1. By Prof. Lydia Ayers

2. Types of Intervals augmented intervals + 1/2 stepaugmented intervals + 1/2 step diminished intervals - 1/2 stepdiminished intervals - 1/2 step perfect intervalsperfect intervals major intervalsmajor intervals (1/2 step) (1/2 step) minor intervalsminor intervals

3. Major and Minor Intervals two kinds of 2nds, 3rds, 6ths and 7thstwo kinds of 2nds, 3rds, 6ths and 7ths major intervals are one half step larger than minor intervalsmajor intervals are one half step larger than minor intervals minor 2nd Major 2nd Perfect 4th perfect intervals have only one sizeperfect intervals have only one size Augmented 4th is one half step larger than Perfect 4th; same size as diminished 5th

4. Major and Minor Intervals major intervals are one half step larger than minor intervalsmajor intervals are one half step larger than minor intervals examples:examples: a major 2rd is one half step larger than a minor 2rda major 2rd is one half step larger than a minor 2rd 1 half step2 half steps3 half steps4 half steps a major 3rd is one half step larger than a minor 3rda major 3rd is one half step larger than a minor 3rd

5. IntervalInterval all intervals of the major scale measured up from the tonic are either MAJOR or PERFECTall intervals of the major scale measured up from the tonic are either MAJOR or PERFECT M2P4 P5 M6M7P8 M3

6. Interval:Interval: all intervals of the major scale measured down from the tonic are either MINOR or PERFECTall intervals of the major scale measured down from the tonic are either MINOR or PERFECT m3 P4 P5 m6 m7 P8m2

7. Complementary Intervals a pair of intervals that make an octavea pair of intervals that make an octave a major 3rd + minor 6tha major 3rd + minor 6th a minor 3rd + major 6tha minor 3rd + major 6th a major 2nd + minor 7tha major 2nd + minor 7th a minor 2nd + major 7tha minor 2nd + major 7th a perfect 5th + perfect 4tha perfect 5th + perfect 4th

8. Complementary Intervals M2 P4 P5 M6 M7 P8 M3 m3 P4 P5 m6 m7 P8 m2

9. Complementary Intervals Multiplying the frequency ratios of two complementary intervals produces an octaveMultiplying the frequency ratios of two complementary intervals produces an octave X+X+ ====

10. Matching Harmonics [ii:45] just consonances have some matching harmonics from the harmonic series[ii:45] just consonances have some matching harmonics from the harmonic series

11. How Can We Measure Interval Size? CentsCents octave divided into 1200 centsoctave divided into 1200 cents frequency ratio for 1 cent = 1200 or 2 1/1200 = 1.0005778frequency ratio for 1 cent = 1200 or 2 1/1200 = 1.0005778 frequency ratio for 100 cents (equal- tempered semitone) = or 2 1/12 = 1.0594631frequency ratio for 100 cents (equal- tempered semitone) = or 2 1/12 = 1.0594631

12. Just Noticeable Difference 1 to 4 cents1 to 4 cents hearing test: write down whether the pitch in the four pairs of tones goes up, down or stays the samehearing test: write down whether the pitch in the four pairs of tones goes up, down or stays the same 1.[ii:50] 2.[ii:51] 3.[ii:52] 4.[ii:53]

13. CentsCents to find cents with a calculator:to find cents with a calculator: calculate decimal ratiocalculate decimal ratio find the natural logarithm (In)find the natural logarithm (In) multiply by 1731.234 (if using "log" multiply by 3986.314)multiply by 1731.234 (if using "log" multiply by 3986.314) example:example: 3/2 = 1.53/2 = 1.5 In(1.5) =.4054651In(1.5) =.4054651.4054651 x 1731.234 = 701.95498 cents.4054651 x 1731.234 = 701.95498 cents 1731.234 = 1200/(ln 2) 3986.314 = 1200/(log 2)

14. Why the magic number 1731.234? Let x be the interval ratio we want to convert.Let x be the interval ratio we want to convert. Let y be the value of x in cents.Let y be the value of x in cents. 2^(y/1200)= x ln(2^(y/1200))= ln(x) (y/1200)*ln(2)= ln(x) y * (ln(2)) / 1200= ln(x) y= ln(x) * 1200 / ln(2)

15. Review Question How many cents difference between 550 and 440 Hertz? How many cents difference between 550 and 440 Hertz? A.701.95498 B.386.3137 C.315.64128 D.407.81999 E.none of the other answers

16. Interval Name Chart RatioName 81/80Syntonic comma 16/15minor second 12/11narrow neutral second 11/10neutral second 10/9narrow major second 9/8major second 8/7supermajor second 7/6subminor third 6/5minor third 11/9neutral third 5/4major third 9/7supermajor third

17. Interval Name Chart RatioName 4/3perfect fourth 11/8super fourth 7/5augmented fourth 10/7diminished fifth 3/2perfect fifth 11/7subminor sixth 8/5minor sixth 5/3major sixth 12/7supermajor sixth 7/4subminor seventh 9/5wide minor seventh 11/6neutral seventh 15/8major seventh

18. Kraig Grady: Opening Invocation for a Shadow Play Kraig Grady: Opening Invocation for a Shadow Play in just tuning on a tubulong instrumentin just tuning on a tubulong instrument harmonic series scale: H n = H n-3 + H n-2harmonic series scale: H n = H n-3 + H n-2 improvised using special melodiesimprovised using special melodies http://www.anaphoria.com/shadow.html

19. [ii:36] Unison the most perfect consonancethe most perfect consonance two (or more) pitches which have the same frequencytwo (or more) pitches which have the same frequency frequency relationship of 1/1 (in the example, both frequencies are 64 Hertz)frequency relationship of 1/1 (in the example, both frequencies are 64 Hertz) with two sine waves in unison, it is difficult to hear two toneswith two sine waves in unison, it is difficult to hear two tones

20. [ii:37] Octave the second most perfect consonancethe second most perfect consonance the interval between two pitches in which one has double the frequency of the otherthe interval between two pitches in which one has double the frequency of the other frequency relationship of 2/1 (in the example, the lower frequency is 64 Hertz, the higher frequency is 128 Hertz)frequency relationship of 2/1 (in the example, the lower frequency is 64 Hertz, the higher frequency is 128 Hertz)

21. OctaveOctave the name "octave" comes from the Italian word for "8" because it is the 8th pitch in the scalethe name "octave" comes from the Italian word for "8" because it is the 8th pitch in the scale 8va means to play a pitch an octave higher than written - used to avoid ledger lines without changing the clef8va means to play a pitch an octave higher than written - used to avoid ledger lines without changing the clef [ii:38] 8va symbol in the first measure raises the notes to the same pitches as in the second measure

22. [ii:39] Perfect Fifth the third most perfect consonancethe third most perfect consonance the interval between two pitches in which one has 1.5 times the frequency of the otherthe interval between two pitches in which one has 1.5 times the frequency of the other frequency relationship of 3/2 (in the example, the lower frequency is 128 Hertz, the higher frequency is 192 Hertz; 192/128 = 3/2 = 1.5)frequency relationship of 3/2 (in the example, the lower frequency is 128 Hertz, the higher frequency is 192 Hertz; 192/128 = 3/2 = 1.5)

23. [ii:40] Perfect Fourth the other intervals, such as the perfect 4th, get their names from the same numbering systemthe other intervals, such as the perfect 4th, get their names from the same numbering system frequency relationship of 4/3 (in the example, the lower frequency is 192 Hertz, the higher frequency is 256 Hertz; 256/192 = 4/3 = 1.333)frequency relationship of 4/3 (in the example, the lower frequency is 192 Hertz, the higher frequency is 256 Hertz; 256/192 = 4/3 = 1.333)

24. ThirdsThirds [ii:41] major 3rd[ii:41] major 3rd frequency relationship of 5/4frequency relationship of 5/4 [ii:42] minor 3rd[ii:42] minor 3rd frequency relationship of 6/5frequency relationship of 6/5 3rds and 6ths are consonant because they do not have beats3rds and 6ths are consonant because they do not have beats

25. SixthsSixths [ii:43] major 6th[ii:43] major 6th frequency relationship of 5/3frequency relationship of 5/3 [ii:44] minor 6th[ii:44] minor 6th frequency relationship of 8/5frequency relationship of 8/5

26. SeventhsSevenths [ii:46] major 7th[ii:46] major 7th frequency relationship of 15/8frequency relationship of 15/8 [ii:47] minor 7th[ii:47] minor 7th frequency relationship of 9/5frequency relationship of 9/5 7ths and 2nds are dissonant because they have beats7ths and 2nds are dissonant because they have beats

27. SecondsSeconds [ii:48] major 2nd[ii:48] major 2nd a whole step or whole tone = two half stepsa whole step or whole tone = two half steps frequency relationship of 9/8frequency relationship of 9/8

28. SecondsSeconds [ii:49] minor 2nd[ii:49] minor 2nd the half step or semitonethe half step or semitone frequency relationship of 16/15frequency relationship of 16/15 the smallest interval in traditional western music the smallest interval in traditional western music the distance between any white and black key, and also the distance between the white keys E-F and B-Cthe distance between any white and black key, and also the distance between the white keys E-F and B-C