Technical Notes for Newbeams

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Technical notes for Newbeams, a xenharmonic album by Andrew Heathwaite:

I created Newbeams in February 2012 for the RPM Challenge. I wanted to create a thoroughly xenharmonic album that was full of interesting-to-me stuff using some newly-acquired hardware and software, pursuing relatively new-to-me scales, temperaments, and rhythmic ideas, that would be exciting and interesting to open-minded, adventurous listeners.

These are my "technical notes" for the album, including compositional details in the domains of tuning and rhythm. I would expect a composer and/or theorist might be interested in this stuff, but I don't consider it necessary for enjoyment of the album. It is not very thorough.

The primary instrument I used was an Axis-49, which is a generalized keyboard with hexagonal buttons, about the size of a small cereal box. For some tracks I played it "live," and for others I used it to select notes for melodies that were otherwise sequenced. So the tunings I used on Newbeams are not only tunings in the abstract, but actually mappings to the keys of the Axis-49. As a rule, if it wasn't in the mapping, it wasn't in the piece.

The tunings I used were mostly, in the language of regular mapping theory, rank-2 temperaments. A large part of the attraction for me was that such systems map perfectly to the two dimensions of the Axis-49's array. Harmonically, my approach reflects an interest in approximations of the higher-limit consonances of Just Intonation, including ratios involving 7, 11, 13 and 17. The harmonic structures can sometimes be labeled "otonal," sometimes "utonal," and sometimes they are more ambiguous (eg. essentially-tempered dyadic chords).

Another thread here was an interest in odd time signatures and divisions of the beat. I find the relationship between odd tuning and odd time a fruitful area to play in.

Some tuning background

All the tuning terminology in the track notes below can be looked up on this wiki, but it may help to give a short overview here.

Regular temperaments are systems for mapping intervals of Just Intonation to other (tempered) intervals by adding or subtracting one or more generating intervals. If the generating intervals are the octave and the meantone perfect fifth, for instance, you have meantone temperament, where four perfect fifths minus two octaves gives you a major third. In this way, every JI interval can be described in terms of the generators. Also, the dimensionality of a system can be reduced -- for instance, from the five dimensions required for 11-limit JI, to the two required for a rank-2 temperament. The rank here tells us the number of generators as well as the number of dimensions. Rank-2 temperaments are frequently associated with Moment of Symmetry scales, in which one generating interval functions as the "period", i.e., the interval at which the scale repeats (typically an octave or fraction of an octave) and the other generating interval is iterated until a scale is arrived at with exactly two sizes for each kind of generic interval (eg. 2-step or 3-step). Tracks from Newbeams that feature MOS scales are Spun in Orwell[13], Hypnocloudsmack 2 in Unidec/Hendec[8], and Tumbledown Stew in Sensi[11]. Elf Dine on Ho Ho uses a subset of Orwell[13]. Tracks that feature MODMOS Scales (MOS Scales that have been modified by chromatically altering some of the tones) are Hypnocloudsmack 1 in a MODMOS of Echidnic[10], and Hypnocloudsmack 3 in a MODMOS of Sensi[8].

A perhaps less common approach to rank-2 temperaments is to freely move through the tempered lattice, without regard to playing in any particular scales. This approach makes composing in a temperament more like composing in JI, where freely moving around in a lattice generates new pitches that are closely related to the ones adjacent to it. The use of a generalized keyboard makes this kind of structure almost trivially easy to explore. Tracks that use this approach are Rats, Shorn Brown, and Jellybear.

Lambdomas or tonality diamonds are JI structures in which one tone can function as any overtone or undertone in the system, and the rest of the tones fall into place around it to complete the chords. Two tracks make use of higher-limit lambdomas: Undernova and Cardinal.

And finally, there are two tracks which use EDO systems (equal divisions of the octave) without any particular temperament intended: Pentaswing and Shimmerwing.

Track notes:


This is one of several pieces in 46edo that I wrote using an Axis-49 key mapping for Sensi Temperament. Sensi offers 8-, 11-, and 19-tone MOS scales, but for Rats, I explored the keyboard freely, without concern for staying "in" a particular key or scale. To my ears, this piece has a tonal center and patterns of tension and release that you might associate with "tonality". It obviously does not take a tour of all 98 keys on the Axis-49, and so somebody could of course analyze it and say, "This is the scale." Although this sort of analysis may be informative, I have not done this and don't plan to.


The mapping I used for this and all other tracks on this album in 46edo Sensi temperament is the one on the right in the diagram above. Likewise, the mapping I used for the two tracks in Orwell temperament is the one on the left. Both of them make MOS scales easy to play by bunching all the tones together in a zig-zag pattern.

Shorn Brown

This piece takes a walk in 37edo through the keyboard in a mapping for Orgone (or Orpheus) temperament. I was playing with 37edo as a system, that is, a 13-limit system with no perfect fifths. I hear this piece going on a journey to harmonically distant places and then suddenly returning, and would consider it as "tonal" as Rats. It is the only "pop song" on the album.


In March, I relearned Shorn Brown to perform live, so I took an opportunity to figure out exactly what tones I happened to use. I was just curious. It turns out that the complete chain of orgone generators looks like:

0 _ _ 3 _ 5 6 7 8 9 10 11 12 13 _ 15 16 17 18 19 20 21 22 23 24 _ _ 27 28 29 _ _ _ _ _ _ _

For a grand total of 24 tones out of 37 (not counting octaves). Although this is most of the tones available in 37edo, I think it makes sense to still call this orgone, as there are lots of contiguous tones in the generator chain and a whole chunk of the chain is not used at all (represented by the _'s at the end of the list above). I was hardly thinking about the generator chain at all as I composed the piece: I was simply developing a chord progression and melody bit by bit, paying attention to the sorts of chords I was producing as I went. It was this particular keyboard and this particular mapping that made my intuitive approach to a complex temperament in a relatively large EDO not only possible but actually pretty easy! No other interface could have generated this song.


Another free walk through the Axis-49 in 37edo -- this time a mapping optimized for Porcupine temperament. As with Rats and Shorn Brown, I was paying attention to the intervals between the instruments and how they relate to 13-limit harmony -- but this piece has a less "tonal" sound overall to my ears. I started with specific constraints regarding the size of different sections and how they repeat, but wound up chopping the whole thing up and rearranging the chunks. In the end, this made it much more interesting to me.

Hypnocloudsmack 1

This is one of three "live" MIDI keyboard improvisations which I recorded and later sped up beyond human playability. Other composers have done this sort of thing; Jacob Barton calls these "hyperimprovisations". All three Hypnocloudsmack tracks explore specific sub-scales of 46edo associated with different temperaments. This one is in a MODMOS of Echidnic[10] and plays with 46edo's 17-limit interpretation. Although the MOS scale Echidnic[10] has the 600¢ half-octave as a period, the MODMOS I'm using here is not symmetrical in this way, but repeats at the octave. Harmonically, I was especially interested in the two different sizes of neutral third that 46edo offers. I also enjoyed having an opportunity to use the half-octave as 17/12 and 24/17 here, rather than 7/5 and 10/7.

Elf Dine on Ho Ho

The tuning here is 53edo, specifically a few tones from Orwell temperament, and the principal harmony is an 11-limit utonal chord. The form, of course, is a round in four parts. The meter consists of five beats per measure, with each beat divided into seven subdivisions. My own shorthand calls this 5x7. EDOHH has an unlikely companion piece -- Pentaswing -- which is in a meter I'd call 7x5: seven beats per measure, with each beat divided into five subdivisions. (Actually, Pentaswing is sometimes more like 7x5x2, as the subdivisions are occasionally subdivided. EDOHH is strictly 5x7.) A beat divided into 7 has the peculiar characteristic that it can't be divided in half; the closest it comes is a 4+3 or 3+4 rhythm, both of which sound a little odd. I don't think the voice part I recorded quite captures the oddness of these asymmetrical subdivisions, but you may be able to hear it in the synth part. The lyrics are a letter palindrome by Marji Gere.


The rhythm here is a simple 7/4 beat with some space at the end of the measure. We also have a highpass filter on the percussion, along with some delay and reverb. Underneath this foundation, there's a synth improvisation using an arpeggiator. The volume on the arpeggiator track is patched to the signal level on the percussion track, which explains the general in-and-out dynamic of the piece. The tuning is a mapping to the Axis-49 keyboard of half of a lambdoma (diamond) based on prime harmonics through the 43rd. The improvisation sticks mainly to high-limit otonal chords within the diamond. These are obviously not the most ideal conditions to appreciate such chords, but nonetheless, that's what's going on.


As I mentioned in the description for Elf Dine on Ho Ho, Pentaswing uses a meter I call 7x5 with extension 7x5x2. This means seven beats per measure, each beat divided into five subdivisions (and, in the case of the extension, each subdivision divided further into two smaller subdivisions). The basic idea of the pentaswing rhythm is to generalize the "swing" or "shuffle" rhythm of jazz and other styles, which features each beat divided into a long first duration and a short second duration, usually with a 2:1 ratio between the durations, allowing for a triplet grid. If the triplet is further divided, it's divided in half. That gives us at most six subdivisions of a beat (3x2). So if we happened to have seven beats in a measure with a standard triplet swing, I'd call that meter 7x3x2 or 7x6. Compare this with 7x5, the basic "pentaswing" meter. In pentaswing, the "swing" grouping is 3+2, rather than 2+1 or equivalently 4+2. Pentaswing is slightly closer to even than standard swing, giving it a "lazier" or "looser" quality. The subdivisions are slower, too: 5 subdivisions in a beat instead of 6. The pentaswing cannot do an even triplet. Uneven triplets with 5 subdivisions are 2+2+1, 2+1+2, and 1+2+2. When the 5's are divided into 3's (5x2 -- decaswing, perhaps), we have more even triplets available: 3+3+4, 4+3+4, and 4+3+3. (Of course, standard triplet swing has the same problem approximating quintuplets. Uneven quintuplets with 6 subdivisions include 2+1+1+1+1, 1+2+1+1+1, 1+1+2+1+1, 1+1+1+2+1, and 1+1+1+1+2; none of those really sound like quintuplets. With twelve subdivisions, we can do better of course: 3+2+2+3+2, 2+2+3+2+3, 2+3+2+3+2, 3+2+3+2+2, and 2+3+2+2+3. These rhythms are exactly analogous to the modes of the black-key pentatonic scale in 12edo. In addition to being MOS rhythms, they are maximally-even and thus also Euclidean rhythms.) Pentaswing (the song) uses both 7x5 and 7x5x2 in different measures. The tuning is 23edo, obviously not all of it, but not a specific intentional subset, either, but something that emerged out of purely melodic and rhythmic considerations.


I created the diagram above to compare triplet swing and pentaswing. I didn't use it as a reference for the album, but it might be useful in future projects that use these types of rhythms. I think the pentaswing rhythm is worth returning to.


Spun also uses a pentaswing rhythm, but at a faster tempo, and without ever subdividing the quintuplets. The overall meter is mixed, rather than periodic in the way that Pentaswing (the song) is. The harmonic backbone of this piece is a sequence of essentially tempered dyadic chords in Orwell[13]. I used this list of chords as a reference and developed this sequence of keyboard patterns as another reference. Rather than choose a large variety of chord qualities, I decided to focus on transpositions of only a few chords. So there's:

  • 0-5-6-7-8 and all transpositions available in Orwell[13]: 1-6-7-8-9; 2-7-8-9-10; 3-8-9-10-11; and 4-9-10-11-12 (5 chords).
  • 0-2-3-5-10 and all transpositions: 1-3-4-6-11; and 2-4-5-7-12 (3 chords).
  • 0-1-3-8-11 and all transpositions: just 1-2-4-9-12 (2 chords).
  • 0-5-7-10-12 -- no transpositions available in Orwell[13] (1 chord).

For a total of 5+3+2+1 = 11 chords. So here's the sequence I came up with for the A section. The number in square brackets is the number of beats in that measure:

0-5-6-7-8 ; [7]

0-1-3-8-11 ; [4]

2-4-5-7-12 ; [5]

2-7-8-9-10 ; [6]

3-8-9-10-11 ; [3]

0-5-7-10-12 ; [7]

1-3-4-6-11 ; [3]

4-9-10-11-12 ; [4]

1-2-4-9-12 ; [6]

1-6-7-8-9 ; [5]

0-2-3-5-10 ; [7]

There's a two-bar introduction and connector piece which comes before and after the A section and just plays the first two chords of the A section (0-5-6-7-8 and 0-1-3-8-11). Then there's another A section with a different melody but exactly the same form. This A section, however, goes right into a shorter B section which introduces new chords:

  • 0-2-7-8-10 and its transpositions: 1-3-8-9-11; and 2-4-9-10-12 (3 chords).
  • 0-2-5-7-10 and its transpositions: 1-3-6-8-11; and 2-4-7-9-12 (3 chords).
  • 0-1-6-7-12, which has no transpositions in Orwell [13] (1 chord).

For a grand total of seven chords. The B section goes like this:

1-3-6-8-11 ; [4]

1-3-8-9-11 ; [7]

0-1-6-7-12 ; [5]

2-4-7-9-12 ; [6]

0-2-7-8-10 ; [3]

0-2-5-7-10 ; [3]

2-4-9-10-12 ; [4]

We then have a repeat of the two-bar connector, a short "piano solo" on the A section, a repeat of the B section, yet another instance of the two-bar connector, and a final repeat of the A section with its original melody. Then the song comes quickly to an end by hitting that hot 0-5-6-7-8 chord a few more times. I developed the melodies one-by-one using this basic form, keeping the notes of the melodies always playing one (or more, in the case of the "piano") of the five tones in the chord at that instant, treating the chord as a pentatonic scale in which to "improvise". In this way, the parts touched on all the tones of Orwell[13] by jumping around between different pentatonic subsets. Adjacent chords with tones in common provided opportunities to carry notes in the melodies over the barline, helping to emphasize the smoothness of the harmonic system while varying the rhythmic gestures. The result is, I think, a pretty hot little jazz piece.

Hypnocloudsmack 2

Another hyperimprovisation. This one plays with the temperament listed on this wiki as Unidec/Hendec. The period is a half-octave and the generator is 7\46edo, about 182.6¢. The bass part is in a strict LLLsLLLs MOS/DE scale, but the keyboard has a little more freedom to explore the lattice. This was physically the case because of the instruments I was using. My left hand played the bass on a Korg PadKontrol, which offers a square grid array of 4x4 = 16 MIDI-assignable velocity-sensitive pads. I decided to give it two octaves of the scale, which takes exactly 16 pitches, leaving no pads left-over. My right hand played the "lead" sound on the Axis-49, which obviously had some keys to spare.


A few times in February, I met with Chris Vaisvil on Skype to organize a collaboration. The result is this open-ended jam in 10edo. We actually did it in only two live passes, starting with a synth track I sent him and ending with a fretless guitar track he sent me (although he gave me a few different takes to choose from). We had been scheming for a 19edo collaboration as well, but I got stuck on that one and it never got off the ground. I'm very happy with Shimmerwing, and I think it adds a nice contrast to the material around it.


What a weird track. The rhythm here is 5x13 -- five beats per measure divided into 13 subdivisions. The tuning is a high-limit lambdoma system, although not exactly the same one that appears in Undernova. I'm not sure quite what to do with a 13-beat or a high-limit lambdoma, but this is a start.

Tumbledown Stew

I originally composed this round in ChucK, but I produced a listenable version of it for the album in Renoise. The tuning is Sensi[11] in 46edo, and I think the song sounds pretty tonal, with the first chord a sensi approximation of an 8:10:12:13 tetrad and all the others some approximation of an otonal chord. The scale pattern is LLLsLLLsLLs, with L = 5\46edo, approximately 130.4¢, and s = 2\46edo, approximately 52.2¢. The meter has a long-measure of 18 beats, divided into measures of 5, 5, 5, and 3 beats. Each measure corresponds to a chord change, so this is a four-chord song.

Hypnocloudsmack 3

The scale here is a MODMOS of Sensi[8] in 46edo. Ordinary ol' Sensi[8] has scale pattern ssLssLsL, with L = 7\46edo ~ 182.6¢, and s = 5\46edo ~ 130.4¢. The difference between L and s is the "chroma", "c", and here it's 2\46edo ~ 52.2¢. Sensi[8]'s "s" is the same as Sensi[11]'s "L", and its "c" is the same as Sensi[11]'s "s". This is to be expected (see Tumbledown Stew above). Anyway, if you add c to L you get "A", an augmented step, and if you subtract c from s you get "d", a diminished step. The form of the MODMOS used in this hyperimprovisation is sssLsssA. Since A is 7\46 + 2\46 = 9\46edo ~ 234.8¢, an approximate 8/7, we have a harmonic seventh above the bottom note in this scale. We also have three intervals of a perfect fifth available, instead of just one, as you'd get in the standard Sensi[8] MOS scale. This improvisation wanders around a lot, but mostly moves between three main chords, each with a perfect fifth above the bass note.