Musical Notes

Musical notes were first seen at the end of Episode 5 on something that appears to be a spell circle for Sayaka. It has been much debate about whether the notes actually mean something, but at least many familiar musical symbols have been identified.

Musical Scores

Sayaka's Spell Circle

Sayaka's spell circles consist of circular staves with stylized musical notes: usually three identical concentric ones, though in some cases just a single circle can be seen. The circles are typically seen spinning, though whether they all spin in the same direction or not seems to differ: in both her Episode 5 and Episode 8 healing scenes they spin in the same direction, but in the Magia Record game, in her base version's Magia attack the first and third circles spin clockwise, while the second one spins counterclockwise. Their colours vary as well: they are typically glowing in light blue, but, for instance, in Episodes 7 and 8 during her fight against Elsa Maria they are black-and-white in a way that maximally contrasts them against their background, and her Magia Record game transformation features a dark blue underwater circle and a white glowing one. The colours appear to have no correlation to utility.

Sayaka has been seen using them for many purposes, including healing (Episode 5, Episode 8, Magia Record Episode 13), transforming (Episode 5 (BD only), Magia Record game), creating a surface from which she can jump upside-down in mid-air (Episode 7), and attacking (Magia Record game). Additionally, they were seen unrelated to any magic usage in Madoka Magica Mobage Story The Hollow Little Mermaid, simply as an aesthetic animation during the event's intro cutscene. They also appeared on her Magia Record Surfer version Doppel's surfing board.

Sayaka's healing circle in Episode 5
Sayaka's Episode 5 Blu-ray transformation displays musical scores.
Continued
Sayaka's score in Episode 7, used to jump
Some of the healing circles in Episode 8
Sayaka's eye in the train scene in Episode 8
Continued
Continued
Continued
The previous staff, straightened
Mobage, intro cutscene to The Hollow Little Mermaid
Magia Record game, transformation
Continued
Continued
Surfer version's Doppel
Doppel sprites
Sprites from which Sayaka's magic circle is composed in her battle animations
Magia Record Anime Episode 13, healing.
Continued
Fan-made recreation of the straight staff, based off multiple instances of her magic circles and Magia Record game sprites

Sayaka's transformation

In Sayaka's Episode 5 BluRay transformation, outside of the aforementioned magic circles, a few straight staves appear. A video on Nico Nico Douga seems to indicate that the musical notes are from Tekla Bądarzewska-Baranowska's A Maiden's Prayer.

Sayaka's Episode 5 Blu-ray transformation displays musical scores.
The score

Oktavia Von Seckendorff

After Sayaka turns into a witch at the beginning of the Episode 9, various sheets of musical notes appear. Instead of forming a circle, these notes flow through the screen forming the witch's barrier, and later in the episode in the "to be continued" screen after the endcard. The notes seen here are to the music played during the fight with Oktavia.

The music notes shown in Episode 9 are actually to Symposium magarum, the song that plays during the fight with Oktavia von Seckendorff. The music can be heard here.

Notes that appeared at the pre-OP scene of episode 9
Sheet music and a MIDI file based on the episode 9 notes, also being episode 9's BGM

Music Collection

The Music Collection features stylized notes from Sayaka's magic circles in its visual design, both as a background to the tracklist with the notes being mostly strewn randomly without any staff, and as actual circular staves on both discs. The tracks from the circles may be nonsensical as the clefs are not at the beginning.

Tracklist of Music Collection features notes in its background
Music Collection Disc 1
Music Collection Disc 2

Rebellion

During the scene with Hitomi being saved from her Nightmare, Tchaikovsky's Reverie's score appears in the background, paired with the word "Reverie" in Witch Runes.

Rebellion, Reverie

Magia Record

Outside of the aforementioned circles used by Sayaka in many places, her regular version's Doppel features a staff with unidentified sword-like symbols.

Alexandra Kurusu also uses stylized musical notes in her attacks including ones forming a magic circle, though her note design differs from Sayaka's. That being said, the very notes she uses are limited to what Sayaka had, going as far as including the combined 10-note symbol.

Sayuki Fumino's attacks feature a magic circle as well, using regular human-readable notes. However, her circle appears to be completely nonsensical, just placing the notes at random heights and angles.

Staff from Sayaka's Doppel.
A comparison between some notes from Sayaka's and Sasha's magic circles, and their Soul Gem shape
Sasha's magic circle
Notes used in Sasha's attacks
Sayuki's magic circle
Notes used in Sayuki's attacks

Musical notation

Notes

Note	Witch Character Symbol	Pic	Notes
Note Whole			May be a whole note
Note 1/2 or 1/4			May be a half- or quarter note
Note 1/16 or 1/8			May be a eighth- or sixteenth note. Having two beams would imply the latter.
Note Grace			May be a grace note attached to a whole note, descending from G to F.

Rests

Symbol	Witch Character Symbol	Notes
Rest 1/4 or 8	(-)	May be a quarter- or eight note rest
Rest 1/4 or 8 with Forte	(-)	May be a quarter- or eight note rest, with a Witch Character letter F (F) above that may mean forte.
Rest 1/8 or 1/16	(-)	May be an eighth- rest or sixteenth note rest

Other

Symbol	Witch Character Symbol	Notes
Forte		A runic letter F (F) that may mean forte. Below it is a symbol that may be a quarter rest
Fortissimo		Two Witch Character F's (F) that may mean fortissimo
Repeat sign	(-)	A thick bar line with two dots on each side that may be a two-way repeat sign
G-clef (?)		G-clef-like symbol. Based on the top, it's more likely to be a treble clef than a French Violin Clef.