Would you like to play the exercise?
Download the PDF file with the sheet music and tab below.
This 4 bar rhythmic study features odd meters, polyrhythms within polyrhythms/nested tuplets, (one being a 3 over 5 with a subdivided nested quintuplet), as well as some 32nd note tuplets. There's also a fun game at the end that could win you a prize or two. I hope you enjoy it!
Would you like to play the exercise?
Download the PDF file with the sheet music and tab below.
We have an official winner! The winning answer came from Instagram. Prize video coming soon, stay tuned!
It's been a while since the last post. We will get back to new learning material soon.
Meanwhile, here's a video of my last recording session at the Red Owl for one of my tunes...
Thanks to Ray Rojo, Eli Marcus, Lorenzo Ferrero, and Evet Park for playing!
Thanks to Biswarup Chattopadhyay for recording and editing!
A common question I get from potential readers of my book is the following:
What is the fundamental difference between a polyrhythm and a polymeter? Could you discuss what makes one different from the other?
If you look in a book, the definition of polymeter will probably look something like this:
Whenever two or more independent rhtyhms are played simultaneously, (where one of these rhythms is an irrational rhythm), a polyrhythm is created.
(Irrational rhythms are defined as unequal note groups. Read,  p.22)
However, you asked me to pinpoint the defining qualities of each one, and not to define the term.
In my opinion, a polyrhythm is probably best and most simply described as a "compound tuplet". This compound tuplet’s micro subdivisions are usually made up of rhythmic figures that form part of a smaller tuplet.
A polymeter is best described as the superimposition of one meter on top of another. It could also be described as the simultaneous performance of two or more independent meters.
To sum it up...
POLYRHYTHM = COMPOUND TUPLET
POLYMETER = MORE THAN ONE METER PLAYED AT A TIME
Hopefully these definitions have helped you identify the most basic difference between a polyrhythm and a polymeter. If you would like to learn how to identify, dissect, construct, practice and apply polyrhythms and polymeters step by step, I would highly recommend that you buy a copy.
Thanks for your reply.
I'm looking at getting your book to help my rhythm. But I'm not sure because it might be too advanced for me. I struggle with the basics of rhythm but am strangely drawn to fast syncopated music in the genre I play (flamenco).
I just have trouble knowing where I am in the rhythm when things get syncopated and fast, so when I play something syncopated and strongly accented I'm not sure that I’m hitting the syncopation accurately. Weirdly, I can feel when the syncopation is out but have no idea where I am in the rhythm, it's all over the place. I've been playing for many years but seem to be stuck at this point. I've talked to good players and some don't count at all and have awesome rhythm and then I've seen people on the net say you need to count at least in the beginning. This problem is not just when I’m playing guitar, if I try to clap straight rhythm over music that has some syncopation my clapping goes out of time.
Here's a specific example of the sort of thing that I have trouble playing tightly;
Imagine 2 subdivisions within a beat (1+, 2+). If I need to hit a chord on the off beats for say 3 or 4 beats at a fast tempo around 200bpm. If I play it slow I can count the subdivisions and play it fine but when I speed it up I'm lost. It's like I can count it but not feel it accurately, obviously when fast counting subdivisions is not easy and probably not a good thing.
One thing I’m guilty of is not using the metronome enough when I practice. Maybe I just need to use it, work with the metronome a lot more. Again I know musicians that have never worked with the metronome and have good rhythm - very annoying!
Hope all that makes sense, and you have some good suggestions to help me.
(New South Wales, Australia)
You mentioned having problems with eighth note upbeats, so I’d recommend that you work on eighth note permutations.
You should practice with the metronome. Not only that, but you need to keep counting as well. If you can’t do it fast yet, that only means that you haven’t practiced it enough at a slower speed. There's always going to be people that will learn some concepts faster than you can. You will be able to learn some other concepts faster than them.
This does not mean that I am against practicing without the metronome. I’ve actually talked in favor of practicing without it on this post...
I am just saying that it doesn't matter what other people need/don't need to use in order to learn a concept. The only thing that matters is what you need to learn it. If it doesn't come easy, then it is time to sit down and try to figure things out slowly with a metronome.
Below this post, you will find two pages worth of eighth note and eighth note rest permutations to use as a guide. You can find them at the end of this post as a PDF file. It is now up to you to write every possible eighth note & eighth note rest permutation inside of a 4/4 bar. Remember to include permutations using ties, dotted notes, accents, etc.
After this step is done, write out all of these permutations for every meter you are interested in playing. If you don’t feel like writing them out, you could also get a copy of:
Best of luck, don't give up...
In the last post you saw a video that displayed metric modulations, polyrhythms and polymeters. Some cool hats were also featured!!!
For this post I will be explaining some of the most interesting rhythmic things that happened during the video. You can download the score as a PDF file at the end of this post. The video has also been embedded at the end for your convenience.
Thanks go to Ray Rojo (drums) for taking the time to transcribe his playing note for note!!!
Here are some of the things to look out for:
Bar 8: There is a 4:3 polyrhythm.
Every four quarter notes inside of the 3:5’s form a phrase:
You can also see the phrase when you look closely at the chords being played by the guitar, as well as the notes being played by the bass on bars 49-52. The beginning of each phrase has been circled in red:
(In case you're wondering, the (Q1) is what I personally use to describe a quartal/stacked fourths voicing. The number 1 indicates that it is built from the 1st degree. Something else that you might have noticed is that rhythmic displacement has also been formed...)
This four quarter note inside of a 3:5 phrase is a polymeter that, among others, could be seen in the following ways:
An explicit polymeter lasting 6 bars where the second meter is a 4/4 that looks like this:
This option would sound the same, and not affect instrument charts. However, I see it as having a big impact on the score. Writing it this way would not display the 3:5's which are at the core of the idea. Let's take another look at Guitar #2 and the Bass on bars 49-52...
The drums also highlight the 3:5 during these bars. Let's look at the attacks that do so. For the most part, they happen on the cymbals...
Another con of notating it as Option #1, would be that if you wrote this out for every single instrument, a metric modulation would have taken place that completely eliminates the 5/4 time signature.
I do not consider this as the best option for my example because the drums emphasize the 5/4, which in this particular case, I used as the "foundation meter"; referring to the meter over which the second meter is superimposed. (You can find more on why I chose the 5/4 as the "foundation meter" under Option #2.)
Take a second to look at the following attacks happening from bars 45-47. Ray's cymbal work is highlighting the 5/4 meter. Can you see it?
The 5/4 time signature would also be eliminated if we took Option 1 and:
Another problem with Options 1A and 1B, is the fact that we are looking for a 1 bar phrase and all of these turn our 4 quarter note phrase into a 4 bar phrase. (The 1 bar phrase problem is explained in more detail under Option #3.)
Couldn't you express the polymeter as an explicit one?
Why didn't you?
The majority of musicians I've talked to prefer playing implicit polymetric notation over explicit polymetric notation. The top reason being that they can constantly track their position in relation to the "foundation meter". I've even seen some musicians go as far as ignoring the second meter altogether, performing it as a syncopated "foundation meter" instead.
I just noticed the accents on the phrase. Don't they suggest a second meter with a numerator of five? Something like a 5/8 or a 5/16 meter?
In this case however, they are the divisions of each one of the four groupings found within one bar of the second meter I selected. As a matter of fact, my choice of accents had the intention of creating ambiguity.
An explicit polymeter where the second meter is comprised of a mixed meter 6/4 + 2/3 | 4 (lasting 6 bars). I say explicit because it would be difficult to identify as an implicit polymeter in comparison to some of the other choices presented, perhaps even a bit impractical.
(Mixed meters can be found in the work of Elliot Carter, Edgar Varèse, and Pierre Boulez among others.)
(The second meter of the mixed meters is a fractional meter. This type of notation can be seen in the work of Pierre Boulez and Roman Haubenstock-Ramati among others.)
Wait... Couldn't you write the mixed meter as the time signature and write the 5/4 as an implicit polymeter?
You could do that with every single option presented here...
Why didn't you do it like that? How do you choose which meter to write as the time signature?
When I'm thinking of ideas, I tend to use my left foot to perform the "foundation meter" over which everything else is superimposed upon. In my particular case, that was the 5/4.
Choosing the mixed meter (6/4 + 2/3 |4 ) as a foundation meter could be an option, but my personal goal when notating something is to capture/describe/translate the idea as best as possible. After all, notation is a means of creating a record for future reproduction. If you use it accurately, you will create a picture of your present thoughts and ideas for your future self to study/introspect/reminisce.
An implicit polymeter with a first meter of 5/4 (lasting 8 bars), and a superimposed fractional meter of
5 [3 eighth note (lasting 24 bars). Each bracket below the staff is identifying the duration of one bar of the fractional meter.
(Notation of fractional meters such as the one below the staff can be found in the work of Charles Wuorinen.)
Something I don't like about this option is that it does not cover the entire phrase. My "one"/downbeat is steadily showing up after the 4th group of notes. If you remember, this is why we considered Option 1 in the first place.
We are getting closer though, we just need a fractional meter that covers the entire phrase...
An implicit polymeter with a first meter of 5/4 (lasting 8 bars), and a superimposed fractional meter of
20 [3 eighth note. (lasting 6 bars). The bracket below the staff is identifying the duration of one bar of the
(Notation of fractional meters such as the one below the image can be found in the work of Charles Wuorinen.)
This last choice is how I view it personally, and what I was thinking at the time. It is easier to notate as an implicit polymeter, and the entire phrase is captured under this second meter.
How do the drums come into play?
Below is a picture of all the drum attacks consolidated into the 20 [3 eighth note fractional meter. I have kept the attacks in groupings of 5 notes as shown on the quintuplets under the 3:5 polyrhythm to make it easier to visualize. No note brackets are shown for the same reason.
Do you see any patterns being formed? The left page features the bars 45-48, (where the drums highlight the 5/4), and the right page features bars 49-52 (where the drums highlight the the 3:5 polyrhythm).
Everything circled in red is identical...
Here are some things to look for in the picture above:
Before we move on, couldn't you just have notated everything as a "normal" implicit polymeter for the
20 [3 eighth note fractional meter? Why did you decide to notate the implicit polymeter as subdivisions of the 3:5 quarter notes?
I personally chose not to because it would not display the 3:5. Again, 3:5 is the figure from where this entire idea is derived from. Let's not forget that our phrase lasts 4 quarter notes inside of a 3:5.
Even though those 4 quarter notes last the same amount of time as a bar of the 20 [3 eighth note fractional meter, I see both notations as being equally important.
The fractional meter:
The 3:5 subdivisions:
Personally, I think that the most practical way for me two show both functions was to display the fractional meter as subdivisions of the polyrhythm.
Other things that might interest you:
There are rhythmic displacements happening in the following bars:
Bars 47-49: In regards to the 5/4, Tenor Sax 1 & Guitar 1 perform a rhythmic displacement that begins on the 1st quarter note of 3:5 then moves to the 2nd quarter note of 3:5 and finally moves to the 3rd quarter note of 3:5.
Here's how it looks on the score. I've eliminated all other instruments so that it is easier to see:
This same displacement is performed by the bass during bars 46-48.
During this section, the soprano sax only plays on bars 49, 51 and 52. When looked at continuously you can also see some rhythmic displacement happening.
Well, I hope that this post has given you some choices to consider when writing your own music.
At the very least you should have enjoyed the hats...
What better way to celebrate the holidays in the "LEARN" section, than with a video postcard that includes metric modulation, polyrhythms and polymeters?
Thanks to Ray Rojo (drums) for inviting me to do this, Lorenzo Ferrero (saxes), Gui Bodi (bass), Gabbs Casanova for recording the bass, and BC Chattopadhyay for recording the rest of the instruments and mixing it.
Happy Holidays and Happy New Year! Enjoy the video!
Exactly one year ago, the "LEARN" section of this site started. As a thank you, I am attaching a tune that features 4:3 polyrhythms at the end of this post.
I look forward to answering more of your questions, as well as reading your e-mails in the future.
Thank you so much for reaching out.
It's encouraging to see how involved you are with your subscribers.
I recently graduated from music school and have started working in a non-music school job. My biggest struggle is knowing what to practice, given the limited amount of time. I try to keep it to the most important things for me: rhythm, harmony, and melody. It's within those themes that I start to stumble, I mostly practice my time with a metronome and recently I have been wanting to introduce new rhythmic concepts into my playing, hence why I purchased your book and signed up for your website.
I try to practice at least two hours a day but I find even when I reach that the practices don't feel very successful.
I've also tried combining different approaches to learning multiple things at once, such as practicing a new voicing and its inversions while playing a specific rhythm over a tune or vamp. But I think that's too ambitious.
Hope this isn't too long, any guidance, advice or positive practice experiences are greatly appreciated.
Deciding on what to practice can certainly feel daunting at times. There are a million concepts and only one of you.
I think that you where onto something when you talked about consolidating concepts. However, I would suggest that you first learn them individually.
In addition, I would suggest that all of your practices focus on applying the concepts that you are working on. A lot of students practice with two folders: the “practice” folder and the “playing” folder; never applying some of the concepts they practice. I’ve even seen students religiously practice concepts without having any idea as to how they are applied.
Below is an example of a practice routine that incorporates all of the elements you mentioned. I have attached it as a PDF file in case you’d like to use it as a starting point.
You might not be able to get through all of this in one day, but the important thing is to go through the full list. As the days go on, everything will be easier to do, and you will be able to practice more concepts.
Eventually, you will just run through the whole tune, practicing a different concept every time you run through the form. Once you feel that you can do this comfortably, start mixing the concepts.
Are you done yet? Grab a new tune, wash, rinse and repeat!
I bought your book, I know it’s going to be super helpful in the future but a lot is too advanced for me right now. I can play a lot of the examples but don't understand them etc..
I was wondering do you have a book that would serve as a good basis before I start on this? Or if you don't, could you recommend me which ones are good?
Man thanks and all the best from London :)
Thanks for writing.
Since I don’t know which concepts you are having problems with, I am going to write a list of books/learning aids that I like best for practicing “primer” or “foundation” concepts you should understand.
With rhythm books, I think that the goal is to find one with as many rhythmic permutations as possible.
Here are a few recommendations:
1.“Encyclopedia of Reading Rhythms” by Gary Hess – It’s almost as if Hess wrote out every possible permutation involving whole notes, halves, quarters, eighth note triplets, quarter note triplets and sextuplets.
2. “Factorial Rhythm” by Mick Goodrick and Mitch Haupers - Very hard to find. The copy I own was bought out of Mick's defunct Mr. Goodchord website. I really like the progression of this book as well as Mick’s tradition of giving you the material and expecting you to do the work.
3. “Practicing Tuplets” and “Practicing Tuplets II” posts on the “LEARN” section archives of this site.
4. “Gateway to Rhythm” DVD by John McLaughlin and S. Ganesh Vinayakram – I’ve mentioned this one quite a bit. It will help you with assigning syllables to rhythmic figures. Some of the material discussed will act as a primer for polyrhythms and implicit polymeters.
5.You should read this post as well…
by Gardner Read: Just about everything you will ever need to know about notating complex rhythm.
Practice Exercise Structure:
Look at how the “Dividing Polyrhythms/Polyrhythms within Polyrhythms Exercise Sheet” is constructed on pages 38-51 of the book. There is an attempt to show all the permutations of each group of rhythmic figures. Practice your rhythmic figures like this. The only thing I would change is that I would place the brackets starting on each possible beat. I didn’t do this in the book because I wanted to give you a constant phrase to keep you grounded. (Training wheels…) When practicing, you should eliminate this crutch...
Always follow I.M.T.R. (from page 89 of the book)
That’s enough for now. Just practice these until you are ready to go into the book.
Avoid "analysis paralysis" at all costs!
PRACTICING TUPLETS PART II: USING METRIC MODULATION TO EQUALLY DIVIDE TUPLETS WITH SYMMETRICAL AMOUNTS OF ATTACKS
After answering the last Q&A, I started thinking about techniques that could be used to practice tuplets with a symmetrical amount of attacks. That is to say, tuplets where the combined number of attacks can be divided into equal groups. (Most of the time, these groups will have 2, 3 or 4 attacks each.)
Below is a step by step guide on how to use metric modulation in order to better visualize the exact duration of these types of tuplets...
A lot of people think about 32nd notes as a metric modulation that looks like this:
However, they fail to use this same type of rationale when thinking about nonuplets, dodecatuplets, and even sextuplets. Let’s take a second to visualize the steps to doing this:
First of all, what is your goal?
For this first example, our goal will be to understand how to equally divide a nonuplet:
Step 1: Identify the amount of equal note groupings you can make out of the tuplet. In the case of a nonuplet, we can make three groups of 3 notes:
Step 2: Identify the rhythmic figure that will serve as an anchor point for your attack groups. This figure should equal the length of an entire group. In the case of a nonuplet, we will use an eighth note inside of an eighth note triplet. Here is the nonuplet with all three anchor points displayed below:
Step 3: Visualize the symmetrical group anchors as quarter notes and the attacks as their subdivisions after a metric modulation has taken place:
Go through the same steps for dodecatuplets and sextuplets. The answers will be provided below. For the purposes of those answers, we will divide the tuplets in the following ways:
Dodecatuplet - 3 groups of four attacks
Sextuplet - 2 groups of 3 atttacks
If you did it correctly, your results should look like this:
You might be asking yourself:
Wait... Can't you divide the tuplets in other groups? Yes.
Why did you select those specific divisions? They happen to be my favorite ones.
What happens when you select other divisions? Your answers will be different, but still correct. Provided you performed all the steps correctly of course...
One last note:
I feel like I should stress that these are all ways of trying to better understand divisions in tuplets. In no way am I saying that you should perform a metric modulation every single time you see a tuplet. This was written to help you better understand/visualize the duration of each one of the attacks.
That being said, visualizing your anchor points will help you perform your divisions in a more precise fashion. This is due to the fact that anchor points serve as a sort of "calibration/reference point". Having several of these points will help you adjust your tuplet in "real time" as you perform them.
I really like your book and am trying to make it a part of my practice time.
I have problems playing tuplets. Do you have any suggestions you could give on how to practice them?
(Bronx, New York)
Practicing tuplets is very important in order to be able to play some of the polyrhythms in the book. You should definitely start with them and work your way up to polyrhythms such as 5:2 and so on. Here are a few things you could try:
Suggestion #1: Use syllables for each attack:
Just like you would use tri-po-let for triplets or 1e+a for sixteenth notes give syllables to your tuplets. Here are some examples of syllables for a quintuplet:
Hippopotamus (I've seen this one used by drummers a lot...)
Da-Di-Gi-Na-Dum (used in Konokol or Konnakol)
Ta-Ke-Ghi-Na-Ton (used in Carnatic rhythm as part of the Gatis)
You can make up your own syllables if you want.
No matter which syllables you decide to use, just make sure that you divide each attack equally throughout the beat!
Suggestion #2: While you get the hang of things, you could look at tuplets as full bars:
If it helps, imagine that each tuplet is a bar where the click only plays the downbeat.
Example: For a quintuplet, imagine that you have a bar of 5/8 where the metronome only plays the downbeat.
Example: For a septuplet, imagine that you have a bar of 7/8 where the metronome only plays the downbeat.
Suggestion #3: Practice your tuplets going in and out of other figures:
Don't just practice playing quintuplets over and over again. When you feel like you can play them, practice going in and out of other rhythmic figures. Here's an example of what a practice rhythmic line could look like:
Well, that's all for now. I hope that these come in handy during your tuplet practice.
Thanks for reaching out!
All my best,
This is a cool way to further my learning process. Actually, the first time I heard about using the metronome more and breaking it down how you have the first two chapters, was when I watched a seminar/class on YouTube from Kenny Werner. I'm so glad to see this laid out in a book. I'm plugging away through the 2nd chapter for about a week now. I find it's very rewarding once I learn how to play the 5 over 2, and 4 over 3. I'm still working on the 3 over 5, and 3 over 7. They can be tricky. I love using the stonekick metronome beats app with these exercises.
Also, I kind of wish you put the chord names over top of the 5 over 2 study. Very nice piece by the way. I feel I've barely scratched the surface in this book. And I've already learned so much.
Thanks for the kind words. It makes me so happy whenever I hear that someone is benefiting from the book. I cannot help but feel humbled when I learn that a person is taking the time to thoroughly study each chapter.
Kenny Werner has some great stuff! I really like his “Effortless Mastery” series, and have both his book and DVD. It’s something to check out if any of you haven’t yet.
About why I didn’t print the chords:
At the time, I thought that if I went into harmony it might distract from the rhythmic aspect of the book. I even wrote about it during the introduction. I’m going to go ahead and post them for you as both a picture, and a downloadable PDF placed below the picture. Note that on bar #11 there is an asterisk. It indicates that the chord displayed over the bar is the recorded chord. However, I prefer the one added below the sheet.
As part of your routine, try to incorporate some of the book concepts into your playing/composing. That way, you will get the most out of them.
Thank you for writing the book and answering my e-mail.
I have problems when I practice implicit polymeters. How do you keep track of both meters? Do you have any practice advice? Is there anything specific that you do?
In order to keep track of both meters I would suggest that you go through the following steps:
Pick your poison:
Decide if you are going to use number or konokol syllables for your odd meter groupings. Once you’ve made a decision, stick with it!
I personally prefer konokol because it is easier for me to pronounce faster. Which one is better for you? You’ll never know until you try…
Note: If you are not familiar with konokol, I would highly recommend purchasing a copy of “The Gateway To Rhythm” by John McLaughlin and Selvaganesh Vinayakram. I’m pretty sure I’ve mentioned this video before, but it is definitely worth a second mention. Once you view it, there will be no mystery as to why this is one of my favorite instructional videos of all time…
Before anything else, reciting odd meter groupings should become second nature to you. Practice groupings like 2+3, 3+2, 4+3, 4+4+3+2, or any others that you can think of to the point where you do not have to focus on reciting them.
Visualize your second meter/ the “Count Von Count Method”:
While keeping time I tap my left foot, as I am sure you do too. While keeping two meters simultaneously, I tend to “pay attention” to the meter my part focuses on. The second meter is always kept/“taken over” by my left leg. I always visualize big, white, “Sesame Street” numbers that change with the beat on top of that leg as well.
I think that this has a lot to do with the fact that I watched a lot of “Sesame Street” as a kid, and loved counting "with" Count Van Count.
Practice alternating second meters over a static meter:
A good exercise you could do is to tap quarter notes with your left foot while indicating the beat number by holding up fingers with your hand. As you do this, recite four bars worth of odd meter groupings. After the four bars of the second meter have passed, start reciting a new odd meter for four bars. Start with one or two meter changes and work your way to as many as you can. The long term goal is to be able to shift at will.
Be thorough with your groupings:
Write a list of different groupings per meter to make sure that you don’t always practice the same groupings.
Have fun and improvise:
After you can do the exercises provided above, grab your guitar and start improvising over a one chord vamp in 4/4 while you do this.
Next, attempt to make your meters meet. End your second meter where the first meter ends, and start a new second meter on the downbeat of a new bar for the first meter.
Once you can do this, practice playing a second meter over chord changes, then attempt to make your meters meet over them.
I hope that these exercises help you. Good Luck!
Q. Dear Jan,
Thank you for answering our questions.
I have a short one for you…
Have you ever seen a special metronome for polyrhythms?
A. Hi Nicolas:
Some years ago, I used to practice with some software called “Bounce Metronome” until my old computer crashed. Back then it was only for PC’s, so I haven’t used it since. (I opted for a Mac when that happened.)
After checking the website today, it looks as though they have a beta version for Mac. I will be giving it a try as soon as possible.
There’s also an app on my phone named "PolyRhythm" that I downloaded some years back. Every once in a while I’ll turn it on and just practice counting polyrhythm subdivisions out loud.
I think that I saw a physical one at one point that went for $350 or something like that. As far I can recall, they only made a limited run.
Although these are nifty little gadgets, remember that they are not necessary and any old metronome will do. At the end of the day, you should be able to perform these rhythms without a special metronome; just like you are able to perform half notes, quarters notes, eighth notes and so on.
I hope this helps...
Q. Hi Jan!
It's very awesome, your concern for the students of rhythmic functions like me...
First of all, my first struggle with the book is... How can i program the metronome?, or what's the best way to make a guide when recording music over a metric modulation pattern?
i.e. if i want to record the example number 2 (on page 2), I can program the metronome at some tempo and record the bar 1, but in the moment of recording bar 2 i have some trouble because I can hit the notes approximately , but i really need to play in the right spots to make the metric modulation sound good.
I hope my point is well explained.
Infinite thanks Jan , Greetings from Mexico, I hope this season is being perfect for you!
A. Hi Daniel,
It's funny how life works out sometimes. I was walking around with this idea in my head for a while, and then your question came along.
Once you find it, you will see that there is a link to a step by step instruction on how to use the calculator. The link is below the "New BPM Value" entry field.
The calculator is based on a formula I found inside a collaborative work written by Richard Delone, Vernon Kliewer, Hoarece Reisberg, Mary Wennerstorm, and Allen Winold.
Winold was the one who adapted the use of the formula to metric modulations, and wrote the rhythm section of the book.
I highly recommend that everyone purchase this book, as the formula is just a tiny paragraph amongst roughly 500 pages of invaluable information.
In fact, here is a picture of mine, so that you can see what the cover looks like if you decide to buy it:
Good luck recording!
All my best,
Answers to the questions of "Meters and News" subscribers, as well as some articles, are posted in this section of the website.