Definition of Basic Terms:

Transcription

1 Definition of Basic Terms: Temperament: A system of tuning where intervals are altered from those that are acoustically pure (Harnsberger, 1996, p. 130) A temperament is any plan that describes the adjustments to the sizes of some or all of the twelve fifth intervals in the circle of fifths so that they accommodate pure octaves and produces certain sizes of major thirds (Donahue, 2005, p. 19). Equal Temperament: The tuning of an octave in 12 equal parts (Harnsberger, p. 50) Pythagorean comma is divided equally among all twelve intervals in the circle of fifths and every fifth is narrower by 1/12 th of the comma (Donahue, p. 24). Well Tempered: This system is classified as having (1) no unusuable intervals (Donahue, p. 25) and (2) a combination of fifths of different sizes (p. 25) Additional Terms Defined: Intervals: Relationship between two notes (Donahue, 2005, p. 4) Semi-tone: Half step; Example is C to C# or E to F

2 How did Pythagoras do it? Pythagoras created a seven-tone scale from a series of consecutive 3:2 perfect fifths (Bain, 2002, p. 1). All the ratios were in whole numbers which follows the Pythagoreans belief in whole numbers (Bain). The ratios of the intervals are: Interval Ration Unison X 2 nd 9/8X 3 rd 81/64x Perfect 4 th 4/3x Perfect 5 th 3/2x 6 th 27/16x 7 th 243/128x Octave 2/1x (Bain, 2002, p. 1) This tuning is not known as a temperament because it does not temper with any of the pitches (Donahue, 2005, p. 21). After finding seven pure octaves and twelve pure fifths, this method ends up being 24 cents off. This difference is known as the Pythagorean Comma (Donahue, 2005, p. 8, 21). Because of the comma, there are only 11 pure fifths with the 12 th fifth being 24 cents off (Donahue, 21). How is Equal Temperament Different? Definition: The distance or intervals between the pitches are split between all the notes. Another way of describing the split is the Pythagorean comma, which is 24 cents, is split equally between the pitches of a scale. The distance was split equally using a logarithm. The result is the 12 th root of 2 for the distance between each pitch. Logarithmic tables were published since the 1620 s and from that time onwards we see the application of logarithms to the calculation of temperament (Christensen, 2002, p. 211). Since all the notes are split equally, you can magically travel all the way around the Circle of Fifths (found later in this handout) and return to the same in tune C. However, in Pythagoras tuning, after traveling around the Circle of Fifths you would be 24 cents off. There are sacrifices though. All semi tone intervals are equal, but there are no longer pure fifths or thirds like in Pythagoras tuning or other tempered scales. The positive side is that all intervals can be used because there are no really bad ones. (Donahue). All Major 5ths are equal at 700 cents & All 3rds are equal at 400 cents (Donahue, p. 40)

3 How many different kinds of Temperaments are there? Regular Temperaments vs. Irregular Temperaments Fifths of only 1 size, with or without Fifths of various sizes wolf interval Equal Temperaments vs. Unequal Temperaments All fifths the same size, no pure fifths Fifths of various sizes Circulating Temperaments vs. Non Circulating Temperaments Allows modulations through all key does not allow modulation signatures (all fifths and major thirds through all key signatures (at are usuable) least one fifth or one major third is unusable) (All the above quotes from Donahue, 2005, p. 39) THEREFORE: Equal Temperament is (1) Regular, (2) Equal and (3) Circulating Just Intonation: A system of tuning where all intervals are derived from the pure fifths and the pure thirds, as opposed to temperament (Harshberger, p. 130). This would also be Pythagors Tuning since the notes are not tempered with in this system (Donahue, p. 21). Mean-Tone Temperament: This system of tuning provided pure intonation for the key of C Major at the expense of many notes outside of that key being out of tune (Harnsberger, 1996, p. 81). Well Tempered: This system is classified as having (1) no unusuable intervals (Donahue, p. 25) and (2) a combination of fifths of different sizes (p. 25) Others: There are other temperaments including Werckmeister III, Kirnberger III, Vallotti temperament, and 1/6 Comma Meantone Temperament (Donahue). There are more, but we will not have enough time to go into all of these.

4 What are the benefits and drawbacks to the different temperaments? Equal Temperament Well Temperament Just Intonation or Pythagoras Tuning Mean-Tone Temperament Benefits Solution to the problem that certain intervals did not remain the same size after modulation (Bain, 2003, p. 6) No unusable intervals (Donahue, 2005, p. 25) It s Pure!! Key of C keeps pure intonation (Harnsberger, 2006, p. 81) Pure Major 3rds at 386 Cents (Donahue, p. 40) Drawbacks Not Pure because fifths and thirds are all equally compromised All keys sound basically the same Fifths of different sizes (Donahue, p.25) Also 3 rd of different sizes. Both of these differences gives keys different sounds and feelings (Donahue, p. 40) 24 cents short Wide Major 3 rds of 408 Cents (Donahue, p. 40) Wolf interval 4 wide 3rds (Donahue, p. 24) How is Cents used to Measure frequencies? Cents is a measurement started by Alexander Ellis in the late 19 th century (Donahue, 2005, p. 5 & Christensen, 2002, p. 210) and is based on the logarithm of an intervals frequency ratio (Donahue, 2005, p. 5). The word cent was chosen because 100 units make up an equal-tempered semitone (Christensen, p. 210) Cent Value = 3986 x log(frequency ratio) Example of Cent Value of an Octave: Frequency Ratio is 2.0 (Ratio of 2/1 = 2.0) Plug into formula Cent = 3986 x log(2) Cent = 1200 (Donahue, p. 5) The size of the interval is cents is also determined by the formula: I = 1,200 x log 2 i with I = Size in Cents & i = Size ratio frequency Or (logi / log2)1200 (Christensen, 2002, p. 210)

5 The Chart below from Donahue (2005) shows how the ratios are turned into decimal and then used in the formula to determine Cent value. Interval Frequency Ratio Decimal Cents P0 2: P5 3: P4 4: M3 5: m3 6: M6 5: (p. 5) Circle of Fifths Johann David Heinichen was the first to describe the Circle of Fifths in 1728 (Donahue, 2005, p. 6 & Kennedy & Bourne, 1996). An arrangement... used by theorists to illustrate relative harmonic remoteness of one key from another (Kennedy & Bourne). The chart moves around the circle by moving up a fifth interval to the next note. By following the chart, the number of sharps and flats will add and subtract in numerical order. This is a key part of theory because a musician can use it to determine the key signature and how keys are related to one another. A 5 th is the distance between two notes. If C = 1, then D = 2, E = 3, F = 4 and G = 5. C is the top key signature on the circle because it does not have any flats or sharps. Since G is a fifth above C, it is the next letter clockwise on the graph. An interactive teaching website to help students understand and memorize the Circle of Fifths is found at Chart below available at

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7 Why did I choose this Topic? I choose this topic because I wanted a math topic that I could easily apply in my instrumental music classroom. I teach both band and orchestra and play scales everyday in the classes. Understanding more of the specifics behind the construction of the scales can help me teach the students why we play them and how to tune pitches, chords, and scales. How can I use this Information in a Middle School Classroom? Circle of 5 th Graphics and Websites would be a great way to show students how key Signatures have a pattern and have them understand and fill out their own blank Circle of 5 th s sheet. Splitting up the String in halves to show the orchestra where octaves and harmonics are found. When teaching positions (where the left hand moves up the instrument to make higher pitches) to the orchestra, I can use the information that the distances become smaller and smaller if one goes higher up on the string (Christiansen, 2002, p. 210). The following shows an example of how a string with a length of 10,000 units can continue to be split into halves to create a higher octave. The length between ends of strings becomes smaller as the octave goes higher. Example: ,000 (Christensen) When tuning chords in the band or orchestra, I can adjust the chords to sound more pure by having the 3 rd degree of the scale tunes lower (or goes flat) while the 5 th degree of the scale tunes higher (or goes sharp). More Questions How do the overtone series help create rich colors and sounds? Pianos are set in their intonation, but other instruments, especially in band, have notes that can be naturally out of tune because of the mechanics of the instrument. What notes are out of tune on each instrument and what is the best way to teach students to tune those hard notes? (Possible discussion: Tuner vs. By Ear) How many types of temperaments have been created?

8 References Bain, R. (September, 2002). A pythagorean tuning of the diatonic scale: A tuning system based on the lines of fifths. Retrieved June 18, 2010 from /atmi02 /pst/index.html Bain, R. (April, 2003). The harmonic series: A path to understanding musical intervals, scales, tuning and timbre. Retrieved June 18, 2010 from fs/bain/atmi02/hs/index-audio.html Christensen, T. S. (2002). The Cambridge history of western music theory. Cambridge: Cambridge University Press. Donahue, T. (2005). A guide to musical temperament. Lanham, Maryland: Scarecrow Press, Inc. Harnsberger, L. (1996). Essential dictionary of music: Definitions, composers, theory, instruments, and vocal ranges, 2nd ed. United States of America: Alfred Publishing Co., Inc. Kennedy, M. & Bourne, J. (1996). "Circle of fifths." The Concise Oxford Dictionary of Music. Retrieved June 20, 2010 from Encyclopedia.com: doc/1o76-circleoffifths.html Learn the Circle of Fifths. Interactive Website. Retrieved from Linkware Graphics Music Images. Music Theory Circle of Fifths for Treble Clef Handout, Study Sheet, Reference Chart. Retieved June 20, 2010 from com/ music/i/circle5ths/ Violin Picture: Equal Temperament: Chromatic Tuner v3.5 PIcture: