musicians of old

Chapter 888 Five Basic Relationships!

Pitch is just a number!
Fanning's voice echoed in the "lecture hall," gradually taking on an unquestionable air of authority.

The blackboard was quickly painted with an octave of "piano keyboard". Fanning started marking from the leftmost Do and stripped each note of the twelve equal temperament from its former identity and function, simplifying it into an integer from 0 to 11.

C is 0, C is 1, D is 2, and so on until B is 11.

The next octave higher, C, is not marked because the octave difference is considered equivalent.

"Pitch set theory. A pitch set is a combination of arbitrary pitch numbers that forms the sequence of intervals within it. We call it a 'vector'—'interval vector'!"

Fanning looked around the room and saw the confusion and excitement in "their" eyes.

"Interval vectors are also very easy to understand."

The chalk in his right hand swept across the white keys of that "piano keyboard" in one swift motion.

“do-re-mi-fa-so-la-xi-do!!”

The crisp sound of the scale, the incredibly simple C major scale, rang out directly in the classroom.

"The major scale that most people can sing off the top of their heads is just a set of notes."

“How did we identify major scales in the past? We looked at the intervals between the internal notes, such as ‘whole-whole-half-whole-whole-whole-half’, or ‘major second-major second-minor second-major second-major second-major second-minor second’. These are all correct.”

"This is actually called an 'interval vector'! This is the 'interval vector' of a traditional major scale! The sequence of these intervals records the 'intervals' of all the pitches within it!"

"But the order of notes in a set of pitches can be shuffled. How can we still identify its uniqueness when it is shuffled? There must be a standard that recognizes only one sequence of 'interval vectors' to avoid errors."

"We usually choose the most compact, leftmost interval arrangement as the identification feature, which gives rise to the concept of the 'basic form'. It is the identity fingerprint of a set of pitches, the mystical portrait of its soul! By comparing the 'basic forms' of two sets of pitches, you and I can determine their relationship!"

Fanning tapped the podium lightly with her index finger, emphasizing each word.

Then he spread his five fingers.

There are five types of relationships between sets of pitch levels!

Another wave of unprecedented knowledge, imbued with a divine aura, hit her in the face. Fanning's chalk swished across the blackboard, writing five phrases side by side—

1. Equality relation! Sets that can be converted to the same type through shift or mirror operations are those with the same 'basic type'!

"2.Z relationship is correct! Or called 'homogeneous heterogeneous tone groups', which is slightly the opposite of the first point, referring to the two being different 'basic types' but having the same 'interval vector'!"

3. Inclusion relationship! Subsets and supersets are used to analyze the relationship between the part and the whole!

4. Complementary relationship! The addition of two sets constitutes the complete twelve-tone scale!

"5. Similarity! Besides the first four points, the rest are the fifth! There's a similarity scale that can measure the closeness between two sets!"

After explaining the five basic relationships, Fan Ning raised his hand and waved.

With a "crack," the wood shattered, and the entire dial of the clock on the classroom wall, whose hands were spinning erratically, was pulled out from the air!
Then, it was attracted to the top of the blackboard.

The twelve dots on the dial form a perfect circle.

"All chords and melodies in music history can be summarized using this theory of scale degrees! All of them!"

Fanning randomly connected three of the points to form two different triangles, one a major triad and the other a minor triad.

"But today, I still want to overturn your perceptions, even those in a traditional context!"

"You usually think that major triads and minor triads are definitely different chords, but I'm telling you, in fact, their variations are extremely limited; in the theory of pitch sets, they are the same!" "As I mentioned earlier, 'equal relations' include operations of transposition and reflection. Transposition is the translation of a pitch set in space, which on a clock face is rotation; reflection is the flipping of them around a certain axis on the clock face."

The C major triad drawn by Fanning, do-mi-sol, forms a triangle connecting the 0, 4, and 7 chords on the dial; the C minor triad drawn by Fanning, do-bow-mi-sol, forms a triangle connecting the 0, 3, and 7 chords on the dial.

"call out--"

A sharp, straight scratch runs through the middle of the 3 and 4 marks, as well as the 10 and 11 marks.

This is the mirrored flip axis of these two closely spaced triangles!
"Therefore, the set of scale degrees of a C major triad and a C minor triad are reflections of each other, belonging to the first type, and are equal!"

A ripple of pale gold "Pureroma" energy, causing a tremendous cognitive shock, spread out from Fan Ning.

"What about major and minor triads with different note names?"

"I'm sorry, but this change is still too rudimentary."

Fanning chuckled.

“I can demonstrate to you clearly that even chords considered ‘distant keys’ in traditional harmony still do not exceed the scope of the first type of scale degree relationship—equal relationship!”

Fanning turned around again and quickly drew a triangle of a different color.

"Because the shifts and reflections can be superimposed and combined, if I can do it once, I can do it two or three times as well!"

He drew a red triangle on the dial to represent the C chord.

The reflection around the axis yielded a brown triangle Fm.

Then, by translating it by a major second, we obtain the purple triangle Em.

The key difference is a full six key signatures, but the transformation is still equivalent!
Another ring of pale golden ripples, resembling "Pure Ruoma," emanated from Fan Ning's body.

This time, the effects not only reached the "lecture hall" but also extended to the corridors and wasteland outside!
Some of the shadowy figures shivered or even convulsed.

However, some of the shadowy figures behaved in the opposite way. The knowledge, which was tempting yet still untouched by their "desire," made them drool. Their limbs swelled and wriggled like suckers, and they even left their "seats" and moved closer to the podium.

"I've discovered that some of the 'people' under your Divine Descent Society are truly stupid."

Fan Ning had used different geometric shapes to quickly demonstrate the variations of seventh chords, ninth chords, augmented and diminished chords, and suspended chords, but upon seeing this, he suddenly shook his head.

"It's just a teaching entry point to connect the past and the future. Is your brain so bad after being infected by the 'worm'? Since the geometric shape of the 'twelve-division dial' can be drawn arbitrarily, can't you draw some other patterns? Don't you know how to draw your own 'mysterious chords'? Sigh, I'm really worried for you."

Since it's essentially a circle anyway, and the dial can be divided into "twelve equal parts," then it can also be divided into "fifteen equal parts," "seventeen equal parts," or even "unequal parts." Playing with more avant-garde differential music like this isn't out of the question.
Or, since we can use numbers to indicate pitch, why not use numbers to indicate rhythm, dynamics, or timbre as well? This would be a more thorough approach to "total sequenceism."
These guys who keep shouting about the "ultimate secret" are still too conservative when it comes to making music.

"Alright then." Fan Ning shook his head indifferently and laughed, "You guys are the dumbest class I've ever taught. Come on, you've learned the stuff, now it's time for the practical experience you've been waiting for. Now put on these new lenses and re-examine those territories that were once considered 'chaotic'!"

Fan Ning reached out his hand again, as was his habit, towards the pile of musical scores and lecture notes.

Then it stopped above a pool of colorful slime.

"Oh, I forgot, you guys have destroyed all the tonal music."

He bent down and took some new sheet music out of his briefcase.

"Schoenberg, Op. 11, Three Piano Pieces, are you satisfied now?"

(End of this chapter)