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Tuesday, September 27, 2022

OFF the BEATen path

I was recently researching various musical settings of William Blake's poem The Lamb. One of my favorites is by the great American song composer Lee Hoiby, and I came across this recording of Hoiby himself accompanying the song.



It's a fine recording of a song which is successful in part because of its simplicity and directness, but there's something un-simple right at the start. Having accompanied and coached this song many times, I know well that Hoiby has laid a subtle trap for the singer (and pianist) in the one-bar piano introduction. As you can see below in bar 2, the left hand is syncopated one sixteenth note off from the right hand melody and vocal line, but Hoiby chose to begin the song with that left hand figure syncopated against...well, against nothing. Or, more specifically, against some unheard downbeat felt by the pianist and perhaps by the singer - but inaudible. So, if the notation is followed precisely, it will likely sound like the singer comes in early. We hear four eighth notes (suggesting something steady) but the voice enters halfway through the final one.


The curious thing about the recording above - with the composer at the piano - is that the singer basically comes in after four full eighth notes as if it had been notated this way:
It would be tempting to say that the performers are mis-reading the pianist's own music. In fact, that is literally true, but in a broader sense, this is a nice example of how notation can be both limiting and - maybe - freeing. Although one could plausibly interpret this situation as the composer simply changing his mind about how the song should start, one could also think of the syncopated notation as more suggestive than literal. I love the idea of suggestive notation. So what is Hoiby suggesting?

When I see something like this with a missing downbeat, it somehow communicates that the music should be a bit off the ground. It suggests a floating sensation which probably means my wrists will come up as the fingers go down. This is as opposed to a grounded sensation in which I'd let my arm weight sink into the keys more. What difference does that make? Well, pianists have been arguing for ages about how touch and weight affect sound, and I don't want to get into the physics of whether any variable other than velocity is in play. But I do think the imagination of this floating sensation can have an effect on how the music comes out...somehow.

It's possible Hoiby also wants a sense that the sound doesn't really have a strong beginning; we're just tuning into a figure which has been going on imperceptibly until it sneaks in pianissimo. Unfortunately, Hoiby's performance, at least as it comes across in recording, doesn't quite fulfill that ideal as the low E-flat is actually quite rich and present. This recording, also featuring the composer at the piano, achieves this gently tuning in effect a bit better, though in this case I hear Hoiby's rhythm as closer to this (it's an important reality that performances of clearly notated music won't and shouldn't always come out mathematically precise):
That version also feels different because it is transposed a full major 3rd lower, which makes a big difference in this low piano register. The intro bar here is also simply slower than what follows so it has an out of time feeling, which perhaps is what Hoiby hoped to suggest? Or, more realistically, what he came to want. A very important consideration here is that sometimes composers don't know exactly what they want - or what they want changes after going to press.

I suppose Hoiby might be astonished that I've spent even this much time thinking about it, but he's asking for....something... with this kind of notation. I've now spent too many years thinking about that unmoored syncopation to be satisfied with something that doesn't approximate the rhythm shown. For me, it should suggest a very gentle rocking motion over which a melody comes floating in from above. [In addition to rocking, which fits with the lullaby character, another image that comes to mind is of a shepherd lightly plucking harp strings nonchalantly against a melody; this image goes well with the pastoral character of Blake's poem.]

The fact that the melody and rocking motion are out of sync should not create tension but rather suggest an equilibrium between two complementary gestures. Below is a recording I made years ago of my arrangement for solo piano. While I can't say definitively that the first low E-flat doesn't sound like a downbeat, I do like the continuity of that figure into the more obviously offbeat left-hand as the song continues:


I also couldn't resist doing a little audioshopping of the first Hoiby recording posted above. Here, I've compressed the space (by about a sixteenth) right before the singer enters, and I prefer it to Hoiby's unaltered performance. So for all my speculation about Hoiby wanting something less literal than he wrote, I still want something less pedestrian than what he settled on. I wonder if he simply found that the ideal of an imagined downbeat works better for the performer than the listener.


Of course I'm always interested in the connections this kind of situation sparks in my memory. In this case, as I thought about the notation, I had a strong sense I'd experienced a similar rhythmic challenge in another well-known art song. I kept thinking it was probably something by Schumann because...well, because that was my intuition and I know that Schumann loved to write in suggestive ways. But I also wondered if I was remembering something by Fauré or Debussy. It took a good night's sleep and a good hour's more thinking about it before I finally remembered the opening to this little song from Schumann's Op. 39 Liederkreis:

You may hear it sung here. I would not say this is one of my favorite Schumann songs, but there's no missing that suspended gesture in the very short intro. I'm sure my mind made this Hoiby  Schumann association because coaching a singer to find the downbeat in one is closely analogous to the other. The Schumann intro is ultimately easier and more natural both because 1) the notes aren't jumping around and 2) coming in on an upbeat doesn't feel quite so off-kilter as the out-of-the-blue downbeat in Hoiby. (The Schumann has still bewildered more than a few students.) Hoiby's choice to begin with such a low note also makes it sound more like a downbeat to begin with than Schumann's mid-range, second inversion harmonies. But I can't help wonder if Hoiby had Schumann's accompaniment in mind (consciously or not) and if he hoped to achieve a similar effect. 

However, I think Schumann was in a class by himself when it comes to this sort of floating above ground rhythmic/metrical gesture. The opening song and final postlude of Dichterliebe achieve this beautifully, even if downbeats are easy enough to find. You can hear the first song here and the impossibly beautiful postlude (see below) beginning here at 29:34. Charles Rosen wrote memorably about Schumann's love for the fragment. Beginning a song with no clear center of gravity is a wonderful expression of that Romantic impulse to break away from rigid frames.


This final example from Schumann doesn't cause any ensemble problems for the performers, but the way the incandescent third movement of his Piano Quartet begins with an accented diminished chord on beat 2 achieves the same suspended effect. Steady downbeats soon follow, but after the gorgeous main tune has been explored, the piano right hand begins a wandering melody at 1:40 which is gently syncopated against the left hand. Soon it's not clear what's a beat and what's an offbeat and we are simply floating. 


As a sort of postscript, I'll mention one other song which came to mind when I was doing a mental search for what turned out to be Schumann. The opening of Hugo Wolf's Ich hab' in Penna has confused many singers-in-training for similar reasons as Schumann and Hoiby, though it is quite different in character. Here, the frantic piano part begins with constant eighth notes starting on the "and of 1" and the singer should begin on the "and of 1" in the next bar. Though this isn't really a syncopation because there's no rhythmic stress on an offbeat, the absence of an initial downbeat helps propel the song's headlong energy. However, some singers can't help but hear the first note as a downbeat (even when the pianist tries to put a subtle accent on beat 3) so this misleads them into starting one eighth note too late. It's not so hard for the prepared pianist to right the ship but that makes for an uncomfortable mental gear shift! Hear here. (No pianist needs the stress of an unsettled start knowing that terrifying postlude is just around the corner.)



Tuesday, September 13, 2022

Smiles of a Sondheim Night

The subject of today's post is definitely on the trivial side of things, and I debated whether it merited a blog post. Then I remember that this sort of triviality is one of the things I care about the most, and the blog exists to document just this sort of thing. A blog devoted to the intersection of "music and the mind" should certainly have time for situations when the mind uncovers some unexpected musical connection. Quite the opposite of some scholar digging out connections from painstaking research, the magic is that I did not seek them. They found me using all those strange musical circuits our brains assemble to...well, to do whatever it is we're doing when we do music. This isn't just words about music - it's neuroscience!

First some quick background, because the swirling worlds and words of social media often play a big part in how these pathways take shape. On Friday, Facebook reminded me of a favorite memory from eight years ago. Here's what I wrote about my then teenage daughter:

[My daughter] just blew my mind. I'd (playfully, of course) told her if she sang "I'm so fancy" one more time, I wouldn't drop her off at home before I went to pick up her siblings. She defiantly starts in wordlessly intoning the first four notes and then, on the fly, switches to "For All the Saints" which, improbably, starts with the same four notes. I was in equal measures appalled and so proud. (Sol, mi re, do...)

If you don't know the past-its-prime Iggy Azalea song or the Ralph Vaughan Williams hymn tune, you may hear both in this little audio illustration I created:



With some help from a Facebook friend, I realized that this admittedly simple melodic formula - a descending 5-3-2-1 - has been used to begin many a famous melody. For example:



Although my friend pointed out the Tchaikovsky connection as one she noticed, it's worth noting that I only found the other tunes by searching on Themefinder.org. 

This in turn reminded me of Leonard Bernstein's wonderful "Infinite Variety of Music" lecture in which he explores melodies beginning with an ascending 5-1-2-3, which he refers to as the "How dry I am" motif.  I once posted my own annotated version of that lecture audio to YouTube. The "How dry I am" part begins around nine minutes in:



As Bernstein says, some of these melodies (whether his 5-1-2-3 family or my 5-3-2-1 family) sound remarkably different due to rhythmic/metric/harmonic factors. I've gone decades never noticing the connections between the 5-3-2-1 tunes in that first set I just posted. Thus, I still believe there's something special going on when someone like my daughter intuits such a link.

So it is that we come to my favorite recent connection, even though it is quite ephemeral and only deals with three little notes. Of course, with some effort, one could easily find a Bernsteinian infinite number of ways in which a three-note motif like this next one occurs in various contexts, but this thematic connection is a bit more magical because of the linked contexts.

For various reasons, I found myself watching and listening to a few different versions of Sondheim's A Little Night Music late last week. This led me to watch Smiles of a Summer Night, the 1955 Bergman film on which Sondheim's 1973 musical is based. The Bergman film is not a musical, though it has an attractive soundtrack by Erik Nordgren. But I'll admit I wasn't paying close attention to the score.

Lo and behold, at a particularly pivotal moment when the young Henrik storms out of a momentous dinner party, my ears were drawn to a little motif in the orchestra. I just made a mental note in the moment, but when I went back to check later, it confirmed what I'd heard. The first three notes of Sondheim's iconic "Night Waltz" theme are played quietly but emphatically at the end of a restless minor key phrase. In a minor key, the notes would be 3-2-1, though in Sondheim's major-key waltz melody, the scale degrees would be 5-#4-3.

Never mind the analysis, I think the connection can pretty easily be heard, and it's no big surprise that it found me because Sondheim uses that motive over and over. I'd surely heard it dozens of times in the days leading up to hearing Nordgren's otherwise quite different music. I'm not suggesting that Sondheim borrowed this idea from Nordgren, though it's not inconceivable that it had somehow stuck in his head. But I enjoyed thinking about this odd little portal from 1955 to 1973.

In this first video, you may see the short original scene. Just as Henrik turns to face the table (with back to us), Nordgren's orchestra plays the motif. We hear it even more intensely as Henrik runs out of the room, this time in a 3-2-1 version. (A flat 2 is a particularly intense way to approach the first scale degree.) 



In the following, I've tried to incorporate the Sondheim theme. Mostly I did this because I enjoy this sort of challenge (which involves adjusting keys, tempi, etc.) It doesn't fit perfectly into the original context because Sondheim's theme is too elegant and charming for this particular high-stakes moment, but I still enjoy the collage. Having written in my previous post about appreciating when directors aren't afraid of soundtrack silence, I'll admit I'm undercutting Bergman's original conception. In his film, the score fades to silence as Anne calls after Henrik and seems as if she might faint. The silence underscores the awkwardness for those left behind.

My new Sondheimed version seems a little out of place at first (though one could hear the waltz's gentility as underscoring Henrik's "forgive me"), but as Anne falters, the lilting waltz theme takes on a new meaning when her husband and maid come to her side. I think that part actually works, first to accompany her dizziness with the dizzying sweep of the waltz theme and then arriving at a climax as her husband and maid arrive to check on her. The theme then flutters away as she exits, leaving her husband to question many things.



I have often thought of such connections as portals - ways in which our own powers of perception can lead from thinking about one musical work to another. My favorites of the portals I've investigated in the past include:

  • oddly similar "held bassoon note" moments in two otherwise completely different works by Mendelssohn and Copland. 
  • the ways in which works by Beethoven and R. Strauss combine into Bernstein's "Somewhere."
    • (that portal from Beethoven to Strauss is effected by knowledge of Bernstein's tune!)
  • the seemingly endless connections between Mozart's three most famous violin concerti
I don't think I'll ever lose interest in this sort of thing, but I can't tell you when I'll post the next one because I won't be looking for it!

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P.S. As you might imagine, I've watched that one-minute Bergman clip many times in the past few days. The intentional framing of the scene is pretty clear, first with the vivid table arrangement of all the guests seated opposite Madame Armfeldt and with how Henrik is framed facing the table (the priest-to be standing as if facing an altar), back to us - but it wasn't until after many viewings that I noticed Bergman flips the room perspective immediately after Henrik leaves. It's odd because it almost seems as if Anne is going the wrong way when she gets up to follow Henrik - he's just left the scene by running away from us, and now she runs towards us. I'm not sure what this means, but it's interesting!

Monday, September 5, 2022

Composing on Camera

[bit of meandering post here...I don't get to my main topic until the tenth paragraph]

I've been thinking about music in movies recently on a variety of fronts. I recently re-watched the 2012 film A Late Quartet, which is a pretty rare example of a mainstream movie set almost entirely within the world of classical music.* Although Philip Seymour Hoffman is amazing as the insecure second violinist, Christopher Walken is fun to watch as a soft-spoken aging cellist, and the story explores some interesting themes (not just musical ones), my main issue here is with the music - not with the late Beethoven quartet which is a main "character," but with the soundtrack by Angelo Badalamenti.

This is not intended as a slight against Badalamenti's skills, but his music has an overtly emotional "here's what you should be feeling" tone which, while suitably autumnal (since the film focuses on a character's journey with age), simply feels too intrusive and sentimental. If they'd asked me (they didn't!), I'd have suggested that the Beethoven quartet featured (Op. 131) would stand out better if the dramatic scenes were allowed to play with little or no music - except for Beethoven. As it is, the drama tends towards a "soap opera" level of melodrama, and Badalamenti's score leans into that very strongly. (You may hear it for yourself here. Basically a late Romantic, Elgar/Delius sort of sound dominated by warm strings and warm woodwinds, especially clarinet. Weeping melodies, poignant harmonies...nothing offensive, but somehow so pointedly emotional as to sound generic. Conventional in a way that is the opposite of late Beethoven.)

Again, this is not to judge the composer too harshly. In this interview, he so much as says his goal was to underline the emotion: "Yaron [the director] and I agreed that the score should emote passion and pain. The characters are beset with a series of hardships which are all very personal. We needed to feel this." But did we need his music in order to feel this? What's been curious for me to realize over the years is that, as much as I love music, I don't love having music tell me what to feel in scenes which are supposed to feel genuine. (Opera and musical theater are something else, of course.) A good script and good acting should be enough, and I wish directors were less afraid of silence. (My own feelings may be colored in part by the fact that my ears are too easily drawn to music, so even background music can be distracting.)

A re-watch of 1967's The Graduate a few weeks ago confirmed this for me. As iconic as the Simon and Garfunkel songs are, they are used only intermittently and quite purposefully, not as window dressing. Yes, it's ironic that its most famous song is The Sound of Silence, but the sound of silence is genuinely important in many scenes, especially some of the most uncomfortable ones. Many of the best, most intense scenes play out with no music.

In the case of A Late Quartet, I believe that letting Beethoven's music be the only music would have better underlined the otherworldly power of late Beethoven. There's something odd for me about being pulled back and forth between two musical worlds in one film, and though Badalamenti is not trying to be Beethoven, I can't help but feel that his music suffers by comparison. (In fairness, it may have been difficult enough to get a movie made about a string quartet, so perhaps they thought it helpful to portray these characters as regular people by using regular film music.)

As it happens, my favorite movie I saw this summer is a tiny little French film, Petite Maman, which boldly uses almost no music, allowing quiet scenes to play out with silence as a powerful frame for the subtleties of the story. About an hour into the 73-minute runtime, two young girls briefly discuss music while taking turns listening through headphones - we see them listen, but we hear nothing. Suddenly, an entire scene plays out scored by a bright, Europop song which....well, it's not really my cup of tea, but it still packs an incredible dramatic punch because of what has preceded. In this case, the music is very emotional and absolutely helps us understand an emotional breakthrough, but its power comes from the fact that there has been so much quiet before this moment.


Of course, there are movies in which wall-to-wall music makes sense. I rewatched the original Star Wars trilogy this summer, and John Williams' space opera score works wonders there. Just try watching these deleted scenes which have no underscoring (or sound effects or significant editing, to be fair) and one can see how much this film would not work if it relied on its screenplay and acting.


A Late Quartet also offers an opportunity to think about the contrast between diegetic music (music actually experienced as music by characters on screen) and any other type of music which is added for the viewer's benefit (?). I'll take this opportunity and any opportunity I can to observe that Christopher Guest's A Mighty Wind is the best example I know of a movie which relies almost entirely on diegetic music - the performances of the three folk groups created for this story, with the original songs sung by the actors, are the key to its magic. The songs are fun and sometimes poignant, and we feel how closely connected they are to the characters.


Of course, the structural simplicity and brevity of folk songs make it much easier to incorporate them in full than it would be with a 40-minute string quartet. It would also be far too much to ask the actors from A Late Quartet to play Beethoven for real, though they do an admirable enough job faking it, and the Brentano String Quartet's actual playing is fantastic. Nonetheless, I wish the director/producers had been brave enough to let the music speak for itself a bit more. Imagine using this frenetic Presto instead of the generic Jogging music Badalamenti wrote. Imagine letting us feel the cold New York scenery in which these lost characters live with this instead of this. As it is, the only really complete Beethoven we get is when the closing scene performance of Op. 131's finale continues all the way through the credits. 

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Now to shift gears to the main thing I wanted to write about: characters writing music on screen. There's a very special subset of diegetic music which occurs when we get to see/hear a composer write the music as part of the story. It requires a good bit of hubris to think that film can really take us inside this mysterious process which, after all, probably happens most often inside a composer's head, but it's surely the kind of thing that interests people. We all would like to look inside that creative process. 

I'm going to cite three favorite examples, though I'd be curious to know if you know of others. First up, I'll steal a scene cited in a memorable sequence from That's Entertainment, Part 2. As absurd and over-the-top as it is, this imaginary carriage ride in which Johan Strauss II writes Tales from the Vienna Woods is kind of brilliant, and it's ridiculously fun to watch. I particularly appreciate how slowly it develops and that we get to see brief moments of the composer feeling stuck and unsatisfied, though these blocks are quickly removed. 



The second scene I have in mind is probably the most famous and acclaimed of its type. Although I have mixed feelings about the overall success of Amadeus as a film (mostly due to some of the acting, and a few stage-y scenes which don't play as well for me on screen), it's hard to resist the drama of a dying Mozart dictating the Confutatis maledictus to his awestruck rival, Salieri. Though Peter Shaffer's original play was enormously successful, this best scene from the film was newly written by Shaffer for the film. The playwright later admitted to regretting a few overly simplistic details (like Salieri being confused by simple tonic/dominant harmonies), but this is a great example of how diegetic music can work both as subject matter and as emotional/dramatic framing. (Also a reminder of how effective it can be to stick with the music of a composer subject for all of the soundtrack as Mozart's music is absolutely the MVP of Amadeus.) 

Now, having gone from the ridiculous to the sublime, it might seem like I'm lazily settling back into the ridiculous, but this final scene is actually the most successful composition scene I can think of, and it comes not from stage or the big screen but from a 1970's sitcom. Somehow, in about three minutes, the brilliant Tony Randall convincingly writes an entire song. A very stupid, silly song, yes, and with more emphasis on the development of the lyrics, but a song nonetheless, with references to leitmotif, melodic direction, and word painting along the way. I, of course, have no business posting this video but I couldn't find it elsewhere and so far YouTube has only notified me that copyrighted content is recognized - it's not taking it down. The songwriting begins around 1:58, although I've included part of the opening scene for context - especially as it lets us hear a sample of Felix's previous songwriting prowess.




Now that's a strange line-up: Cartoonish Johann Strauss, Melodramatic Mozart, and giddy Tony Randall, but I think it shows an interesting variety of ways in which the composing process can be shown to viewers. Let me know what I've missed!

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* P.S. There are a few high-profile classical music movies on the horizon. Tár, starring Cate Blanchett as a fictional composer/conductor is coming out soon and Maestro, Bradley Cooper's biopic about Leonard Bernstein, is set to arrive in 2023.

P.P.S. Another famous movie/theater song supposedly written on the spot is Do-Re-Mi, though it arrives too fully formed to seem like something in process. (Although maybe with more time Maria could have done better than "LA, a note to follow SO," which seems lazy and also messes up the solfège since in this case SO is sung on TI.)

P.P.P.S. One could make the case that this documentary scene shows a new song, Get back, coming to life before our eyes. Of course, this vernacular style lends itself more naturally to "writing out loud" than do many other styles of music. 

P.P.P.P.S. I once wrote here about my then Top 13 Movie list with brief discussion on the role of music in each. 


Saturday, August 6, 2022

An Aug 6 for Aug 6 in an August 6tet

Once again, Aug 6 has snuck up on me and I didn't get you anything. Well, at least not anything new or original, but I'll see if I can reignite the blog with a couple of observations. First of all, it happened to dawn on me that today is August 6 not long after dawn while on a long morning walk. And quite by coincidence, I had already listened to perhaps my favorite-ever Augmented 6th (Aug 6) chord while walking as I'd been revisiting a recording from last summer of the first movement of Brahms' Sextet No. 1 in B-flat. I even re-listened to the marvelous Aug 6 moment a couple of times while unaware that today is Aug 6.

Actually, let's back up and remember that I first "discovered" Augmented Sixth Day exactly ten years ago today when I saw that a student's harmonic analysis entered into an Excel spreadsheet had the abbreviation "Aug 6" automatically converted by Excel's internal logic to "8/6/12." The fact that this discovery happened on August 6 still blows my mind. This also reminds me of a clever bit of memery I saw just this past week online:


Yes, Excel will interpret anything it can get its hands on as a date, and this can actually be quite annoying at times, but it's worth it to know about Augmented Sixth Day.

Since I'm short on time here, I'll just quote from this post from last October about Aug 6th Chords in Mendelssohn, Brahms, and Haydn. Have a great Augmented Sixth Chord Day!

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from the post Augmented Reality, 10/16/21, beginning after a discussion of an Augmented Sixth chord in an early Mendelssohn string symphony:


...So it was that a few weeks after starting my re-exploration of Mendelssohn's augmented-sixth-based modulation in this Andante, I heard my 14-year-old cellist son play the extraordinary first movement of Brahms' first sextet at chamber music camp. (To be honest, this music was a bit over the heads of these campers musically and instrumentally, but they did a great job with the challenge, and managed to present it with confidence and a convincing overall shape.) It's music I've heard often enough, but didn't know super well, yet as the Recapitulation approached, I remembered that something special was going to happen. (Side Note: this kind of musical memory-based recognition/anticipation is one of the most satisfying things about listening to familiar music; this is surely one reason people who love classical music enjoy returning to old favorites.)

As the dramatic texture unfolded and suddenly the opening theme returned in a harmonically arresting way, it occurred to me that this sounded a lot like what happens in Mendelssohn's Andante. The two musical passages (Mendelssohn and Brahms) are different in so many ways, but that makes the analogical connection all the more meaningful. (I've been writing about my love for and fascination with analogical thinking since the blog began.) Hopefully I'll get around to writing about the Brahms sextet more in a future post as there is SO much to say, but for now just listen to the way the music moves from G-flat Major to the home key of B-flat Major. These are not closely related keys (they only share three common pitches), but Brahms uses a German Sixth chord as a sneaky way to get from one key to the next. The video below should begin at 6:35, where the cresc. is marked in the score.



The most important thing to listen for is that, in m.233, the first violin (highest instrument) goes from E-natural to F (up a half-step) at the same moment the 2nd cello (lowest instrument) goes from G-flat to F (down a half-step). G-flat and E-natural are an augmented sixth apart - not an interval that shows up if sticking to one key - and the resolution of those distant notes outward to an octave is the fundamental thing about how augmented sixth chords work. (There's something else unusual going on in this case which I'll get to in a later post.) 



Back here in 2022, I'll leave you with this glorious recording, and of course I recommend listening to the whole sextet, but the BIG MOMENT starts at around the 10:38 mark.


Sunday, March 20, 2022

Slicing Pi

It's highly likely I'm the only one who's been bothered by this, but my 3/14 post stretching a familiar 3/4 minuet into 3.14/4 time has stayed on my mind since the notation system I devised was a bit sneaky. I knew each 3/4 bar needed 0.14 extra beats, and I found a way within Lilypond to multiply note values in a way that achieved that (at least for playback purposes). But I cheated with the time signature. I just made my own time signature sign that showed a 3.14 over a 4 and then used Lilypond's "cadenza" function, which allows the user to place barlines wherever one wants. Essentially, this means the internal calculations act as if there is no time signature, so as long as the various parts line up rhythmically, the musical output will sound right, even though the barlines aren't generated by time signature calculations.

During idle moments this week, I tried to imagine a way to create a time signature which is precisely three beats plus 0.14 beats long. At some point, the old use of 22/7 as a fraction which approximates Pi came to mind, and that helped me to think about this a different way. After flirting a little with the idea of an irrational time signature like....well, 22/7, I realized that what I really need to do was add a single septuplet note to the three quarter notes in each 3/4 bar. 

It's a curious and sometimes forgotten (by me, at least) fact that our names for note values are really just ways of dividing a whole note. So a quarter note means nothing more specific than that it is 1/4 the duration value of a whole note. I finally realized that dividing four quarter notes into septuplets means that it would take 28 septuplet notes to equal one whole note, so a septuplet value could also be called a 28th note.* One doesn't see a lot of 28th notes out in the wild, though curiously the name sounds a lot like a 128th note, which is a thing. (There are 32 128th notes in a quarter note.) But that's basically just a coincidence. 

For my initial "Minuet in Pi" video, I used a special notehead with a pi symbol inside to indicate values which need to be stretched. It's now clearer to me this symbol could be defined as meaning the basic note value is stretched by 1/7. (This is similar to how a dotted note value extends the duration by 1/2.) This makes the notation pretty simple, but that's because I invented a sign to smooth things out.

Anyway, I finally realized that using Lilypond's wonderfully logical design, I could create a composite time signature of 3/4 + 1/28. Since each quarter note beat contains seven 28th notes, the signature could also be written as 22/28, but that doesn't show the additive process as clearly. Of course, one could notate more or less the same thing using 22/16 time, but that wouldn't allow the use of quarter notes to show simple beat values. Using 28 as denominator allows the use of quarter notes more or less as they'd be used in 3/4 time except on the third beat which requires stretching. (I don't think there's a truly elegant way to handle the ties required for that stretched beat.)

Is this all a little more than you want to think about? That would be fair, but I have found this a good challenge to help me think more clearly about how time signatures work. The two videos below are thus more precise than what I posted last week, though also a little less elegant. The first one is probably a little more proper, though the second one is maybe a little easier to read as it preserves more of a 3/4 look. [UPDATE (3/21): See also important footnote version in 22/7 below!]

Minuet in 3/4 + 1/28 Time, version 1



Minuet in 3/4 + 1/28 Time, version 2



* If you're like me and have flirted occasionally with irrational time signatures (which basically have denominators that aren't multiples of 2), you might be tempted to think a time signature with 7 on the bottom is the way to use septuplets as a primary division; but remember that a 7 would just mean a whole note is divided into 7, so that would provide a pulse which would be notated as quarter note septuplet. This would not fix my problem of needing a nice way to show standard quarter note beats which look like the way this minuet is usually notated. I needed a septuplet division of the quarter note itself, which is how we end up with 4 x 7 = 28. 

In other words, although a 22/7 time signature would look cool, it would only work if the primary beats in each bar were indicated as whole notes like below. And, whaddya know? Having added this as a "day after" footnote, I now think this is my favorite way of notating this meter!

Minuet in 22/7 Time