[paaskanama]

Friends,

The last few days' thoughts have been consumed with the topic of metric modulation - a compositonal tool for adding an accelerando or retardation feel to a piece of music.

I'd like to say I have the basic concept under my belt, but some of the foundational characteristics are still eluding me whenever I apply additional thought. Being of a highly mathematical mind, this is frustrating me greatly.

If the meter changes from QUARTER (note) = QUARTER to QUARTER = HALF (note), it is a simple way to mark a change to double time. Similarly, if the change is from QUARTER = QUARTER down to QUARTER = EIGHTH, the tempo is cut in half.

Even following metric steps like this:
INITIAL TEMPO: QUARTER = 96 bpm
ALTERATION: QUARTER-NOTE TRIPLET = DOTTED EIGHTH
NEW TEMPO: QUARTER = 108

Those I can understand.

It's when a QUARTER NOTE TRIPLET = EIGHTH that I get a little lost.

Can somebody boil down to layman's terms and explain to me what either side of the "equal" sign is relating to. I may have remembered this before, but the continued thinking helps me escape the once-seemingly-simple solution.

I hope any explanations can help other composers as well.

[Nate]

Although I've never really dealt much with this technique (or even heard much about it), I think it should be just a simple matter of using fractions to convert from one tempo to another.

Originally Posted by paaskanama

Even following metric steps like this:
INITIAL TEMPO: QUARTER = 96 bpm
ALTERATION: QUARTER-NOTE TRIPLET = DOTTED EIGHTH
NEW TEMPO: QUARTER = 108

For example:

In this situation, you are starting off with quarter note triplets in 96 bpm, then converting them equally to dotted eighths in some new tempo. Here's the way to do this, mathematically: First, realize that a single quarter note triplet takes up precisely two-thirds of a beat (exactly twice as much as an eighth note triplet). Then, realize that a dotted eighth note takes up exactly three-quarters of a beat. Now, realize that you're just converting two-thirds of a beat at some tempo into three-quarters of a beat at another tempo. The form of this equation is (2/3)x=(3/4)y... where X equals your initial tempo of 96. So....

Wait a minute.... mine comes out different (85.something), but I know why.

When you say "quarter note triplet = dotted eighth", do you mean that a quarter note in the OLD tempo equals a dotted eighth in the NEW tempo?? Or do you mean the opposite???

When I hear an expression like that, I think of the former of my two answers (that the first note duration is from the old tempo, and the second is from the new). I don't remember the proper way to write these, but I think you were thinking the other way... where the quarter note triplets were supposed to be in the new tempo. If this is the case, the equation should be reversed: (3/4)x=(2/3)y, where x is your initial tempo. Let's see..... *does calculations*... yep.

I think this former understanding of the equation is incorrect, though... because you had it reversed in your three prior (and simpler) examples regarding quarters, halves, and eighths.

So... the equation should be (2/3)x=(3/4)y... which would SLOW THE TEMPO DOWN to 85 or so.

It's when a QUARTER NOTE TRIPLET = EIGHTH that I get a little lost.

This is the same deal, mathematically speaking. This time, the equation would be (using the proper understanding of the equation, where the quarter note triplet is relating to the old tempo and the eighth is relating to the new tempo): (2/3)x=(1/2)y, where x is your initial tempo.

Can somebody boil down to layman's terms and explain to me what either side of the "equal" sign is relating to.

The side on the left of the equals sign refers to a beat (or fraction thereof) in the initial ("old") tempo; the side on the right refers to a beat (or fraction thereof) in the new tempo. The equation sign sets the proper ratio between the two tempi by demarcating exactly how the two are supposed to relate to each other.

I hope that makes sense.

In His love,
Nate

[paaskanama]

Originally Posted by KeyboardFreak

When you say "quarter note triplet = dotted eighth", do you mean that a quarter note in the OLD tempo equals a dotted eighth in the NEW tempo?? Or do you mean the opposite?

It's means that what used to be played in the span of a quarter-note triplet's time (2 beats) now occurs in the space of a dotted eighth (3/4 of a beat) - therefore, the tempo increases greatly. (2 and 2/3 times faster)

Originally Posted by KeyboardFreak

The side on the left of the equals sign refers to a beat (or fraction thereof) in the initial ("old") tempo; the side on the right refers to a beat (or fraction thereof) in the new tempo. The equation sign sets the proper ratio between the two tempi by demarcating exactly how the two are supposed to relate to each other.

The equal sign, I think, is to give the player an "example" of what used to be played up to that point must now be condensed or expanded to fit within the new rhythmic designation of the meter. A rough estimation, as it were.

It still bothers me that I can't grasp the mathematical part of it where I can see the rhythmic designation and ascertain the numerical designation.

Thank you for your input.
Anyone else? I know you've been reading...


-paaskanama

[Nate]

Originally Posted by paaskanama

It's means that what used to be played in the span of a quarter-note triplet's time (2 beats) now occurs in the space of a dotted eighth (3/4 of a beat) - therefore, the tempo increases greatly. (2 and 2/3 times faster)

By a quarter note triplet, do you mean THREE quarter note triplets or ONE??? If the answer is three, then you are correct; but if the answer is one, then my previous reply is still correct.

Usually, when a rhythmic equality is notated between tempo changes, it refers to INDIVIDUAL notes.... so the norm would be for "quarter note triplet" to refer to ONE such note, which would take up exactly 2/3 of a beat.

When 2/3 of a beat in one tempo becomes the equivalent of 3/4 of a beat in a new tempo, the tempo will slow down... not speed up.

The equal sign, I think, is to give the player an "example" of what used to be played up to that point must now be condensed or expanded to fit within the new rhythmic designation of the meter. A rough estimation, as it were.

It is not a rough estimation, and it is not for condensation or expansion purposes.... it is an exact equality that defines how one tempo relates to the next.

Using your "double time" (AKA "cut time") example, let me explain:

When you are moving along in 4/4, a quarter note (when notated on a piece of music) takes up exactly one beat AT ANY TEMPO. It could be fast, it could be slow, but the quarter note will always be one full beat.

When you see a notation that says that a quarter note now equals a half note (quarter = half), it means that the old quarter note and the new half note are going to take up EXACTLY the same amount of time. If you were moving at 60 bpm beforehand, the quarter note would span the space of one second of time... then if you saw the Q=H sign, then, for any music written after the equality, the half note would take up that same one second of time. This would have the effect of doubling the effective tempo... in other words, this would have an effect such that, if you were to copy the music from before the equality and place it after the equality, every note's duration (time-wise) would be cut in half. Whereas a half note BEFORE the equal sign took up two full seconds of time, a half note AFTER the equal sign only takes one full second of time.

Moving now to the quarter note triplet equals dotted eighth case:

A quarter note triplet is the note that takes up exactly one-third of two beats... such that three fit in the space of two normal quarter notes. This works out to exactly two-thirds of one beat. If your tempo was 90 bpm, this would work out to exactly one full second of time for each quarter note triplet.

Now, if you equate one of these quarter note triplets to a dotted eighth note in a new tempo/metre, all you are saying (as in the former example) is that any dotted eighth notes written after the equality should take up exactly as much time as the quarter note triplets each did before the equality. This means that the dotted eighth notes should take up exactly one full second.

But what effect does this have on the tempo? Is it still 90 bpm to the quarter note? No. Let's figure out why:

What you have now is a tempo wherein a dotted eighth note takes up one second of time. There are 60 dotted eighth notes in a minute... if the basic beat was the dotted eighth note, the tempo would be 60. But, the basic beat is not a dotted eighth note, but a quarter note... which is exactly 4/3 as long (rhytmically) as a dotted eighth note. So... multiply 60 by 4/3, and you have the new effective tempo..... 80 bpm.

The steps look something like this:

1) Find original tempo (base it off of the quarter note in simple metre, and off of the dotted quarter note in compound metre). In this case, it was 90 bpm.

2) Find the fraction of a beat that is taken up by the note value given on the left-hand side of the equation. In this case, it was 2/3... the length of a quarter note triplet.

3a) Multiply this fraction by the original tempo. In this case, the result was 60.

3b) The number you obtain here shows you how many notes of the note value given on the left-hand side of the equation will occur in a minute. You can divide this number by 60 to obtain the length (in seconds) of each note. In this case, the result was one full second per quarter note triplet.

4) Find the fraction of a beat that is taken up by the note value given on the right-hand side of the equation. In this case, it was 3/4... the length of a dotted eighth note.

5) Multiply the number obtained in step 3a by the reciprocal of the number obtained in step 4. In this case, the product was 80.

The number produced by step 5 is the new tempo (again, based off of the quarter note or dotted quarter note).

Hope that made sense.

It's really not as difficult (or abstract) as you are making it out to be. The equal sign is there only to obtain the proper relationship between the two tempi. In this case, the second tempo was exactly 8/9 of the original tempo.... this would be the case NO MATTER what the original tempo was. Had the original tempo been 180 bpm, the quarter note triplet to dotted eighth equality would still have reduced the tempo to 8/9 of its original value... or 160 bpm. This is simple ratio-nal arithmetic.

In His love,
Nate

[paaskanama]

Originally Posted by KeyboardFreak

By a quarter note triplet, do you mean THREE quarter note triplets or ONE???

A quarter-note triplet means the grouping of "three in the space of two" quarter notes; therefore, the expanse of two beats.

Originally Posted by KeyboardFreak

If you were moving at 60 bpm beforehand, the quarter note would span the space of one second of time... then if you saw the Q=H sign, then...the half note would take up that same one second of time. This would have the effect of doubling the effective tempo

Right. Similarly, if a quarter note were equated to a dotted quarter note (not a "swing feel" notation), you would go from a 4/4 time "signature" to a 12/8, right? So, where 8 eighth-notes were played before, now 12 must fit. Therefore, the tempo increases because more has to be fit in that space.


In the quarter-note triplet example (and using a broader example):

Think of the "events" that occur within the space that the triplet covers: for example, two quarter notes and two eighth notes - all occuring within the space of 2 beats.

When what was a quarter-note triplet is now equaled to a dotted eighth, the "events" are now condensed to occur within a dotted eighths' (3/4 of a beat) time.

If the BPM was 96 in this case, it now equals 108.


-paaskanama

[Nate]

Originally Posted by paaskanama

A quarter-note triplet means the grouping of "three in the space of two" quarter notes; therefore, the expanse of two beats.

I know what it means, the question is how many of the three notes included in a set of quarter note triplets you are counting. Usually, on scores of the classical persuasion, you will see exactly one note on either side of the equality.

If, however, there were THREE quarter note triplets (meaning, one set of three quarter notes, that takes up a full two beats of music) on the left side of the equality, the tempo would change drastically... much more so than you are proposing.

If there were two beats in the old tempo that were equated to three-fourths of a beat in the new tempo, the equation would be like this: 2x = 3/4y, where x is the original tempo and y is the effective new tempo. In this case, it would work out to 256 bpm for the new tempo.

What you did was equate a single quarter note triplet (one note out of a set of three) in some new tempo to a dotted eighth note in the original tempo. I know this is not what you intended to do, but it is, nevertheless, what you did. Your equation was 3/4x = 2/3y, which turned 96 into 108. This isn't correct... by either of our understandings of the term "quarter note triplet". You take the term to mean the entire set of three notes, I take it to mean one member of said set... but either way, your results don't match up.

I'm not trying to bicker with you, just trying to clear things up.

Similarly, if a quarter note were equated to a dotted quarter note (not a "swing feel" notation), you would go from a 4/4 time "signature" to a 12/8, right?

Usually, yes. More specifically, you would be going from a simple metre (2/4, 3/4, 4/4, 2/8, 4/16, etc) to a compound metre (6/8, 9/8, 12/8, etc). The basic pulse would no longer come in sets of two eighth notes (one quarter note), but in sets of three (one dotted quarter note).

So, where 8 eighth-notes were played before, now 12 must fit. Therefore, the tempo increases because more has to be fit in that space.

Again, this depends on your definitions... and the way that you choose to measure tempo. My answer to your analysis would be, "No, the tempo does not change at all."

Typically, you measure tempo in a simple metre piece of music from the quarter note (or the eighth note if it's 4/8 or something, and so on); in compound metre, however, you measure tempo from the dotted quarter note. You see that in either case, you're simply measuring the frequency of the basic pulses. If a quarter note in simple metre is equated to a dotted quarter note in compound metre, the tempo really remains the same. The basic pulses (the quarter in simple, the dotted quarter in compound) are still coming at exactly the same rate, it's just that there is more stuff (eighth notes, in particular) happening in the span of the basic pulse in the compound metre.

Does that make sense?

When you play a set of eighth note triplets in simple metre, does the tempo change? No; you simply play three notes in the span of time that you would usually play two notes. Think of the change from quarter note pulses to dotted quarter note pulses as a GIANT set of eighth note triplets. It's as if you suddenly switched, halfway through the song, to playing three notes to every pulse instead of two. The tempo doesn't change, only the division of the beat does.

In His love,
Nate

[paaskanama]

Originally Posted by KeyboardFreak

The question is how many of the three notes included in a set of quarter note triplets you are counting. Usually, on scores of the classical persuasion, you will see exactly one note on either side of the equality.

Yes, I should have clarified earlier that this was in relation to contemporary writing. FAR from typical classical behavior.

The bracket I've been looking at indicates that A quarter from a quarter-note triplet would then equal a dotted eighth. So, just the single note, not the grouping of three.

Originally Posted by KeyboardFreak

In either case, you're simply measuring the frequency of the basic pulses. If a quarter note in simple metre is equated to a dotted quarter note in compound metre, the tempo really remains the same. The basic pulses (the quarter in simple, the dotted quarter in compound) are still coming at exactly the same rate, it's just that there is more stuff (eighth notes, in particular) happening in the span of the basic pulse in the compound metre.

It makes sense, but I'm not thinking that way. I'm determining the BPM for the quarter (or whatever initial subdivision) on both sides. That being: if Q=120 and the modulation is Q = dotted Q, then what does Q equal in BPMs? (180) That's where the tempo increase is noticable.

Originally Posted by KeyboardFreak

When you play a set of eighth note triplets in simple metre...you simply play three notes in the span of time that you would usually play two notes.

The implication I'm addressing, though, is not dealing with triplets at all. It's a distinct change from a simple to compound meter where ALL instruments adopt the modulation.

Using a section from Dream Theater's "Hell's Kitchen" as an example: The latter sections of the song alternate 12/8 and 10/8 signatures, but there's 1 bar that is marked 4/4. When the rhythm goes from the 12/8 groove to the 4/4 bar, it feels like the piece speeds up momentarily.

What really happens is: 4 eighth notes get cut out! In this case, what would be the metric modulation of the dotted eighth?



This is really making my head hurt all over again, but thanks for taking the time to give me feedback on this, Nate. The whole thing started with Steve Vai and a song of his called "Incantation." His exact words about (the metric modulation within) the song is "the quarter note triplet becomes 'one'...the eighth note."

Here's the link to a page on his website where he discusses subdivisions of the beat, metric mods, polyrhythms, et al.

http://www.vai.com/LittleBlackDots/tempomental.html

I want to find a website or two that break this analysis down more directly and conclusively, but I've been coming up empty so far. Any references?


-paaskanama

[Nate]

Originally Posted by paaskanama

Yes, I should have clarified earlier that this was in relation to contemporary writing. FAR from typical classical behavior.

Actually, they're identical, judging from what you said in your next paragraph.

The bracket I've been looking at indicates that A quarter from a quarter-note triplet would then equal a dotted eighth. So, just the single note, not the grouping of three.

This is what I have been assuming all along. I'll check my math again, I could be messing up somewhere.

In His love,
Nate