In music, the interval (from the Latin intervallum 'space' , actually "space between bulwarks", from Latin vallus " bulwark ") is the pitch between two at the same time (simultaneous, with the harmonic interval) or one after the other (successively, with the melodic interval Interval) sounds .
The most important of these intervals, the octave, underlies all historically developed tone systems. The pitch of any octave interval can be divided into one or the other diatonic - heptatonic scale . The pitches of this ladder are named after the Latin ordinal numbers : "Prime" (from Latin prima , "the first"), "second" (from secunda , "the second"), "third" (from tertia , "the third") etc. The steps form intervals with the starting tone of the ladder, each with the same name as the step. The starting tone itself has the number 1. This is why the intervals are based on an inclusive counting system: Prime refers to the interval that the starting tone (or any tone) forms with itself, i.e. the distance from the first to the second second tone, i.e. a distance of 1 tone level, etc. If the term does not mean the interval, but the relevant tone level, the clearer terms third tone , fifth tone etc. are sometimes used.
Example : Major third f 'a', fourth f 'b', fifth f 'c' and octave f 'f' '.
Intervals correspond to certain quotients (ratios, proportions): originally string length ratios, generally frequency ratios . Nowadays, their size is often measured in cents . When adding intervals (one after the other), the units of cents are added, but the frequency ratios are multiplied.
In conventional European music, the smallest interval used is the small second , also called a semitone . In the equal temperament , it measures 100 cents. All other intervals used in this music can be given as numbers of semitones.
Execution of intervals one after the other
→ Main article: The additive interval space
The successive execution of intervals can be described by addition or subtraction. The associated frequency ratios are multiplied or divided.
- Addition: minor third + major third = fifth or subtraction: fifth - minor third = major third.
- In cents : 316 cents + 386 cents = 702 cents or 702 cents - 316 cents = 386 cents.
- Frequency relationships: 6 / 5 • 5 / 4 = 3 / 2 or 3 / 2 : 6 / 5 = 5 / 4 .
The frequency ratios of the intervals are exponential. Therefore the size of an interval is calculated logarithmically.
interval In cents Frequency ratio 1 octave 1200 2: 1 2 octaves 2400 4: 1 3 octaves 3600 8: 1 5th = log 2 (3/2) octave≈7 / 12 octave 1200 • log 2 (3/2) = 702 cents 3: 2
Main article → Music theory in ancient Greece → The tone families
- Octave (frequency): 2: 1 (octave up when halving the length)
- 5th (frequency): 3: 2 (5th upwards at two thirds of the length)
- Fourth (frequency): 4: 3 (octave 2: 1 up, then fifth 3: 2 down, so: 2 ⁄ 1 : 3 ⁄ 2 = 4 ⁄ 3 )
- Whole tone (frequency): 9: 8 (fifth 3: 2 up, then fourth 4: 3 down, so: 3 ⁄ 2 : 4 ⁄ 3 = 9 ⁄ 8 )
He did not take into account the major third (5: 4), but an interval consisting of two major whole tones, by the syntonic comma (81:80): the ditone (81:64). If the ditone was subtracted from a pure fourth, the leimma remained (256: 243). With these intervals, no stable harmonic triad could be formed, so that ancient Greek music did not yet develop harmonics in the later European sense. Only Archytas and Didymos determined the major third (5: 4), Eratosthenes the minor third (6: 5).
The Pythagoreans only allowed intervals to be calculated as integer ratios. They did not find a quotient whose doubling results in 9: 8, so that they could not divide the whole tone into two equal semitones, but only into a smaller ( diesis ) and a larger ( apotome ) semitone. For them, an octave was not mathematically exactly the same as the sum of six whole steps or twelve semitone steps, because twelve perfect fifths in a row result in a slightly higher target tone than the seventh octave of the original tone. The difference is known as the Pythagorean comma .
For the first time, Philolaos converted added musical intervals into multiplied acoustic proportions. This method was optimized after 1585 by Simon Stevin using an exponential function and around 1640 by Bonaventura Francesco Cavalieri and Juan Caramuel y Lobkowitz using the logarithmic inverse function. Euclid hypothetically understood interval proportions as frequency ratios without being able to measure them.
In contrast to the Pythagoreans, Aristoxenus did not define intervals mathematically, but acoustically as an audible "space" (diastema) between two tones of a continuous melody , as was the case in Greek music practice. Accordingly, he assigned a certain number of fixed pitches (tones) to each interval that it encompasses. The fourth contained four successive notes, a so-called tetrachord . Its outer tones were later also briefly referred to as the interval, so that the term henceforth meant the distance from the first to the last tone of such a tone sequence.
Aristoxenus divided the whole tone practically into two, three or four equal sub-intervals. The various combinations of semitones and whole tones within a tetrachord resulted in its genus ( tone type : diatonic, chromatic or enharmonic). Two tetrachords following one another at a distance of a whole tone resulted in different scales (modes) within the framework of an octave.
Interval names and scale levels
Various tonal systems emerged in Europe, of which the major-minor tonal system prevailed over the alternatives known today as church modes in Central Europe . All of these European tonal systems are based on heptatonic scales ; H. Scales with seven pitches per octave, which are specially designed diatonic with five whole-tone and two semitone steps . The well-known diatonic interval names arose from the Latin ordinal numbers of these pitches ( prima “the first”, secunda “the second”, tertia “the third” etc.). The following interval names can be found in the literature:
|7th||Seventh or sept|
|13||Tredezime or third decimal|
|15th||Fifths or fifths or double octaves|
Types of intervals
The intervals second, third, sixth and seventh occur in two types each, as a large and small interval. The difference is a semitone in each case.
In addition, each interval can be excessive or diminished . This also means enlarging or reducing by a semitone. The excessive fourth (also called tritone ) and the diminished fifth can already be found in the root tone series : F - B or B - f , and accordingly in every major scale between the fourth and seventh degree and every minor scale between the second and sixth degree. These two intervals sound the same in the equal tuning . In all other cases, excessive or reduced intervals result from alteration , i.e. increasing or decreasing a tone by a semitone step.
Primes, fourths, fifths, and octaves that are neither excessive nor diminished are said to be pure . (The word "pure" has nothing to do with pure mood here.)
As an abbreviated notation for intervals and for pitches in chords (three or more tones of different pitch in harmony) has become common:
- Arabic numbers for the interval sizes or pitches: 1 = prime, 2 = second, 3 = third, etc.
- + = large
- - = small
- > = diminished (like decrescendo symbol)
- <= excessive (like crescendo symbol)
In musical practice, double excessive and double diminished intervals are rare .
As a complementary intervals , supplements intervals or reversal intervals one ever called two intervals in the octave that complement each other to an octave. The complementary interval is created by shifting the upper tone one octave down or the lower one an octave up in the given interval (basic form). Each are complementary:
- Primes and octaves,
- Seconds and sevenths.
- Thirds and sixths,
- Fourths and fifths.
Pure intervals remain pure, large ones are supplemented with small ones, diminished ones with excessive ones and vice versa. Intervals that go beyond the octave are not supplemented separately, but viewed as addition to an octave: A decime corresponds to an octave plus a third; a sixth is then also complementary to it.
Consonances and dissonances
If the tones of an interval sound simultaneously, they are divided into consonant (" sounding together") and dissonant ("diverging") sounds . Intervals are called consonant, the tones of which are perceived as merging with each other, well-fitting, harmoniously relaxed, calm and stable. Intervals are considered dissonant, the tones of which have a strong friction against each other and sound restless and therefore create the listener's desire to dissolve into a consonance.
Which intervals are or are perceived as consonant or dissonant depends primarily on culturally influenced listening habits. In general, however, the following applies: the degree of consonance is higher, the smaller the integer the ratio (the proportion) of the oscillation numbers (frequencies) of the two tones of an interval can be expressed. This discovery is attributed to Pythagoras . In antiquity and throughout the Middle Ages , only the octave (frequency ratio 1: 2), the fifth (2: 3) and the fourth (3: 4) were considered consonances. Since around 1500, thirds and sixths have gradually been perceived as consonances. All seconds and sevenths as well as all excessive or diminished primes, fourths, fifths and octaves are considered dissonances. The fourth had a special position since about the 16th century: in the theory of composition and counterpoint it was considered a dissonance if it was formed from three or more voices by the lower voices in a polyphonic setting.
The possibilities for using consonant intervals have expanded over the centuries of development of polyphonic music in Europe. According to the traditional theory of harmony in European art music, dissonant sounds are mainly used in musical composition to create harmonic tension on unstressed beats and especially to create cadence at finals or internal caesuras . A particularly typical example of this is the dominant seventh chord , which has the minor seventh as a dissonant tone. In the functional harmonics of European music, this sound has the function of increasing the harmonic tension before the consonant final sound. The functionally harmonious listener hears a clear striving tendency of the seventh ( leading tone ) - it has to be resolved down a semitone.
The use of dissonances for heightened harmonic tension became increasingly prevalent in Romantic and Late Romanticism . Already the music of Richard Wagner , Max Reger or Gustav Mahler showed tendencies that almost every tone belonging to the scale or outside the scale could be used as a leading tone that could be resolved upwards or downwards, so that the tonality began to dissolve (see also: diminished chord , excessive chord ).
In atonal music of the 20th century, but e.g. B. with jazz one can then speak of an emancipation of dissonance. In the composition technique of twelve-tone music, dissonances are preferred. As a result, consciously set consonances in these pieces of music appear “unstable”; Because of this attraction, for example, the triad in twelve-tone music could be used as a special means of expression in the form of a motif.
Diatonic intervals in the octave space have integer vibration ratios and therefore each have a characteristic sound, so that they can be recognized and distinguished even if they are slightly out of tune. That is why they appear under the same name in different moods .
In the just intonation all intervals from the root of a major or minor scale are tuned from the exact and heard it, based optimal: the small and large second with the frequency ratio 16 / 15 and 9 / 8 or 10 / 9 , the small and large Third with 6 ⁄ 5 and 5 ⁄ 4 , the fourth and fifth with 4 ⁄ 3 and 3 ⁄ 2 , the minor and major sixth with 8 ⁄ 5 and 5 ⁄ 3 and the minor and major seventh with 16 ⁄ 9 and 9 ⁄ 5 and 15 ⁄ 8 . The triads (thirds and fifths) of the tonic, the dominant and the subdominant are pure. With modulations (in addition to a change of sign in the notation) there is a pitch difference of a syntonic comma . If you tune a 12-step keyboard to a pure scale, other keys can only be used to a limited extent, which severely limits the harmonic possibilities.
Therefore, so-called temperatures with small detunings have been common since the Renaissance in order to be able to use more keys. Special moods are named after the special intervals that characterize them. In the mid-tone tuning , many major thirds are tuned in (the fifths are therefore about 5 cents too small) and the syntonic comma is distributed evenly over other intervals. In the case of the well-tempered tunings , the deviations from the pure tuning were extended so that all keys of the circle of fifths - albeit with different characteristics - became playable.
The unit cent is used to “measure” the fine changes in the intervals in the various moods . With equal tuning , all twelve semitones of the octave are precisely tuned to 100 cents, so that the Pythagorean comma is distributed over all pitches. All other intervals are slightly out of tune, but they sound the same in all keys.
Tables of fifths and thirds in all pitches and in the different tunings can be found in the section Comparison of tuning systems .
Table of intervals
|Prime||1 ⁄ 1||Prime||0 cents||0 cents|
25 ⁄ 24
135 ⁄ 128
small chromatic semitone
large chromatic semitone
256 ⁄ 243
16 ⁄ 15
|Leimma (Pythagorean tuning)
diatonic semitone (pure tuning)
10 ⁄ 9
9 ⁄ 8
|small whole tone (pure tuning)
large whole tone (pyth. and pure tuning)
32 ⁄ 27
6 ⁄ 5
|minor third (Pythagorean tuning)
minor third (pure tuning)
5 ⁄ 4
81 ⁄ 64
|pure major third
ditone (Pythagorean tuning)
|Fourth||4 ⁄ 3||pure fourth||498 cents||500 cents|
45 ⁄ 32
7 ⁄ 5
729 ⁄ 512
1024 ⁄ 729
64 ⁄ 45
10 ⁄ 7
|Fifth||3 ⁄ 2||perfect fifth||702 cents||700 cents|
|small sixth||8 ⁄ 5||pure minor sixth||814 cents||800 cents|
|major sixth||5 ⁄ 3||pure major sixth||884 cents||900 cents|
16 ⁄ 9
9 ⁄ 5
7 ⁄ 4
|pyth. and smaller pure (octave - large whole tone)
larger pure (octave - small whole tone)
|major seventh||15 ⁄ 8||diatonic pure||1088 cents
|octave||2 ⁄ 1||pure octave||1200 cents||1200 cents|
Detailed interval tables of the Pythagorean, mean-tone, pure and equal tuning:
The beginnings of well-known popular song melodies are often used to make the most important diatonic intervals easier to remember. However, this method is only reliable to a limited extent, as the same intervals can sound differently in different musical contexts - depending on, among other things, the position of the scale, type of tone, timbre, expression. For example, the minor third from E to G in C major (for example in "Olé, olé, olé") sounds different than the same interval in the key of E minor (for example in " O Heiland, tear the heavens up ", EG 7). The major third usually arouses a major association from the lower note upwards, but can also sound somber when played downwards: for example in the unison opening motif of Beethoven's “Fateful Symphony” (GGG-Es) . Here it is not yet audible whether this interval is to be classified as part of a C minor or E flat major sound.
|small second (semitone step)||
" If a bird comes flying ..."
" Snow - e - flakes, white skirts, when do you come with snow? ... "
" From Him mel high, I Come ..." ( Martin Luther )
" When I get older ..." (the beginning of When I'm sixty-four , The Beatles )
Fur Elise by Beethoven
|great second||" Al - le my ducklings"||
" Sleep, child , sleep"
" Yes - the day ..." (Lennon / McCartney - The Beatles)
" A Vo - gel wanted - te be married
..." " power up the door
..." " A - read my loue, ye do me wrong, ..."
" Häns - chen small ..."
" Kuk - kuck , Kuk - kuck , it's calling from the forest ..."
" Al - le birds are already there ..."
" And in the snow-ge-ge-bir ..."
" Mor - ning has bro-ken ..." (Cat Stevens)
" Kum - ba -ya, my Lord ..."
“ Innsbruck , I have to let you” ( Heinrich Isaac )
Leitmotiv of the 5th symphony by Beethoven (fateful symphony) : GG- G - Es (indifferent, see introduction)
“ Straw - ber -ry Fie - lds for-ever ... " (major) (The Beatles / John Lennon)
" Ce ci -lia you're brea-king my heart ... "(major) (beginning of Cecilia, Simon & Garfunkel )
Ta - tü ( Martinshorn )
" When all the little fountains flow ..."
" O Tannenbaum , ..."
Beginning of the Eurovision anthem based on Marc-Antoine Charpentier
" A - ma -zing Grace"
" Mor - gen , children, will give it that ..." (Melody of Carl Gottlieb Hering )
Little Night Music by WA Mozart, G - D - G - D - G - D - G -HD
|Tritone||" Ma - ri -a ..." (Maria from West Side Story )
" The Simp -sons ..." (beginning of the theme song of the Simpsons )
|In flown If a bird : "... by the lovers - most a greeting ..."
"... The marches the farmer the Röß-lein one-spans ..."
"By Mon - sun ..." ( Tokio Hotel )
|Fifth||" Wake up , my heart's beautiful ..."
Chariots of Fire by Vangelis (the first two notes of the keyboard surface sound)
" Tomorrow Santa Claus is coming ..."
" Ick Heff mol en Ham-borg-he Veermaster see ..." (Shanty)
" Now is the day has ended ..." (after Adam Krieger )
|small sixth||" When Israel was in Egypt's land ..."||"... even tight around the hand." (2nd end of Zum Tanze there goes a girl )
" Schick - sals-me -lodie" / " Where do I begin" (Soundtrack Love Story by Francis Lai )
|major sixth||" This image nis is enchantingly beautiful ..." (The Magic Flute, Mozart)
" The Christmas tree is the most beautiful tree ... "
" I know it will gescheh'n once a miracle
..." " Ma co -me ballistic bella bim-ba ..." (Italian folk song)
" My Bon nie is over the ocean ..."
" And now the end is near ...", My Way
" No bo dy knows the trouble I've seen, ..." (Gospel)
" Win - de weh'n , ships - fe go ..."
|minor seventh||" There's a place for us ..." (Somewhere from West Side Story )
"We put down with tears and ru -fen you ..." (repeated in the final chorus of the St. Matthew Passion , JS Bach )
" Sing, sing, what happened? ... "(beginning of the refrain of Zogen once five wild swans )
" The win - ner takes it all" (ABBA)
|"... and the He- inherit be -ginnt." From ferrous are already the woods|
O terra, addio, final duet from Aida
" Take on me" (A-ha)
Introduction to "Gash in Your Subversive Idyll" (Ec8or)
|The hut on chicken feet from pictures at an exhibition by Mussorgsky|
" Some - where over the rainbow ..." (Wizard of Oz)
" I'm singing in the rain ..."
"... even tight around the hand ." (1st end of Zum Tanze there goes a girl )
"of ... he did not read -sen can." Finally the canon Caffee ( Carl Gottlieb Hering )
Beethoven 9th Symphony 2nd movement (beginning)
Another possibility, which completely dispenses with donkey bridges, is to divide the intervals heard into consonances and dissonances. In the next step, a further differentiation can be made into perfect and imperfect consonances as well as soft and sharp dissonances. Finally, the size of the interval is roughly determined. With this method, the selection of possible solutions is limited to one or two intervals if the classification is correct. The great advantage of this method is that it enables the student to understand his solution and thus possible errors can be corrected more efficiently. One problem is the subjective perception of the categories consonance and dissonance.
See: Main article tone structure (mathematical description)
Frequency ratios can be assigned to intervals. The frequency ratios of multiples of intervals increase exponentially. The cent measure is a logarithmic measure of the frequency relationships. This is proportional to the size of the interval. Cent is a sub-unit of the octave with the definition 1200 cents = 1 octave (or 1 semitone of equal importance = 100 cents ).
|1 octave||2||1200 cents|
|2 octaves||4th||2400 cents|
|3 octaves||8th||3600 cents|
|k octaves||2 k||1200 k cents|
|minor third||6 ⁄ 5||1200 log 2 ( 6 ⁄ 5 ) cents = 315.641 cents|
|major third||5 ⁄ 4||1200 log 2 ( 5 ⁄ 4 ) cents = 386.314 cents|
|Fourth||4 ⁄ 3||1200 log 2 ( 4 ⁄ 3 ) cents = 498.045 cents|
|Fifth||3 ⁄ 2||1200 log 2 ( 3 ⁄ 2 ) cents = 701.955 cents|
When adding intervals (one after the other), the units of cents are added, but the frequency ratios are multiplied.
- Fifth + fourth = 702 cents + 498 cents = 1200 cents = octave. (Frequency ratios: 3 ⁄ 2 × 4 ⁄ 3 = 2 ⁄ 1 )
- Minor third + major third = 316 cents + 386 cents = 702 cents = fifth. (Frequency ratios: 6 ⁄ 5 × 5 ⁄ 4 = 3 ⁄ 2 )
An interval space can be viewed as an additive ordered computational area. The addition corresponds to the execution of intervals one after the other.
The most important intervals are:
|Name of the interval space||Basic intervals||Interval space|
|The twelve-level interval space Interval
space of equal-level mood
|Basic interval: The semitone H with 100 cents||All intervals are multiples of H.|
|The system of fifths
interval space in Pythagorean tuning
|The basic intervals are the octave Ok and the fifth Q||All intervals are multiples of Ok and Q|
|The fifth-third system
interval space of pure tuning
|Basic intervals are the octave Ok, the fifth Q and the major third T.||All intervals are multiples of Ok, Q and T|
|The all-encompassing interval space||The intervals can be divided as required.||All intervals are (real) multiples of the octave
. The unit cent = 1/1200 ok should be located here.
- Sigalia Dostrovsky, John T. Cannon: Development of musical acoustics (1600-1750). In: Frieder Zaminer (ed.): History of music theory. Volume 6. Darmstadt 1987, ISBN 3-534-01206-2 , pp. 7-79.
- Mark Lindley: Mood and Temperature. In: Frieder Zaminer (ed.): History of music theory. Volume 6. Darmstadt 1987, ISBN 3-534-01206-2 , pp. 109-332.
- Wilfried Neumaier: What is a sound system? Frankfurt am Main / Bern / New York 1986, ISBN 3-8204-9492-8 .
- Frank Haunschild: The new theory of harmony. Volume 1. AMA-Verlag, Brühl 1998, ISBN 978-3-927190-00-9 , pp. 32-42 ( intervals ) and 104 ( general information on chords).
- Wieland Ziegenrücker: General music theory with questions and tasks for self-control. German Publishing House for Music, Leipzig 1977; Paperback edition: Wilhelm Goldmann Verlag, and Musikverlag B. Schott's Sons, Mainz 1979, ISBN 3-442-33003-3 , pp. 63-77 ( Die Intervalle ).
- Bernd Alois Zimmermann : Interval and Time: Articles and writings on the work. Edition Schott, Mainz 1974, ISBN 3-7957-2952-1 .
- List of frequency ratios and their German interval names ( Memento from July 11, 2007 in the Internet Archive )
- GNU Solfege, free hearing training software
- Another interval trainer
- Lissajous curves: Simulation for the graphic representation of musical intervals, beats, vibrating strings
- Joachim Mohr: Tones and intervals
- Ulrich Kaiser: Intervals and chords OpenBook for children
- On the etymology cf. Helmut KH Lange: General music theory and musical ornamentation. A textbook for music schools, conservatories and conservatoires. Franz Steiner Verlag, Stuttgart 1991, p. 57; in addition Douglas Harper: interval . In: Online Etymology Dictionary .
- M. Honegger, G. Massenkeil (ed.): The great lexicon of music. Herder, 1976, Volume 4, p. 194.
- HJ Moser: General music theory. 3. Edition. Verlag de Gruyter, 1968, p. 42.
- Walter Opp: Handbuch Kirchenmusik , Volume 1. Merseburger, 2001, ISBN 3-87537-281-6 , pp. 216, 225, 235.
- With the frequencies for f '(352 Hz ), a' (440 Hz), b '(469.33 Hz), c' '(528 Hz) and f' '(704 Hz) the frequency ratio of the intervals is calculated as follows : Major third = 5: 4 (f 'a': 440 Hz: 352 Hz = 5: 4), fourth = 4: 3 (f 'b': 469.33 Hz: 352 Hz = 4: 3), fifth = 3: 2 (f 'c' '528 Hz: 352 Hz = 3: 2) and octave = 2: 1 (f' f '': 704 Hz: 352 Hz = 2: 1)
- In tunings that are not of the same size there are several semitones of different sizes, e.g. B. in the pure tuning a total of three. The counting of the semitones of an interval is only an approximate description.
- 12 fifths ≈7 octaves, i.e. 1 fifth ≈ 7/12 octaves. (The difference to the exact value is the Pythagorean comma ). With an octave = 1200 cents the result is: fifth ≈ 700 cents.
- The calculation is carried out here in the modern version with the frequency ratios (intervals upwards greater than 1, intervals downwards less than 1). The reciprocal values of the frequency ratios correspond to the length ratios of the string.
- Arnold Schering: Handbuch der Musikgeschichte , Georg Olms Verlag, Hildesheim 1976, p. 23.
- Peter Schnaus: European Music in Spotlights. Meyers Lexikonverlag, Mannheim a. a. 1990, ISBN 3-411-02701-0 , p. 28.
- Peter Schnaus: European Music in Spotlights. P. 25.
- Helmut KH Lange: General music theory and musical ornamentation. A textbook for music schools, conservatories and conservatoires . Franz Steiner, Stuttgart 1991, ISBN 978-3-515-05678-6 , pp. 59 ( limited preview in Google Book search).
- Gottfried Weber: General music theory for teachers and learners . Carl Wilhelm Leske, Darmstadt 1822, p. 58 ( limited preview in Google Book search).
- Mark Levine: The Jazz Piano Book . Advance Music, Petaluma 1992, ISBN 3-89221-040-3 , pp. 33 .
- Helmut KH Lange: General music theory and musical ornamentation. A textbook for music schools, conservatories and conservatoires . Franz Steiner, Stuttgart 1991, ISBN 978-3-515-05678-6 , pp. 24 ( limited preview in Google Book search).
- online music theory according to Everard Sigal
- Hermann Grabner: Allgemeine Musiklehre , p. 84.
- There are two whole tones in pure tuning: the large (Pythagorean) whole tone with the frequency ratio 9 ⁄ 8 and the small whole tone with the frequency ratio 10 ⁄ 9 . Executing these two whole tone intervals one after the other results in the major third with the frequency ratio 5 ⁄ 4 .
- Corresponding to the two big seconds (whole tones) there are two small sevenths with the frequency ratios 16 ⁄ 9 and 9 ⁄ 5 .