ABRSM Grade 5 Theory Transposition by Numbers

One of the most difficult parts of the grade 5 theory exam is to get the intervals correct OK we can probably tell that it's a fifth but is it a perfect fifth or a diminished fifth or what?  Similarly with transposition again yes transposing a piece up or down by a minor 3rd say means that the notes are displaced up or down a line or space but what about the accidentals, even worse what if it asks for a key signature change. A while ago when I was first studying music with the OU I hit on the idea that a lot of the confusion could be clarified by using numbers instead of letters. This works really well for intervals and transposition

First number the notes of the Chromatic scale starting with middle C as zero

So we have the table

              Db     Eb        Gb     Ab     Bb
          C C# D D# E F F# G G# A A# B 
       
          0  1   2   3   4 5 6    7  8   9 10 11

Each number is the number of semitones above middle C that the corresponding note is.


Then for transposition for grade 5 the main ones are

Up or down a major 2nd (2 semitones) simply add or subract 2 modulo 12 to the numbers then use the first line to work out the notes


Up or down a minor 3rd (3 semitones) simply add or subtract 3 modulo 12 to the numbers then use the first line to work out the notes

Finally up or down a perfect 5th (7 semitones)

So for example suppose we want to transpose a given melody down a perfect 5th we subtract 7 then add
12 if the new number is less than zero or subtract 12 if the new number is greater than 12. 


             Db     Eb        Gb      Ab     Bb
          C C# D D# E F F# G G# A A# B 
       
          0  1    2  3   4 5 6    7   8  9 10  11

Performing the arithmetic gives for C 0 -> -7 then add 12 to get 5 (you only have to do this once)
then just start from 5 to give



              Db     Eb       Gb      Ab      Bb
          C C# D D# E F F# G G# A A# B 
       
          0 1  2 3  4  5  6   7   8  9 10 11
          5 6  7 8  9 10 11 0   1  2 3  4

So that we see eg that C goes down to F (5) or Ab goes to D (2) and so forth.

So writing this table out and doing the appropriate arithmetic will give you a fail safe method of
transposing accurately especially for those tricky accidentals.

Of course you have to remember if the original melody has a given key signature to take into account
the key notes. Thus for Bb major the accidentals are Bb and Eb so every time you see a B or an E remember
these are Bb and Eb

This also works for key signature changes so suppose I start in Eb and I want to go up a major second
Eb is 3 according to the table add 2 to give me 5 the key signature is now that for F major ie simply with Bb

In the next post I will show how a similar technique can be used for intervals