How to notate complex metric modulations... er, uh, tempo equations

On episode 5 of "hey man how do you do that?," Brian shows you exactly how to add some elasticity to your stale beats with tempo equations. We look to Elliott Carter, who was preoccupied with elongating, abbreviating, abstracting, and generally f*cking time in his music.
FYI, the underscoring consists of two of my electronic tracks ("Memphis" and "I am a worm and no man"), both of which utilize metric modulations, and neither of which is available commercially. [You can download them on my Patreon page tho.]
2:12- defining Metric Modulation
3:13- some Brass Tacks
5:20- exceedingly complex example
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Elliott Carter 'Collected Essays and Lectures' amzn.to/3gZJVr2
Elaine Gould 'Behind Bars' amzn.to/2Wsflgx
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Sources:
Carter, Elliott. Collected Essays and Lectures, 1937-1995. 'The Time Dimension in Music.' 1965.
Carter Elliott. ibid. 'String Quartet No. 1 and No. 2.' 1970.
Gould, Elaine. 'Behind Bars.' 2011.
Hannaford, Marc. 'Elliott Carter’s Rhythmic Language: A Framework for Improvisation.' 2012.
Mead, Andrew. 'Time Management: Rhythm as formal determinant in certain works of Elliott Carter.' 2012.

[@BrianKrock]

Oh, guys, also: every score in the Big Heart Machine store is 20% until Dec. 31st with the promo code "happyholidays": www.bigheartmachine.com/shop (When I was growing up, score were legit my favorite xmas gift, just saying...)

[@ozzycarmona]

Do my ears deceive me, or was that Adam Neely reading the Thomas Mann quote?

[@BrianKrock]

@@ozzycarmona trust your ears :)

[@AndrewGordonBellPerc]

Percussionist here and as such I've spent a lot of time with Carter's solo pieces for timpani, which are also a great example of metric modulations. One thing I'd like to point out is part of the reason Carter used the term metric modulation was because he used them as structural markers of form since traditional tonal modulations were no longer really a part of music. (or at least not in a way which could be easily perceived by a listener) For that reason I think it still makes sense to call this device a "metric modulation" within the context of Carter's music, while in the broader context of music it probably would be better to call them tempo equations. So all metric modulations are tempo equations but not all tempo equations are metric modulations.

[@BrianKrock]

Interesting! Thank you for your perspective! I agree with you whole-heartedly- your last sentence puts it very succinctly.

[@jordanrosemanmusic]

Brian thank you for the suggestion to listen to Elliott Carter’s first string quartet. The first metric modulation in the Carter is coincidentally the exact same I was trying to notate in my piece!!! Solved my problem instantly haha!

[@jonasgolland1020]

Thank you for a clear and well researched guide. In college in 2010 when I learned what a metric modulation is, we were taught the old 'Precedente' style of marking the new subdivision first followed by the old, without the Precedente version. It was a pop school where we did some jazz and it was okay, but not the sort of school (Tech Music Schools which is now BIMM) that would have you nurturing a culture of time changes really outside of musical theatre and nostalgic prog tunes because they made a lot of money. I agrere that we should adopt tempo equations into our usage, as one of the many ways that music notation should be able to adapt to more practical use.

[@jonasgolland1020]

So I've applied this to my new solo I'm writing, and notationally I was able to do it in Sibelius, as well as anchor the new system text to the bar line, and the playback doesn't actually read it. There were 16th note triplets and they were to become 16th notes, but when it crosses the line it slows down to the original tempo of the tune. The issue must be with that. I'm using First but that shouldn't be the reason :)

[@drinkpen]

This was really great, thanks - I really need to explore Elliott Carter properly. This is a lovely way to cleanly notate this stuff...with the delivery to match, great stuff

[@francissadleir9805]

Wow, this is awesome! thanks so much for making a video about this, I wasn't expecting that! Excellent content and happy holidays.

[@BrianKrock]

haha Francis Four, I feel as though I know you because I said your name so many times recording this! Your question really just made me stop and think... so thank YOU. I'm actually recording a new composition next week in which I used this exact metric modulation, because I couldn't stop thinking about it, I guess...

[@francissadleir9805]

@@BrianKrock well I must also say thank you too. your channel is the perfect niche for me as a clarinettist/composer myself.

[@bharathondrums]

This was so helpful man! Thank you!! I've been trying to understand the metric modulation in 'Story' by Chon at 3:10 and how to put it on paper. Cheers!

[@tomasdesouza2847]

Yes, Modal equations. I'm all for it.

[@Drumhead10100]

Procrastinating from writing a paper for an Elliott Carter class by watching this. I wish the piece I chose had more metric modulations...

[@BrianKrock]

What piece did you choose??

[@Drumhead10100]

@@BrianKrock Two Fragments for String Quartet. I was looking at the Cello and Piano Sonata (theres some fun metric stuff in the first movement) but the teacher recommended a later piece that was heavier on set theory. Just wanted to say your videos are awesome! I saw Carters Harmony book on the shelf last video and wondered if you'd do a video on his techniques, didn't expect it to be the next week though lol

[@GoldenScarab45]

A man who shares Sibelius hotkeys is a man after my heart

[@Javi7Tron]

I´m a jazz drummer and a matematician and I´m not sure that if the nomenclature "tempo equation" is appropriate. We should try to define matematically what is a "metric modulation". I guess that it is a bijective function f:R+ -------> R+ associated to the tempo change that comes from the metric modulation and it´s related with an algebraic equation with one unknown. An algebraic equation with one unknown is an expression of the form f(x)=0 where f(x) is a polynomial over the real field. So if I have a metric modulation (for example "quarter note=half note" ) this is equivalent to give an algebraic equation, the unknown "x" of this equation is the new tempo, so if I know the old tempo, the equation associated to the metric modulation allows me to calculate "x" i.e. the new tempo. At the same time this is equivalent to the existence of a bijective function f: R+ ------>R+ that for the old tempo of the piece ( a positive real number "t" ) gives me f(t) : the new tempo after the metric modulation. So your nomenclature "tempo equatión" makes sense but also we could say "tempo transformation" or "tempo bijection" or "tempo isomorfism" ( a bijective function is an isomorfism on the category "Set" of Sets and functions , this is a particular case of an area of matematics called "Category theory and functors" and I have done some research on that area). I should write this with rigor and develop some definitions and theorems and apply them to musical examples of metric modulations.

[@EliahNebb]

Thanks for including the key commands; that might be the biggest headache of notation software. One example of metric modulation that I love happens in Jennifer Higdon's The Singing Rooms, between mvmts 1 and 2. The last measures of mvmt 1 have similar content, but the time signature decreases each measure while keeping the beat the same. So it goes from 4/4 to 3/4 to 2/4, giving the feeling of increased speed, then ending up in a much faster 4/4 than where it started. (I don't have the score right in front of me, but this is basically what happens.) Also, I've heard about the 103rd birthday concert. Is it true that they played a piece that Carter finished while sitting in the audience?

[@BrianKrock]

Haha that’s a great story, but no, I was sitting a few rows behind Mr. Carter. He was just smiling from ear to ear the whole time- but not composing! One of the newest pieces that year- “a sunbeam’s architecture”- is one of my favorite pieces of music, ever. My old teacher, Jim Pugh, was playing trombone on a couple of the world premieres, and he was also playing with steely Dan later in the week, so he brought Donald and Walter with him and they loved the concert, too! What a night!

[@smallsignals]

You have such a delicate voice, it's so lovely.

[@smallsignals]

That is actually curious to me, now that I think about my music degree.

[@smallsignals]

Let's NOT adapt that lexicon change, please? Metric Modulation is more consistent to me. But we do need to know the formulae for tempo equations. It's the bane of my existence figuring out the tempo change between 4/4 and 6/8. Technically the BPM should be the same, but it somehow works out different. My brain so ouchy dumb.

[@Gusrikh1]

This is very, very educational.. thank you.

[@robertblumschein6201]

That was a blast

[@hesambani4943]

Hey Brian, Well explainded but not enough, could you please make a tutorial about it?

[@marcosfischer_]

Young Walter White from an alternative universe hahahaha that’s a compliment 😁

[@Yiddish_Harpsitech]

I'm still trying to figure out what you mean when you say 'adding arrows like this' and then don't show the hotkey to do it. In my case, the metric modulation right click does not include the arrows, so I am using a separate text/right click to show arrows and then having to drop them over the metric modulation text.

[@cypressbartlett9083]

Great video! I'm orchestrating a thing, and the tempo modulates to triplet crotchets (one bar = 3 crotchet triplets). Is there a way to modulate this? Or is it more practicable to change the time signature from 4/4 to 2/4 and notate the bars with crotchet triplets. Many thanks if you're able to help :)

[@Jack1Roland]

What website do you use for the sheet music?

[@clearbrookprogram5924]

Academic Overture by J. Brahms. The entire work.

[@johnnytwo-shoes9798]

I have a question about the difference between the 2 (or 3) poly terms and I have already seen your previous video defining these terms (kzbin.info/www/bejne/sHq3gGOhh8yUbLs) based on the definition of Henry Cowel. However I question the acceptance and validity of his definition against the definitions of metre and rhythms. I'm not assuming I'm correct just furthering the discourse, and questioning the blind acceptance of a past academic's take. By definition a simple 3:2 polyrhythm as you've described it (and is the accepted term for this) contains in the same duration of a measure, a 3/4 measure which is simple triple metre, over 2/2 , a simple duple metre. Hence 2 simultaneous metres, duple over triple. So by definition wouldn't that be polymeter? ... and with 5 vs 4 crotchet note grouping using accented rhythms to possibly imply different metre feel but measured at 4/4, wouldn't that technically still be polyrhythm under the same metric of simple quadruple metre with an implied metric difference by the rhythm but since its still measured against 4/4 it remains with a static metre and a polyrhythm..? And lastly in contradiction to keeping the concepts separate and some of my other points, doesn't having a 2 metres eg duple and triple, happening simultaneously, implied by 2 sets of rhythms mean they are in fact the same aural concept, which only has separation of meaning depending on the perspective when written i.e. 3:2 is 3/4 over 2/2 poly metre, but when written under 4/4 also 3 triplet crotchets over 2 minims (polyrhythm) and lastly back to polymeter 3/4 over 2/2 where the a crotchets would not have the same duration value so poly tempo? I also see the idea of 2 rhythms implying 2 metres (duple over triple) in the same metric, being defined as poly rhythm (different rhythms, same measure) but it's a sonic concept that holds relevance only in his it's written and which way the metric is applied, so 1 measure with 2 rhythms of the implied 2 metres of duple and triple (polymetre) taking up the same duration but the duration value of each crochet (rather than the semibreve) being different (polytempo) using a different time signature. Hence the confusion. Is it just an academic way of trying to quantify and label essentially the different perspective framing and measuremenf of the same sonic principle? Apologies in advance but this has been plaguing my thoughts since your last poly tempo vid (which was awesome). In context I'm a guy who questions the pronunciation of divīsive (devic-ive) based on the word divide when it would be less ambiguous in spoken language to pronounce it div-i-sive based on division. Also the use of past (rather than passed) to mean going beyond something e.g. having past the shop (outside a chronological context) - knowing the accepted grammar and understanding that "passed” in this use is not the past tense of the verb but still think that passed would work better as an exception and past be relegated to implying time.. .. so being that guy, I ask the above and question the accepted definitions of poly* when attached to , metre and rhythm ( more so than tempo). Thoughts are appreciated.

[@smallsignals]

Okay, but why have decided the time signature changed to 20/8?

[@smallsignals]

Sorry, I understand why. But I think it's a gap in your explanation process.

[@smallsignals]

Also, umm, just play the music from the sheet music? How else do expect people to get a feeling for it? Ffs. Woah.

[@seanbeadles7421]

I’ve recently written a song where one half is subdivided in quintuplets at 80 bpm and the other in sixteenths at 100 bpm, so the quintuplets and sixteenths are the same length. Kinda took a second to figure out how to notate it tho. I decided to use tuplets rather than a big denomination time sig. Makes for a really disorienting transition and I love the effect it makes.

[@GokhanCebecioglu]

🤘🤘🤘🤘

[@onesyphorus]

not a single line of irrational time siggies ?

[@robertwilkscomposer3726]

I call them "tempo modulations", because that's what they are.

[@marshallgrey2159]

How is this got only 2k