This is a research done by a group of high school students concerning Golden Ratio found in different categories of Music. Please refer to the 'notes' for our script. If you want to download the presentation, you are welcome to contact me: krubiz[at]gmail[dot]com. Do comment on our work. Thank you!
25. C A B 1-x x 1 x 2 + x - 1 = 0 Solve by using quadratic formula a = 1, b = 1, and c = -1
26. C A B 1-x x 1 x 2 + x - 1 = 0 Solve by using quadratic formula a = 1, b = 1, and c = -1 -b ± √ (b 2 – 4ac) 2a x =
27. C A B 1-x x 1 x 2 + x - 1 = 0 Solve by using quadratic formula a = 1, b = 1, and c = -1 -b ± √ (b 2 – 4ac) 2a x = -(1) ± √ ((1) 2 – 4(1)(-1)) 2(1) =
28. C A B 1-x x 1 x = -(1) ± √ ((1) 2 – 4(1)(-1)) 2(1)
29. C A B 1-x x 1 x = -(1) ± √ ((1) 2 – 4(1)(-1)) 2(1) -1 + √5 2 = 0.6180339887 x =
30. C A B 1-x x 1 x = -(1) ± √ ((1) 2 – 4(1)(-1)) 2(1) -1 + √5 2 = 0.6180339887 Or x = -1 - √5 2 = -1.6180339887 x =
31. C A B 1-x x 1 x = -(1) ± √ ((1) 2 – 4(1)(-1)) 2(1) -1 + √5 2 = 0.6180339887 Or x = -1 - √5 2 = -1.6180339887 (rejected) x =
Winnie: Okay, hands up. How many of you have songs stored in computer or ipod? Sure, we often hear music in everywhere. Some end up as songs we probably never want to hear again, while others end up on our iPod as masterpiece compositions we want to listen to over and over. Ruby: Why do we like certain songs more than others? Are they catchier? Are they more varied? Or is there another factor no one’s aware of? It may be because of the Golden Ratio. In the following time, our group is going to discuss what Golden Ratio is and how is it related with music. Please enjoy!
Priscilla: Let’s begin with numbers!
Priscilla: Do you know what it is?
Priscilla: Phi?
Priscilla: Yes. The golden ratio.
Priscilla: And it’s approximately equal to 1.618…
Priscilla: So…
Priscilla: How can we get this number?
Alison: Let’s talk about Fibonacci Sequence.
Alison: Can you figure out the relationship in these numbers?
Alison: Probably many of you can see this relationship. Each term is the sum of the two previous terms.
Alison: How about this?
Alison: Yes. As you go farther and farther to the right in this sequence, the ratio of a term to the one before it will get closer and closer to the Golden Ratio.
Alison: Any other way to get the golden ratio?
Sally: Sure! And since we are in Science class. Let me briefly explain how to divide a segment in Golden ratio in Mathematics. Let’s Consider line CAB.
Sally: If the ratio of the lengths BC to AB is equal to the ratio of the lengths AB to AC, then the segment has been cut in the extreme and mean ratio, or in a golden ratio. Let’s assume the length of segment BC is 1. We can call AB “x” and AC “(1 – x).” We can solve for x.
Sally: We can solve this using our quadratic formula. a = 1, b = 1, and c = -1.
Sally : After the easy calculation, we can get two results of x. Since we are calculating the length, the negative number is rejected.
Sally: Therefore, that’s the way for us to obtain phi, the golden number.
Sally: And so, We can get BC to BA is in golden proportion.
Fiona: So, what does it mean?
Fiona: We are going to tell you how to make use of this ratio.
Fiona: Which rectangle do you feel the most comfortable?
Fiona: Probably this one. Why?
Fiona: Because this is a golden ratio rectangle!
Fiona: Actually, Golden ratio is in everywhere. For example, we can find this in artworks.
Fiona: Architecture.
Fiona: Fibonacci Numbers are very common in plants. For example, the number of petals. Is nature playing game with us?
Winnie: Moreover, golden ratio is all around human body. Do you know that?
Winnie: Such as…
Winnie: Human Faces!
Winnie: Look! We can find this magic number in our faces.
Winnie: What about…
Winnie: I n MUSIC?
Tiffanie: Do you know that golden ratio is also used in the design of violins?
Tiffanie: Let’s divide it in different sections. Let’s see how golden ratio can be found in this magical instrument.
Tiffanie: This section is divided in 1:1.618, which is in golden ratio.
Tiffanie: And so on.
Sally: Wow. That’s fantastic. Can we find it in other instrument as well?
Ruby: Sure, for example, in piano.
Ruby: In piano, there are 13 notes that separate each octave of 8 notes in a scale.
Ruby: There are eight white keys
Ruby: and five black keys.
Ruby: The black and white keys are separated into groups of three
Ruby: And two
Ruby: What can we discover? 2, 3, 5, 8, 13…
Ruby: They are Fibonacci Numbers!
Sally: Actually, a pure musical tone is characterized by a fixed frequency and amplitude.
Sally: The standard tone used for tuning is A, which vibrates at 440 vibrations per second.
Sally: A major sixth can be obtained from a combination of A with C, the latter note being produced by a frequency of about 264 vibrations per second.
Sally: The ratio of the two frequencies 440/264 reduces to 5/3, which is the ration of two Fibonacci numbers.
Priscilla: A minor sixth can be obtained from a high C with 528 vibrations per second.
Priscilla: and an E which is 330 vibrations per second.
Priscilla: The ratio in this case, 528/330, reduces to 8/5, which is also a ratio of two Fibonacci numbers and already very close to the Golden Ratio!
Sally: Can we apply gold ratio in the composition of music?
Tiffanie Sure. For example, the Music for Strings , Percussion and Celesta.
Tiffanie: This piece is composed by Bartok, who is known for his chromatic technique of obeying to the laws of Golden Section in every moment.
Tiffanie: In Music for Strings , Percussion and Celesta , the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1. Let’s listen to the opening of the following music carefully. Please pay attention to the rhythm.
Tiffanie: Besides, in the piece , 89 bars of the movement are divided in 2 parts. However, it’s not being divided in a usual way.
Alison: They are actually in the form of Golden Ratio. The two parts are i) with 55 bars and ii) 34 bars by the pyramid peak of the movement. The Pyramid peak in defined by loudness. Further divisions are marked by the mutes for the instruments and by other textural changes. It’s amazing that all the nos. of measures are with close to the Golden Ratio.
Alison: Moreover, In Sonata for Two Pianos and Percussion , the various themes develop in Golden Ratio order in terms of the no. of semitones. It adds a bit of mysticism and science to composition and music.
Fiona: So, does it mean all great music are composed in the form of golden ratio?
Fiona: In 1995, a mathematician had investigated w hether Mozart had used the Golden Ratio in the 29 movements from his piano sonatas that consist of 2 distinct sections.
Fiona: The 2 sections are..
Fiona: i) Exposition, which introduce the musical theme , and we will name it E.
Fiona: And the second one is the Development and Recapitulation, we call it DR, which revisited and further developed the main theme.
Fiona: DR has 62 bars
Fiona: And E has 38 bars.
Fiona: 62 divided by 38, we get 1.63… which is approximately equal to phi. However, after a full analyze of Mozart's work, we find that Mozart does not use golden ratio in his work. It just happens that these two sections fit in the golden ratio.
Winnie: However, after a full analyze of Mozart's work, it was found that Mozart does not use golden ratio in all of his work. It just happens that these two sections fit in the golden ratio. Therefore, good music may not be composed with Golden ratio as well.
Winnie: Now let’s hear some soundtracks from video games., see whether the golden section is properly applied. Winnie: As you can read from the passage, the “Chamber of Sages” theme begins right at the Golden Section, right when you start hearing the harp and bells. You’ll hear them at the 18-second mark in the clip there. 18/29 is approximately equal to the golden ratio, the climax begins there. Ruby: As we listen to more examples, we can find out that most of the clips show significantly how golden section is applied. Climax are usually begins at two-thirds of the songs. Besides, we can see the reason why the writer chooses game songs for analyzing. It’s because they generally sounds more interesting and musical, and the Golden Sections are a little more impressive.
Priscilla: The golden ratio is thought by many to represent a ratio of natural beauty. Buildings, human faces and many other things have golden ratio. Musical compositions that exhibit this ratio tend to be considered “beautiful,” regardless of whether or not the listeners realize the presence of the Golden Ratio. In music, the golden ratio can be the measure that’s located at .618 the length of the song or the melody, or whatever. The fact that something significant lies at this point, dividing the musical piece or part of it into golden proportions seems to have a subconscious effect on the listener. The song seems to be a little closer to perfection, and the golden section is that “something” about the song that people aren’t sure why they love it. Tiffanie: For us, normal students, although it may not be easy to decide whether a song have Golden ratio or not, we do encourage you to look at the music on your own and discover more as what we have suggested may be controversial. The progress will certainly be interesting. You may become a great composer in the future if you can understand and apply the golden ratio. Sally: Golden Ratio is undoubtedly a great finding. However, is 1.6180339887498948482… really a magic number? Does it really make the music much more beautiful? Or we just have the thought that songs which claim to have golden ratio are better? The answer is up to you.
Winnie: Okay, hands up. How many of you have songs stored in computer or ipod? Sure, we often hear music in everywhere. Some end up as songs we probably never want to hear again, while others end up on our iPod as masterpiece compositions we want to listen to over and over. Ruby: Why do we like certain songs more than others? Are they catchier? Are they more varied? Or is there another factor no one’s aware of? It may be because of the Golden Ratio. In the following time, our group is going to discuss what Golden Ratio is and how is it related with music. Please enjoy!
