QUADRIVIUM Assignment 10a EQUAL TEMPERAMENT AND TEST PREPARATION -Annandale- Tuesday, April 9nd -Fishkill- Wednesday, April 17th Exam on residue, greatest common divisor, tuning, etc. There is a practice exam below. The actual exam will be nearly identical to the practice exam, only shorter. You may use a calculator and you may make a CheetSheet™, which is one 8.5" x 11" sheet of paper with anything you want on it. Here is a two-octave "Pythagorean" system using "T" for tone (9/8) and "s" for semitone (256/243), imposed onto a keyboard. -Perfect fourths (4:3, P4s or tetrachords) are made up of two Tones and a semitone. Here are all the possibile combinations for a perfect fourth: TTs, TsT, or sTT. -Perfect fifths (3:2, P5s) are made from three Tones and one semitone. There are 4 different combinations. -Perfect octaves (2:1, P8s) are made from 5 Tones and 2 semitones. Lots of combinations. How many perfect fourths can be found in this 2-octave system? Answers found on p. 10 How many perfect fifths? How many perfect octaves? The keyboard diagram above is totally anachronistic. Historical examples of this layout (white and black keys with 12 keys to an octave) begin in the 15th century. The conceptual problem with the modern keyboard is that Tones and semitones are all equally spaced from low to high. They are all the same width. This reinforces the notion of equal temperament since intervals are all physically the same size whether low or high. Practically speaking though, piano keys are finger-sized for convenience. HW-10a p1 The fret layout of a guitar is a better conceptual example for how pitch was perceived before modern keyboards. Lower intervals are obviously bigger than those same intervals higher up in pitch. The lower octave is twice the size of the higher octave. Two to one. 2:1. Measure this diagram. A modern major scale is made up the following intervals. [...where W = whole step and h = half step]. Do - re - mi-fa - sol - la - ti-do W - W - h - W - W - W - h Each whole step in the modern system is made from 2 half steps. They are all "equally tempered," meaning that all of the half steps, 12 in all to an octave, are the same. This is not the case for "Pythagorean" tuning. Premodern intervals were based on superparticular ratios, derivable from the Platonic Lambda, which emphasized perfect fifths [3/2], forths [4/3] and octaves [2/1]. E.g. Ptolemy's Diatonic Syntonon (often called Pythagorean tuning) is very close to our modern major scale, but instead of whole steps and half steps, it is constructed from Tones and semitones: 9/8 and 256/243. This allows a tuning to have lots of perfect fifths, fourths, octaves. [See exercise above.] ! ! ! ! ! ! ! ∙ ! ∙ !"# = ! ∙ ! ∙ ! = ! ! = ! = ! = �4, the perfect fourth, ! ! !"# !! !! !! !!!! !! ! ! ! ! ! ! ! ! ∙ ! ∙ !"# ∙ ! = ! ∙ ! ∙ ! ∙ ! = ! ! = ! = �5, the perfect fifth, ! ! !"# ! !! !! !! !! !!!! ! ! ! !"# ! ! ! !"# !! !! !! !! !! !! !! !!"!!" ∙ ∙ ∙ ∙ ∙ ∙ = ∙ ∙ ∙ ∙ ∙ ∙ = = 2 = �8, octave [major]. ! ! !"# ! ! ! !"# !! !! !! !! !! !! !! !!"!!" The Pythagorean Tones and semitones should not be confused with Whole steps and half steps. ! Boethius considered the idea of "half" of a Pythagorean Tone, , but realized that it was not possible ! because "half" of ! was not a ratio of numbers. Here's why. ! Another way of putting this question is, What times itself equals ! ? Whatever that is, will be "half" of a ! Pythagorean Tone. Using algebra it is put like this, �! = !. ! And it is solved like this, �! = ! = ! = ! = �. That is a "half" of !, but that 2 is irrational and ! !∙! ! ! ! therefore not considered a number in premodern music theory. HW-10a p2 ! ! = ≅ 1.060660172, approximately. It's irrational... unthinkable. ! ! ! In decimal form it goes on forever without pattern. The semitone, !"#, on the other hand, is rational. It can be written as the ratio of two integers. !"# It also produces a long decimal (ca. 1.053497942386 …) but it is not irrational. !"# ! is close to , but not exactly the same. [They differ by about 0.007] !"# ! !"# ! th But does something that cannot do. When "added" to two Tones it completes a perfect 4 . !"# ! ! ! ∙ ! ∙ !"# = ! = �4, the perfect fourth. ! ! !"# ! And if another 9/8 Tone is put on top of the fourth, you get a perfect 5th. ! ∙ ! ∙ !"# ∙ ! = ! = �5, the perfect fifth. ! ! !"# ! ! And if you put a perfect fourth onto a perfect fifth you get a perfect octave. ! ∙ ! = !" = ! = P8, the octave. ! ! ! ! "Pythagorean" tuning is quite elegant. All those mathematically perfect intervals are a joy to ponder, but this tuning system doesn't play well with music written in multiple parts. A single-line melody sounds great, a melody with fifths, fourths, and octaves decorating it is divine, but anything much more complicated (like chords and key changes) sounds awful. Enter Equal Temperament The story of equal temperament is long, complicated, and at times very unclear, but the result is mathematically interesting. For starters, let's just look at a comparison of equal temperament with so-called "Pythagorean" tuning. As you can see, it's damn close. [I put an enharmonic tuning in there to show you just how different that one is.] To be equally tempered all intervals must be the same... they must be equal. The modern system divides an octave into 12 half-steps. The Pythagorean semitone is sort of a half step. Here is what happens if you use the semitone to make an octave. !"# !"# !"# !"# !"# !"# !"# !"# !"# !"# !"# !"# !"#!" ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ = ≈ 1.87. That's not even close to an octave. !"# !"# !"# !"# !"# !"# !"# !"# !"# !"# !"# !"# !"#!" HW-10a p3 So, how do you divide the octave into 12 equal intervals? Put another way, what times itself 12 times equals 2/1... an octave. Put in algebraic language... �!" = 2, where x is the interval we want. Solve for x to find the half-step interval by taking the 12th root of both sides.... !" �!" = !" 2 !" � = 2. There you have it. That's a half-step in equal temperament. !" !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 = !" 2 = 2. That's an octave. Unfortunately for the Pythagoreans, this is not a rational number. It also doesn't make perfect fourths and fifths. !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 ≅ 1.335. A perfect fourth should be ! = 1. 33. ! !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 ∙ !" 2 ≅ 1.498. A perfect fifth should be ! = 1.5. ! It's equally tempered, but it's bad-tempered. There are no perfect fourths or fifths in the equally tempered system. There are only perfect octaves. All other intervals are imperfect... irrational... blasphemous! But, it should be noted, the subtle differences between perfect and imperfect are barely noticable. Modern music theorists like to further divide the equally tempered half-step into 100 even smaller pieces called "cents." This is done in the same way that we divided the octave into 12 equal pieces. This time we are dividing the half-step into 100 equal pieces, or the entire octave into 1200 pieces. The "c" in the equations below is the incredibly small "fraction" that represents this tiny interval. �!"" = � ∙ � ∙ � ∙ � ∙ � … ��� = !" 2 = ℎ��� − ����. Solve for c to find the interval of 1¢. !"" !"" !"" !" !/!"" �!"" = �!"" !/!"" = �!"" = �! = � = 2 = 2!/!" = 2!/!"## � = 2!/!"## ≅ 1.00057789506553 If approximated as a superparticular ratio � ≈ !"#$. !"#$ This all smells of logarithm. ! Identity: � = � is the same as y = log! �, � = 2!/!"## log � = ! → 1200log � = 1 ¢ ! !"## 2 So remember, c is essentially the tiny tiny fraction that represents a 1¢ interval. On our 24" monochord strings, this interval would be 2 ten-thousandths of an inch off 0. That white line at the far left of the diagram is a crude estimation of where 1¢ would be. One cent is tiny. HW-10a p4 Long story short, here are the forumlae for converting ¢ents-to-ratios and ratios-to-¢ents. ��¢/���� = ����� Cents-to-ratios E.g. What ratio is 702¢? 2!"#/!"## ≅ 1.5 … . �� !. A fifth ! � ���� ∙ (�. ���)��� ≅ �¢ Where ! is the input ratio. �� � ! Ratios-to-cents E.g. A perfect fifth is !. What is that in cents? ! 1200 ∙ (3.322)log ! ≅ 702¢ !" ! The Ratio-to-cents equation has been converted to log10 to that you can use your calculators. Many calculators don't have log2. Exercises on ¢ents Convert the cents to ratios (usually meaning a decimal) and ratios to cents. Round answers to 1 or 2 places when appropriate. E.g. 1 What is the ratio for 600¢? 2!""/!"## ≅ 1.41 [which is 2.] 6 E.g. 2 How many cents is 6/5? 1200 ∙ 3.322 log ≅ 315.6¢ !" 5 1. What is the ratio for 1200¢? 2 2. How many cents is 2/1? 1200¢ 3. What is the ratio for 500¢ 1.335... 4. How many cents is 256/243? about 90¢ 5. What is the ratio for 100¢? 1.06... 6. How many cents is 9/8? about 204¢ 7. How many cents is 4/3? about 498¢ 8. How many cents is 3/2? about 702¢ 9. How many cents is 4/1 2400¢ 10. What is the ratio for 2400¢? 4/1 Note: All equally tempered intervals will be multiples of 100¢. I.e. 100¢, 300¢, 900¢, 1500¢, etc. HW-10a p5 Practice Exam Convert this Fibonaccian mixed fraction into a modern decimal number. Remember, you work right to left with these. [From 7a] Example 1. ! ! 4 = !" ! 1 4 4 + + = 4.44444. .. 3 3 ∙ 12 1. ! ! 6 = !" ! 6.65 2. ! ! 5 = !" ! 5.4615... 3. ! !"#$ 13 = !" !"#$ 13.333412... Fibonaccian Multiplication and Residue. [From 7a] You don't need to multiply like Fibonacci, but you need to find the 3 residues and show your work. ? ��� E.g. 2 8 The result is.... 8 . 43 43 The residue can be found in 3 ways: a) 8 4 + 3 = 56. Then subtract 9s until you have the residue of 2. 56-9-9-9-9-9-9=2 b) 344 → 3 + 4 + 4 = 11. Subtract 9s to find residue of 2.