These are the modes of the western major scale along with the interval formula.  For more information, see Lesson 21

For playing in real time, this representation of our interval numbers is better because the fifth interval can now be used as a reference point to find the 4, -5, +5 and 6. This representation is only used when describing melodies as this entire course is focused on music theory  which improves your real time playing. A -5 is less confusing and easier to locate than a #4 with your real time playing. When you're playing in real time, you're not thinking about modes, you just want to locate the notes easier.

The mode order going from darkest to brightest along with the associated chord in the key of C. This table is handy when choosing chord progressions. If you're not sure how these are derived see Lesson 22. Observe the first and third columns are key independent, while the middle column is in the key of C.

The Lydian is the brightest because it's equivalent to an Ionian #4. Observe the column in the middle is in the key of C,  the Mode Name and Chord Number columns are key independent.

A bright sound is felt when rising in pitch, a sad sound is felt when we falling in pitch. This is an objective feeling we all feel when listening to interval movement. If you're writing a melody, rising in pitch will give the melody a life, falling in pitch gives the melody a sad or left bright feel.