Discover and read the best of Twitter Threads about #TheoryThursday

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Today is #TheoryThursday ๐Ÿง!

The most important question in all of Computer Science is probably whether P equals NP. That is, are all problems easy to validate also easy to solve?

How can we even start to ponder this question? ๐Ÿงต๐Ÿ‘‡
๐Ÿ”– In case you're just hearing about this, here's a short refresher on what we are talking about:

Let's think now about what would it mean to properly answer this question. ๐Ÿ‘‡
Say we want to answer "yes". How can we do that? Do we have to check that *all* NP problems are actually in P?

๐Ÿ‘‰ To begin, there are infinitely many problems in Computer Science.

There must a better way...
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Today is #TheoryThursday ๐Ÿง!

I want to talk about what is, possibly, the most important question in all of Computer Science: Does P equal NP?

If you have heard of this and want to learn a bit more, read on... ๐Ÿงต๐Ÿ‘‡
Computer Science is all about finding clever ways to solve difficult problems.

We have found clever algorithms for a bunch of them: sorting stuff, finding shortest paths, solving equations, simulating physics...

But some problems seem to be way too hard ๐Ÿ‘‡
One example is the Travelling Salesman problem.

โ“ Find a cycle starting in your home city to visit all major cities in your country and return home with the least fuel cost.

This is the kind of problem we expect computers to solve easily, right? That's what computers are for!
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On musical form; sixth instalment: Sonata Form

What is sonata form? For many people, the term sounds perhaps daunting, associated with the loftiest flights of imagination as it is, and with musical minds of the greatest renown, like Beethoven, Brahms, Haydn and Mozart.

In a structural sense, however, sonata form is not too hard to explainโ€”especially seeing that I have already spent so many words on other forms of a similar construction.

The sonata form is a ternary form. In essence, therefore, it differs not significantly from any other ternary form; not from the simple ternary form, not from the minuet or scherzo and neither from the rondo forms.

Read 41 tweets
Hey, guess what, today is #TheoryThursday ๐Ÿง!

A silly excuse I just invented to share with you random bits of theory from some dark corner of Computer Science and make it as beginner-friendly as possible ๐Ÿ‘‡
Today I want to talk about *Algorithmic Complexity *.

To get started, take a look at the following code. How long do you think it will take to run it?

Let's make that question more precise. How long do you think it will take to run it in the worst-case scenario?
We can see that the code will run slower if:

๐Ÿ‘‰ your computer is older;
๐Ÿ‘‰ the array is longer; or
๐Ÿ‘‰ x happens to be further to back, or not present at all.

Can we turn these insights into an actual formula? We will have to get rid of ambiguous stuff like "old computers".
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On musical structure; fifth instalment: Rondo Forms

To outline the structure of the rondo forms, we will need to resort to discussing its technical trappings. A general portrait of the rondo forms can be painted, however, by defining a few key characteristics.

The rondo forms capitalise upon one of musicโ€˜s most fundamental principles, namely repetition. They are primarily defined by the repetition of the primary idea, its secondary sections acting as digressions from this primary idea.

The repetitions of the rondo's main idea, often varied, provide the development of the musical idea. As outlined in Schoenbergโ€˜s words on the form I shared last week, the subordinate sections are merely there to prevent the monotony of hearing repetitions all the time.

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For today's #TheoryThursday, a few words from Arnold Schoenberg as a prelude to the discussion of Rondo form.
Schoenberg disputed the popular notion that the musical rondo is more than superficially an imitation of the poetic โ€œrondeauโ€ (i.e. a repeated refrain and couplets)

He notes that, in the poetic form, the refrain underlines the common factor of juxtaposed elements, while in the musical form, the repeated section is the main ideaโ€”and intervening sections only subordinate digressions.

The difference is thus, that the stanzas of the poetic โ€œrondeauโ€ develop the idea stated in the refrain, while the digressions from the refrain in the musical rondo serve as complements, to save the repetition of the refrain from the danger of becoming monotonous.

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Today, on #TheoryThursday, (I know it is Friday but as I wrote the text yesterday, I refuse to change it) allow me to say a few words on musical theory itself, on the textbooks which have been used throughout the ages, and on their influence on the music of the masters.

Schoenberg denounced theorists and proclaimed the Living Art as the source of ever-evolving musical lawsโ€”'tis not for the theorist to decide which rules music should adhere to. Debussy said that โ€œLes oeuvres d'art font les rรจgles, les rรจgles ne font pas les oeuvres d'artโ€.

In part, I subscribe to this view, in that theorists should not be lawmakers, that they do not decide the course of musical historyโ€”but they can document it, and in this regard their work is very valuable, as long as their advice is not heeded like tables of the law.

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๐Ž๐ง ๐ฆ๐ฎ๐ฌ๐ข๐œ๐š๐ฅ ๐ฌ๐ญ๐ซ๐ฎ๐œ๐ญ๐ฎ๐ซ๐ž; ๐Ÿ๐จ๐ฎ๐ซ๐ญ๐ก ๐ข๐ง๐ฌ๐ญ๐š๐ฅ๐ฆ๐ž๐ง๐ญ: ๐ฅ๐š๐ซ๐ ๐ž๐ซ ๐ญ๐ž๐ซ๐ง๐š๐ซ๐ฒ ๐Ÿ๐จ๐ซ๐ฆ๐ฌ

Onwards to more extensive forms, we come to larger ternary forms, the most prevalent of which are the dance forms; the minuet and the scherzo.

Both the minuet and scherzo can either be independent pieces or movements of a larger structure, like a suite, sonata or symphony.
They are essentially identical in form, though one can speak of differences in a general sense, especially in character and tempo.

The minuet, generally in 3/4 time, has a more or less stately character and an average metronome marking of 60-70 to a bar in Haydn and Mozart. Still, the variations in mood are myriad, as proven by the difference between this minuet by Haydn...

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๐Ž๐ง ๐ฆ๐ฎ๐ฌ๐ข๐œ๐š๐ฅ ๐ฌ๐ญ๐ซ๐ฎ๐œ๐ญ๐ฎ๐ซ๐ž; ๐ข๐ง๐ญ๐ซ๐จ๐๐ฎ๐œ๐ญ๐ข๐จ๐ง ๐š๐ง๐ ๐Ÿ๐ข๐ซ๐ฌ๐ญ ๐ข๐ง๐ฌ๐ญ๐š๐ฅ๐ฆ๐ž๐ง๐ญ: ๐“๐ก๐ž ๐Œ๐จ๐ญ๐ข๐ฏ๐ž

In musical masterpieces, it is impossible to see a single part or fragment as independent of any of the other parts.

Even where musical phrases are simply juxtaposed, such as in a Bach chorale, they could not have been presented in any other way.

For the musical work comes to the master as a whole, a composer, then, has not the freedom to pick a theme and vary it here and thereโ€”the master must carry through the consequences inherent in the theme his mind conceives.

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๐–๐จ๐ซ๐๐ฌ ๐ข๐ง ๐ฆ๐ฎ๐ฌ๐ข๐œ

Throughout musical history, there have been countless forms which composers have employed in order to express their musical ideas, like symphonies, concerti and operas for orchestral forces and quartets, trios and sonatas for smaller ensembles.

These are only a few of a vast array of musical forms, and I have only mentioned a few predominant mainly in the Classical and Romantic eras; there are a host of others!
To illuminate their merits would take countless hours and countless Tweets.

Today I would therefore like to direct your attention to a musical form which has been almost a constant throughout the entirety of history.
When humanity first made music, they did so using the instrument born within virtually every human body; the voice.

Read 37 tweets

The resources a composer can use to communicate their music are not confined to notes and markings alone. Often, words are written over their music, especially at the beginning, in order to convey more clearly the musical thought.

There are pieces which have only a metronome marking, like the pieces in Schumannโ€˜s โ€œKinderszenenโ€. The character and mood of the pieces is to be discerned in the notes themselves.

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Before the invention of the metronome, composers had to communicate a tempo to the performers by means of words. Through a shared frame of reference between composer and performer, the tempo could be fairly exactly communicated.

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On this fine #TheoryThursday , I would like to share with you a little bit of insight into something that many consider one of musicโ€˜s most important components, namely MELODY.


A melody is nothing more than a linear succession of pitches conditioned by rhythmic factors, yet we tend to ascribe great value to melody. Why is this? What makes this aspect of music so important that people judge music on the perceived absence or presence of a melody?

The concept of melody is quite intimately linked to the possibilities and limitations of the human voice. This is, after all, our primeval musical instrument. This immediately explains too why we value melody so highly.

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The title of a musical composition can be descriptive and refer to the musicโ€™s extramusical influences, like the title of a symphonic poem by Liszt which tells us basically what the music is about, or that we should keep in mind a certain programme.

More often, a compositionโ€˜s title denotes form, like with the sonata or rondo, or implies a certain character without presenting an explicit programme.

Other times still, the title refers to the instrumental forces for which the piece is arranged (a symphony being, almost invariably, a work for orchestra, while a piano sonata speaks of both the form and the instrument employed).

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Today I would like to talk about something in music which is utterly laughable, but nonetheless of such an influence that we must take it seriously, although we would want to declare it ludicrous.
This is the influence of notation on music, especially on HARMONY.
In conventional harmony, there is an assortment of chords which are treated as โ€œchance harmoniesโ€, as sonorities which occur because of the voice leading or as appoggiaturas, and these are not classed with traditional chordsโ€”giving rise to the notion of โ€œadded-tone chordsโ€.
Compare the following two harmonies; neither is traditionally classified amongst the consonances.
The first is a seventh chord, and therefore has a name and description in harmonic theory. The other is, at first glance, a c major triad with a d added above it.
Read 15 tweets

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