What’s that scale?? 1

Note Grades should be available on some computer somewhere. The numbers are based on the total number of correct answers, so 100% = 40. When I review the numbers, this may change. We will get your individual results to you shortly.. Be forgiving, I don’t have the foggiest idea how to do this stuff either. Let’s click… 2

The test was? (any answer = 2 points) A. Easy B. Ok C. Hard D. Very Hard E. Extremely difficult!!! 3

How much did you collect for a hitman? A. $500 B. $1,000 C. $10,000 D. $50,000 E. It doesn’t matter … you are still here. 4

However you did, the test was A. Fair B. Unfair C. Very unfair 5

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Important Definitions From last week.. 7 The PERIOD, T is the time it takes to go from one condition to the next time that exact condition is repeated. The frequency, the number of oscillations per second, is given by: Example: If T=2 seconds F=1/2 (sec -1 )=0.5 per second

Question 8 What is a tone and how do you prove it??

9 Remember Helmholtz ?  Today The SINE curve

Remember Helmholtz’s Results Note from Middle CFrequency C264 D297 E330 F352 G396 A440 B496 10

Today 11 Let’s start looking at how this scale developed. It is mostly arithmetic. This material is in Measured Tones. Readings: Chapter 1pages 1-11 oRead pages for the “flavor” o Chapter 2 – All: Don’t worry about the musical notation.

The Guitar Strings - Review 12

Consider Two Situations 13 For the same “x” the restoring force is double because the angle is double. The “mass” is about half because we only have half of the string vibrating.

So… 14 For the same “x” the restoring force is double because the angle is double. The “mass” is about half because we only have half of the string vibrating. k doubles m -> m/2 f doubles!

Guitar Pressing the fret that is in the middle of the string doubles the frequency~ Walla … the octave In general … the frequency is proportional to the length of the string. The violin works in a similar way. 15

Now …. lets look briefly at the MONOCHORD 16

What did Pythagoras do? 17 He compared sounds from different pressure points and listened to see which sounded the best.

Octave 18 f 2f SUM Time  The sum has the same basic periodicity as The original tone. Sounds the “same”

Violin works in the same way! 19

The Violin 20 L We will make some measurements based On these lengths.

Play an octave on one string Volunteer to watch where the finger winds up on the finger board. Measure the length of the string. How close is it to ½ the length? 21

Let’s Listen to the Violin 22 1) Let’s listen to the instrument, this time a real one. The parts  One tone alone.. E on A string  E on the E string  Both together (the same?)  A Fifth A+E open strings  Consecutive pairs of fifths – open strings.  A second? Third? Fourth? Seventh?

23 The ratios of these lengths Should be ratios of integers If the two strings, when struck At the same time, should sound “good” together.

Remember this argument? 24 For the same “x” the restoring force is double because the angle is double. The “mass” is about half because we only have half of the string vibrating.

Pythagoras Noticed that the sound of half of a string played against the sound of a second full string, both with the same original tone, sounded well together. This was called the octave (we discussed this last time). He then noticed that a very melodious tone also came when the string was divided into 1/3 – 2/3. When the larger portion of the string was played against the original length, it was called the fifth. In particular, the tone was “a fifth above the original tone”. 25

So… 26 For the same “x” the restoring force is double because the angle is double. The “mass” is about half because we only have half of the string vibrating. k doubles m -> m/2 f doubles! Octave

The keyboard – a reference 27 The Octave Next Octave Sounds the “same” Middle C

The Octave octave 12 tones per octave. Why 12? … soon. Played sequentially, one hears the “chromatic” scale. Each tone is separated by a “semitione” Also “half tone” or “half step”. Whole Tone = 2 semitones 28

Properties of the octave 29 Two tones, one octave apart, sound well when played together. the same note In fact, they almost sound like the same note! A tone one octave higher than another tone, has double its frequency. Other combinations of tones that sound well have frequency ratios that are ratios of whole numbers (integers). It was believed olden times, that this last property makes music “perfect” and was therefore a gift from the gods, not to be screwed with. Pythagoras This allowed Pythagoras to create and understand the musical scale.

The Octave As we determine the appropriate notes in a scale, we will make use of the fact that two tones an octave apart are equivalent. We can therefore determine all of the equivalent tones by doubling or halving the frequency. This process is used to build up the scale. 30

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Fifth 32 C G C f 1.5f 2f A fifth is a span of 5 whole tones on the piano. It also spans 7 semitones.

Let’s look at the “fifth” Formed with 2/3 of the original length. Considered to be a “perfect” sound because of the small number ratio in lengths. We can form many of the notes of a scale using this ratio. The scale so formed sounds great but has problems. 33 2/3 L m=2/3 M (smaller) k=3/2 K (larger)

The Perfect Fifth … Sounds Good! 34 frequency f 1.5f 2f fifth Octave

Other Fifths – also pretty good! 35 Beethoven’s Fifth

The Intervals: The fifth is 7 semitones above the fundamental tone, f. Since f and 2f are an octave apart, the interval from G to C should also be melodic. This interval consists of five (5) semitones. This “special interval” is referred to as a FOURTH. Let’s see how much of a scale we can create using these two musical intervals. 36 C G C f 1.5f 2f fifth fourth

37 1/43/4 reference This is a nice ratio of small integers that will also harmonize with the cosmos.

OK … Let’s build a scale! 38

Pythagorean Fifths Scaling the Scale Fifths and fourths sound good together so we try to make a scale with as many of these harmonies as possible. We start with Middle C at frequency f (264 Hz ) We will actually add the numbers later. First tone is a fifth: 1.5f  G Last tone is the octave: 2f  C above Middle C. 39 C G C f 1.5f 2f

P’s 5 Question: Are there any other intervals between 1f and 2f that correspond to singable intervals? Pythagoras Rule: Take an existing ratio. Multiply by 1.5 to get a fifth above the ratio. If the number is greater than 2, reduce it by an octave (divide by 2) If the number is less than 1, increase it by an octave by doubling the number. 40 Ratio1/14/33/22/1 Decimal

Another tone: 41

More of the same … 42

So Far From CRatioFrequency C264 D297 E330 F352 G396 A440 B496 We could start with the A below middle C and get the 440 right.

Tones together We discussed that a scale should be made up of tones that sound well together. Even for a scale that is put together as we have just done, some tones will sound a bit bad together; but not terrible. Let’s see why some of the better combinations sound well. 44

The original sound A:440 Hz. 45 time

The Octave: A PERIODIC sound and our brains accept this as a “nice” tone.

The fifth 47

The Third f 0 48

Longer period of time 49

50 A New Phenomenon T~ seconds estimate

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This phenomenon is called BEATS 52 The beat frequency between two similar frequencies the difference between the frequencies is found to be the difference between the frequencies The beat frequency between two similar frequencies the difference between the frequencies is found to be the difference between the frequencies

53 Max Min

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Beats Low beat frequencies (1-20 Hz) can be heard and recognized. Faster beat frequencies can be annoying. Two frequencies an octave apart but off by a few Hz. will also display beats (difference between the frequencies as well) but they are harder to hear and somewhat unpleasant to the ear. 55

Problems The system of fifths to generate a scale works fairly well BUT if you start on a different note (F instead of C), the frequencies of the same notes will differ by a slight amount. this means that an instrument usually must be tuned for a particular starting mote (key). Modulation doesn’t work well. One interesting problem is the octave over a large range. 56

The Octave Problem Seven octaves represents a frequency range of 2 7 =128 The same distance is covered by 12 fifths: (3/2) 12 = Some people can hear this difference … a problem, Many other tones wind up being slightly different. 57

Problems.. You can create scales using different sets of “primitive” combinations … thirds, sixths. Each yields a specific scale. They are not the same (read chapter 1 in MT). One can’t change “keys” easily using these schemes. Something had to be done. Solution: Equal Tempered Scales. The frequency difference between two consecutive semitones is set to be: 58

 Keeps the octave exactly correct  Screws up all of the other intervals ◦ But we can’t easily hear the difference  One tuning will work for all keys 59

60 Interval Ratio to Fundamental Just Scale Ratio to Fundamental Equal Temperament Unison Minor Second25/24 = Major Second9/8 = Minor Third6/5 = Major Third5/4 = Fourth4/3 = Diminished Fifth45/32 = Fifth3/2 = Minor Sixth8/5 = Major Sixth5/3 = Minor Seventh9/5 = Major Seventh15/8 = Octave (fourths, fifths and sixths)

Back for some physics 61