Scale of the Day: B Dorian diminished 4 mapped to the Square-root-of-2

The B Dorian diminished 4 mapped to the Square-root-of-2 Scale.

Melody Study: Melody #1 - Just Transposition and Retrograde Melody

Proceeding forward with a study of melodic transformative theory let's return to the initial hymn melody:


Example 1: Hymn Melody

Just Transposition

Here is the 5-limit tuning worked out in a previous post:


Example 2: Hymn Melody in 5-limit Just Intonation

Transposition involves moving each pitch in a melodic line by a fixed interval. In just transposition we simply use a fixed just interval. To calculate the correct frequency ratio one only needs to multiply by the fixed interval ratio. For example, if we were to transpose this 5-limit hymn melody up by an 8/7 major second we would have the following:


Example 3: Hymn Melody transposed by 8/7.

Now this 5-limit melodic line is shifted into a 7-limit space, even though the melodic line itself adheres to a 5-limit sound. It is only 7-limit relative to the E Flat 1/1 pitch.

Retrograde Melody

Another relatively straight forward function for melodic transformation is retrograde melody. For example, this hymn melody played in reverse would look like this:


Example 4: Hymn Melody - Retrograde.

A couple of variations on the retrograde function to consider involve separating the pitch sequence and rhythm and reversing them independently.


Example 5: Hymn Melody - Pitch order Retrograde.


Example 6: Hymn Melody - Rhythmic Retrograde.

These are relatively simple operations that could be combined with transposition functions (or any other functions) to form counterpoint lines with the original melodic line for a unified polyphonic texture or used sequentially for melodic variation. Next time we return to this melody we'll take a look at some possible inversion strategies.

Scale of the Day: E Flat Dorian

Synthesized audio sample of an E Flat Dorian Scale.

See also:
E Flat Dorian Scale notated.
E Flat Dorian Interval Analysis.

Happily Ever After

I find Happily Ever After by Randy Hostetler in my ears as I periodically approach this outstanding work from time to time to experience this epic text piece that transcends the limitations and frustrations of spoken language to reveal the rich potential of text-sound. This piece uses a great idea realized with talent and an ear for detail.

Happily Ever After is a 43-minute work consisting of manipulated recordings of Randy Hostetler and 64 of his friends and families telling stories. Each story begins with "Once upon a time..." and ends with "... happily ever after." Everything in between is up to the story teller as a loose improvisation in narration. These performances are manipulated largely by temporal and thematic placement and the actual language is never obscured beyond some careful overlaps between storytellers (and even these are panned comfortably within the stereo field so that one can pick out a story line at any moment).

This work accomplishes something extremely rare in text-music compositions. Less than 20 minutes into this work it eliminates my resistance to text-sound and the pervasive consciousness of tangling sound with literal meanings vanishes and a carefully woven texture reveals something beautiful and human. One becomes emerged within the attractive din of a family gathering in a warm home. For a work limited to the sound of recorded voices there's a rich texture that emerges from the range of age, gender and varying degrees of enthusiasm, humor, melancholy, remorse, regret, irony, sarcasm, seriousness and comfort level with being recorded. The piece opens with a brilliant overdub of multiple "Once upon a time" along with another massive wave of "okay, is this thing on?" before opening up into a wide polyphony of human experience. There's a surprising dynamic range that keeps the experience vibrant as one strains to hear a lone, quiet story telling before being swept up in a wave of activity.

What emerges from this texture, and this experience, is a pervasive sense of a meta-story of the multiple components of human mortality. No single story slides into focus. Some episodes slide into the foreground but the overall "story" is more impression than narration. This is an unusual, fantastic work that illuminates an unpretentious vulnerability that works on so many different levels.

Scale of the Day: G Dorian diminished 4

The intervallic content of the G Dorian diminished 4 Scale.

Scale of the Day: B Pythagorean Dorian

The B Pythagorean Dorian Scale. Note the symmetry between the +3.91 cent 9/8 major second with the -3.91 cent 16/9 minor seventh. Or the -5.87 cent 32/27 minor third to the +5.87 16/9 major sixth. The sound of the Dorian scale owes much to the perfect symmetry of inverse interval classes contained within the octave.

Scale of the Day: F Sharp Aeolian mapped to the Square-root-of-2

The F Sharp Aeolian mapped to the Square-root-of-2 Scale.

Baseball Post: Shigetoshi Hasegawa Retires

Shigetoshi Hasegawa announced his retirement yesterday. I first noticed this guy when he was working out of the bullpen for the Angels. He seemed like an under-rated relief pitcher and anytime I saw him come out against the Mariners I knew that he would snuff out any rallies as they just couldn't do anything with his fastball. His mechanics were really interesting to watch as well. I was sure glad to see him move over to "our side" when he joined the Mariners in 2002. He was outstanding in 2003 when his career best 1.48 ERA got him an All-Star nod that season.

Easily my fondest Shiggy memory will be teaching my son how to pronounce his name. That's a lot of syllables for a 3-year-old just learning language skills to negotiate. But he would say it with a lot of enthusiasm whenever he came out of the bullpen and we were both fans of his impressive talent.

Scale of the Day: C Sharp Aeolian

The intervallic content of the C Sharp Aeolian Scale.

Over-notation

Kyle Gann has been blogging about over-notation. I'd read his online essay some time ago. He's right on the mark on this subject.

I'm a chronic over-notator. I learned early from a "professional" academic composer that there was no limit to the amount of information one should provide in a written score and it eventually folded into my personal style. I don't see over-notation as being inherently better or worse in terms of compositional craft. Not every piece calls for it. Indeed, my ears are often drawn strongest to things that aren't notated at all. I think some of these academics look at over-notation as some kind of evidence that the composer has taken multiple parameters into account and has worked out a fully formed idea. It's a valuable trait to consider every parameter as part of the final sound. Otherwise, one might think composing is about notes on paper as opposed to sound in all the rich ways one can perceive and slice it.

One of the largest pitfalls of over-notating - and Gann singles it out as well - is the tendency to artificially inject drama through extensive variation along every notated parameter. The most obvious example is dynamic markings. Not every idea is served by contrasting dynamics. Morton Feldman finds vast universes within the space between ppppp and ppp. It takes some measure of courage to find a notation that serves the idea and not the other way around.

Scale of the Day: F Sharp Aeolian diminished 4

The F Sharp Aeolian diminished 4 Scale as one would find it on any conventionally tuned, equal tempered instrument.

Melodic Study: Melody #2 - Transposition and Just Intonation Strategies for Non-tonal Melodic Lines

(Part 2 in a continuing series of posts regarding melodic manipulation and development) Let's turn our attention to a second melodic line from an early composition of mine:

example 1: Melody #2

Transposition

Transposition is one of the simpler - more immediate melodic manipulations. Each pitch-class is either raised or lowered by a single fixed interval. For example, here is Melody #2 raised by a minor third:

example 2: Melody #2 Transposed by a minor third

Just Intonation Strategies for Non-tonal Melodic Lines

Aesthetically, non-tonal melodies are well suited for a free style approach to intonation as articulated by Lou Harrison. In this manner it is possible for a composition to roam freely in tuning space without being bound by a fixed scale. And as non-tonal melodies are already unbound by a fixed "tonic" or scale structure to begin with they make a great candidate for free style intonation strategies. There are a couple of different ways of going about this.

The first approach is to tune each note relative to the note that preceded it. So a possible 5-limit realization of this melody might look like this:

example 3: Melody #2 In 5-limit free style

The frequency ratios quickly make use of extremely large integers as this melodic fragment becomes unhinged from any sort of tonal center or scale reference. The linear tuning logic from note to note is actually fairly simple even as things move quickly away from the original E Flat (1). The C natural comes in as a 5/3 major sixth relative to E Flat. The augmented fourth that follows is a 45/32 augmented fourth relative to the 5/3 (with frequency ratios, one adds intervals by multiplying, therefore, (5/3)(45/32) = (75/64)). The 25/16 is a 4/3 perfect fourth relative to the 75/64. And so on until the E Flat "returns" as a 3805/32768 after cycling through so many intervals. Sonically, this is harmonically intelligible and unsettling at the same time.

The second approach involves isolating phrases and tuning each note relative to a significant pitch found within each phrase. Using this approach, an alternate 5-limit realization of this melody might look like this:

example 4: Melody #2 In 5-limit free style, using phrase approach

In this example the phrasing marks are used to delineate discrete gestalt units (as opposed to performance indicators). The first note of each "phrase" unit, or gestalt, is tuned relative to the first note of the gestalt that preceded it. Each subsequent note is tuned relative to the first note of the gestalt. In this manner the rate of travel through tuning space is slowed down considerably compared to example 3 even though this technique generally amplifies the "unhinged" qualities of non-tonal melody with such a liquid approach to intonation.

5-limit just intonation is best suited for triadic, diatonic, tonal music. A non-tonal melody such as this one is not a particularly good fit for traditional 5-limit just intonation. The 5-limit system features good major thirds/minor sixths (produced by the prime factor 5) and perfect fifths/perfect fourths (produced by the prime factor 3) prominently. Meloldy #2 makes prominent use of major 7ths and augmented 4ths. So this suggests a system built on prime factors that represent these intervals. One could "swap" out the 5 and 3 of the 2-dimensional tuning lattice with a 31 (for the 31/16 major seventh) and 11 (for the 11/8 augmented fourth). So, melody #2 using a just intonation system limited to the prime factors of 2 (the octave), 11 and 31 might look like this:

example 5: Melody #2 In 11, 31-just intonation

Now this melody begins to take on an inner logic as it carves out a distinct, and unfamiliar harmonic vocabulary. This could easily spin outward from here into a full composition that explores the sonic qualities of an 11, 31-space.

Other possible strategies for applying just-intonation to non-tonal melodies include:

• 3 or greater dimensional lattice spaces.
• Single dimension systems (such as a Pythagorean system).
• An unlimited, multi-dimensional lattice system that uses single primes for each interval type.
• Cyclical "1/1" values assigned to a new member each time through a tone-row.

I'll need to flesh these out in a future posting.

Scale of the Day: F Phrygian

The F Phrygian Scale as one would find it on any conventionally tuned, equal tempered instrument.

Scale of the Day: F 11-axis, Construct No. 1, Lydian Mode

The F 11-axis, Construct #1, Lydian Mode - Scale.

Scale of the Day: E 5, Cube-root-of-2, Construct no. 1, Lydian Mode in Square-root-of-2-space

The E 5, Cube-root-of-2, Construct #1, Lydian Mode in Square-root-of-2-space Scale.

Baseball Post: January Predictions for 2006

It's about one month until catchers and pitchers report to spring training. Just for fun, here are my current predictions for the season ahead based on educated guessing and gut feeling:

AL West final standings:

(1) Los Angeles Angels
(2) Oakland Athletics
(3) Texas Rangers
(4) Seattle Mariners

Yes, I predict a repeat of the 2005 standings in the AL West. Texas has improved its pitching this off season. But not enough to catch Oakland. I would love to see Seattle finish ahead of Texas, but I don't think that will happen at this point.

AL Central final standings:

(1) Chicago White Sox
(2) Cleveland Indians * Wild Card
(3) Minnesota Twins
(4) Detroit Tigers
(5) Kansas City Royals

Another duplicate of 2005's results. But this time I think Cleveland will take the Wild Card berth. I expect the Royals to stink it up for another year.

AL East final standings:

(1) New York Yankees
(2) Toronto Blue Jays
(3) Boston Red Sox
(4) Tampa Bay Devil Rays
(5) Baltimore Orioles

The AL East has seemed set in stone over the past few years. New York will take the division yet again. But I see Toronto being more competitive this year as Boston sinks. And I see crash and burn for Baltimore. So much so that the Devil Rays will move out of dead last for the first time in franchise history.

Predicted AL Postseason:

ALDS
Los Angeles Angels defeat the Chicago White Sox in 5
Cleveland Indians defeat The New York Yankees in 4

ALCS
Los Angeles Angels defeat Cleveland Indians in 6

Pennant goes to the Angels.

NL West final standings:

(1) Arizona Diamondbacks
(2) San Diego Padres
(3) San Francisco Giants
(4) San Diego Padres
(5) Colorado Rockies

Still a weak division, I see the D-backs rising above on the acquisition of Orlando Hernandez alone. I'd like to try and catch one of his starts this season on the off chance of seeing his eephus pitch. (The eephus is a particular obsession of mine. In my next life I'd like to be a left-handed starting pitcher with a good eephus pitch. I'm only aware of 3 active pitchers who throw such a pitch: Casey Fossum, Livan Hernandez and Orlando Hernandez. The unbalanced schedule should bring Orlando out my way more frequently than the other guys.)

NL Central final standings:

(1) St. Louis Cardinals
(2) Houston Astros
(3) Chicago Cubs
(4) Milwaukee Brewers
(5) Pittsburgh Pirates
(6) Cincinnati Reds

I'm pretty confident with first and last in this division. The stuff in the middle may vary.

NL East final standings:

(1) New York Mets
(2) Atlanta Braves * Wild Card
(3) Philadelphia Phillies
(4) Washington Nationals
(5) Florida Marlins

I'm going out on a limb on this one. For the first time in 14 seasons the Braves will not take the division. But they'll ride in on the Wild Card anyway for an unbroken chain of playoff appearances.

Predicted NL Postseason:

NLDS
St. Louis Cardinals defeat the Atlanta Braves in 3
New York Mets defeat the Arizona Diamondbacks in 3

NLCS
St. Louis Cardinals defeat the New York Mets in 5

Pennant goes to the Cardinals

World Series prediction:

St. Louis Cardinals defeat Los Angeles Angels in 7 games.

World Championship awarded to the Cardinals.

Actual results may vary. I may have to update my predictions every month or so.

Scale of the Day: E 5, Square-root-of-2, Construct No. 1, Lydian Inversion

The E 5, Square-root-of-2, Construct No. 1, Lydian Inversion - Scale.

Melodic Study: Melody #1 - Transposition and 5-limit Tuning

After numerous postings and scale examples focused on the subject of harmony, it's time to add some material on melody. In particular, I'd like to focus on my current thinking on manipulating melodic content and applied harmonic and rhythmic theory using specific melodic examples. In this particular post I would like to look at transposition and 5-limit just intonation applied to a single melodic example.

Let's begin with the following hymn tune as a simple melodic basis:

Example 1: Hymn Melody

This melodic line has the following qualities:

• Diatonic, tonal melody in E Flat Major (Ionian mode)
• 8-bars in length (4/4 meter) with a pick-up of one quarter
• Rhythmically dominated by the long, short, short pattern of half-note followed by two quarters, emphasizing beats 1 & 3 (with all rhythmic momentum driving toward the first beat)
• Rhythmically dominated by quarters and half notes - the dotted quarter followed by an eighth note in measure 4 is the only exception and is largely heard as an ornamentation on the half note duration

Transposition

One of the simplest operations to perform on any melodic line is straight transposition. For example, by raising each pitch in this melodic sequence by an augmented fourth we can transpose our melody into A Major.

Example 2. Hymn melody transposed.

With the exception of being in A Major, this melodic sequence retains all other harmonic and rhythmic qualities found in the original example.

Transposition is a conceptually simple operation to perform and compositionally adds a small degree of variation (and some "dramatic" effect) without perceptually obliterating the source melodic sequence. It is an effective manipulation. Aesthetically, it's not terribly exciting.

5-Limit Just Intonation

Simple diatonic melodies such as this hymn excerpt are well-suited for 5-limit just intonation. The emphasis and scalar gravity of the tonic, major 3rd and perfect fifth of the root major triad are strongly suggestive of the 4th, 5th and 6th partials found in the overtone series and these are well represented by 5-limit just intonation. A typical 5-limit E flat Major (Ionian) Scale would be tuned as follows:

Example 3. 5-limit just intonation E Flat Major Scale

The numerical expressions on the bottom are the frequency ratio. This is said to be 5-limit just intonation because the largest prime factor in either the numerator or denominator of these ratios is 5. The +/- expressions on the top are the deviation, in cents, from equal tempered tuning. Each equal tempered semitone is 100.00 cents wide. The 9/8 major second is 3.91 cents larger than a 200.00 cent major second (for a total size of 203.91 cents). The 5/4 major third is 13.69 cents smaller than a 400.00 cent major third (for a total size of 386.31 cents). One of the qualities of a 5-limit just intonation scale such as this is that the root position major triad will have very little roughness as the overtones of each chord member will overlap exactly leading to an astonishing fusion of chord into timbre. You don't get that with equal tempered triads.

Applied to our original melody, you get the following:

Example 4. Hymn Melody in 5-limit Just Intonation.

This is only one possible way to tune this melodic sequence. But for a simple hymn melody, this is the intonation that most closely matches how one would naturally sing these intervals.

In future postings I will return to this melody and examine alternate methods for tuning and transforming this particular melodic sequence. I will also look at non-diatonic melodies and the kinds of intonation schemes that could be applied to them. Also, I'd like to examine my personal theory of texture classes in an effort to articulate a larger conceptual approach to music.

Scale of the Day: D Sharp Locrian 1% narrow

The D Sharp Locrian 1% narrow Scale.

Scale of the Day: D Sharp Locrian 2% wide

The D Sharp Locrian 2% wide Scale.

Scale of the Day: D Sharp Locrian diminished 4 1% wide

The D Sharp Locrian diminished 4 1% wide Scale.

Just Guitars

I've had a chance to give Just Guitars a few spins now. In an email from Hucbald at A Monk's Musical Musings comes this vivid description: "...justly tuned fret boards look like somebody puked up strips of pasta all over them." They are odd. And it's hard to stop looking at the pictures of fret boards included with this disc. The overall sound is even more intriguing. John Schneider does a fantastic job with this material. This is a great repertoire of music in just intonation that sheds some new light on some familiar works and opens my ears to some new possibilities for these specialized acoustic guitars.

The high point of this listening experience comes toward the end as Schneider performs two movements from The Harp of New Albion by Terry Riley. I've spent a lot of time with the Terry Riley recording of this work for solo piano (in 5-limit just intonation) and I've had a chance to hear it performed live. As a composition it's a personal favorite and this arrangement for just guitar is beautifully done. It's great to hear those familiar chords that open "New Albion Chorale" on acoustic guitar. Schneider captures the mesmerizing quality of this composition and sustains exactly the right energy level throughout.

John Schneider also proves to be a great interpreter of the music of Harry Partch. As an important figure in just intonation and a composer of some of the most idiomatic music of the 20th Century -- the fact that Harry Partch was also a great guitarist is often overlooked or forgotten. Schneider brings a great understanding to this material and brings out the troubadour qualities of the Partch oeuvre.

"Letter from Hobo Pablo" is performed for voice and just guitar alone. I'm so familiar with the original Partch recording for voice, Adapted Guitar and Kithara that I notice just about every detail where Schneider's vocal inflections differ from Partch. The guitar arrangement is a dead ringer of the original version harmonically and so many of these large chords seem impossible - yet fully represented here on this newer version. The humanity of the hobo account in the letter comes through with a bittersweet mix of melancholy, humor and sincerity.

"December, 1942" by Harry Partch is a setting of three texts by Shakespeare, Tsurayuki and Ella Young that is new to me. They are similar to The 17 Lyrics of Li Po. I enjoy them a great deal and feel compelled to seek out the Partch recordings of this work.

Rebekah Raff and Gene Sterling perform with Schneider on the Kithara and Diamond Marimba for this recording of "Three Intrusions." These Harry Partch compositions just sound better with multiple interpretations like these.

Also included on this disc is one of the last works composed by Lou Harrison: "Scenes from Nek Chand (2002)." This is an outstanding work for Steel Guitar in just intonation that draws sonic influences from all sides of the Pacific Ocean. John Schneider performs five other Lou Harrison works as well: "Tandy's Tango (1992)", "Cinna (1957)" , "Palace Music (1971/88)", "Plaint & Variations on 'Song of Palestine'" and "Serenado por Gitaro (1952)." I need to get more familiar with the vast Lou Harrison catalogue. This is wonderful stuff.

The opening work on Just Guitars is an arrangement of "Rhythmicon I" by Carter Scholz for 17 justly tuned guitars. This is a work I reviewed in its digital realization last year. 8 Pieces is a disc I keep coming back to time and again because of pieces like "Rhythmicon." The formal ideas behind it are eerily similar to my own. This work bears a strong resemblance to "Septet" by James Tenney for 6 electric guitars and electric bass (performed by Seth Josel on Go Guitars). But while the digital realization of "Rhythmicon" and the amplified guitars of "Septet" are successful I find this multiple acoustic guitars version of "Rhythmicon I" disappointing. It's a nice sound, but this particular algorithmic compositional idea really calls for electricity and amplification.

Scale of the Day: D Sharp Locrian mapped to the 3/2

The D Sharp Locrian mapped to the 3/2 Scale.

Scale of the Day: D Sharp Locrian diminished 4 mapped to the Triative

The D Sharp Locrian diminished 4 mapped to the Triative Scale.

Scale of the Day: D Sharp Pythagorean Locrian mapped to the Square-root-of-2

The D Sharp Pythagorean Locrian mapped to the Square-root-of-2 Scale.

Scale of the Day: D Sharp Locrian Major 2

The D Sharp Locrian Major 2 Scale as one would find it on any conventionally tuned, equal tempered instrument.

Baseball Post: Circle Me Bert! ...and other offseason observations

3,701 career strikeouts. That's 5th all-time and that's Bert Blyleven's total for 22-seasons. It was 3rd all-time when he retired in 1993. Why isn't this guy in the Hall of Fame yet? There's more here on why he should be inducted. I'm inclined to agree.
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I'm all for parity in the MLB as having 30 reasonably competitive teams is more interesting than having the Yankees make the playoffs like clockwork. However, there's an unintended consequence of the "luxury tax" that allows ownership to gut teams like the Florida Marlins. After their recent fire sale that sent some outstanding talent to other teams these guys have a projected starting lineup that should disqualify them from receiving compensation from teams like the Red Sox and Yankees. They have Pokey Reese slated to start at second base. He's an excellent defensive short stop and that's just about the full extent of his selling point. With Miguel Olivo working behind the plate they essentially have at least two offensive blanks in the lineup. Three rookies in the outfield? Another in the starting rotation? And then two more at the corners? The fan base in South Florida should be insulted. At this rate they should have Dontrelle Willis batting clean-up every fifth game.

Dontrelle Willis is a lot of fun to watch and it looks like he'll be the only thing going for this team. And even then he'll have to supply his own run support just to get the wins. The Marlins had this 2nd-in-the-Cy-Young-votes pitcher (he should have won it) for a bargain price last season and they reward him by surrounding him with inexperienced, cheap players? Look for Dontrelle to change uniforms soon as the Marlins to be dead last in the NL East with at least 90 losses. This wasn't such a bad team last year and there should be a penalty for such gross mismanagement.
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The most interesting rumor I've heard has Jeff Weaver joining the Angels starting rotation. I watched his shutout complete game performance against the Rockies this past September. I don't know what kind of zombies are running the Dodgers to let this guy go but I'd love to see him pitch against American League hitters (again) and he seems like he'd be a great fit in the Angels' rotation.

Scale of the Day: C 5, 3 Construct #1, Lydian Mode

The C 5, 3 Construct #1, Lydian Mode - Scale.

Scale of the Day: B Whole-tone 1% wide

The B Whole-tone 1% wide Scale.

The Great Learning

My ears are hopelessly hooked on "Paragraph 2" of The Great Learning by Cornelius Cardew. In particular, the 1968 Scratch Orchestra performance conducted by Cardew as found on the digitally remastered Deutsche Grammophon CD. What a great, prolonged sonic texture! The initial impact of sounds like these has often been the initial attraction to the music of Cornelius Cardew for me. The radical ideas behind them soon follow and I find myself riveted by this elusive figure of the 1960s European avant garde.

The Scratch Orchestra was an important creative outlet for Cornelius Cardew and it was assembled specifically for The Great Learning. He sought to establish a large, democratic ensemble made up of musicians from a wide spectrum ranging from professional musicians to amateurs. He then applied non-traditional, non-specific notation and written instructions that treats all performers as equals toward achieving a focused sonic ideal. The resulting sound is energetic, rough around the edges and beautiful. "Paragraph 2" uses a combination of drums and choir to build a soundscape of ragged pulse and wordless vocals. There's a fantastic immediacy to this sound that grows from the improvisatory nature of this work.

After peeling myself away from repeated listenings to "Paragraph 2" I finally turn my ears to "Paragraph 7" for an a capella atmosphere of sustained voice with the words "swept away" woven in at various layers. The low volumes of this choir adds an intensity from having such a large ensemble working within such a quiet space. The result feels like a communal meditation. A church choir without the trappings and social order of a church.

The Great Learning is a 7 hour work in total. Only "Paragraph 2" and "Paragraph 7" are found on this recording. It is exactly the kind of large scale work that begs to be lived within for the full duration. If a complete recording were available it would be tempting to take long vacations immersed in such a soundscape. Aesthetically (and socially), Cornelius Cardew was onto something that feels lost to this era.

Scale of the Day: B Whole-tone mapped to the Triative

The B Whole-tone mapped to the Triative Scale.

Scale of the Day: E Flat Whole-tone mapped to the Square-root-of-2

The intervallic content of the E Flat Whole-tone mapped to the Square-root-of-2 Scale.