predator approaches and evade it.  

Complex beings process complex stimuli for all manner of information, so the cognitive effort after the raw sensation can take time.   This necessary delay, for human beings at least, means experiencing sense-data-in-itself even while reflecting on the question "What is it?"  For music this gives rise to a duality: even as we surrender to an immersive experience we might parse it and deduce some of its structure (or at least experience the mere presence of structure through its artifacts).  

Perhaps "duality" make this too simple: we can also reflect on the feeling of surrender, we can also reflect on the feeling of deductive experience.  We can "hear that" a sound got made in a certain way, we can hear that a sound is similar to different from some other sound, and we can "hear that" a variety of sounds enjoy some order or pattern.   Music is not just how it sounds, it is at least(!) all that the stratified listening mind does.

To Hear and Hear-That, as a kind hypercanon (a canon on a canon on canon on a canon) depends on "transposition up the steps of the overtone series (as opposed to up or down a fixed interval).  

From another but equivalent viewpoint, the many chords here correspond to solutions of the problem of expressing any given beat number, if possible, as the sum of four distinct fourth powers (a.k.a. biquadrates).   From the choice of that problem, and for mathematical reasons, a kind of rhythmic steady state emerges, a flow that neither thins nor thickens on the average, but nevertheless projects predictable/unpredictable variation (depending on the computational power of the agent who would predict).   In contrast to the rhythmic steady state, as the piece progresses the larger beat numbers require solutions with larger biquadrates, so the timbre of the piece slowly evolves from consonance to some kind of resonant dissonance.

For an ending, both a bit arbitrary but also a bit natural, the piece goes on only as long as 32 harmonics suffice.  



To Hear and Hear-That

Another hypercanon piece.  

This time, from the mathematical perspective, beats get filled when their beat number equals the sum of a square, a positive cube and 6th power.  

Because the powers vary, the music keeps the counts on different instruments. 

The numbers to square get heard on the piano (with 1 the top key and then descending chromatically).

The numbers to cube get heard on pizzicato strings tuned in 19-TET, using an ascending scale that divides three octaves into 19 equal intervals which sound as nearly 10:9 rations.

The numbers to raise to the 6th power, never more than 4, map to a small gamut of tuned drums.

The complexities of the sets of sums of just two kinds of powers asserts itself as the irregular staggered entrances of the canon on the instrument associated with other power.  

If this is too much information for you, the bottom line is an intense listen which, white it has a hypersimple underlying concept, also feature a rather complex surface regardless, and because of the density of the beat numbers with a "yes" answer, intense but also unfamiliar syncopation. 

Big thank you to Peter Thoegersen, who literally wrote the book on polymicrotonal (and polytempic) music.  He's a genius and you should check out his music.  While his music and also his working methods are quite different than mine, he has opened me up to one of his great discoveries, namely that the listening mind can instinctively decouple the layers of a complex music according to the tuning systems girding each layer (at a given moment).  The effect is like passing to an experience in a higher dimension, akin to the discovery of perspective in visual art.   I'm not sure Peter would even count this as "poly" since two of the layers are 12TET and my treatment of 19TET avoids actual microintervals.  But I'd never have come to this way of projecting my ideas without his influence.

Higher Powers, for Piano, Percussion and Pizzicato Strings

This is a kind of polytempic work in the sense of Peter Thoegersen (well, if he thinks so).  The whol time span gets divided into 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 equal parts with each tempo supporting a layer consisting of only 7th chords that have one particular root.  Since I make chords in the slower tempi sound louder, the interaction of dynamics and periodicity tends to balance the importance of all the different roots.  With the next slower layer always carrying the given layers dominant, a certain amount of root progression emerges.  But it's complicated!

Those Fibonacci Fractal Four-Chord False Relation Blues, for piano four hands

Said the poet: "Euclid alone has looked on Beauty bare."

Push back on "alone" if you like, but beauty may dwell in the immaterial, and from thus the possibility of art formed purely out of ideas. Or not purely: a sensual experience may, by its structure, simultaneously carry ideas. Two art objects, one immersive, the other cognitive, can simultaneously inhabit the same experience.

Varèse framed music as "structured sound," and I would add, alternatively, "sonified structure." Mathematics is the science of structure, so, if you like, the part of music theory robust (to some extent) against the contingencies of sonic materials. I have a long interest both in bringing to my composition ready-made mathematical objects and mathematical phenomena, and in letting the same emerge from sonic considerations themselves. But truly to infuse music with at least the ghost of "Beauty bare" requires evoking mathematical argument.

So consider communicating something by drumming. Fix a small unit of time. Now say you decide to work with only two durations between strikes, namely 2 units and 3 units. Even with the best coding, that choice limits your bandwidth—limits the rate at which you can transmit information.  Now one can quantify this in a sophisticated way with information theory, but we don't need to get into that now. But here's an interesting and not quite trivial fact: if your durations are 1 unit and 5 units, then you have exactly the same bandwidth as with 2 and 3 units! 

A long and short duration might remind you of Morse code. Well, if you "glue" the "dash", the long duration, of the 2–3 system to the following duration, you will get a drumming pattern in a 2–5–6 system. And that translation is reversible. But if you glue the "dot" of the 1–5 system to the following duration, then you also get a pattern in the 2–5–6 system, and also in a reversible way. So you can get from 2–3 to 1–5 via the trick of interpolating between them with another system you might not originally have thought to consider. There's an episode in the piece that makes precisely this transition/translation audible.

But it doesn't stop there! All the following systems also have the same bandwidth as the three mentioned above:
                                                                1–6–11–16–20, 
                                                                1–6–10, 
                                                                1–7–10–12–16, 
                                                                2–6–7–10–11, 
                                                                3–4–5, 
                                                                3–4–8–9–10, 
                                                                3–5–7–8–9, 
                                                                4–5–6–7–8.
Now hopefully the required "bridge" between, say, 1–6–11–16–20 and 3–4–8–9–10 doesn't seem obvious at all! Nevertheless one will experience it if one attends very closely to the sonic details of the piece.

Yes, you can forget all this and you still have an intense experience reminiscent of ritual drumming, albeit without any periodic underpinning. By the way, he piano part works just as the drums do, but also correlates pitches to the durations.

The title: if you wish to transmit a long random "message" in the 2-3-system, a certain ratio between the frequency of 2's and 3's will emerge.  The famous golden ratio would emerge from the 1-2-system and goes by the name "phi."  Hence "high phi" as one old and mostly forgotten proposal for a name of our ratio here.

High Phi, for drums and piano obbligato

The music can speak for itself, sonically, if you prefer.

But if you care to understand what's going on, here it is in brief.  The durations in the cello are 2, 4, and 8 8th notes; the durations in the bass line of the piano part are 3,5,6, and 7 8th notes.  Any rhythm of any duration of the former sort admits a translation into a rhythm of the latter sort, and vice-versa, indeed a translation that preserves the overall duration (to within a fixed few beats anyway).   I expect that this fact hardly seems obvious!  But the violin part is the intermediary, along with the other voices in the piano part.   Attend to the relationship between cello and violin; and between piano and violin, and, with some concentration, the logic indeed reveals itself.

Elegy for Piano Trio (High Phi II)

In both movements of this quartet, each instrument gets assigned a repeating rhythmic pattern—sequences of notes and rests of equal length (say, all eighth notes). 

The first violin plays a pattern of 11 beats (6 notes, 5 rests), the second violin a 7-beat pattern (4 notes, 3 rests), the viola 6 (3+3), and the cello 5 (3+2). These patterns cycle independently.

One full cycle  takes 2310 beats—the least common multiple of 11, 7, 6, and 5.

But—crucially—beat durations vary.

Indeed, the duration of a beat depends on its density—meaning how many instruments are sounding at that moment (0 through 4). Each density appears a fixed number of times within the full cycle, independent of pattern contents assuming the constraints above. 

The twist: beat durations get adjusted so that each density—whether a silence, solo, duo, trio, or a full quartet—occupies exactly one-fifth of the total time of the movement. Rare textures are stretched, common ones compressed.

Thus the interaction of the parts effects an emergent large-scale rhythm.

Out of millions of possible combinations of patterns, one movement employs the set that yields the most unpredictable overall rhythm (in a precisely defined mathematical sense). The other, the most predictable.

As for pitch: each instrument traces a fragment of the cycle of fifths—in near standard tuning. The quartet plays 17TET, so with the octave divided into 17 equal steps, which makes a "perfect" fifth span 10 steps. Each instrument repeatedly travels through this cycle fragment, alternating between octaves. Individually, the lines sound diatonic; together, they form unfamiliar and highly dissonant verticalities—often stranger than even the most dissonant music in 12-tone systems.

Despite this microtonality, no specialized technique is required. If each player tunes their instrument to a slightly different “A,” then standard fingerings suffice—as long as they resist the instinct to tune to one another.

This is a composite of 12 tracks: 4 strings, 4 pianos, and 4 drumsets all in four different tempos and 4 different tunings. All instruments are playing the original Hypercube score! No added notes. 

Hypercube Version III (for Chamber Ensemble: Strings, Pianos, and drumsets)

To all concerned: 
I am announcing publication of my new book, NOW AVAILABLE:
https://www.routledge.com/Maqam-Melodies-Pitches-Patterns-and-Developments-of-Music-in-the-Middle-East-and-other-Microtonal-Writings/Thoegersen/p/book/9789815129052
My book, Polytempic Polymicrotonal Music, is available on Amazon. 
https://www.routledge.com/Polytempic-Polymicrotonal-Music-The-Road-Less-Traveled/Thoegersen/p/book/9789814968294

I have a new album out through Magic&Unique Group Records on Spotify and other digital streaming platforms:
https://open.spotify.com/album/5qWHmcvyAemvVn2TJH1bio?si=OD5TmPptQ-ypr8WOI0EZKA&nd=1

https://whitecentipedenoise.com/products/peter-thoegersen-facebook-cantos-cdr-flea
My CD via Flea label has been out since April, 2021, featuring the wonderful Lore Lixenberg singing over my microtonal piano accompaniment
Hope you enjoy it!

https://www.newworldrecords.org/collections/vendors?q=Peter%20Thoegersen

From the liner notes on my CD now available on AMAZON and New World Records:
"The author of these notes has spent his life explaining radical music, and the music on this disc may be the most radical I've ever written about. Peter Thoegersen (b. 1967) is not yet a name known to the music world; not for any lack of connection to other famous musicians, but because he came to composition late, and because his artistic aims are so broad and complex that they have taken years to evolve. His aesthetic is well defined, and he is upfront about having a name for it: "Polytempic Polymicrotonal Music." That means he has musical layers simultaneously moving in different tempos, plus, even more radically, that he has different microtonal scales playing at the same time. Few composers have aimed at such fervent and heterogeneous multidimensionality. And yet, while Thoegersen's music can seem overwhelming at first, running through it is a level of charm and simplicity that belies the initial impression.

In total, these three pieces emphasize a particular side of Thoegersen's output. The image his music creates of ornately mathematical structures filled with intuitively friendly melodies maps on to the personality of the man himself: outspoken and acerbically ruthless in his musical opinions, Thoegersen is also marked by a jolly sense of humor and an inexhaustible love of cats and other animals that make him ultimately less imposing and more personable than he might seem at first. The same can be said for his music. And by going further than anyone else has gone in terms of this particularly American concept of fusing the polytempic with the polymicrotonal - extending and combining the conceptual worlds of Ives, Cowell, Nancarrow, Partch, Carter, even Frank Zappa and many others - he has created a special place for himself within American music."                                                             

-Kyle Gann (from the liner notes)

Hypercube Version III (for Chamber Ensemble: Strings, Pianos, and drumsets)

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