MofA Week 3
Contents
- 1 Tuesday - Announcements
- 2 Music Practicum and theory intro
- 3 Review: Arab, Arab Music, and Dananeer
- 4 Theory and practice in Arab music
- 5 Setup
- 6 Medieval Arabic theory of music (science, metaphysics)
- 7 Basic acoustical concepts underlying Arab music theory
- 8 Pythagorean scales - summary
- 9 Tonal theory in medieval period in Arabic-speaking regions
- 10 Theoretical approaches: scale and mode
- 11 Touma's interpretation of scales and maqam
- 12 Contemporary concept of maqam in Egypt and Levant
- 13 Informal discourse about music
- 14 Musical practice
Tuesday - Announcements
- Greetings: Ahlan wa Sahlan! - "Ahlan Biik!" ("Ahlan Biiki" for female, "Ahlan Biikum" for group)
- CSL: See http://www.csl.ualberta.ca/ to sign up via their Portal, then print, sign, and return form to them directly.
- Zotero; : please join our new AMP group. Check out Zotero and become familiar with how it works as a collaborative tool, to support our CSL projects.
Music Practicum and theory intro
- Ear training
- maqam Rast, Sika, Hijaz: scale degrees, qafla
- Durub: wahda, maqsum, MuhajjarListen to muwashshah: Badat Min al-Khidri[1] Badat Min al-Khidri
- Dulab Rast
- Theoretical representations of pitch collections: maqamat and ajnas
Review: Arab, Arab Music, and Dananeer
Theory and practice in Arab music
- consider the relation among
- formal (music) theory (e.g. philosophical treatises)
- informal discourse (about music) (e.g. work of Scott Marcus on intonation, modulation)
- (musical) practice (composition, improvisation) (e.g. analysis of performances by Dr Jihad Racy)
- In relation to practice, theory can be
- Prescriptive (and then often ideological, polemical): streamlined (Safi al-Din al-Urmawi) vs. Descriptive (ethnographic): accumulative (al-Farabi)
- Some "theory" is very informal, more like a mnemonic for performers ("finger modes")
- More practical vs. more philosophical
- More musical vs. autonomous and metaphysical (often venturing far beyond music per se, a metaphor for cosmic relationships: e.g. cosmological theory, music of the spheres, relation of modes to zodiac, seasons, bodily humors...we'll talk about this next week)
- Why read music theory?
- Mainly, because we have no other access to music in the region prior to the 19th century
- Theory projects ideologies of musical aesthetics, helps to attain an emic understanding
- Theory condenses broader cultural values
- Theory reflects cultural politics (e.g. Abbasid inclusiveness)
- Theory reflects broader historical and discursive trends and provides historical insights (e.g. influence of Greek philosophy, Western practice)
- Theory may also be useful to you as a musician!
- Arab music theory includes:
- Focus on urban "art" music synthesis (a complex mix of Arabian, Persian, Byzantine, and other musics, a fusion catalyzed by Islamicate civilization), but not music of rural areas, local music, music of ethnic minorities, or religious music
- High value of music (from Greeks) that "music" is a key subject for philosophy, part of the medieval "quadrivium" (arithmetic=number, geometry=number in space, music=number in time, astronomy=number in space/time) traceable to the Greeks (Pythagoras, Plato) via Roman Boethius (6th century) {contrasted with the lower trivium: study of rhetoric, grammar, logic}
- Borrowings from Greek theory (from Euclid to Aristoxenus via Bayt al-Hikma), but adapted to new musical practices (word "musiqa" itself comes from Greek and designates at first music theory)
- Scientific/descriptive theory
- Acoustical and psychoacoustical theory of
- pitch, interval, tetrachord, scale
- rhythm and meter
- Modal theory of maqam
- Theory of melody, composition
- Theory of musical instruments
- Acoustical and psychoacoustical theory of
- Objective (cosmological) theory: music's connection to the cosmos
- Subjective (microcosmological) theory: music's influence on the human being
- Music therapy
- Music, emotion (tarab), spirituality (sama`)
- Music and ethos
- Musical polemics vis a vis Islam
- Malahi
- Sama`
Setup
Setup:
Medieval Arabic theory of music (science, metaphysics)
- Early period: "finger modes" - mnemonics related to the oud - no mathematical precision, only enough to specify modality; not placed in a broader theory (and there are ongoing debates about their content).
- Impact of Greek thought changed all this.
- Caliph Ma'mun (r. 813-33) and Bayt al-Hikma (House of Wisdom; actually founded by his father, Harun al-Rashid, r. 786–809), in Baghdad (9th to 13th centuries, until destruction by Mongols).
- Range of disciplines: medicine, history, physics, chemistry, astronomy, zoology, cartography, geography, geometry, arithmetic, music...
- Translations from other languages (Greek, Persian, Syriac, Sanskrit)
- New scholarship
- Influence of Greek philosophical-scientific treatises on Arab music theory
- Word "musiqi" ("musiqa") enters Arabic from Greek, comes to imply theory
- Centrality of number
- Pythagorean tuning (stacked 5ths, or ratios of 3:2)
- Double octave system
- Tetrachords
- Consonance and dissonance
- General notion of systematic exposition
- Music theory as a principal philosophical-mathematical pursuit, part of what was later called the quadrivium (music, arithmetic, geometry, astronomy), studied after the trivium (grammar, logic, rhetoric), and together constituting a liberal education in preparation for the study of philosophy
- Importance of music in any philosophical oeuvre
- Music theory as also applied and mnemonic
- Key figures in Arabic language music theory: philosophers (ethnicity is a dubious construction here...!)
- al-Kindi (d. 870) ("the philosopher of the Arabs")
- Ibn al-Munajjim (d. 912) - documents "classical" style of Ishaq al-Mawsili (d. 850)
- al-Farabi (d. 950)
- Ikhwan al-Safa (late 10th c) [Epistle 5: On Music]
- Ibn Sina (d. 1037)
- Safi al-Din al-Urmawi (d. 1294)
- Two kinds of theory:
- Sonic, mathematical, and aesthetic (we'll talk about this today)
- Metaphysical and spiritual (we'll talk about this in another week)
- Difficulties in interpreting medieval theory
- Relation of theory and practice? Theory may be prescriptive, descriptive, or independent.
- Theory as mnemonics, understood in conjunction with oral tradition - often details of the written tradition are incomplete and impossible to interpret unambiguously (see Wright on Ibn al-Munajjim)
- Theory of earlier period is filtered by later ideologies
- Many works and all sound is lost
- Components of theory of sound in Arabic writings
- Rhythmic theory of cycles or modes: iqa`, usul, or darb
- Tonal theory (our focus today)
- pitches, intervals
- gamut: full pitch or interval set, a “tuning system”
- ajnas (singular: jins): tetrachords
- Scales: structured subsets, usually 7 tone, with additional structure added (tonic, dominant, starting, ending tones)
- Melodic modes: melody-generators – scales with additional structure.
Basic acoustical concepts underlying Arab music theory
(use Audacity, tone generator, spreadsheet and associated audio examples, and digital monochord to illustrate) Basic ideas about sound:
- Time signal: music is number in time
- Periodic signal: a signal repeating exactly, cycling - over all time (cps=cycles per second, or Hertz)
- Locally periodic signal: a signal repeating exactly, for a while...
- Period: shortest duration over which signal repeats (e.g. 0.0023 seconds)
- Frequency: reciprocal of period, the number of periods per second (e.g. 1/0.0023 seconds = 440 Hertz)
- Sound: pressure wave as time signal
- Pitched sound: locally periodic signal
- Constant pitched sound: fixed period or frequency (e.g. A4 = 440 Hertz)
- DEMONSTRATION: Audacity
- Musical intervals: fixed frequency ratio (e.g. octave = 2/1, fifth = 3/2, major 2nd = 9/8)
- To stack intervals, multiply ratios (e.g. up two octaves = 4/1; up two fifths = 9/4)
- For intervals, reciprocals are equivalent but oppositely directed:
- Ratio > 1 is ascending
- Ratio between 0 and 1 is descending
- ex: 3/2 = "up a fifth", 2/3 = "down a fifth"
- String length ratio is the reciprocal of frequency ratio. The Pythagoreans referred to strings (the monochord) because they did not understand frequency.
- dividing a string in half raises the open string pitch by an interval 2/1 or an octave
- dividing a string into 1/3 and 2/3 and plucking the longer segment raises the open string pitch by an interval of 3/2 or a fifth
- DEMONSTRATION: with tone generator (from A 440 illustrate: ratios 2/1, 1/2, 3/2, 9/4, 9/8 = 880, 220, 660, 990, 495). Monochord.
- Octave equivalence and reducing intervals to a single octave
- Two pitches that are an integral number of octaves apart (ratio: 2/1, 4/1, 8/1, 16/1...) sound "sort of" the same (same "chroma", different "height").
- That is, multiplying or dividing the frequency by a power of two doesn't change the pitch "chroma"
- A single octave can be represented by all ratios between 1 and 2 multiplied by the lower octave frequency (e.g. 440 Hertz to 880 Hertz)
- All pitches can be mapped into this octave without changing chroma, as follows:
- if the ratio is >2, then divide it by 2, and repeat until it's between 1 and 2
- if the ratio is <1, then multiply it by 2, and repeat until it's between 1 and 2
- ex: "up two fifths" = 9/4, but 9/4 > 2, so reduce it to 9/8
- ex: "down a fifth" = 2/3, but 2/3 < 1, so reduce it to 4/3
- Equal divisions of the octave and cents
- Tempered semitones (octave divided in 12 equal parts, S, such that S x S x S x S x S x S x S x S x S x S x S x S = S^12= 2
- S = 12th root of 2
- Cents (tempered semitone divided in 100 equal parts)
- Each tempered semitone comprises 100 cents.
- Thus there are 1200 cents to the octave.
- All intervals can be measured in cents (or any other interval)
- Measuring frequency ratios in intervallic units (log scales)
- Given a frequency ratio R, how many octaves does it contain?
- Need to determine how many 2s in the sequence: 2x2x2x...x2 = R
- Answer: log(R) base 2
- E.g.: how many octaves are in the frequency ratio 32?
- Given a frequency ratio R, how many tempered semitones does it contain? How many cents?
- Need to determine how many tempered semitones in the sequence: S x S x S x ... x S = R
- Answer: Number of SEMITONES = log(R) base S. Number of CENTS = 100 * # semitones
- E.g.: how many tempered semitones are in the frequency ratio 3:2? Take log(3/2)base (12th root of 2) = 7.02 semitones or 702 cents.
- An easier way to think about it, perhaps: Take log(3/2) base 2 - this gives us the number of octaves. Then simply multiply by 1200.
- Given a frequency ratio R, how many octaves does it contain?
Pythagorean scales - summary
- (Use virtual keyboard to illustrate)(use Audacity, tone generator, spreadsheet and associated audio examples to illustrate)
- Music and ancient Greek thought, from Pythagoras to Plato
- To Pythagoras (6th century BC) is attributed a mystical interpretation of number. He came from Samos, close to modern Turkey, but founded his school in Croton (present-day Italy).
- The basic numbers were positive integers, used for counting, and their ratios. These they considered "rational".
- Pythagorean thought influenced successors, Socrates, Aristotle, and Plato, the Neoplatonists, and down to the medieval period.
- Pythagoreans regarded number as sacred, especially small whole numbers.
- They favored numbers which were ratios of whole numbers as more harmonious
- Certain numbers were key. The numbers 1, 2, 3, 4 (sum=10) written as a triangle were called tetractys, representing the dimensions (0, 1, 2, 3), respectively:
- 1=the monad (creator), 0 dimensions (a point)
- 2=dyad (power), 1 dimension (2 points determine a line)
- 3 = triad (harmony), 2 dimensions (3 points determine a plane)
- 4=tetrad (cosmos), 3 dimensions (4 points determine a space), the four humors of the body, 4 elements, etc.
- Tetractys could represent ratios among these values and hence basic musical intervals; there was even a Prayer to the Tetractys.
- They disliked numbers that could not be represented as a ratio of integers (e.g. pi, square root of 2), and called them "irrational", "proving" the square root of 2 is rational
- Numbers and musical intervals
- The Pythagoreans recognized that string length is related to pitch: shorten the string, and the pitch goes up. They observed particular intervals: octave (halve the string), fifth (two thirds). (They didn't know about frequencies.) and that many "consonant" intervals were related to small integer ratios.
- Naturally they sought to apply their focus on integer numbers.
- Musical distances (intervals) were understood as equivalent to numbers
- Music thus reflected Divine Principles.
- The ancient Greeks sought just intonations: frequency (string) ratios always whole numbers (ideally small ones!). (Only in modern times were tempered intonations introduced, subdividing the octave into equal pieces.)
- They also believed in the Harmony of the Spheres.
- Your ear hears the octave (diapason) as a kind of "sameness" (different pitch "height", but same "chroma") - an octave doesn't present "new" pitch content, from a certain standpoint.
- Thus multiplying or dividing the interval by 2 doesn't change this basic "chroma" quality of the pitch.
- To map onto a single octave: multiply or divide by 2 until you have a number between 1 and 2.
- Basic ratios: 1/1 (unison), 2/1 (octave)...and the all-important 3/2 (generating "new" pitches with respect to the octave). Going down we have: 1/2, 2/3 (=4/3). (Also 5/4, 6/5, etc.)
- What's important to recognize is that a sequence generated by 3/2 never repeats, even when wrapped back into a single octave (by dividing by 2). There's a clash between the "new" of 3/2 and the "same" of 2/1.
- Greeks used "monochord" as a scientific instrument to illustrate intervals (whereas Arabs would use the `ud, the tambur, and other instruments). Try this Monochord simulator
- Observe graph on spreadsheet and listen to accompanying examples: construction of the diatonic scale.
Theory:
- Consider a series of fifths (3/2) always reduced to a single octave (by moving pitches up and down an integral number of octaves when necessary)
- let a fifth be represented as @ - an undirected tonal distance
- from any pitch: let an ascending fifth be written: @^1, descending fifth: @v1 (or ^@1, v@1, respectively)
- Let's start at @0
- ascending series: @^1, @^2, @^3, @^4...
- descending series: @v1, @v2, @v3, @v4...
- Then:
- @^2 = whole tone up = (3/2)*(3/2)*(1/2) = 9/8 (about 204 cents), while @v2 = whole tone down
- @v1 = fourth up = (2/3) * 2 = 4/3
- @v5 = limma up = (2/3)^5 * 2^3 = 256/243 = L (about 90 cents), while @^5 = limma down
- @^12 = Pythagorean comma up = (3/2)^12 / (2^7) = 3^12/2^19 = 531441 / 524288 = ^C (about 24 cents), while @v12 = Pythagorean comma down
- See Tuning and Scales in Greek, Arab, and Western Theory, circle graph (download for full flexibility)
- Creating a whole tone out of limmas and commas
- What is @^7? @^7 = up seven fifths = up twelve fifths, down 5 fifths = @^12 - @^5 = comma + limma = ^CL
- @^2 = @^7 - @^5 = ^CLL = whole tone = ^LLC
- OR: @^12 = @^5 + @^5 + @^2 = -L -L + W = C, thus W = C+L+L
- The Pythagorean diatonic (heptatonic scale)
- Fundamental plus 6 ascending fifths creates a scale: @^0, @^1, @^2, @^3, @^4, @^5, @^6
- Taken in frequency order: @^0, @^2, @^4, @^6, @^1, @^3, @^5
- Now start with @^1: @^1, @^3, @^5, @^0, @^2, @^4, @^6, repeating to @^1
- In fifths: up 2, up 2, down 5, up two, up two, up two, down five, etc.
- ..."up 2" = whole tone, or LLC
- ..."down 5" = limma, L
- Scale becomes: LLC LLC L LLC LLC LLC L
- Note this scale contains two small intervals (limmas): @^5 to @^0, and @^6 to @^1. But they're non-adjacent (separated by whole tones).
- Aside: Why seven tones?
- If we added an eighth - @^7 - it'd fall between @^0 and @^2 - we'd have @^0, @^7, @^2, @^4, @^6, @^1, @^3, @^5
- and thus three small intervals in a row:
- @5 to @0 = limma (5 fifths)
- @0 to @7 = limma + comma (7 fifths)
- @7 to @2 = limma (5 fifths)
- Aside: Naming
- Since we use just 7 tones, we can name them with 7 letters (from @1): C D E F G A B
- Note that @0 is F, and @6 is B.
- If we shift everything up by a fifth, then we drop F=@0 and add @7, which is higher than F by LC, between F and G. So we can call it F#. Shift everything up another fifth and the note C drops out, replaced by C# (you can now see how key signatures are formed). The interval between a letter name and the same letter name shifted up or down by adding or subtracting a single flat or sharp is equivalent to a movement of 7 fifths; it's LC.
- The limma interval @^5 corresponds to the small intervals -- 7th to tonic interval, and 3rd to 4th -- in the diatonic scale; it's embedded in the scale, so its two tones - differing by a limma - must have successive letter names, e.g. E to F (@5 to @0), F# to G (@7 to @2)
- Divide a whole tone F-G as follows: From F v@5 moves us to the note for which F is leading tone, which we can call Gb, a limma above F. From F, ^@2 moves us to G. Moving up ^@5 from G gives us G's leading tone, F#, a limma below G. These two leading tones are separated by @12 which is a comma. So the whole tone interval F-G (^@2) is now divided as follows: F (L) Gb (C) F# (L) G.
- Thus: Three intervals in the chromatic series: F - Gb - F# - G = L C L (F to Gb and F# to G are leading tone relations, while Gb to F# corresponds to 12 fifths - a Pythagorean comma). So ^C corresponds to respelling the note using the next lower letter.
- Compare: Pythagorean, Just, and Tempered scales (see spreadsheet, hear Audacity files)
Refer to spreadsheet and associated audio examples
Tonal theory in medieval period in Arabic-speaking regions
- Theory is closely linked to instruments, particularly chordophones (ud and tanbur), providing flexible visual representation (monochord was Greek theoretical instrument)
- Most often the `ud serves as reference
- 5 strings (low to high): bam - mathlath - mathna - zir - hadd (mix of Arabic and Persian terms)
- 4 "frets": sababa - wusta - binsir - khinsir (names of the fingers: index, middle, ring, pinky) (debate as to whether these are theoretical or real frets)
- 5 notes per string (but some are variable)
- Each string provides a tetrachord (jins)
- Jins species (anwa`) - four notes
- First degree fixed (mutlaq)
- fourth degree fixed (khinsir) - perfect 4th
- third degrees is variable (sababa OR wusta, not both)
- Jins combine to form scales, basis for modes
Theoretical approaches: scale and mode
In Farmer's terminology:
- Theorization of Old Arabian school (Hijazi, practical but with retroactive Greek and prescriptive influence): e.g. Ibn al-Munajjim's version of Ibrahim al-Mawsili's practice, in early Abbasid period, supposedly linked to the earlier Ibn Misjah's 8 "finger modes"
- Theory of the Philosophers: e.g. Ibn Sina, and al-Farabi's Kitab al-Musiqi al-Kabir (Greek influence, with multicultural ethnographic approach), mid-Abbasid period
- The Systematists: e.g. Safi al-Din al-Urmawi's Kitab al-Adwar (prescriptive systematizer), late Abbasids
- Modern theorists from the 19th c onwards (e.g. Michel Mashaqa and move towards equal temperament; Westernization/systematization for transposability)
Refer to spreadsheet and associated audio examples
Touma's interpretation of scales and maqam
- Critical reading: attempts to differentiate the "pure Arabian" from Turkish or Persian, on the one hand, and Western music, on the other.
- Pure Arabian scales vs. mixture with Greeks
- Temperament as "western"
- Space/time distinction: Arabian as "spatial" vs European as "temporal".
- Maqam vs. taqsim
- There are many mistakes and inaccuracies in these sections, perhaps partly due to faulty translations.
Contemporary concept of maqam in Egypt and Levant
Review from Tuesday:
- Music is mathematics, number in time - part of the quadrivium
- Review of basic pitch, frequency, interval concepts
- Examples from http://maqamworld.com
Tonal system (maqam): based on 24 tone gamut out of which are extracted 7 tone scales, adding pitch function (structure). Mode (maqam) is scale with additional melodic information.
- maqam:
- collection (set) of pitches or intervals
- tonal functions defined on the set: tonic (qarar), dominant (ghammaz), stops (marakiz), start (mabda`), etc. on those pitches or intervals
- mode - melodic tendencies, network pathways, materials ("licks") based on the set
- pitch or interval set
- pitch scale degrees (hertz)
- intervals (ratios, cents)
- Just vs equal tempered vs. "musical" (practical) intonation
- e.g. Pythagorean theory/Arab theory vs. 24 tone vs. musical intuition
- scale: structured set, double octave
- decomposition: genres (ajnas)
- pitch (tonal) functions
- tonic (qarar)
- dominant (ghammaz)
- subdominant
- points of repose (marakiz)
- leading tone
- final tone
- mode: network defined on the set
- tonal ornaments
- context-sensitive intonation, allotones
- melodic patterning, material
- scalar direction
- progression of melodic development
Informal discourse about music
- Intonation
- (Marcus article)
- Sami Abu Shumays on microtones
- Modulation (Marcus article)
Musical practice
- of Dr Ali Jihad Racy, analyzed in Taqsim Nahawand (Nettl and Riddle article)
- video of Dr `Atif `Abd al-Hamid (Cairo)
- examples at www.maqamworld.com
For next time: select one maqam from www.maqamworld.com. Study the maqam's structure as presented there, and listen to all the examples. Using these examples as models, develop your own composition or improvisation in the same maqam.