Risposte

Cosa si intende per “Sezione Aurea”.

Se dividiamo un segmento (per semplicità di lunghezza 1) in due parti non uguali in modo tale che la parte più corta sta alla parte più lunga come la parte più lunga sta al segmento intero, otterremo che la parte più lunga misurerà 0,618033… e quella più corta 0,381966… Abbiamo così costruito una “Sezione Aurea”.

La “Sezione Aurea” quindi è un rapporto: il rapporto tra 1 e 0,618033… e tra 0,618033… e 0,381966…
(Un po’ come succede quando si parla del rapporto tra i due lati del rettangolo formato dalla pellicola 24X36 che hanno un rapporto di 1,5).

Questo rapporto si esprime con il reciproco di 0,618033… ovvero 1/0,618033… che dà come risultato: 1,618033…
Strano numero questo no? La serie di decimali rimane invariata!

Per precisione la “Sezione Aurea” si esprime come: 1 più, radice quadrata di 5 meno 1, fratto 2.

Da questa regolina possiamo costruire altre figure geometriche più complesse come il "Rettangolo Aureo"
Come si può notare (vedi allegato) ogni rettangolo che deriva da ogni passo successivo, è simile ovvero ha le stesse proporzioni tra i lati lunghi e quelli corti.

Oppure il "Triangolo Aureo"
Anche in questo caso, il triangolo derivato avrà sempre due angoli di 72° e uno di 36°.

File allegati

 Aurea.jpg ( 86.63k ) Numero di download: 9538
 Rap.jpg ( 32.2k ) Numero di download: 6480

 

http://www.wired.com/news/culture/reviews/...ml?tw=rss.index
From Crypto to Jazz

By Alexander Gelfand
02:00 AM Aug, 31, 2006

To the uninitiated, modern jazz can sound like a secret language, full of unpredictable melodies and unexpected rhythms. For alto saxophonist Rudresh Mahanthappa, however, the idea of jazz as code is more than just a metaphor.

Mahanthappa is best known for combining avant-garde jazz with Indian classical music. But for his latest release, Codebook, from Pi Recordings, the artist looked instead to cryptography and number theory for inspiration. (The album's title pays homage to The Code Book, a history of cryptography by the British science writer Simon Singh.)

The very first track, "The Decider," is a groovy primer on how to turn math into music. Its bristling melody (.mp3) is derived from the Fibonacci sequence, an infinite series of integers that governs the structure of everything from pineapples to the Parthenon.

Fibonacci's fingerprints can be found in the work of classical composers from Bach to Bartok, but intentionally basing a composition on the series is hardly standard practice in jazz. What's most striking about "The Decider," however, is how closely its written melody resembles one of Mahanthappa's improvised solos, a correspondence that reveals just how deeply the saxophonist has internalized what might have remained an abstruse, pencil-and-paper exercise.

Later on in the piece, drummer Dan Weiss spells his own name in Morse code, using short durations to represent dots and long ones to represent dashes. ("Play It Again Sam" begins in similar fashion, with every member of Mahanthappa's quartet dotting and dashing (.mp3) his name.)

Returning to the realm of number theory, the tune "Further and In Between" is based on the cyclical number 142857. Like all cyclical numbers, this one has some very strange properties; for example, if you multiply it by 2, 3, 4, 5 or 6, you get the same digits in a different configuration (for example, 2 x 142857 = 285714).

By mapping particular musical pitches to each digit and running through his multiplication tables, Mahanthappa came up with a winding, circuitous melody (.mp3) that makes a surprising amount of sense. That's partly because he wedded it to a strong, swinging rhythm, and partly because he gave himself permission to fudge things a bit in order to prevent the math from overwhelming the music.

"Frontburner," based on a heavily encrypted form of John Coltrane's classic "Giant Steps," demonstrates a similar balance between musicality and mathematical rigor.

Cryptonerds will be pleased to know that Mahanthappa used a portion of the "Giant Steps" melody as a musical keyword in conjunction with several different scales to encipher the original tune (.mp3). He used a similar method to generate the melody for "Play It Again Sam," further complicating matters by throwing in a biblical Hebrew cipher known as "atbash".

In cryptographic circles, this is known as a polyalphabetic substitution cipher, and it was the preferred form of military encryption right up through World War II.

In this particular case, it may have been too effective: The first, properly encrypted form of "Frontburner" didn't quite work from a musical perspective, so Mahanthappa massaged the results until he got something that did. The end result (.mp3) is a tune that will keep both sides of your brain buzzing happily away.

Making avant-garde jazz accessible to the general public is no mean feat. Making math-based music easy on the ears is even harder. Yet somehow Mahanthappa has managed to do both. And that's a code many musicians would doubtless like to crack.