ByBy

Prof. Lydia AyersProf. Lydia Ayers

Types of IntervalsTypes of IntervalsTypes of IntervalsTypes of Intervals

• augmented intervalsaugmented intervals + 1/2 step + 1/2 step

• diminished intervalsdiminished intervals - 1/2 step - 1/2 step

• perfect intervalsperfect intervals

• major intervalsmajor intervals

(1/2 step)(1/2 step)• minor intervalsminor intervals

Major and Minor IntervalsMajor and Minor IntervalsMajor and Minor IntervalsMajor 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 major intervals are one half step larger than

minor intervalsminor intervals

minor 2nd

Major 2nd

Perfect 4th

• perfect intervals have only one sizeperfect intervals have only one size

Augmented 4th is one half steplarger than Perfect 4th;

same size as diminished 5th

Major and Minor IntervalsMajor and Minor IntervalsMajor and Minor IntervalsMajor and Minor Intervals• major intervals are one half step larger than major intervals are one half step larger than

minor intervalsminor 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 step 2 half steps 3 half steps 4 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

IntervalIntervalIntervalInterval• all intervals of the major scale measured up from all intervals of the major scale measured up from

the tonic are either MAJOR or PERFECT the tonic are either MAJOR or PERFECT

M2

P4

P5

M6

M7

P8

M3

Interval:Interval:Interval:Interval:• all intervals of the major scale measured down all intervals of the major scale measured down

from the tonic are either MINOR or PERFECTfrom the tonic are either MINOR or PERFECT

m3

P4

P5

m6

m7

P8

m2

Complementary IntervalsComplementary IntervalsComplementary IntervalsComplementary 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 4th a perfect 5th + perfect 4th

Complementary IntervalsComplementary IntervalsComplementary IntervalsComplementary Intervals

M2

P4P5

M6M7

P8

M3

m3P4

P5m6

m7P8

m2

Complementary IntervalsComplementary IntervalsComplementary IntervalsComplementary Intervals• Multiplying the frequency ratios of two Multiplying the frequency ratios of two

complementary intervals produces an octavecomplementary intervals produces an octave

X+

==

Matching HarmonicsMatching HarmonicsMatching HarmonicsMatching Harmonics

• [ii:45][ii:45] just consonances have some matching just consonances have some matching harmonics from the harmonic seriesharmonics from the harmonic series

How Can We Measure Interval How Can We Measure Interval Size?Size?

How Can We Measure Interval How Can We Measure Interval Size?Size?

• CentsCents• octave divided into 1200 octave divided into 1200 centscents• frequency ratio for 1 cent = frequency ratio for 1 cent = 1200 1200 or or

221/12001/1200 = 1.0005778 = 1.0005778• frequency ratio for 100 cents (equal-frequency ratio for 100 cents (equal-

tempered semitone) = or 2tempered semitone) = or 21/121/12 = = 1.05946311.0594631

Just Noticeable DifferenceJust Noticeable DifferenceJust Noticeable DifferenceJust Noticeable Difference

• 1 to 4 cents 1 to 4 cents • hearing test: write down whether the hearing test: write down whether the

pitch in the four pairs of tones goes up, pitch in the four pairs of tones goes up, down or stays the same down or stays the same

1.1. [ii:50][ii:50]

2.2. [ii:51][ii:51]

3.3. [ii:52][ii:52]

4.4. [ii:53][ii:53]

CentsCentsCentsCents• 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 multiply by 1731.234 (if using "log" multiply

by 3986.314)by 3986.314)• example: example:

• 3/2 = 1.5 3/2 = 1.5 • In(1.5) = .4054651 In(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)

Why the magic number Why the magic number 1731.234? 1731.234?

Why the magic number Why the magic number 1731.234? 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)2^(y/1200) = x= x

ln(ln(2^(y/1200)2^(y/1200))) = = ln(ln(xx))

(y/1200)*ln(2)(y/1200)*ln(2) = ln(x)= ln(x)

y * (ln(2)) / 1200y * (ln(2)) / 1200 = ln(x)= ln(x)

yy = ln(x)= ln(x) * 1200 / ln(2) * 1200 / ln(2)

Review QuestionReview QuestionReview QuestionReview Question

How many cents difference between 550 How many cents difference between 550 and 440 Hertz?and 440 Hertz?

A.A. 701.95498 701.95498

B.B. 386.3137386.3137

C.C. 315.64128315.64128

D.D. 407.81999407.81999

E.E. none of the other answersnone of the other answers

Interval Name ChartInterval Name ChartInterval Name ChartInterval Name ChartRatioRatio NameName81/8081/80 Syntonic commaSyntonic comma16/1516/15 minor secondminor second12/1112/11 narrow neutral secondnarrow neutral second11/1011/10 neutral secondneutral second10/910/9 narrow major secondnarrow major second9/89/8 major secondmajor second8/78/7 supermajor secondsupermajor second7/67/6 subminor thirdsubminor third6/56/5 minor thirdminor third11/911/9 neutral thirdneutral third5/45/4 major thirdmajor third9/79/7 supermajor thirdsupermajor third

Interval Name ChartInterval Name ChartInterval Name ChartInterval Name ChartRatioRatio NameName4/34/3 perfect fourthperfect fourth11/811/8 super fourthsuper fourth7/57/5 augmented fourthaugmented fourth10/710/7 diminished fifthdiminished fifth3/23/2 perfect fifthperfect fifth11/711/7 subminor sixthsubminor sixth8/58/5 minor sixthminor sixth5/35/3 major sixthmajor sixth12/712/7 supermajor sixthsupermajor sixth7/47/4 subminor seventhsubminor seventh9/59/5 wide minor seventhwide minor seventh11/611/6 neutral seventhneutral seventh15/815/8 major seventhmajor seventh

<w1:32><w1:32> Kraig Grady:Kraig Grady:Opening Invocation for a Shadow Opening Invocation for a Shadow

PlayPlay

<w1:32><w1:32> Kraig Grady:Kraig Grady:Opening Invocation for a Shadow Opening Invocation for a Shadow

PlayPlay• in just tuning on a tubulong instrumentin just tuning on a tubulong instrument

• harmonic series scale: Hharmonic series scale: Hnn = H = Hn-3n-3 + H + Hn-2n-2

• improvised using special melodiesimprovised using special melodies

http://www.anaphoria.com/shadow.html

[ii:36][ii:36] Unison Unison[ii:36][ii:36] Unison Unison• the most perfect consonancethe most perfect consonance• two (or more) pitches which have the same two (or more) pitches which have the same

frequency frequency • frequency relationship of 1/1 (in the example, both frequency relationship of 1/1 (in the example, both

frequencies are 64 Hertz) frequencies are 64 Hertz) • with two sine waves in unison, it is difficult to with two sine waves in unison, it is difficult to

hear two toneshear two tones

[ii:37][ii:37] Octave Octave[ii:37][ii:37] Octave Octave• the second most perfect consonancethe second most perfect consonance• the interval between two pitches in which the interval between two pitches in which

one has double the frequency of the other one has double the frequency of the other • frequency relationship of 2/1 (in the frequency relationship of 2/1 (in the

example, the lower frequency is 64 Hertz, example, the lower frequency is 64 Hertz, the higher frequency is 128 Hertz) the higher frequency is 128 Hertz)

OctaveOctaveOctaveOctave• the name "octave" comes from the Italian the name "octave" comes from the Italian

word for "8" because it is the 8th pitch in the word for "8" because it is the 8th pitch in the scalescale

• 8va means to play a pitch an octave higher 8va means to play a pitch an octave higher than written - used to avoid ledger lines than written - used to avoid ledger lines without changing the clefwithout changing the clef

[ii:38] 8va symbol in the first measure raises thenotes to the same pitches as in the second measure

[ii:39][ii:39] Perfect Fifth Perfect Fifth[ii:39][ii:39] Perfect Fifth Perfect Fifth• the third most perfect consonancethe third most perfect consonance• the interval between two pitches in which one the interval between two pitches in which one

has 1.5 times the frequency of the other has 1.5 times the frequency of the other • frequency relationship of 3/2 (in the example, the frequency relationship of 3/2 (in the example, the

lower frequency is 128 Hertz, the higher lower frequency is 128 Hertz, the higher frequency is 192 Hertz; 192/128 = 3/2 = 1.5) frequency is 192 Hertz; 192/128 = 3/2 = 1.5)

[ii:40][ii:40] Perfect Fourth Perfect Fourth[ii:40][ii:40] Perfect Fourth Perfect Fourth• the other intervals, such as the perfect 4th, the other intervals, such as the perfect 4th,

get their names from the same numbering get their names from the same numbering system system

• frequency relationship of 4/3 (in the frequency relationship of 4/3 (in the example, the lower frequency is 192 Hertz, example, the lower frequency is 192 Hertz, the higher frequency is 256 Hertz; 256/192 the higher frequency is 256 Hertz; 256/192 = 4/3 = 1.333) = 4/3 = 1.333)

ThirdsThirdsThirdsThirds

• [ii:41][ii:41] major 3rd major 3rd • frequency relationship of 5/4 frequency relationship of 5/4

• [ii:42][ii:42] minor 3rd minor 3rd • frequency relationship of 6/5frequency relationship of 6/5

• 3rds and 6ths are consonant because 3rds and 6ths are consonant because they do not have beats they do not have beats

SixthsSixthsSixthsSixths

• [ii:43][ii:43] major 6th major 6th • frequency relationship of 5/3frequency relationship of 5/3

• [ii:44][ii:44] minor 6th minor 6th • frequency relationship of 8/5frequency relationship of 8/5

SeventhsSeventhsSeventhsSevenths

• [ii:46][ii:46] major 7th major 7th• frequency relationship of 15/8frequency relationship of 15/8

• [ii:47][ii:47] minor 7th minor 7th • frequency relationship of 9/5frequency relationship of 9/5

• 7ths and 2nds are dissonant because they 7ths and 2nds are dissonant because they have beats have beats

SecondsSecondsSecondsSeconds

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

SecondsSecondsSecondsSeconds• [ii:49][ii:49] minor 2nd minor 2nd

• the half step or semitone the half step or semitone • frequency relationship of 16/15 frequency relationship of 16/15 • the smallest interval in traditional western the smallest interval in traditional western

music music • the distance between any white and black key, the distance between any white and black key,

and also the distance between the white keys and also the distance between the white keys E-F and B-CE-F and B-C