Ethan Hein discusses the mystery of why the bottom number in a time signature is what it is, and why there seem to be no good explanations for what that number actually indicates

https://www.ethanhein.com/wp/2024/the-bottom-number-in-time-signatures-has-always-confused-me/#more-29476

Explaining time signatures is hardly easy and why the bottom number is what it is is does seem stranger than the top number.  If you have a measure of 7/4 then you can say there are 7 beats and that they are quarter notes.  That's simple, but Ethan Hein points out that no one easily explains why that bottom number is what it is. I could say that the bottom number indicates, often, the foundational metrical unit that establishes pulse but the reason that becomes circular quickly is because there are so many ways to notate musical passages it's not a given that notation itself tells you what meter you're in.

Danuta Mirka, in her book Metric Manipulations In Haydn and Mozart : Chamber Music For Strings, 1787-1791, points out that in the 18th century composers would stick with the established meter no matter how many times or in what ways they actually changed meters in a written work.  In other words, you cannot assume that a composer won't shift out of 4/4 into 5/8 or 6/8 or 3/4 or even potentially 11/16 and 13/16 in a work just because they stay resolutely in 4/4 on the written page.  Thanks to accent marks and other articulations, as well as notational conventions, it was possible to create metrical changes in a work despite never changing the written meter.  Music in the 18th century was resolutely monometric in notation even though metrical changes and outright changes of meter happened in practice.  I'm still early into the book enough I can't share examples cited beyond the fact that she works with two of the biggest usual suspects in the history of music theory, Haydn and Mozart.

I had to go with a Theodore Front link because the Oxford University Press link kept having issues

https://global.oup.com/academic/product/metric-manipulations-in-haydn-and-mozart-9780195384925

Speaking from personal experience the other thing about notating meter is that not all composers feel it's worth it to notate something in, say, 11/4.  Shostakovich didn't always write out grooves in 5/4 or when he could switch back and forth between 2/4 and 3/4 as he does in the third movement of his Third String Quartet.  

https://www.youtube.com/watch?v=G27ERARI4z4&t=743s

Or you could have a passage where a cello barrels along in 2/4 while other strings in Bartok's String Quartet No. 4 shift into 3/4 before the music moves from meters like 2/4 and 3/4 (at the same time) to 5/8.  This part of Bartok's Third String Quartet has a killer pizzicato cello solo. 

https://www.youtube.com/watch?v=Knd04_iYTGc&t=290s

But you can see, if you read along with Bartok's score, he'll have whiplash meter shifts between 3/8 and 2/4.  My memory may be fuzzy here but I vaguely recall the composer Paul Creston, in his book Principles of Rhythm, complained that there were huge spans of music in Bartok's string quartets where he could have notated things in 4/4 and used judicious accent marks rather than endlessly changing meters in every stave for every instrument.  Good luck finding a copy of the Creston!  If I hadn't read it decades ago I couldn't even just allude to it now.  Creston's point stands, there are all kinds of ways you can notate a single passage and some composers go for the hard bordering on inscrutable way to write out a score and others stay four-on-the-floor in their scores even if they introduce meter changes.

That the bottom number in a meter conveyed tempo information would generally true right up to about the 20th century but from probably 1930 onward all bets are off.  For instance, take 3/2.  I know from my experience in choral singing, the Alice Parker arrangement of "Hark! I hear the harps eternal", which is scored in 3/2 but does not feel all that slow, at least for a choral work.  You can see the score in the link below.

https://www.youtube.com/watch?v=fglrDoLN7nE

The arrangement has a light, airy element to it yet it doesn't sound slow, and it sure doesn't feel slow if you try singing it. 

Parker packs in contrapuntally dense call and response passages just barely over a minute into an already short arrangement.  There's such a thing as going with the larger numeric value (the half note) because you don't want to scare your sight-reading singers to death with needlessly small note values.  And if you want the music performed with swing (whether or not that happens varies from ensemble to ensemble and conductor to conductor and Parker's arrangement withstands either approach because she's that great a choral arranger), then going with the larger note value lets you gauge how much you want (or even can) swing in the arrangement depending on the exigences of how good your singers are.

For 5/8, I know he's hardly beloved even among fans of classical music but one of the more memorable entries in 5/8 I've heard is the Fuga in G from Paul Hindemith's Ludus Tonalis.

https://www.youtube.com/watch?v=F4PATHcm5EM

But don't think he was the first or only composer to write a 5/8 fugue because Anton Reicha's Op. 36, No. 20 Fugue in A major predates Hindemith by more than a century.

https://imslp.org/wiki/36_Fugues_pour_Piano%2C_Op.36_(Reicha%2C_Anton)

https://www.youtube.com/watch?v=9ZYx_KHReQU

Reicha's fugues were so weird Beethoven reportedly said "the fugue is no longer a fugue".  Reicha also advocated the use of quarter-tones to embellish melodies in his Treatise on Melody in 1814

https://imslp.org/wiki/Trait%C3%A9_de_m%C3%A9lodie_(Reicha%2C_Anton)

For 11/8 I recommend a few pieces by contemporary Bulgarian guitarist composer Atanas Ourkouzounov, who has a Youtube channel that features his music, in some cases with read-along scores.  So if you want 11/8 here's a Chaconne.

Ourkouzounov - "Chaconne" d'après une improvisation de Petar Ralchev (youtube.com)

hear also ... 

Ourkouzounov - Chaconnitsa (2022) (youtube.com)

hear also Asymmetrical Improvisations where the first movement is in 11/8 and the second is in 7/8

https://www.youtube.com/watch?v=Rsoep3f-VM0

But the thing about 11 anything is you could also go 4+3+4 over 8 or 3+3+3+2 over 8 or you could notate it as 4/4 and 3/4 and 4/4 or three measures of 3/4 followed by a measure of 2/4.  Because asymmetrical meters break down into smaller units (i.e. Brubeck going 2+2+2+3 within 9/8) some composers just don't use the less conventional meters when they can (they hope) inspire players to play their music by staying with simpler meters for the sake of reading.  It's possible to cram 3/8, 6/8, 5/8, 13/16 and 11/16 into a work that's officially 4/4 for most of its run.  

It's that reality, and examples I have semi-randomly grabbed from the classical repertoire that I think helps explain why the bottom number in a meter can seem so ambiguous.  Maybe back in the day when people used the concept of tactus it was possible to talk about a stable undergirding rhythm without attempting to use notated music to confine performers to what we call measure but the horse left the barn a long time ago on that front.  

Ethan Hein on Led Zeppelin and the folkloric integrity of the blues and revisiting Anton Reicha's old 1810 treatise extolling quarter tone deviations from diatonic scales as part of a singer's art

http://www.ethanhein.com/wp/2022/led-zeppelin-and-the-folkloric-integrity-of-the-blues/

... 

But why do harmonica players bend their notes in the first place? Blue notes are a profound music-theoretic mystery. Why should it sound good to play certain notes so far out of tune? Performers in every kind of Western music inflect their pitches for emphasis and feeling, but outside of the blues and its descendants, no one intentionally plays notes that are a quarter-tone flat on purpose. It’s customary to explain blue notes by saying that they are harsh, weird, or dissonant. But this is not the way I and other blues lovers experience the music. Blue notes sound unusual and ear-grabbing, but they don’t sound “bad” the way that an extremely flat note would in a performance of, say, Mozart. How can this be?

...
Ever since I began to dig into writings by and about the history of microtonality in Western and semi-Western music it has seemed that even the idea that microtonal inflections in Western music might be more of a scholastic super-myth than something testified to within the literature of the time.  Kyle Gann mentioned that Anton Reicha (who taught Lizst) openly speculated about quarter-tones.  I had to do my own digging around to find that Reicha extolled the use of quarter-tones in aria performances in his Treatise on Melody going back to the earlier 19th century.  Since Pendragon Press is no more (!) I have to get a little help to show you what Reicha wrote.

Ted Gioia's history of music as Europe vs. Africa traffics in an artificial binary--exploring the Native Hawaiian ancestry of slide guitar and Anton Reicha's advocacy of microtonal melodic ornament in 1814

… No matter what the political structure one advocates, the wrong music can apparently send it toppling. Perhaps the strongest aspect of this evolution is its reversal of the ancients’ philosophical dichotomy. The guitar, a kind of modern lyre, is now the dangerous source of disorder, while the flute is seen as a prim, subdued instrument evoking respectability and orderliness.  This is one of the most striking examples of the dialectic at play in music history in which things turn into their opposites.

 This is not mere happenstance. The guitar shook things up in the twentieth century because it was the focal point for the African diaspora’s attack on the Pythagorean paradigm that had dominated Western music since 500 BC. As blues guitar techniques entered the mainstream, they validated the use of bent notes and introduced sounds outside conventional scales. This assault on Pythagorean notions of playing in tune—which for centuries had required musicians to keep within the boundaries of carefully delineated scales and maintain a proper tone—threatened to overturn all the hierarchies of established music, both social and aural. [emphasis added] …

 … This African counter-paradigm represented the most forceful attack in more than two millennia on the codification of music into a system of discrete notes. The knife could bend a guitar note in a way that a hand or finger could not. With a weapon in hand, the subversive performer could rediscover music’s origins in unconstrained sound. As such, the modern guitar represents the exact danger that Plato had perceived in the flute more than two millennia before: an untuning of the universe. Music: A Subversive History, page 106

 The rise of the blues may offer the most powerful test case we could devise for assessing a key thesis of my book, namely, the assertion that musical innovation comes from the underclass. This hypothesis, if it could be proven, would show how different music is from other art forms.  Innovation in painting and sculpture, for example, has almost always happened in close proximity to rich patrons—even today, a couple hundred affluent collectors set the tone for the visual arts economy. The rise of the novel drew momentum from the wealth-creating enterprises of the Industrial Revolution. The new artistic mediums of today—video games, virtual reality, and probably whatever is coming next—appear inextricably connected to profit-generating businesses in Silicon Valley. Only music plays by different rules, almost defiantly so. Music: A Subversive History, pages 335-336

Being a guitarist I’m flattered that Gioia thinks the guitar has been so revolutionary in music in the last century. I’m tempted to agree but …   is Ted Gioia imagining that all post-Pythagorean bids at tuning system represent repressive norms?  Did blue notes really defy the established order by not obeying “the rules”, whatever those are of “properly” ordered scales and notes?  Are we talking about the rules of sixteenth and seventh century species counterpoint grounded in the performing traditions of acapella vocal music? Those rules emerged for some practical as well as theoretical reasons. 

Or is Ted Gioia trying to say that blues riffs with notes a quarter-tone away from the notes E flat or A natural on a keyboard is subversive?  String players who perform Haydn string quartets already know that E flat and D sharp are not the same pitch, which was why when Haydn went to the trouble of saying to play a passage scored in E flat starting from the pitch of D sharp that was because he felt he needed to. 

some doubts about Richard Taruskin's presentation of Anton Reicha in his Oxford History of Western Music

https://www.nytimes.com/2018/09/07/arts/music/anton-reicha-ivan-ilic-beethoven.html

Taruskin presents the composer Anton Reicha's treatment of sonata form as a "conservative" one but compared to whose definition of sonata would Reicha's definition be considered conservative or traditional?  Wasn't this the Anton Reicha who wrote thirty-six fugues in his Op. 36 in which there's a fugue in A major in 5/8 that gets an answer to its subject in E flat?  Who else in the zone of 1803 was composing fugues in 5/8 with a subject being answered at the tritone rather than the fifth?  Reicha also, as happily recounted by Kyle Gann over at this blog post, was willing to endorse the theoretical utility of quarter tones.

Don't believe me, go over here to IMSLP and see for yourself!
https://imslp.org/wiki/36_Fugues_pour_piano_%28Reicha%2C_Anton%29

Thanks to Gann's blog post I learned of the Tiny Wirtz recording and I got hooked.  I eventually went on to pick up recordings of all of Reicha's woodwind quintets, which are all pretty glorious!

Not too coincidentally I was also immersing myself in the guitar sonatas of another Bohemian composer, Wenzel Matiegka.  I've written enough thousands of words about that I don't want to belabor that too much ... but I'm going to mention it again anyway because I proposed near the end of my long-form analysis of early 19th century guitar sonatas by guitarist composers that if you look at the themes composed or chosen by the Bohemian, Spanish and Italian composers who played guitar in the late 18th and early 19th century who were writing guitar sonatas it takes very little adjustment to transform a theme in a guitar sonata into a strain for a ragtime.  This is, admittedly, an academic point of consideration (if I were officially an academic!) but let's remind ourselves that in the ragtime literature we've got "Bohemia" by Joseph F. Lamb and "Solace (A Mexican Serenade)" (aka a Mexican serenade). George L. Cobb wrote a "Russian Rag" that adapted a prelude by Rachmaninoff.
And if you'd like to hear a version of it, go over here.

Another presentation. This one opens with a performance of the prelude in C sharp minor followed by the Cobb ragtime adaptation.  Both of these performances seem, to me, too fast to fit a ragtime style but I can't help admiring the chops required to play ragtime too fast even when I find it troublesome in terms of the really explicit warnings in published scores about how it is never right to play ragtime fast.

See, as a guitarist composer if I write a bunch of ragtimes and specify that the quarter note be at sixty beats per minute on a tempo I'd probably have guitarists asking me if I can slow things down.

So how does this connect to a new set of recordings of keyboard music by Anton Reicha?  Well, I've made this case in the past that while ragtime can be understood as a style pioneered by great African American pianists like Scott Joplin not everyone who composed ragtime was African American and, more to the point, the influences tend to get name-dropped in titles and subtitles.  So we get pretty explicit notices about Bohemia and Mexico and Russia as well as references to Africa and the African American experience.  Where this might be germane to musicology and new musicology discussions is that there's something Spanish language nation states (whether Spain or Mexico, etc) might have in common with Bohemia (aka what we'd now consider something like Czech music)--it's not-German.

In my own compositional activity I realized that it took so little work to transform themes from my favorite Italian, Spanish and Bohemian guitarist composers' sonatas into ragtime strains I figured, why couldn't I try to reverse this process and transform ragtime themes into the basis for sonata forms?

I'm not saying this can't be done with German music, but I am thinking in a blog post for the record about how curious it is that rags have been published that name-drop Mexico, Bohemia and Russia as places of inspiration for ragtime as a style.  Rag was evidently a more fusionist style than might be presented in a ssurvey if we only looked at Scott Joplin and James Scott.  Joplin, if memory serves, had some exposure to the "big B" composers, too, so it's not that there could be no German influence of some kidn of ragtime but that's a musing--ragtime is pretty likely not on the radar of German music spieclaists for having any shot at being "profound".

Because ragtime si obviously dance music and uses recursive forms it tends to be thought of as lightweight  It may be regarded as lightweight but that doesn't mean dance music can't evolve intos oemthing more "substantial". Practically the whole span of music hsitory we now call baroque would seem like a possible proof of that point.

Anyway ... back to Reicha ... it seems like Taruskin's Oxford History presented Reicha as being more conservative than I think anyone with more than a passing familiarity with his actual compositional (let alone theoretical) output might agree to.

And if it might seem Reicha's theorizing about what a sonata form may be seems conservative I saw a scholar (name escapes me at the moment) point out that many an earlier treatise on composition were designed to facilitate composing and were not designed to facilitate analysis.  So in that sense a Reicha treatise on composition wouldn't necessarily be a treatise you should take as giving you a roadmap to analyze what he or other composers had already done as much as providing a schematic outline of things you "could" do if you wanted to.

Its one of the things about treatises on counterpoint that I suspect can get lost in debates over whether X or Y has the "real" grasp of contrapuntal idiom X or Y.  The idea with these treatises was to encourage and catalyze actual composition.  Notoriously analysis seemed to come up short on explaining what a J. S. Bach or a Haydn was doing and there can be some complaints about how treatise A or B doesn't seem to fit with what composer C or D was actually doing.  Well ... at the risk of pointing out the obvious, maybe it wasn't supposed to and nobody who read these sorts of treatises labored under that misconception.

Bruce Haynes, in his book The End of Early Music, mentioned how in what we'd now call ancient music treatises there's an idea that theoretical writings about music are to music what a recipe is to the finished culinary creation--no one would mistake the recipe for the food for being the food.  It doesn't mean you don't care about the recipe and it certainly doesn't mean you don't care about the relationship the recipe has to the food, just that in your enjoyment of a good recipe you don't forget that the good recipe doesn't become food until you "perform" the recipe.

Finding the theoreticians who wrote about music in the past wanting because their work didn't give us the tools to analyze the music that was written by the composers who were the contemporaries of those theorists forgets that those books were probably not being written for the people who already knew how to compose according to the conventions of the day.  Almost every theory book I can remember that has tried to describe how "classical form" works has laid out a variety of rules only to throw in the seemingly inevitable and inevitably funny caveat of "except for Haydn".  It's funny every time I see it, that caveat.

I finished reading Richard Taruskin's Oxford History in the last few years and I recently read a lengthy critique/rebuttal of the master narrative approach of Taruskin's Oxford History written by Franklin Cox.  I have my own reservations about some of Cox's criticisms with the topic of new music in mind, because i find through my own experience that when people say that a composer like Ferneyhough is hard to understand but that you can understand his work an objection that Taruskin seems to have a narrative where he takes down elites can inspire people to do what I've previously called a shifting of elite definitions--the "elite" becomes the fiscal elite who control the finance sector as though cognitive elites who dominate academies somehow don't count as being elite or at least being elite in some more forgivable way.

If music can only be  appreciated by a cognitive elite then that's still, arguably, elite music.  The recently passed composer George Walker once said that classical music is music by elites for elites.  Maybe the nature of what and who that elite is changes from age to age but the elite part has stayed pretty constant.  Over the years I have gotten the impression that the musicians who can admit to themselves they're working at some kind of cognitively advantaged position can also look at writing music in a way that doesn't compel any and all would-be members of a possible audience to have to do as much work sa the composer or musicians do to get the music performed.  Another way of putting this is I like Haydn because as abstract and lofty as his music can get he doesn't set aside making music hat could be appreciated by people who won' get all of the esoteric tricks of craft.  He may have written a lot of music for court and gentry but he was the son of a hweelwright, wasn't he?

Reicha's music can get esoteric and weird but not in a way that, in my experience, loses sight that music should in some way be fun.

And I just don't see how when I read through the Op. 36 fugues or listen to them that they could be construed as conservative in the late 18th or early 19th century.

Taruskin did seem to traffic in a kind of master narrative about master narratives.  My take away from his history was that there's no non-imperialist patronage system.  The nature of the imperialism or colonialism changes across the coarse of history, for want ofa better way to put it, but there's no arts patronage scene that is innocent of the trappings of power and money--all our love for th bbeautiful art that emerges from each of those cultural dynasties of patronage does not have to blind us to those historical observations.

Where some critics of Taruskin take issue with what they regard as a kind of neoliberal triumphalist narrative seem to me to be missing a possibility--that Taruskin has not exactly said that this new era is innocent of power or ideology.  He'd rather live in this particular era and, as he's joked, vote for Adlai Stevenson endlessly

I also have some issues with the ease with which Taruskin blithely declared the guitar has not been part of the literatre music tradition.  Never mind that we've got scores going back for centuries from composers like Diabelli, Carulli, Giuliani, Molitor, Matiegka, Sor, and others.  I'm also surprised Taruskin had no word for the guitar since he's known as a specialist in Russian music and yet the Russian seven-string guitar literature goes back for centuries, too!  I've had fun listening to recordings Oleg Timofeyev has made of the Russian literature in the last few years and look forward to hearing more of the recording he does in that direction.  Just because Taruskin hasn't familiarized himself with Diabelli's Op. 29 guitar sonatas doesn't mean there's no F major guitar sonata (which, honestly, much as I've enjoyed Diabelli's guitar sonatas over the last ten years the F major is the one that I think holds up best, you could totally skip the others).

There's a sense in which, if critics of Taruskin's perceived (by them) vendetta toward German music were taken further they could point out that his depiction of Reicha seems skeletal and even inaccurate, and that Reicha may be a case study of a capable, even brilliant composer and theorist whose work got sidelined in the wake of 19th century battles (all I know at the moment, I'm afraid, is that there was reportedly some battle between Reicha and Cherubini that, so the sketchy reports go, had something to do with whether the sonata or the fugue was most likely "the" compositional path forward).

This could be a point at which those who are tired of the stolid German canon might want to reappraise the field.  Reicha's work has been around for centuries and if you want an alternative to the German hegemony new musicology seems worried about, Reicha seems due for rehabilitation or renewed recognition. Sure, "we" know that one of his students, Lizst, came to define the "new German" sound ... but I don't feel like digressing down all of that. 

Now I get that in his long-form criticism of Taruskin's Oxford History that Franklin Cox might believe Taruskin is unfairly dismissive of New Complexity and of non-tonal musical developments.  It is disappointing there was no room in the five-volume narrative for Ligeti or Takemitsu and a range of others.  Taruskin's approach could be considered the start of other approaches. 

But there are other things that can be said in criticism of Taruskin's sprawling history besides that he can't hide a disdain for the New Complexity.  He didn't manage to hide his disdain for Paul Hindemith's music, either, seeing as I read his essay asking where any of the good Hindemith music might actually be.  Now as a Hindemith fan I could get upset that Taruskin can't enjoy anything about Hindemith but there's no need for that. 

But Taruskin's description of Reicha, when I read it in the Oxford History, seemed too cursory and, well, inaccurate.  Even on the point of whether Reicha's definition of sonata could be considered conservative, that depends on what else he was writing and proposing at the time. 

Guitarists, I think, would benefit from soaking themselves in Reicha's work.  The Op. 36 fugues are begging to be arranged for guitar ensemble.  Actually ... several of them have.

https://www.youtube.com/watch?v=TjMONlto1pI&list=PL04CskJvgSLMXI4L20dMDoSkgcTd35KJa

If you want to go observe the glory of Reicha's woodwind quintets ... go over here.
https://www.idrs.org/scores/Lehrer2/Reicha/Introduction_Reicha.html