# Scratching my head over Tempo map and Fermatas

I can’t get a grip on the math behind how fermatas effect tempi. I’ve a attached a small project that I’ve been experimenting with.

Can someone explain what’s going on here?

I’ve set my initial tempo to Q=200 to make the math easy;

I’ve added a “Normal” fermata over the half note in m3.

I’ve set the default values in Library|Playback Options|Timing for Normal Pauses to “Hold duration % = 50” and “Gap duration % = 50”, and checked "Play back pauses (These are the default settings in the “Solo piano” Project Template that I used to create this file).

At the fermata in on the downbeat of bar 3, tempo changes to Q=67 from beat 1 to 2 .

On beat 2 the tempo goes back to Q=200, at which point the note is cut off (aka the “gap”).

I’d like to understand the math regarding the relationships between the “Hold duration %” and “Gap duration %” values, and the current tempo?

I love that I can change these values on an individual basis in the properties panel, but until I can understand the math, I’m getting tempo maps that don’t do what I want, which is to achieve the pauses and gaps that I hear in my head (let’s not talk about the “voices” I hear in my head).

Experiment 2

In bar 4 I’ve added a “Long” fermata on beat 1, and the default values are “Hold duration % = 100” and “Gap duration % = 100”. The resulting tempo changes here are Q = 50 (at the pause, beat 1), and Q = 100 (at the gap, beat 2). But try changing the “Gap duration %” to 75, and the gap tempo shows Q = 134; change “Gap duration %” to 40, and the gap tempo shows 200(!)

Ok, math-heads, have at it.

Sorry for the long-winded post.

Thanks.
Fermatas Last Theorem.dorico (688.6 KB)

Let t represent the given tempo, h represent the hold duration % and g represent the gap duration %.

The tempo during the first half of a note with a fermata is (t / 2) * (100 / (100 + h)). This is when you hear the note.

The tempo during the second half of a note with a fermata is (t / 2) * (100 / g). This is when the gap occurs.

For the first fermata in your example, t = 200, h = 50 and g = 50. So the tempo during the first half of the note is (200 / 2) * (100 / (100 + 50)) = 100 * (2 / 3) = 67 and the tempo during the second half of the note is (200 / 2) * (100 / 50) = 100 * 2 = 200.

For the second fermata in your example, t = 200, h = 100 and g = 100. So the tempo during the first half of the note is (200 / 2) * (100 / (100 + 100)) = 100 * (1 / 2) = 50 and the tempo during the second half of the note is (200 / 2) * (100 / 100) = 100 * 1 = 100.

If the gap duration of the second fermata is changed to 75%, then the tempo during the second half of the note is (200 / 2) * (100 / 75) = 100 * (4 / 3) = 133.

If the gap duration of the second fermata is changed to 40%, then the tempo during the second half of the note is (200 / 2) * (100 / 40) = 100 * (5 / 2) = 250.

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Thanks John! That is impressive… and makes my brain hurt!

I’m still mulling this over, and would appreciate advice on how people use these values in their scores, but I’d love further clarification on why the formula for the Hold h, is different than the formula for the Gap g? This reminds me of the Martin Gardner puzzle books my brother bought me that leave me feeling extremely inadequate…

When a fermata is added to a note, its duration is increased by h percent of its original duration. The gap which follows lasts g percent of the note’s original duration. Using the default values for hold duration and gap duration, if a note’s original duration is two seconds and a normal fermata is added, the new duration is three seconds and the gap lasts one second. If a long fermata is added, the new duration is four seconds and the gap lasts two seconds.

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All you need to worry about is that 50% means “half the note’s duration”, so a 50% hold is like a dotted note.

At a tempo of 120, the hold goes down to 40 bpm, which is 1/3 of 120. Without the fermata, let’s say our note plays for 1 second at 120. So at 40 bpm, it would be 3 times as long. However, in order to make space for the gap, Dorico only plays the note for half its duration. Half of 3 seconds is 1.5 seconds! And that’s ‘half as long again’.

But we still have time for ‘the silent second half of the note’. (E.g. the gap.) At normal tempo, half the note is 0.5 seconds. That’s a 50% gap of the note’s duration.

In short: you don’t really need to know what the tempi are. Just think of the hold and gap in fractions of the note’s duration.

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Thanks John and Ben for your very kind and helpful replies! Sorry for this long follow-up post, but I’ve become a bit obsessed over this. I’m trying to help a friend get his playback just how he wants it in order to publish some scores.

After possibly getting my head around fermata playback, I have some observations. Feel free to correct me where I’m off base.

If I understand how fermatas work correctly in Dorico, and display in the Tempo editor:

Regardless of the note value used in the tempo equation, Dorico displays the tempo as a quarter note Beat unit in the Tempo editor.

If you change the Beat unit of a metronome mark in your score via the Properties panel, it doesn’t update in the Tempo editor. Changing the bpm does get updated, but doesn’t correct a change in Beat unit… You need to delete the metronome mark and recreate it to update the Tempo in the Tempo editor.

I’m rephrasing what John and Ben so kindly explained, for my own obfuscation:
To calculate how a fermata over a note affects it’s timing, you must, in effect, think of that note as divided into 2 equal parts, the first part is the Hold, and is assigned a timing based on the “Hold duration %”; the 2nd part is the (non-sounding) Gap, and is assigned a timing based on the “Gap duration %”.

Here are formulae from a non-math guy, for the resulting tempo change(s) that will last the DURATION OF THE NOTE UNDER THE FERMATA (whether eighth note, quarter, half, etc.):

Hold duration % is added to the entire note’s (100%) value. In other words, it lengthen’s the note by this percentage. 0% means “no lengthening”. Negative values are not allowed.

(Current Tempo) divided by 2, divided by (100 + Hold duration percentage) * 100

Gap duration % is multiplied by the entire note’s value.

(Current Tempo) divided by 2, divided by (Gap duration percentage) * 100

A value of 0 is allowed here, and might surprise you if you compare what happens to your tempo map if you change between a value of “0” and “1”. I think this could be handled more elegantly; namely, don’t allow a value of “0”, but instead, toggle the Gap duration % to “off”.

A couple example values for a fermata over a half note:

A Hold % of 0 and a Gap % of 100 will each result in the same tempo change (each will last a quarter note at the prevailing tempo).

Similarly, 50 and 150 will be the same length, etc.

Neither % value normally affects the other, except sometimes they do - try entering Hold duration %=0, and Gap duration %=0, then change the Gap duration % to “1”).

I’m assuming the full length of a note under a fermata can not be of shorter duration than the note without a fermata?

The % values are relative to the length of only half of the actual note value. So the value of the note (16th, 8th, quarter, half) under the fermata acts as a multiplier of length of the fermata. Again, the fermata consists of 2 parts, the “hold” and the “gap”.

I’ve found that is some scenarios changing the note value under a fermata fails to change the resulting tempo map intermittently. See attached video.

In the tempo map, the blue area shows the part of the bar affected by the fermata?

The “Range max.:” value in the Tempo lane seems to constrain the upper limit of the tempo changes ( and consequently that the fermata can impose). So if you enter an arbitrarily low number here it can alter the effect of a fermata on the tempo, possibly inadvertently. What is the use case for entering a number here? If it’s for display convenience only, it probably shouldn’t change the actual tempo values…

I’ll be quiet now…

Your most recent post raises several issues that were not mentioned in your original post. Allow me to start by simplifying the formulas I gave before while handling the case of a fermata without a gap.

Once again, let t represent the given tempo, h represent the hold duration % and g represent the gap duration %.

If g = 0 or the property Hold only is activated, then the tempo during the entire written duration of a note with a fermata is 100 * t / (100 + h).

Otherwise, a note with a fermata is heard during the first half of its written duration and the gap occurs during the second half of its written duration.
The tempo during the first half is 50 * t / (100 + h) and the tempo during the second half is 50 * t / g with a maximum tempo during the gap of 500 quarter notes per minute.

Since h is not allowed to be less than zero, the played duration of a note cannot be shortened be adding a fermata to it.

The beat unit in the tempo editor always seems to be a quarter note, and lowering the range maximum only clips the displayed tempos without affecting the actual tempos.

As your video shows, if you have multiple notes with different written durations which start at the same rhythmic position and a fermata is applied to any of them, the tempo changes only apply to the note with the shortest written duration. The longer notes will continue to sound during the gap following the shortest note, so this situation is probably not very useful. The most common situation when you have multiple notes with different written durations is to have them all end at the same rhythmic position and apply the fermata to the note with the shortest written duration. Then none of these notes will sound during the gap.

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