If done correctly, the start ring will be red (which is blue if playing a normal song).

Song	How to unlock
CONFLICT	Chapter R: Tap 13 times the word 'conflict'.
CYTUS ALIVE SONGS	Chapters 1-10: On the chapter selection screen, press the green (for easy) or red (for hard) song count indicators.
ENTRANCE (two songs)	Chapter 2: Slide the shade of either left or right wing towards the center. Left wing for Ver. A, right for Ver. B.
FIRST GATE	Chapter 9: Tap on the words 'FIRST GATE', the word 'OVERDRIVE' will appear in red.
L (two songs)	Chapter 7: Press and hold the big letter 'L', wait until it turns red, when you can play ver. A (or blue for ver. B).
MUSIC. THE ETERNITY OF US	Chapter K: Clear "Where you are not". Song is unlocked permanently.
PRECIPITATION	Chapter 2: Slide 2 fingers down between the title. Repeat for hard mode.
SAIKA	Chapter 3: Tap the 2 Japanese characters until it turns blue. Tap center to start. Repeat for hard mode.
SLIT (two songs)	Chapter 8: Slide diagonally down/left or diagonally down/right ('SLIT O' or 'SLIT I').
VERVERG	Chapter 1: Wait for the candlelight to turn orange then tap. Repeat for hard mode.
WHERE YOU ARE NOT	Chapter K: Play through all songs in Chapter K. The song will be permanently unlocked.