Projects and Activities Fall 2019
As I enter my 14th year as a professor of Mathematics at Lehman College CUNY, the year of 2019 has been a busy one. Along with the usual research, teaching, collaboration and conferences that fill a mathematician’s life, I’ve had several opportunities to publicly merge my musical and mathematical dimensions. Especially satisfying has been my collaboration with saxophonist, mathematician, composer and educator Marcus Miller. In April I joined Marcus for one of his regular Quadrivium presentations at the National Museum of Mathematics in New York City as part of his Math Plus Music project (http://www.marcustheartyst.com/math-plus-music/). We played quartet with drummer Andrew Atkinson and bassist Noah Jackson, and discussed similarities between the dynamics of mathematics research and musical improvisation. Earlier in the year Marcus and I brought an expanded similar performance/presentation to the Willson Center for Humanities and Art at the University of Georgia. I’m looking forward to further collaborations with Marcus, and you can keep an eye out for his upcoming TED Talk…
Over the summer I had a reunion with Grammy-winning trumpeter, composer, producer Brian Lynch when we played duets at Mezzrow jazz club, including music from “Tone Twister”, my first recording in over 10 years, on Brian Lynch’s Hollistic Musicworks label. Tone Twister, which you can stream and download at https://robschneiderman.bandcamp.com/releases, features the magic chemistry of Brian’s trumpet and Ralph Moore’s tenor, with the deep groovin energy of Gerald Cannon and Pete Van Nostrand on bass and drums. You can read more about Tone Twister in the extended liner notes which are pasted in just below. These notes also include some of my thoughts on the mathematics-music mystery, and a sketch of how I got entangled with mathematics. I hope you dig it!
Extended Liner notes for Tone Twister:
These reflections on my new Hollistic Musicworks release Tone Twister come almost ten years after the recording of Glass Enclosure (RSR CD 193), the last of my ten sessions for the Reservoir Music label, dating back to 1988. Over the past twenty years the fabric of my music has been tightly intertwined with my development as a mathematician, and these past ten years have been particularly intense as my deepening journey into the world of mathematics has led me to the point where the continuous stream of music that irrigates my being is now fully merged with a perpetual flow of mathematical contemplations. The internal musical sounds and mathematical thoughts seem to commingle through some sort of mysterious subconscious filtering system that allows me to consciously to tap into one or the other. So perhaps the occasional epiphany that provides a key piece in a mathematical proof could be related to the melodic fragment that suddenly emerges and blossoms into a new musical composition?
Since music and mathematics are curiously both “opposite” and “similar” depending on who you are asking or on how you are looking at things, I try on occasion to step back and put into words how this paradoxical picture appears from my viewpoint. An early brief description can be found on the Reflections page of my JazzCorner website http://robschneiderman.com/cms17. A more in-depth essay written for mathematicians entitled Can One Hear the Sound of a Theorem? was published in the Notices of the American Mathematical Society 2011, and republished in The Best Writing on Mathematics 2012 (Princeton University Press). In celebration of this release of Tone Twister, whose musical content has been shaped by my mathematical explorations, it feels right to open these liner notes with some further informal ruminations on the mathemusical dilemma.
Why Mathematics is (not) like Jazz:
Mathematics is not only about “solving for x”. Everyday, researchers discover new interesting facts about a wide variety of mathematical structures, including models of different kinds of spaces in every dimension. Interest is guided not only by the desire for useful applications, but also by purely esthetic motivations.
For instance, it is surprising but true that there is only one way to “do calculus” in every dimension except for dimension 4, where there are infinitely many different versions of calculus! This unexpected fact is related to my field of research, known as “Low-dimensional Topology”, which studies 3- and 4-dimensional spaces and their subspaces. It turns out that understanding 3- and 4-dimensional spaces is in many ways much more difficult than understand- ing spaces of dimensions 5, 6, 7, … and greater!
The rough idea is that in higher dimensions there is more “wiggle room”, so the geometry of subspaces can be unfolded and untwisted to correspond to algebraic descriptions; whereas in low dimensions the analogous subspaces can crash into each other and twist around themselves in ways that are unavoidable and much more complicated. To explain further what this “rough idea” is would require pencil and paper and more time than we have available at the moment… So in the meantime you can listen to Tone Twister and perhaps absorb indirectly some low-dimensional geometric topology filtered from my subconscious through music.
I agree with jazz emissary Christian McBride’s goal of working to dispel any general uninformed fear of “jazz as some sort of calculus 4 class”. Soulful swinging does not need to be defined, and the stories that the sounds tell do not depend on any grammatical analysis! The appreciating listener of music does not need to know anything about the technical language that musicians use to describe, organize and prepare music.
On the other hand, mathematics is like “music that only musicians can hear”: Mathematics can transmit beauty, surprise, and even inspiration, but the appreciator does indeed need to speak the language of the land. Mathematics is built from definitions, using logic; and the ongoing discovery of the mathematical landscape involves proving statements by constructing arguments which often get quite complicated, as well as recognizing relationships between already proven statements. This process can involve intuition and creativity, but only after definitions and arguments have been thoroughly internalized by dedicated study and deep concentration. And there is no getting around the fact that the most rewarding mathematics lies way beyond college calculus!
However, there is still plenty of mathematical fun before Calculus 4 for a curious person who is willing to invest some concentrated reading from among the many currently available expository books and webpages. See for instance http://www.ams.org/samplings/math-sites/math-sites
Here are two of my recommendations: For a concise, no-nonsense peek inside the mathematical realm, try A Very Short Introduction to Mathematics, written by Fields Medalist Timothy Gowers. And for the ultimate in further expository reading there are the epic Princeton Companion to Mathematics, and Princeton Companion to Applied Mathematics.
Hopefully the mathematics community’s growing diversity and recent surge in public outreach will help lead to a dissipation of the general mathphobia that has been reinforced by media stereotypes of mathematics as the language of antisocial psychotic geniuses. In truth, a randomly chosen mathematician will be dedicated to mathematics, but will also likely have other enthusiastic interests, such as art, sports, nature, etc. So engage the next mathematician you meet in conversation about mathematics, but also about anything else on your mind, … including music!
However different music and mathematics may be in their accessibility, I believe that there are nevertheless some abstract similarities. For instance, the process of creating or discovering mathematics can involve dynamics that are analogous to a small-group jazz performance: It is very common that mathematical research is carried out by several mathematicians working together. Collaborators will get together for a real-time exchange of ideas, spontaneously alternating lead and accompaniment roles, guided by a thematic issue, developing material statement by statement, pursuing tangential ideas, adapting to mistakes, being ready for unexpected results, and never knowing for sure if the original goals will be achieved. I believe that this analogy with musical improvisation is stronger than any picture of the mathematician as the solitary composer (although the most vital composers do capture the spirit of improvisation in their works), as there is a sense in which the nonperforming composer can rework the landscape to “force his or her theorems to be true” (but not necessarily “interesting”), whereas the improvisor must face the unforgiving judgment of the moment while traveling without a seatbelt. The analogy also extends to the researcher working alone as a solo improvisor, simultaneously playing lead and accompaniment roles as the devil’s advocate, and even to the processes of understanding mathematics and interpreting composed music. (Further elaboration on his analogy, including potential applications to education, can be found in the above-mentioned essay Can One Hear the Sound of a Theorem?.)
For example, at a recent conference, I heard a leading mathematician give a presentation outlining a novel approach to solving a fundamental and notoriously difficult outstanding problem. Since part of the approach involved ideas that I have some expertise in, we got together with a collaborator the next day to examine the relevant details (conference talks typically suppress devilish details). Over the course of a good four or five hours of listening, questioning, answering, realizing, drawing diagrams, and calculating, the full brilliance of the approach gradually emerged along with the remaining challenges that remained to solve the problem. It reminded me of an after-hours jam session at a jazz festival!
How a Musician got tangled up with Mathematics (short version):
The first time I traveled for mathematics I was still a graduate student at UC Berkeley and Paul Kirk (one of my early mathematical mentors, and a jazz guitarist himself ) had invited me to Indiana University Bloomington to speak in the topology seminar. I was surprised when the first question from the audience at the end of my talk was “Why would you try to get a PhD in mathematics when you’ve played with Billy Higgins?”. (I think the question came from a frustrated grad student who happened to be aware of my recording Smooth Sailing (RSR CD 114) featuring Billy and my musical uncle bassist-composer Rufus Reid.) I don’t remember my response, probably because I was too focused on getting the details correct in my first road talk, but the inclusion of mathematics into my life has been a natural complement to the musical path I have followed since discovering jazz as a teenager.
This personal entanglement of music and mathematics traces back to metaphysical conversations with master alto saxophonist Charles McPherson during my early years of jazz playing in San Diego. After settling in New York, further after-hours philosophical musings with musicians led me to the realization that if I was ever going to understand such mysteries as “why time would slow down if you could move at close to the speed of light”, I would need to first learn this thing called “calculus”.
Since it had been over ten years since I had left high school (early, to play music), my math skills needed quite a few late night self-study sessions on algebra, geometry and trigonometry before I was ready to buy two used calculus texts from a street vendor on Broadway near 110th street and start my new hobby. This was in 1988, and I managed to decipher about a semester’s worth of introductory calculus without the ready-made tutorials that are available via today’s internet.
At this point I had a huge stroke of good luck. I was living on 136th street near City College CUNY (CCNY), and through a chain of coincidences I ended up enrolling at “City” to play on the soccer team under new coach, Wilson Egidio. My fascination with soccer goes back to seeing Pelé’s Santos team on television when I was 9 years old. (When Wilson was with the Santos junior team, he actually trained with Pelé, as Pelé was preparing to come out of retirement to play with the New York Cosmos.) So the prospect of playing samba-spirited soccer led me into a college calculus class, and upon being exposed to a real mathematician I immediately saw that there was way more to mathematics then I had been comprehending from my unguided reading of texts. (This experience reminded me of when I was still a high school student in San Diego hearing “live jazz from the outside world” for the first time when Freddie Hubbard give a classroom workshop, playing tunes from his recently released Red Clay album with Junior Cook, George Cables, and Lenny White.) It was also my good fortune that the professor of that first calculus class was Hironori “Tako” Onishi, a serious research mathematician and music appreciator who would become my first mathematical mentor. The wonderful mathematics department at CCNY was very supportive and tolerant of my frequent breaks for gigs on the road, as I worked through the undergraduate mathematics canon.
My experience as a musician definitely helped me learn mathematics, as I immediately realized the importance of “practicing” (studying), as well as immersing myself in the existing new and old literature, paying close attention to the masters, and actively doing mathematics with others. My self-motivation was strong, and the environment was encouraging, so I never looked back. I also never made anything like a “decision” between music and mathematics; I just kept doing both. It certainly helped that I was good enough at mathematics to get departmental teaching jobs at CCNY while I was still a student. Eventually my original interest in physics faded as I became increasingly aware of the beauty of mathematics on its own. I still remember the moment that my second mathematical mentor, Hamish Short (now at the University of Marseille), brought geometric topology into my life by showing me how to cut out a solid torus (“a doughnut”) from 3-dimensional space and glue it back in with a twist! After 6 years I graduated with a Mathematics degree, and got accepted into the UC Berkeley Mathematics PhD program.
At first I was hesitant to leave New York, the jazz capital of the universe, but I remember thinking that “if you’re gonna leave New York for somewhere else in the USA, then Berkeley is a great place to go.” During seven years at UCB under the supervision of award-winning low-dimensional topologist Robion Kirby, I transitioned from student of mathematics to budding mathematician, and at the same time enjoyed being part of the vibrant San Francisco Bay Area jazz community. Kirby’s lectures were particularly fascinating to me, as he had the habit of, well, frequently not preparing any lesson plan ahead of time, so it quickly became clear that improvisation was central to his presentation style. At times he would temporarily get stuck trying to explain some especially tricky material, and I found it very helpful to follow his thought processes in such situations, as he might momentarily back-track to try a different angle of approach, or zoom out to look at the bigger picture in search of a guiding principle. Although I have since incorporated an improvisational flexibility into my own undergraduate teaching style, I admit that I have not been able to successfully reproduce Kirby’s spontaneity in presenting advanced mathematics.
My first postdoctoral job was an NSF-funded year of research with Peter Teichner at the Max Planck Institute for Mathematics in Bonn, Germany. Peter has been my main collaborator over the years, and the MPIM is an amazing institution which is essentially designed to facilitate “jam sessions” among mathematicians from all over the globe. I have had the privilege of returning to MPIM for many research visits since, and always note how, like music, mathematics is truly an international language.
Among the many different forms of meetings and conferences hosted by MPIM, the annual Mathematische Arbeitstagung (literally “Mathematical Workshop”) has an especially improvisational format which dates back to its founding in 1957 by the legendary mathematician, Friedrich Hirzebruch: While a typical mathematics conference will have each day’s presenters and their topics scheduled ahead of time (usually several talks per day), on the first day of the Arbeitstagung all the invited mathematicians gather to decide, in open discussion, which among them will speak on which topics for the rest of the week. Imagine a jazz festival where on the first day the musicians and the audience (not the sponsors and power-brokers) come to an agreement on who will play what with who for the week’s performances!
After postdoctoral stints at NYU’s Courant Institute and UPenn, I have been a mathematics professor at Lehman College CUNY in the Bronx since 2006. The rewarding parts of my work involve helping students learn mathematics as well as the shared exploration of the mathematical realm with an international research community.
The Tone Twister musicians:
The Tone Twister tunes are definitely “drum music”, and drummer Pete Van Nostrand’s creative pulse runs right through the center of this recording. Well-established in the New York jazz scene, I’ve had the pleasure of working with Pete in Brian’s “Unsung Heroes” project, including two volumes of studio recordings and trips to Brasil and Indonesia, as well as on gigs around the city. I knew Pete would enjoy playing the Tone Twister tunes, and his fingerprints and footprints are all over the arrangements. There were no written percussion parts, so the beautiful rhythms and textures you hear are all born from Pete’s imagination over the two days of developing the music from the tune sketches.
Also radiating from the core of this music are bassist Gerald Cannon’s fat sound, seriously buoyant groove and infectious optimism. Gerald needs no introduction, as his resume includes stints with so many major figures in Jazz, including his current position as musical director for piano icon McCoy Tyner. Besides Gerald’s talents as a composer and band-leader (look for his latest release Combinations on Woodneck records), he is also an accomplished visual artist, having recently debuted his paintings in a New York City exhibition.
The first word that comes to mind when thinking about Ralph Moore’s music is “magical”. His unique tone and phrasing somehow manage to evoke both simplicity and mystery, both soulful comfort and inspirational surprise — truly something very special. (Ralph and I first met in a practice room in Boston when we were both still teenagers, so among the Tone Twister musicians, I’ve know him the longest.) Ralph’s extended musical partnership with Brian, including years with Horace Silver and many other projects, has evolved into a special chemistry that makes possible Tone Twister’s emphasis on spontaneous arrangements and simultaneous blowing. Ralph and Brian also feature together on two of my most satisfying previous recordings: Radio Waves (RSR CD 120) with Gary Smulyan, Todd Coolman, and Jeff Hirshfield, and Dark Blue (RSR CD 132) with Peter Washington and Lewis Nash.
From birth to maturity, Tone Twister’s existence is the fruit of Brian Lynch’s multidimensional talents. Brian’s one-of-a-kind blend of harmonic sophistication and funky intuition has earned him a global reputation as a musician, band leader, composer and arranger. And his visionary production skills have been recognized by a Grammy award for his recording Simpático, in collaboration with Eddie Palmieri. Many parts of the Tone Twister arrangements sprang directly from Brian’s horn; while the recording, mixing, editing and post production were guided by his meticulous ear and patience in the studio control room.
Brian and I first met over 35 years ago in San Diego, at a Monday night jam session I was co-hosting with saxophonist Hollis Gentry III (who tragically left the planet way too soon). San Diego had a fertile live music scene at the time, and Brian extended his visit from his hometown Milwaukee to play some local gigs, including with a fusion band I led, as well as regular playing with Charles McPherson, who had moved to San Diego around the same time. From after hours boom-box listening sessions at all-night taco stands, and a common goal of moving to New York, a bond was forged between Brian and me. It wasn’t long before Brian made the NYC move, and I was right behind him. Over the years I have always appreciated the chance to play with Brian, including touring, recording, concert and club performances, as well as private duets in his Chinatown loft.
The Tone Twister music:
The following free associations on the Tone Twister tunes are all decidedly after-the-fact, and can be safely ignored for musical enjoyment purposes. Please listen first, and read the musings after!
The opening tune Footloose Freestyle stylistically reflects the influence of two Eddies: One of my main musical mentors, the late great Eddie Harris, legendary saxophonist, composer, multi-instrumen- talist, vocalist, and comedian. And my musical hero Eddie Palmieri, pianist, composer, arranger and the still blazing sun of the Latin music solar system. The song title refers to the dance-like spirit of my favorite sport, soccer, or in most parts of the world “football”. Soccer is the sport that, at its best, can come closest to the dynamic of jazz in its improvisational nature, abstract use of space, and subtly pulsating rhythm. Soccer’s substructures are sometimes similar to basketball, but emerge more unpredictably, since they sit inside a much larger space, and since possession is less subject to planning and control. You can confirm this analogy with soccer-jazz experts like Ugonna Okegwo or Gerald Clayton. Even the technique of soccer is itself an art, and “Freestyle Soccer” is the form of self-expression that involves keeping a soccer ball dancing around the body using every part of the body except arms and hands, from the bottoms of the feet to top of the head. (I still get to regularly brush up on my own skills thanks to soccer club FC Harlem New York having kindly granted me guest elder status at their weekly coaches-parents-kids pick up game.)
The iconic Nat Cole rendition of Unforgettable, with daughter Natalie’s update, had me believing that Irving Gordon’s beautiful song was more for listening than playing. But after playing it on a gig in Berlin a few years ago with bassist composer Goldsby, the tune stayed with me. I was fascinated by John’s slight tweaking of chords in first 4 bars, and the tune haunted me to the point where I had to start playing it. Eventually the tag of repeating major chords (but with minor sound relative to rest of song) emerged. I loved seeing how in the studio this tag developed into the group composition Distant Memory that you hear as the second half of the medley, and spun off as its own track on the album. The complete, integral performance of the “medley” appears as a extra track on this digital release.
Left Coast Lullaby is an affectionate tribute to my California roots, which include attending primary school near San Francisco and high school in the San Diego area, where I started playing music professionally, as well as my graduate school years in Berkeley. Although I was born in Boston and have lived longest in my adopted hometown of New York City, I always enjoy returning for a visit to the “left” side of the country and soaking in some of the Pacific free spirits. May the blue-lightenment emanating from our coastal states lead us out through the toxic clouds of ignorance and into a more tolerant and truthful future! (continued on next page)
The harmonic structure of Windblown is essentially the same as the Rodgers and Hart classic song Lover. Originally a waltz, jazz musicians have traditionally played Lover as a 4/4 uptempo burner, and one such version closing the last set of the night stayed with me for several days, resonating and rotating until eventually morphing into the first half of the waltz you hear here. Then, during the playing of a duet version with Brian the final sequence of major chords turned into a tag that (after several more days of marination) took on a life of its own around several motif sketches. This, similarly to the procedure in Unforgettable/Distant Memory, became its own track on the album Tailspin. It is also preserved in combined form as the extra track Windblown Tailspin on this digital version of the release.
The attentive listener may recognize how the first ending of Windblown deviates from the usual Lover chord changes by replacing a typical turnaround starting with E minor 7 (essentially the tonic C Major with the third in the bass) by an unexpected E Major tonality. You could think of this harmonic surprise as a sneak preview to the upcoming bridge that starts in the same E Major as the original Lover, but it’s also been a sonic fascination of mine how superimposing the III Major 7 chord over the tonic I major sound (or over the III minor 7) creates a special kind of tension. You can hear this explored for instance in the track Fine and Dandy from my Glass Enclosure Reservoir Music release. There is a subtle melodic/harmonic “explanation” of this tension as an “extension of an extension”: I learned from the inspirational pianist, composer, arranger and multi-instrumentalist Jaki Byard that “the extension of any 7th chord is the 7th chord of the same type starting on the 9th degree”. So for instance, the extension of C Major 7 would be D Major 7, which at the top gives you an unexpected “flatted 9th” as an extension of C Major 7! This unexpected new color note is very cool, but why stop there? Since E Major 7 is the extension of D Major 7, it follows that E Major 7 is the “double extension” of C Major 7, which gives you not only the sharped 5th of C as the 3rd of the E Major extension but also at the top the sharped 9th of C as the 7th of E Major 7!! So somehow this sharped 9th, or minor 3rd, of C is coexisting with the C Major 7 sound! You can keep on going, extending extensions of extensions until returning to where you started, but of course putting too much weight on this kind of “theoretical” approach to music is like playing with fire. For me, such methodical approaches to organizing sound are only useful as tools for experimentation, and never provide “rules” or proven “explanations”. After all, the story told by music depends more on the plot than the characters: It’s tension and resolution unfolding and interacting in the moment rather than fixed forces sending sounds along preordained paths to meet an ultimate fate.
My “intensional confusion” of major and minor got me into trouble on my very first gig in New York when I was still a teenager in the mid 1970s. I had a VW van that I had driven from California, and being able to transport the band’s gear was probably why I was playing electric piano on this gig in a small bar in Brooklyn. Anyway, we were playing Ton Jobim’s sensual bossa nova Once I Loved (O Amor em Paz), and after the closing statement of the melody the saxophonist took a really nice lyrical cadenza which he ended on the 9th degree of what would usually be a tonic minor final resolution. For some reason I was inspired to come in with a luscious Major 9th chord that I thought sounded surprisingly beautiful. The saxophonist, who was also the band leader, did not appreci- ate the surprise, and hollered at me: “Don’t make me major when I’m minor!!#%!!”. (You can ask San Diego guitar guru Peter Sprague about this one, he was there.) By the end of that winter I was back in California, and it took a few more years to build up the momentum for my permanent move to New York in late 1981.
Slapdance-Tapstick, the only original on the session that wasn’t written recently, dates back to a stint of mine with a New York City tap dance troupe during the 1980s (including the honor of some performances with the late tap great Buster Brown of Copasetics fame). The tune’s harmonic structure is roughly I Got Rhythm in Db, with an Eternal Triangle bridge. Gerald finally gets a chance to cut loose and walk up a storm, and re-listening to Pete’s coda chorus still puts a smile on my face. Ralph and Brian’s interactive exchanges and combinational blowing here (and throughout) are exhilarating. Along with some evidence of Thelonius Monk’s happy dissonances, I hear in the piano parts the influences of Nat Cole, Duke Ellington and Bud Powell. A different arrangement of Slapdance-Tapstick appeared on my 1991 release Radio Waves (RSR CD 120), and I’m not sure why this tune recently re-inserted itself into my current rotation, but it sure is a fun one to play!
The title track, Tone Twister, is a tippin’ blues with some chordal nuance. We maintain contact with the just slightly twisted chord changes throughout the blowing, and Gerald’s beautifully thematic solo provides an inspirational launch for all to enjoy stomping rides. Stevie Wonder fans may notice the harmonic similarity between bars 9-12 of Tone Twister and the analogous turn-around in Stevie’s hit track Superstition from his 1972 classic album Talking Book.
The closing Lion’s Tale is a churning, indulgent minor blues in 3/4-time. The harmonic structure, especially the enchanting sequence of four (“minor-sounding”) ascending major seventh chords in bars 9-12, is lifted straight from Brian’s composition Unsung Blues on his Hollistic MusicWorks album Unsung Heroes Volume 1. A city-slicker like me can only imagine what tale a lion might spin, but something about the mood of this take triggers a memory of an urban outback moment from my past: Sometime in the early 1990s I was playing a few nights engagement at a club in Toronto called the Top o’ the Senator with Akira Tana and Rufus Reid’s group TanaReid. (It was during this gig that I composed my “other lion tune” The Lion’s Mane from the CD Dark Blue.) Anyway, I remem- ber stepping out of the club at the end of the night, expecting to hail a taxi back to the hotel but finding the street completely deserted. A light rain was falling and instead of going back inside to arrange a ride (this is before cell phones), I just impulsively started running through the streets in the direction of the hotel. Perhaps high from the evening’s music, I felt energized and maintained a nice loping pace despite wearing a suit and my nice gig shoes. The dark misty concrete streets and buildings felt somehow like my natural habitat. Although I didn’t know Toronto particularly well, I’ve always enjoyed exploring cities by foot, so I forgot about finding a taxi and galloped all the way back to the hotel. The voicing of the final chord of this take of Lion’s Tale leaves it to the listener to decide if the resolution is major or minor, a natural place to end the set.