Musical Modes and Mathematics
Advancing musical theory through maths and help you understand modes on the go!
I Do Play Light Music Aesthetically Loud!
Haha you might be wondering what does this have to with modes? That’s just an attempt to adapt the mnemonic pattern for each musical mode.
Ionian > Dorian > Phrygian > Lydian > Mixolydian > Aeolian > Locrian
Now you don’t need to rote learn the name if you can understand the logic behind this.
The Foundation
Modes are essentially the starting point that you play a scale. Generally, you start C major scale at C and then end at C. But if you chose to start the C major scale at D then you will get D E F G A B Ch Dh (‘h’ refers to higher octave), which corresponds to the second mode of C major known as D Dorian.
Notice how the scale name now changes to reflect the starting note instead.
How many modes are there in a scale?
Depending on a scale, there will be additional N-1 modes apart from the original, where N is the total unique keys in an octave of the scale.
For example, if we take G major scale G A B Ch Dh Eh F#h Gh then you get N = 7 unique keys, therefore the additional modes will be N-1 = 6 modes.
If you use the C Major Blues scale C D E♭ E♮ G A Ch then you will get N = 6, hence the additional modes will be equal to N-1 = 5 modes.
If we include the original scale, we can say that there are a total of N modes available for us.
How to play different modes in a particular key?
Apart from the C Ionian or C Major scale, to be able to identify C Dorian, C Phrygian,…, till C Locrian requires us to do quite a bit of mental somersault to shift the scales and then identify different keys that were changed. Plus the mode names does not help make this easier as well haha.
Convert to Numbers
That’s why to understand this more intuitively, I first convert the tones to numbers from 1 to 12. This way we can look at the relationships between modes and compare them better.
Please note that 1 does not always have to start at C as shown above. We can start 1 at the first note of whatever scales we are playing.
So let’s write down how C Major scale looks like in the number format.
C Major:
- C D E F G A B Ch >>> 1 3 5 6 8 10 12 1
Notice how the gap differences change from 2 to 1 at E to F and then back to 2 again. This will be important in understanding the characteristics of each scale and its mode.
For now let’s map out the remaining modes of C Major scale by listing all scales that have C as part of the scale. You will soon see why some major scales are dropped out since they do not have C as part of their scales.
Don’t stress too much about the modes name and their values. Our aim is to list down all the modes in the number format.
C Dorian (or B♭ Major scale starting at C):
- C D E♭ F G A B♭ Ch >>> 1 3 4 6 8 10 11 1
Just putting it out here in case it helps. I got confused by this a lot and the thing that I find helped was to rephrase the sentence to the following C Dorian = C is a Dorian (or the 2nd note) in what scale)? The answer is B♭ Major scale where C is the 2nd note.
C Phrygian (or A♭ Major scale starting at C):
- C D♭ E♭ F G A♭ B♭ Ch >>> 1 2 4 6 8 9 11 1
C Lydian (or G Major scale starting at C):
- C D E F# G A B Ch >>> 1 3 5 7 8 10 12 1
C Mixolydian (or F Major scale starting at C):
- C D E F G A B♭ Ch >>> 1 3 5 6 8 10 11 1
C Aeolian (or E♭ Major scale starting at C):
- C D E♭ F G A♭ B♭ Ch >>> 1 3 4 6 8 9 11 1
C Locrian (or D♭ Major or C# Major scale starting at C):
- C D♭ E♭ F G♭ A♭ B♭ Ch >>> 1 2 4 6 7 9 11 1
Understanding the hidden patterns
Now that we have all the modes converted to numbers, I will put them in a simple table format for easier mapping.
From this table, we can see that there might be some hidden pattern behind this which could be uncover further. To uncover different patterns, I experimented with the following layout to see what insights we can gather.
From this ladder-like table, it might be seem a bit difficult to decipher the pattern, but here is the key insight I found.
The blue boxes are the inverse progression of C Major scale
If we were to play the scale backward. Now this may sound counter-intuitive since the reverse of C Major scale should be the same as C major scale no?
You see by reverse progression, I mean we take the same gap going forward as well as going backward. This way it seems like you are playing the C Major scale in the mirror world!
C Major: we plus 2 every times except for 3rd & 7th times where we plus 1
1 3 5 6 8 10 12 1
Inverse C Major: we minus every times except for 3rd & 7th times
1 11 9 8 6 4 2 1
Note: Don’t worry! The reason we are able to minus 1–2 = 11 in this world is because in this music world we only have 12 tones. The easier way to look at this is to count backward on the keyboard to get the same result as well.
If we look closely at C Inverse Major Scale
Ch B♭ A♭ G F E♭ D♭ C
We now recognise this as basically the C Phrygian in reverse!
What does this have to do with modes?
Well if you want to play the Dorian mode of C Major scale, you just go to the second position on the C Inverse Major scale (in this case it’s B♭ Major scale) and play that scale or add different tones from that scale.
In summary, if you are playing in any particular key with root at ‘1’ then use the following guide to find the desired mode:
11 —or original key minus a whole tone — Dorian mode scale
9 — or original key minus 2 whole tones — Phrygian mode scale
8 — or original key minus 2 and half whole tones — Lydian mode scale
6 — or original key minus 3 and half whole tones — Mixolydian mode scale
4 — or original key minus 4 and half whole tones — Aeolian mode scale
2 — or original key minus 5 and half whole tones — Locrian mode scale
1 — or the original key — Dorian mode scale
Note: Since the C Inverse Major scale is the same as Reverse Phrygian mode, this means that just remembering the Phrygian mode and scale will also be enough to get you to shift to any other modes! Each position of the reverse Phrygian mode correspond to the scale needed for each mode.
Is this applicable to other types of scales?
Yes! This method look at the characteristics of the scale and their progression to determine the reverse order of that scale so that we can find different modes. If the scale progression is different, it’s inverse scale will also be different. Hence, the modes will be different as well.
A couple of simple examples could be…
- Whole tones scales
Whole tones scale like the C Whole tones scale C D E F# G# B♭ Ch can be denoted as 1 3 5 7 9 11 1. This means that the progression between each tone is always 2. When we find the Inverse C Whole tones scale then the result will be 1 11 9 7 5 3 1 which is the same as the original scale but in reverse!
This means that whatever mode we choose for whole tone scales, the notes will still be the same! Truly one of the most wonderful scales to study about modes (as well as practising pitch identifications).
- Chromatic scales
What about the chromatic scales? e can also denote the scale as follows: 1 2 3 4 5 6 7 8 9 10 11 12. Hence, the progression between each note is always 1. The reverse is thus the same: 1 12 11 10 9 8 7 6 5 4 3 2 1. This is the same as the whole tones scales except that now you have more number of modes available for you but seemingly with little variations among themselves.
- Blues Major scales
Blues scales can be denoted as 1 3 4 5 8 10 1. This scale has different progression gap between each note. Therefore, the reverse or inverse Blues major scales will mirror the same way backward: 1 11 10 9 6 4 1.
So if we are looking at C Blues Major scale: C D E♭ E♮ G A Ch then its Inverse C Blues Major scale will be: Ch B♭ A A♭ F E♭ C. What this suggest is that the C Dorian Blues scale will be based on the B♭ Major Blues scale.
C Dorian Blues scale:
- C D♭ D F G B♭ Ch >>> 1 2 3 6 8 11 1
Additional Observations
1) The number of different keys is related to the modulo of modes
The number of different notes from the original C Major scale in each column corresponds to (the different note number) mod (N-1).
For example:
In column 2, C Major original value is 3 or D. In other modes there are 2 other modes where the values are different (i.e. 2 or C#) and the number of 2 that appears equal to 2 mod (7–1) or 2 times.
This applies to other column as well.
- 3rd column — original value = 5 or E; the different note = 4 and the number of times the different notes appear = 4 mod 6 = 4
- 3rd column — original value = 6 or F; the different note = 7 and the number of times the different notes appear = 7 mod 6 = 1
- etc.
2) The sum of the values might have the potential to determine the mood
This is just a hypothesis that needs to be validated but if you sum up the total of all tones in each mode, we might be able to classify them by mood.
So based on this we have the Ionian and Lydian modes with the highest sums & Phrygian and Locrian modes have the lowest sums. Not sure how accurate this might be given all the subjectivity involved.
There are also a flaw apart from the subjectivity that I can see in this model which is if we have scale that prioritise higher values notes only or what if a scale just have B (denoted by 12), would not this makes this model inaccurate?
Conclusion
Hope this guide help you (as well as me haha) to understand modes and not resort to remember them and forget them when you really need them during practice.
Update: I realised that as a kind I used to count the keys up by 1.5 whole tones to transpose a given key scale to a minor scale since I could not remember all the scales. Given that minor scales are also a kind of a mode, it made me appreciate the profound pattern that encompasses music even more.
