Hi John,

I apologize for being contrary but with all due respect, your calculation is not quite correct. Because the note-value of a quarter note is smaller than the note-value of the dotted quarter note, it is not possible to have fewer quarter notes occurring in the same amount of time as dotted quarter notes if the overall time is to remain the same.
Quote:

If a 1/4 + 1/8 note = 100 then you have = 3 1/8th notes and at 96 tempo each is a value of 32.
Thus if you want to express the tempo in a 1/4 note 96-32= 64.


Since tempo is "beats per minute", to re-map the tempo from one beat type to another, it is necessary to consider that number of different kinds of equivalent beat that occur in one minute and to work through the note type that is equal in each tempo. In this instance, the note type common to both is the eighth note.

A dotted quarter note at a tempo of 96 means that 96 dotted quarter notes occur within 1 minute.

So, since each dotted note contains three eighth notes, then there must be (3 x 96 = 288) 288 eighth notes that occur in 1 minute if 96 dotted quarter notes were played in 1 minute.

Now since it takes two eighth notes to make a quarter note, then that means that there must be (288/2 = 144) 144 quarter notes in minute.

Thus, the above tempo can be expressed a number of ways that are equivalent.

1) dotted quarter note = 96 beats per minute (if the dotted quarter note is the beat unit)

2) quarter note = 144 beats per minute (if the quarter note is the beat unit)

3) eighth note = 288 beats per minute (if the eighth note is the beat unit)

Basically, as the value of the beat note gets smaller, the equivalent tempo must increase because more of the smaller value notes are required to be played in the same amount of time as the larger note. The relationship between note value and tempo is an inverse one.

As example consider marching...

Let's say that in one minute, your left foot "walks" 20 times. This tempo could be written as "left foot = 20 walks per minute"

If we want to convert this to the walking of any foot, the right foot must have also "walked" 20 times if the that's what the left foot has done. Otherwise we wouldn't be marching. This means that the total number of "foot movements" have been 20 x 2 = 40

So a tempo of "left foot = 20 walks per minute" is equal to a tempo of "any foot = 40 walks per minute".

The measured length of time between "any" foot's movement is smaller than the measured length of time between "left foot only" movements. This is the inverse relationship that holds. As the time-value of the defining unit gets smaller, more of that unit must occur if the same amount of time is to be covered as defined by the larger unit.

Regards,
Noel

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