|
|
| Line 1: |
Line 1: |
| '''Pythagorean tuning''' ({{lang-el|Πυθαγόρεια κλίμακα}}) is a tuning of the [[syntonic temperament]]<ref name=Milne2007>{{cite journal
| | Laminate vs. Hardwood. Hardwood will actually improve the value of your home. Laminate flooring normally does not. With that said claimed, allow's discuss exactly what makes sense. If you mount hardwood and laminate into an average energetic American home with 3 children, there's a sporting chance that in 5 years the hardwood may look worn, but the laminate needs to essentially still look great. Now which residence truly held its worth? Yet, I can redecorate the hardwood and right away switch out the worth. But keep in mind, that refinish will certainly costs cash. So we are type of walking around in circles. Laminate flooring is a lot more difficult to scrape, will not fade from sunshine, and is a lot more water resistant than hardwood. Water is not a friend to laminate flooring whatsoever, yet water can spoil hardwood flooring also. If you treasured this article so you would like to obtain more info pertaining to [https://www.youtube.com/watch?v=mru-9LutmHU garage flooring phoenix] generously visit our own web page. Laminate flooring is a snap to fix; and when you fix the board, you will certainly never ever know it was fixed as it does not transform appearance in time. Hardwood is even more of a tiresome repair service and you may discover the repair work was made until it blends in with wear. Animal urine could harm both items. Your pet dog's claws will be far more disturbing to the hardwood finish. It takes a whole lot to scrape laminate flooring, yet it could be damaged and it is not bulletproof. Regardless of what the finish on a hardwood floor has, abrasions will certainly be discovered. The scrapes usually do not obtain to the lumber, however will certainly disrupt the finish coat layer. I have actually hardwood flooring and my huge pugilist pet dog does periodically put a mark into the wear surface. An additional choice of hardwood is a troubled or rustic look. Hand scraped or rustic products will certainly disappoint these scrapes and in fact include in the character. If you have a major scrape in the finish, you simply massage some Aged English into the scrape and it will essentially disappear. The outcomes will not be the same with a smooth or more formal finish. Laminate flooring will take a great deal of misuse, yet it is a photo of a hardwood and attempts its ideal to copy a hardwood or tile flooring. It will not appear, look, or smell like an actual hardwood. Hardwood is real and can be collected from different parts of the globe. Unique woods are now regulated by what is called the Lacey Act. Suppliers today must abide by the act guaranteeing that hardwood foresting is done with a prioritized issue for the environment. This has actually removed unlawful foresting in various other components of the world which has actually translated into greater exotic lumber costs. Woods come from real trees giving the item unparalleled elegance and elegance. Laminate flooring on average is cheaper than hardwood flooring. Ask one of our representatives to reveal you the distinctions between laminate and hardwood flooring which should allow a considerably far better choice on what is ideal for you and your family members. However remember, it is eventually around you. |
| | first1 = Andrew
| |
| | last1 = Milne
| |
| | authorlink =
| |
| | coauthors = [[William Sethares|Sethares, W.A.]], Plamondon, J.
| |
| |date=December 2007
| |
| | title = '''Invariant Fingerings Across a Tuning Continuum'''
| |
| | journal = Computer Music Journal
| |
| | volume = 31
| |
| | issue = 4
| |
| | pages = 15–32
| |
| | url = http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15
| |
| | accessdate = 2013-07-11
| |
| | doi = 10.1162/comj.2007.31.4.15
| |
| }}</ref> in which the [[generator (music)|generator]] is the ratio [[sesquialterum|3:2]] (i.e., the untempered [[perfect fifth]]), which is 702 [[cent (music)|cents]] wide (see the figure labelled "The syntonic tuning continuum" below). [[Image:Syntonic Tuning Continuum.jpg|right|250px|thumb|The syntonic tuning continuum, showing Pythagorean tuning at 702 cents.<ref name=Milne2007 />]]
| |
| | |
| Hence, it is a system of [[musical tuning]] in which the [[frequency ratio]]s of all [[interval (music)|intervals]] are based on the ratio [[sesquialterum|3:2]], "found in the [[harmonic series (music)|harmonic series]]."<ref name="B&S">Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p. 56. Seventh Edition. ISBN 978-0-07-294262-0.</ref> This ratio, also known as the "[[Five-limit tuning#The justest ratios|pure]]" [[perfect fifth]], is chosen because it is one of the most [[consonance and dissonance|consonant]] and easy to tune by ear.
| |
| | |
| [[Image:Pythagorean diatonic scale on C.png|thumb|330px|Pythagorean [[diatonic scale]] on C {{audio|Pythagorean diatonic scale on C.mid|Play}}.]]
| |
| [[Image:Diatonic scale on C.png|thumb|330px|Diatonic scale on C {{audio|Diatonic scale on C.mid|Play}} 12-tone equal tempered and{{audio|Just diatonic scale on C.mid|Play}} just intonation.]]
| |
| [[Image:Pythagorean major chord on C.png|thumb|Pythagorean (tonic) major chord on C {{audio|Pythagorean major chord on C.mid|Play}} (compare{{audio|C major triad.mid|Play}} equal tempered and {{audio|Just major triad on C.mid|Play}} just).]]
| |
| | |
| The system had been mainly attributed to [[Pythagoras]] (sixth century BC) by modern authors of music theory, while [[Ptolemy]], and later [[Boethius]], ascribed the division of the [[tetrachord]] by only two intervals, called "semitonium", "tonus", "tonus" in Latin (256:243 x 9:8 x 9:8), to [[Eratosthenes]]. The so-called "Pythagorean tuning" was used by musicians up to the beginning of the 16th century. Leo Gunther<!-- or Benward and Saker??? reference needs to be clarified -->{{clarify|date=September 2013}} wrote, that "the Pythagorean system would appear to be ideal because of the purity of the fifths, but other intervals, particularly the major third, are so badly out of tune that major chords [may be considered] a dissonance."<ref name="B&S"/><ref>Gunther, Leon (2011). ''The Physics of Music and Color'', p.362. ISBN 978-1-4614-0556-6.</ref>
| |
| | |
| The '''Pythagorean scale''' is any [[scale (music)|scale]] which may be constructed from only pure perfect fifths (3:2) and octaves (2:1)<ref>Sethares, William A. (2005). ''Tuning, Timbre, Spectrum, Scale'', p.163. ISBN 1-85233-797-4.</ref> or the [[Diatonic and chromatic#Diatonic scales|gamut]] of twelve pitches constructed from only pure perfect fifths and octaves, and from which specific scales may be drawn (see [[Generated collection]]). For example, the series of fifths generated above gives seven notes, a [[Diatonic scale|diatonic]] [[major scale]] on C in Pythagorean tuning, shown in notation on the top right. In Greek music it was used to [[Tetrachord#Pythagorean_tunings|tune tetrachords]] and the twelve tone Pythagorean system was developed by medieval music theorists using the same method of tuning in perfect fifths, however there is no evidence that Pythagoras himself went beyond the tetrachord.<ref>{{cite web |first=Peter A. |last=Frazer |url=http://midicode.com/tunings/Tuning10102004.pdf |title=The Development of Musical Tuning Systems |date=April 2001 |archiveurl=https://web.archive.org/web/20060506221411/http://www.midicode.com/tunings/Tuning10102004.pdf |archivedate=2006-05-06 |accessdate=2014-02-02}}</ref>
| |
| | |
| ==Method==
| |
| | |
| Pythagorean tuning is based on a stack of intervals called [[perfect fifth]]s, each tuned in the ratio 3:2, the next simplest ratio after 2:1. Starting from D for example (''D-based'' tuning), six other notes are produced by moving six times a ratio 3:2 up, and the remaining ones by moving the same ratio down:
| |
| | |
| :E♭—B♭—F—C—G—'''D'''—A—E—B—F♯—C♯—G♯
| |
| | |
| This succession of eleven 3:2 intervals spans across a wide range of [[frequency]] (on a [[piano keyboard]], it encompasses 77 keys). Since notes differing in frequency by a factor of 2 are given the same name, it is customary to divide or multiply the frequencies of some of these notes by 2 or by a power of 2. The purpose of this adjustment is to move the 12 notes within a smaller range of frequency, namely within the interval between the '''base note''' D and the D above it (a note with twice its frequency). This interval is typically called the '''basic octave''' (on a piano keyboard, an [[octave]] encompasses only 13 keys ).
| |
| | |
| For instance, the A is tuned such that its frequency equals 3:2 times the frequency of D — if D is tuned to a frequency of 288 [[Hertz|Hz]], then A is tuned to 432 Hz. Similarly, the E above A is tuned such that its frequency equals 3:2 times the frequency of A, or 9:4 times the frequency of D — with A at 432 Hz, this puts E at 648 Hz. Since this E is outside the above-mentioned basic octave (i.e. its frequency is more than twice the frequency of the base note D), it is usual to halve its frequency to move it within the basic octave. Therefore, E is tuned to 324 Hz, a 9:8 (= one [[epogdoon]]) above D. The B at 3:2 above that E is tuned to the ratio 27:16 and so on. Starting from the same point working the other way, G is tuned as 3:2 below D, which means that it is assigned a frequency equal to 2:3 times the frequency of D — with D at 288 Hz, this puts G at 192 Hz. This frequency is then doubled (to 384 Hz) to bring it into the basic octave.
| |
| | |
| When extending this tuning however, a problem arises: no stack of 3:2 intervals (perfect fifths) will fit exactly into any stack of 2:1 intervals (octaves). For instance a stack such as this, obtained by adding one more note to the stack shown above
| |
| | |
| :A♭—E♭—B♭—F—C—G—'''D'''—A—E—B—F♯—C♯—G♯
| |
| | |
| will be similar but not identical in size to a stack of 7 octaves. More exactly, it will be about a quarter of a [[semitone]] larger (see [[Pythagorean comma]]). Thus, A{{music|b}} and G{{music|#}}, when brought into the basic octave, will not coincide as expected. The table below illustrates this, showing for each note in the basic octave the conventional name of the [[interval (music)|interval]] from D (the base note), the formula to compute its frequency ratio, its size in [[Cent (music)|cents]], and the difference in cents (labeled ET-dif in the table) between its size and the size of the corresponding one in the equally tempered scale.
| |
| | |
| {| class="wikitable" style="text-align: center"
| |
| ! Note
| |
| ! Interval from D
| |
| ! Formula
| |
| ! Frequency<br />ratio
| |
| ! Size<br />(cents)
| |
| ! ET-dif<br />(cents)
| |
| |-
| |
| | A{{music|b}}
| |
| | [[diminished fifth]]
| |
| | <math>\left( \frac{2}{3} \right) ^6 \times 2^4</math>
| |
| | <math>\frac{1024}{729}</math>
| |
| |style="text-align: right"| 588.27
| |
| |style="text-align: right"| -11.73
| |
| |-
| |
| | E{{music|b}}
| |
| | [[minor second]]
| |
| | <math>\left( \frac{2}{3} \right) ^5 \times 2^3</math>
| |
| | <math>\frac{256}{243}</math>
| |
| |style="text-align: right"| 90.22
| |
| |style="text-align: right"| −9.78
| |
| |-
| |
| | B{{music|b}}
| |
| | [[minor sixth]]
| |
| | <math>\left( \frac{2}{3} \right) ^4 \times 2^3</math>
| |
| | <math>\frac{128}{81}</math>
| |
| |style="text-align: right"| 792.18
| |
| |style="text-align: right"| −7.82
| |
| |-
| |
| | F
| |
| | [[minor third]]
| |
| | <math>\left( \frac{2}{3} \right) ^3 \times 2^2</math>
| |
| | <math>\frac{32}{27}</math>
| |
| |style="text-align: right"| 294.13
| |
| |style="text-align: right"| −5.87
| |
| |-
| |
| | C
| |
| | [[minor seventh]]
| |
| | <math>\left( \frac{2}{3} \right) ^2 \times 2^2</math>
| |
| | <math>\frac{16}{9}</math>
| |
| |style="text-align: right"| 996.09
| |
| |style="text-align: right"| −3.91
| |
| |-
| |
| | G
| |
| | [[perfect fourth]]
| |
| | <math>\frac{2}{3} \times 2</math>
| |
| | <math>\frac{4}{3}</math>
| |
| |style="text-align: right"| 498.04
| |
| |style="text-align: right"| -1.96
| |
| |-
| |
| | D
| |
| | [[unison]]
| |
| | <math>\frac{1}{1}</math>
| |
| | <math>\frac{1}{1}</math>
| |
| |style="text-align: right"| 0 .00
| |
| |style="text-align: right"| 0.00
| |
| |-
| |
| | A
| |
| | [[perfect fifth]]
| |
| | <math>\frac{3}{2}</math>
| |
| | <math>\frac{3}{2}</math>
| |
| |style="text-align: right"| 701.96
| |
| |style="text-align: right"| 1.96
| |
| |-
| |
| | E
| |
| | [[major second]]
| |
| | <math>\left( \frac{3}{2} \right) ^2 \times \frac{1}{2}</math>
| |
| | <math>\frac{9}{8}</math>
| |
| |style="text-align: right"| 203.91
| |
| |style="text-align: right"| 3.91
| |
| |-
| |
| | B
| |
| | [[major sixth]]
| |
| | <math>\left( \frac{3}{2} \right) ^3 \times \frac{1}{2}</math>
| |
| | <math>\frac{27}{16}</math>
| |
| |style="text-align: right"| 905.87
| |
| |style="text-align: right"| 5.87
| |
| |-
| |
| | F{{music|#}}
| |
| | [[major third]]
| |
| | <math>\left( \frac{3}{2} \right) ^4 \times \left( \frac{1}{2} \right) ^2</math>
| |
| | <math>\frac{81}{64}</math>
| |
| |style="text-align: right"| 407.82
| |
| |style="text-align: right"| 7.82
| |
| |-
| |
| | C{{music|#}}
| |
| | [[major seventh]]
| |
| | <math>\left( \frac{3}{2} \right) ^5 \times \left( \frac{1}{2} \right) ^2</math>
| |
| | <math>\frac{243}{128}</math>
| |
| |style="text-align: right"| 1109.78
| |
| |style="text-align: right"| 9.78
| |
| |-
| |
| | G{{music|#}}
| |
| | [[augmented fourth]]
| |
| | <math>\left( \frac{3}{2} \right) ^6 \times \left( \frac{1}{2} \right) ^3</math>
| |
| | <math>\frac{729}{512}</math>
| |
| |style="text-align: right"| 611.73
| |
| |style="text-align: right"| 11.73
| |
| |}
| |
| | |
| In the formulas, the ratios 3:2 or 2:3 represent an ascending or descending perfect fifth (i.e. an increase or decrease in frequency by a perfect fifth), while 2:1 or 1:2 represent an ascending or descending octave.
| |
| | |
| The [[major scale]] based on C, obtained from this tuning is:<ref>Asiatic Society of Japan (1879). ''[http://books.google.com/books?id=DX0uAAAAYAAJ&pg=PA82&dq=pythagorean+interval&hl=en&ei=aNrlTpK7BorWiALk1eyZBg&sa=X&oi=book_result&ct=result&resnum=10&ved=0CF0Q6AEwCQ#v=onepage&q=pythagorean%20interval&f=false Transactions of the Asiatic Society of Japan], Volume 7'', p.82. Asiatic Society of Japan.</ref>
| |
| | |
| {| class="wikitable" style="text-align:center"
| |
| !Note
| |
| |colspan="2" | '''C'''
| |
| |colspan="2" | '''D'''
| |
| |colspan="2" | '''E'''
| |
| |colspan="2" | '''F'''
| |
| |colspan="2" | '''G'''
| |
| |colspan="2" | '''A'''
| |
| |colspan="2" | '''B'''
| |
| |colspan="2" | '''C'''
| |
| |-
| |
| !Ratio
| |
| |colspan="2" | 1/1
| |
| |colspan="2" | 9/8
| |
| |colspan="2" | 81/64
| |
| |colspan="2" | 4/3
| |
| |colspan="2" | 3/2
| |
| |colspan="2" | 27/16
| |
| |colspan="2" | 243/128
| |
| |colspan="2" | 2/1
| |
| |-
| |
| !Step
| |
| |colspan="1" | —
| |
| |colspan="2" | 9/8
| |
| |colspan="2" | 9/8
| |
| |colspan="2" | 256/243
| |
| |colspan="2" | 9/8
| |
| |colspan="2" | 9/8
| |
| |colspan="2" | 9/8
| |
| |colspan="2" | 256/243
| |
| |colspan="1" | —
| |
| |}
| |
| | |
| In equal temperament, pairs of [[enharmonic]] notes such as A{{music|b}} and G{{music|#}} are thought of as being exactly the same note — however, as the above table indicates, in Pythagorean tuning they have different ratios with respect to D, which means they are at a different frequency. This discrepancy, of about 23.46 cents, or nearly one quarter of a semitone, is known as a ''[[Pythagorean comma]]''.
| |
| | |
| To get around this problem, Pythagorean tuning ignores A{{music|b}}, and uses only the 12 notes from E{{music|b}} to G{{music|#}}. This, as shown above, implies that only eleven just fifths are used to build the entire chromatic scale. The remaining fifth (from G{{music|#}} to E{{music|b}}) is left badly out-of-tune, meaning that any music which combines those two notes is unplayable in this tuning. A very out-of-tune interval such as this one is known as a ''[[wolf interval]]''. In the case of Pythagorean tuning, all the fifths are 701.96 cents wide, in the exact ratio 3:2, except the wolf fifth, which is only 678.49 cents wide, nearly a quarter of a [[semitone]] flatter.
| |
| | |
| <!-- Image with unknown copyright status removed: [[media:Wolf fifth.ogg|Wolf fifth.ogg]] (33.1KB) is a sound file demonstrating this out of tune fifth. The first two fifths are perfectly tuned in the ratio 3:2, the third is the D#—Eb wolf fifth. It may be useful to compare this to [[media:Et fifths.ogg|Et fifths.ogg]] (38.2KB), which is the same three fifths tuned in [[equal temperament]], each of them tolerably well in tune.
| |
| -->
| |
| If the notes G{{music|sharp}} and E{{music|flat}} need to be sounded together, the position of the wolf fifth can be changed. For example, a C-based Pythagorean tuning would produce a stack of fifths running from D{{music|flat}} to F{{music|sharp}}, making F{{music|sharp}}-D{{music|flat}} the wolf interval. However, there will always be one wolf fifth in Pythagorean tuning, making it impossible to play in all [[key (music)|keys]] in tune.
| |
| | |
| == Size of intervals ==
| |
| The table above shows only intervals from D. However, intervals can be formed by starting from each of the above listed 12 notes. Thus, twelve intervals can be defined for each '''interval type''' (twelve unisons, twelve [[semitone]]s, twelve intervals composed of 2 semitones, twelve intervals composed of 3 semitones, etc.).
| |
| | |
| [[Image:Interval ratios in D-based symmetric Pythagorean tuning (powers for large numbers).PNG|frame|right|Frequency ratio of the 144 intervals in D-based Pythagorean tuning. [[Interval (music)#Quality|Interval names]] are given in their shortened form. [[5-limit tuning#The justest ratios|Pure interval]]s are shown in '''[[boldface|bold]]''' font. [[Wolf interval]]s are highlighted in red.<ref name="Wolf"/> Numbers larger than 999 are shown as powers of 2 or 3. Other versions of this table are provided [http://commons.wikimedia.org/wiki/File:Interval_ratios_in_D-based_symmetric_Pythagorean_tuning_(powers_of_2_%26_3).PNG here] and [http://commons.wikimedia.org/wiki/File:Interval_ratios_in_D-based_symmetric_Pythagorean_tuning.PNG here].]]
| |
| | |
| [[Image:Size of intervals in D-based symmetric Pythagorean tuning.PNG|frame|right|Approximate size in cents of the 144 intervals in D-based Pythagorean tuning. [[Interval (music)#Quality|Interval names]] are given in their shortened form. [[5-limit tuning#The justest ratios|Pure interval]]s are shown in '''[[boldface|bold]]''' font. [[Wolf interval]]s are highlighted in red.<ref name="Wolf"/>]]
| |
| | |
| As explained above, one of the twelve fifths (the wolf fifth) has a different size with respect to the other eleven. For a similar reason, each of the other interval types, except for the unisons and the octaves, has two different sizes in Pythagorean tuning. This is the price paid for seeking [[just intonation]]. The tables on the right and below show their frequency ratios and their approximate sizes in cents. [[Interval (music)#Quality|Interval names]] are given in their standard shortened form. For instance, the size of the interval from D to A, which is a perfect fifth ('''P5'''), can be found in the seventh column of the row labeled '''D'''. [[5-limit tuning#The justest ratios|Strictly just]] (or pure) intervals are shown in '''[[boldface|bold]]''' font. [[Wolf interval]]s are highlighted in red.<ref name="Wolf">Wolf intervals are operationally defined herein as intervals composed of 3, 4, 5, 7, 8, or 9 semitones (i.e. major and minor thirds or sixths, perfect fourths or fifths, and their [[enharmonic equivalent]]s) the size of which deviates by more than one [[syntonic comma]] (about 21.5 cents) from the corresponding justly intonated interval. Intervals made up of 1, 2, 6, 10, or 11 semitones (e.g. major and minor seconds or sevenths, tritones, and their [[enharmonic]] equivalents) are considered to be [[consonance and dissonance|dissonant]] even when they are justly tuned, thus they are not marked as wolf intervals even when they deviate from just intonation by more than one syntonic comma.</ref>
| |
| | |
| The reason why the interval sizes vary throughout the scale is that the pitches forming the scale are unevenly spaced. Namely, the frequencies defined by construction for the twelve notes determine two different [[semitone]]s (i.e. intervals between adjacent notes):
| |
| | |
| # The minor second ('''m2'''), also called diatonic semitone, with size <br><math> S_1 = {256 \over 243} \approx 90.225 \ \hbox{cents} </math> <br> (e.g. between D and E{{Music|b}})
| |
| # The augmented unison ('''A1'''), also called chromatic semitone, with size <br><math> S_2 = {3^7 \over 2^{11}} = {2187 \over 2048} \approx 113.685 \ \hbox{cents} </math> <br> (e.g. between E{{Music|b}} and E)
| |
| | |
| Conversely, in an [[Twelve tone equal temperament|equally tempered]] chromatic scale, by definition the twelve pitches are equally spaced, all semitones having a size of exactly
| |
| :<math> S_E = \sqrt[12]{2} = 100.000 \ \hbox{cents}. </math>
| |
| As a consequence all intervals of any given type have the same size (e.g., all major thirds have the same size, all fifths have the same size, etc.). The price paid, in this case, is that none of them is justly tuned and perfectly consonant, except, of course, for the unison and the octave.
| |
| | |
| For a comparison with other tuning systems, see also [[Interval (music)#Size of intervals used in different tuning systems|this table]].
| |
| | |
| By definition, in Pythagorean tuning 11 perfect fifths ('''P5''' in the table) have a size of approximately 701.955 cents (700+ε cents, where ε ≈ 1.955 cents). Since the average size of the 12 fifths must equal exactly 700 cents (as in equal temperament), the other one must have a size of 700−11ε cents, which is about 678.495 cents (the wolf fifth). Notice that, as shown in the table, the latter interval, although [[enharmonically equivalent]] to a fifth, is more properly called a [[diminished sixth]] ('''d6'''). Similarly,
| |
| * 9 [[minor third]]s ('''m3''') are ≈ 294.135 cents (300−3ε), 3 [[augmented second]]s ('''A2''') are ≈ 317.595 cents (300+9ε), and their average is 300 cents;
| |
| * 8 [[major third]]s ('''M3''') are ≈ 407.820 cents (400+4ε), 4 [[diminished fourth]]s ('''d4''') are ≈ 384.360 cents (400−8ε), and their average is 400 cents;
| |
| * 7 diatonic [[semitone]]s ('''m2''') are ≈ 90.225 cents (100−5ε), 5 chromatic semitones ('''A1''') are ≈ 113.685 cents (100+7ε), and their average is 100 cents.
| |
| | |
| In short, similar differences in width are observed for all interval types, except for unisons and octaves, and they are all multiples of ε, the difference between the Pythagorean fifth and the average fifth.
| |
| | |
| Notice that, as an obvious consequence, each augmented or diminished interval is exactly 12ε (≈ 23.460) cents narrower or wider than its enharmonic equivalent. For instance, the d6 (or wolf fifth) is 12ε cents narrower than each P5, and each A2 is 12ε cents wider than each m3. This interval of size 12ε is known as a [[Pythagorean comma]], exactly equal to the opposite of a [[diminished second]] (≈ −23.460 cents). This implies that ε can be also defined as one twelfth of a Pythagorean comma.
| |
| | |
| == Pythagorean intervals ==
| |
| {{Main|Pythagorean interval|Interval (music)}}
| |
| Four of the above mentioned intervals take a specific name in Pythagorean tuning. In the following table, these specific names are provided, together with alternative names used generically for some other intervals. Notice that the Pythagorean comma does not coincide with the diminished second, as its size (524288:531441) is the reciprocal of the Pythagorean diminished second (531441:524288). Also ''ditone'' and ''semiditone'' are specific for Pythagorean tuning, while ''tone'' and ''tritone'' are used generically for all tuning systems. Interestingly, despite its name, a semiditone (3 semitones, or about 300 cents) can hardly be viewed as half of a ditone (4 semitones, or about 400 cents). All the intervals with prefix ''sesqui-'' are [[just intonation|justly]] tuned, and their [[Interval ratio|frequency ratio]], shown in the table, is a [[superparticular number]] (or epimoric ratio). The same is true for the octave.
| |
| | |
| {| class="wikitable"
| |
| ! rowspan="3"| Number of<br>semitones!! colspan="4"|Generic names !! colspan="3"|Specific names
| |
| |-
| |
| ! colspan="2"| [[Interval (music)#Interval number and quality|Quality and number]] !! colspan="2" rowspan="2"|Other naming conventions !! rowspan="2"|Pythagorean tuning !! rowspan="2"|[[5-limit tuning]] !! rowspan="2"|[[Quarter-comma meantone|1/4-comma<br>meantone]]
| |
| |-
| |
| ! Full !! Short
| |
| |-
| |
| | 0 || || || colspan="2"| [[comma (music)|comma]] || [[Pythagorean comma]] <br/> (524288:531441) || rowspan="2"| ||rowspan="2"| [[diesis]] (128:125)
| |
| |-
| |
| | 0 || [[diminished second]] || d2 || colspan="2"| || (531441:524288)
| |
| |-
| |
| | 1 || [[minor second]] || m2 || rowspan="2"| semitone,<br>half tone,<br>half step || diatonic semitone,<br>minor semitone || [[Pythagorean limma|limma]] (256:243) || ||
| |
| |-
| |
| | 1 || [[augmented unison]] || A1 || chromatic semitone,<br>major semitone || [[Pythagorean apotome|apotome]] (2187:2048) || ||
| |
| |-
| |
| | 2 || [[diminished third]] || d3 || colspan="2" rowspan="2"| tone, whole tone, whole step || || ||
| |
| |-
| |
| | 2 || [[major second]] || M2 || colspan="2" style="text-align: center" | [[sesquioctavum]] (9:8) ||
| |
| |-
| |
| | 3 || [[minor third]] || m3 || colspan="2"| || [[semiditone]] (32:27) || [[sesquiquintum]] (6:5) ||
| |
| |-
| |
| | 4 || [[major third]] || M3 || colspan="2"| || [[ditone]] (81:64) || colspan="2" style="text-align: center"| [[sesquiquartum]] (5:4)
| |
| |-
| |
| | 5 || [[perfect fourth]] || P4 || colspan="2" | diatessaron || colspan="2" style="text-align: center" | [[sesquitertium]] (4:3) ||
| |
| |-
| |
| | 6 || [[diminished fifth]] || d5 || colspan="2" rowspan="2"| tritone || || ||
| |
| |-
| |
| | 6 || [[augmented fourth]]|| A4 || || ||
| |
| |-
| |
| | 7 || [[perfect fifth]]|| P5 || colspan="2" | diapente || colspan="2" style="text-align: center" | [[sesquialterum]] (3:2) ||
| |
| |-
| |
| | 12 || (perfect) [[octave]] || P8 || colspan="2" | diapason || colspan="3" style="text-align: center" | duplex (2:1)
| |
| |}
| |
| | |
| ==History==
| |
| Because of the [[wolf interval]], this tuning is rarely used nowadays, although it is thought to have been widespread. In music which does not change [[key (music)|key]] very often, or which is not very [[harmony|harmonically]] adventurous, the wolf interval is unlikely to be a problem, as not all the possible fifths will be heard in such pieces.
| |
| | |
| Because most fifths in Pythagorean tuning are in the simple ratio of 3:2, they sound very "smooth" and consonant. The thirds, by contrast, most of which are in the relatively complex ratios of 81:64 (for major thirds) and 32:27 (for minor thirds), sound less smooth.<ref>However, 3/2<sup>8</sup> is described as "almost exactly a just major third." Sethares (2005), p.60.</ref> For this reason, Pythagorean tuning is particularly well suited to music which treats fifths as consonances, and thirds as dissonances. In [[European classical music|western classical music]], this usually means music written prior to the 15th century.
| |
| | |
| From about 1510 onward, as thirds came to be treated as consonances, [[meantone temperament]], and particularly [[quarter-comma meantone]], which tunes thirds to the relatively simple ratio of [[sesquiquartum|5:4]], became the most popular system for tuning keyboards. At the same time, syntonic-diatonic [[just intonation]] was posited by [[Zarlino]] as the normal tuning for singers.
| |
| | |
| However, meantone presented its own harmonic challenges. Its wolf intervals proved to be even worse than those of the Pythagorean tuning (so much so that it often required 19 keys to the octave as opposed to the 12 in Pythagorean tuning). As a consequence, meantone was not suitable for all music.
| |
| | |
| From around the 18th century, as the desire grew for instruments to change key, and therefore to avoid a wolf interval, this led to the widespread use of [[well temperament]]s and eventually [[equal temperament]].
| |
| | |
| In 2007, the discovery of the [[syntonic temperament]] <ref name=Milne2007 /> exposed the Pythagorean tuning as being a point on the syntonic temperament's tuning continuum.
| |
| | |
| ==Discography==
| |
| *[[Bragod]] is a duo giving historically informed performances of mediaeval Welsh music using the [[crwth]] and six-stringed [[lyre]] using Pythagorean tuning
| |
| *[[Gothic Voices]] – ''Music for the Lion-Hearted King'' (Hyperion, CDA66336, 1989), directed by [[Christopher Page]] (Leech-Wilkinson)
| |
| *[[Lou Harrison]] performed by [[John Schneider (guitarist)|John Schneider]] and the Cal Arts Percussion Ensemble conducted by [[John Bergamo]] - ''Guitar & Percussion'' (Etceter Records, KTC1071, 1990): ''Suite No. 1'' for guitar and percussion and ''Plaint & Variations'' on "Song of Palestine"
| |
| | |
| ==See also==
| |
| * [[Enharmonic scale]]
| |
| * [[List of meantone intervals]]
| |
| * [[List of musical intervals]]
| |
| * [[Regular temperament]]
| |
| * [[Shí-èr-lǜ]]
| |
| * [[Musical temperament]]
| |
| * [[Timaeus (dialogue)]], in which Plato discusses Pythagorean tuning
| |
| * [[Whole-tone scale]]
| |
| | |
| ==References==
| |
| | |
| ===Footnotes===
| |
| {{Reflist}}
| |
| | |
| ===Notations===
| |
| *[[Daniel Leech-Wilkinson]] (1997), "The good, the bad and the boring", ''Companion to Medieval & Renaissance Music''. Oxford University Press. ISBN 0-19-816540-4.
| |
| | |
| ==External links==
| |
| *[http://www.music.sc.edu/fs/bain/atmi02/pst/index.html "A Pythagorean tuning of the diatonic scale"], with audio samples.
| |
| *[http://www.medieval.org/emfaq/harmony/pyth.html "Pythagorean Tuning and Medieval Polyphony"], by Margo Schulter.
| |
| | |
| {{Musical tuning}}
| |
| | |
| {{DEFAULTSORT:Pythagorean Tuning}}
| |
| [[Category:Greek music]]
| |
| [[Category:3-limit tuning and intervals|*]]
| |
| [[Category:Pythagorean philosophy]]
| |