Numeric Formulas for Scales and Modes (All You Need To Know)

2. The Major scale as point of reference

The major scale is the reference point to compare other scales against. Formulas like 1-b2-b3-4-5-b6-b7 show you how different this scale is compared to the regular major scale.

C-Major in generic numeric representation which describes all major scales:

    C D E F G A B
    1 2 3 4 5 6 7
    

3. Meaning of numbers 1 through 7

3.1. Very specific intervals of the major scale

    1 = perfect unison (0 half-steps)
    2 = major second   (2 half-steps)
    3 = major third    (4 half-steps)
    4 = perfect fourth (5 half-steps)
    5 = perfect fifth  (7 half-steps)
    6 = major six      (9 half-steps)
    7 = major seventh  (11 half-steps)
    

3.2. The count of alphabet letters spanning the interval that it represents

E.g., in the key of C–Major, scale degree number 3 spans 3 letters: C–D–E. Scale degree 5 spans 5 letters: C-D-E-F-G

3.3. Scale degrees of the major scale

For example, the 3rd scale degree of the C–Major scale is note E, and in E–Major that’s note G#.

4. How to read formulas of modes and scales?

We take the Key of any major scale and apply the formula. The b sign lowers the note it is associated with by one half-step. The # sign raises the note by one half-step.

Example A: 1-b2-b3-4-5-b6-b7 (Phrygian)

We lower 4 notes as prescribed by the formula. In the key of C-Major as the base the result is C-Phrygian C-Db-Eb-F-G-Ab-Bb. In the key of Bb-Major (Bb-C-D-Eb-F-G-A) as the base the result is Bb-Phrygian Bb-Cb-Db-Eb-F-Gb-Ab.

Example B: 1-2-b3-#4-5-b6-7 (Hungarian Minor)

In the key of C-Major as the base the result is C-Hungarian Minor C-D-Eb-F#-G-Ab-B.

5. How to properly map numbers to note names/letters?

E.g. Ultraphrygian mode 1-b2-b3-b4-5-b6-bb7. C-Ultraphrygian maps to C-Db-Eb-Fb-G-Ab-Bbb. There’s a direct correlation between numbers and letters. Each number equals the number of alphabet letters it spans. If numbers do not repeat then letters do not repeat, and vice versa. The maximum number in this example is 7. So the 7th letter in a row after C is B. The double-flat sign (bb) tells us that we must lower the 7th degree by 2 half-steps down. We write Bbb and not A because the notes of the scale must be spelled correctly, as per numeric formula - without repetition (or with if numbers do repeat). In our case, it would not be correct to write this scale as C-Db-Eb-Fb-G-Ab-A and it would not be correct to write Fb as E.

6. Systematic mode naming scheme

The idea is to use a well-known mode name as a base (implying its formula) and add alterations on top of it without introducing a new unheard name. The mode becomes the reference point and with alteration signs (b and #) we alter the mode’s formula and not the major scale’s.

6.1. Explicit alterations

E.g., Lydian #5 means, you take normal Lydian formula (1-2-3-#4-5-6-7) and add #5 alteration on top. And you’ll get 1-2-3-#4-#5-6-7 and not 1-2-3-4-#5-6-7.

6.2. Multiple perspectives

E.g. the 2nd mode of Melodic Minor can be viewed as both Dorian b2 and Phrygian #6

    2nd mode of Melodic Minor 1 b2 b3 4 5 6  b7
    Dorian b2                 1 b2 b3 4 5 6  b7
    Dorian                    1  2 b3 4 5 6  b7
    

    Phrygian #6               1 b2 b3 4 5 6  b7
    Phrygian                  1 b2 b3 4 5 b6 b7
    

6.3. Multiple alterations

Examples:

• Ionian #2 #5 (1-#2-3-4-#5-6-7)
• Dorian b2 #4 (1-b2-b3-#4-5-6-b7)

Just apply alterations one by one to the base mode's formula (Ionian and Dorian in these cases)

7. Special words in the names of modes

7.1. Augmented

E.g. Lydian Augmented. In the context of scale names this word refers specifically to the interval of a perfect 5th and means that it must be widened by 1 chromatic half-step. Result is 1-2-3-#4-#5-6-7

7.2. Harmonic

E.g. Harmonic Minor (1-2-b3-4-5-b6-7). In western tonal theory, a scale requires something called a leading tone to achieve what is known as resolution. The word leading tone refers to the natural seventh degree which is only 1 half-step away from the tonic and thus creates a lot of tension for resolution. Classical tonality also wants to have a major V-chord instead of a minor v-chord, and to create a major V-chord you need a natural 7th. However, not all scales labeled as Harmonic have a major V-chord because the primary reason for the word is the presence of a leading tone.

7.3. Natural <X>

E.g., Phrygian Natural 6, or Phrygian ♮6. In this case it tells you that you need to cancel the alteration of a sixth and use a natural 6th resulting in formula 1-b2-b3-4-5-6-b7 (regular Phrygian is 1-b2-b3-4-5-b6-b7). Word Natural can point to any degree, not just 6th. And the alteration it cancels can be b or #.

7.4. Dominant

E.g. Phrygian Dominant (1-b2-3-4-5-b6-b7). This word refers to 2 scale degrees at once: the natural 3rd and flat 7th. These two degrees form a tritone interval which is the main feature of the Dominant 7th chord which is usually played on the 5th scale degree called the Dominant degree. But here we have the components of the Dominant 7th chord (1-3-5-b7) on the 1st degree of the scale which hints us that we can choose to play this scale over a dominant V-chord of another scale. For example, if I play a song in C-Harmonic Minor scale (1-2-b3-4-5-b6-7) I can play G-Phrygian Dominant over its V7-chord.

7.5. Composite names

E.g., Lydian Augmented Dominant (1-2-3-#4-#5-6-b7). Lydian scale name tells us there's a #4, Augmented signifies that there's a #5, and Dominant tells us that there's natural 3rd and flat 7th.

7.6. Mixed-base ambiguity

E.g., Locrian #2 (1-2-b3-4-b5-b6-b7). Original Locrian mode (1-b2-b3-4-b5-b6-b7) contains a b2. When you see Locrian #2, you might get confused. The apparent difference between a b2 and a #2 is 2 half-steps (b2 → ♮2 → #2). Do you play what's in the middle, the natural 2, or what is above, the #2? It depends on what you consider the base—the major scale, or Locrian in this case. Playing #2 from above means we are mixing bases, i.e. everything from Locrian, except for the 2nd degree. But playing natural 2 means we are using only one base—the named mode—which is Locrian in this case, and apply alterations to that scale. So #2 applied to a b2 gives a natural 2 (♮2). And that's how most of the formulas are written—with one base, not mixed bases. You can rewrite this as Locrian ♮2 (or explicitly Locrian Natural 2) to avoid confusion and ambiguity. Another option is Aeolian b5.

7.7 Pros & Cons

Explicit alterations

(+) reduced guess work (if you are not really sure what special words are referring to)
(-) mixed-base ambiguity

Special words

(+) friendly scale names
(+) suggests application/function
(+) highlights feature
(-) must know the meaning behind them

8. Exceptions

8.1. Double-flat sign & mixed bases

E.g. Superlocrian bb7 (1-b2-b3-b4-b5-b6-bb7). The scale it alters is Superlocrian (1-b2-b3-b4-b5-b6-b7) which has a b7, so following common conventions, Superlocrian bb7 would have to have 3 flats (b7 + bb7 = bbb7). But instead, bb7 is written in relation to the major scale, thus mixing bases. If you see a bb sign on any degree in some formula, more likely than not, it's relative to the major scale.

8.2. Chromatic note

E.g. the Minor Blues scale (1-b3-4-b5-5-b7 : C-Eb-F-Gb-G-Bb). The formula contains duplicate numbers, the 5th. As a consequence, the letters also have duplicates. The reason for this is the presence of a chromatic note between degree number 4 and 5. There's simply no other way than to repeat the number and the letter, because we have 3 chromatic notes, but only 2 numbers and 2 letters to represent them.

8.3. 8+ notes in the scale

E.g. Bebop Dominant scale (1-2-3-4-5-6-b7-7 : C-D-E-F-G-A-Bb-B). Apart from having the chromatic note between degrees 6 and 7, here we have an octatonic scale, which consists of 8 notes. The major scale which is used as a reference for all these numeric formulas has 7 notes, and we cannot represent 8 notes with 7 numbers without repeating at least one number.

8.4. Inversion-oriented

E.g., the Tritone scale (1-b2-3-b5-5-b7 : C-Db-E-Gb-G(♮)-Bb). The reason for repeating numbers in this case has to do with the so-called Petrushka chord, which is a combination of two major triads, one of which is in root position C-E-G and the other one is in 2nd inversion Db-Gb-Bb (notice 2 letters are skipped between D and G). And writing it this way is obviously a reference to this interesting dissonant chord.

If we were to rewrite this scale as 1-b2-3-#4-5-#6 then the equivalent notes with the root of C would be C-Db-E-F#-G-A#. All perfectly fine, except that if we now extract the letters which represent both triads, then we get C-E-G (correct) + Db-F#-A# (incorrect spelling of 2nd inversion, notice only 1 letter is skipped now - this spelling maps to the root position) and hence we cut the reference to the Petrushka chord and do not outline the fact that it's a sum of 2 major triads in specific inversions.

9. Quiz

• What's the next letter after F? → G
• How many half-steps between 5 and 6? → 2
• How many half-steps between b5 and 6? → 3
• What are equivalent letters for degrees 5 and 6 for the root of C? → G and A
• What are equivalent letters for degrees b5 and 6 for the root of C? → Gb and A
• For the root of C, what are equivalent letters for degrees b6 and bb7? → Ab and Bbb
• How many half-steps are there between b6 and bb7? → 1
• How many half-steps are there between bb3 and b4? → 2