EDT: Difference between revisions
m Inter |
mNo edit summary |
||
| Line 4: | Line 4: | ||
|de = Edt | |de = Edt | ||
}} | }} | ||
EDT: Equal Division of the Tritave (3/1, perfect twelfth). Also sometimes written as | '''EDT''': Equal Division of the Tritave (3/1, 3rd harmonic, just perfect twelfth). Also sometimes written as ED3. | ||
== Introduction == | == Introduction == | ||
Western music generally revolves around the principle of '''octave equivalence''': notes an octave apart are often perceived in western music as being the same ''chroma'' but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the "tritave". | Western music generally revolves around the principle of '''octave equivalence''': notes an octave apart are often perceived in western music as being the same ''chroma'' but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the "[[tritave]]". | ||
It has been argued that pitches a tritave apart can never truly be heard as equivalent in all of the ways that octaves are, with some claiming that the [http://www.mmk.ei.tum.de/persons/ter/top/octequiv.html tonotopic representation of the mammalian auditory system] is inherently biased towards octave-equivalence. With proper context, experience, and training, however, at least some people find that they can experience some degree of tritave equivalence especially using timbres restricted to odd harmonics. It is not known whether odd harmonics actually facilitate the ability to hear in tritave-equivalence. Either way, it is certain that musically valuable organizations of pitch can arise through the equal division of non-octave intervals, regardless of whether the period is perceived as being truly chroma-equivalent, and as such the multitude of equal divisions of the tritave are rich and ripe for exploration. | It has been argued that pitches a tritave apart can never truly be heard as equivalent in all of the ways that octaves are, with some claiming that the [http://www.mmk.ei.tum.de/persons/ter/top/octequiv.html tonotopic representation of the mammalian auditory system] is inherently biased towards octave-equivalence. With proper context, experience, and training, however, at least some people find that they can experience some degree of tritave equivalence especially using timbres restricted to odd harmonics. It is not known whether odd harmonics actually facilitate the ability to hear in tritave-equivalence. Either way, it is certain that musically valuable organizations of pitch can arise through the equal division of non-octave intervals, regardless of whether the period is perceived as being truly chroma-equivalent, and as such the multitude of equal divisions of the tritave are rich and ripe for exploration. | ||
| Line 14: | Line 14: | ||
== Rank two temperaments == | == Rank two temperaments == | ||
If factors of two are eliminated, the simplest possible triad is (1):(3):5:7:(9), with 1, 3 and 9 in parentheses as they're all tritave-equivalent to 1. Hence, 3:5:7 can be viewed as the fundamental consonant triad of no-twos music. The linear temperament that best approximates these chords in the moderate complexity range is the Bohlen-Pierce linear temperament eliminating 245/243, which has a no-twos mapping of [ | If factors of two are eliminated, the simplest possible triad is (1):(3):5:7:(9), with 1, 3 and 9 in parentheses as they're all tritave-equivalent to 1. Hence, 3:5:7 can be viewed as the fundamental consonant triad of no-twos music. The linear temperament that best approximates these chords in the moderate complexity range is the Bohlen-Pierce linear temperament eliminating 245/243, which has a no-twos mapping of [{{val|1 1 2}}, {{val|0 2 -1}}] and a pure-tritaves TE generator which is a sharp 9/7 of 440.488 cents. It possesses [[MOSScales|MOS]] of the forms 4L1s (pentatonic) and 4L5s (nonatonic), and larger MOS of size 13, 17, 30, 43, 56, 69 and 82. This temperament serves a function analogous to meantone in the 5-limit. | ||
At higher complexities, the rank two 3.5.7 temperament tempering out 16875/16807 called [[Canopus]] begins to predominate. This has a mapping [ | At higher complexities, the rank two 3.5.7 temperament tempering out 16875/16807 called [[Canopus]] begins to predominate. This has a mapping [{{val|1 3 3}}, {{val|0 -5 -4}}] and a pure-tritaves TE generator a slightly flat 7/5 at 581.512 cents. This has MOS of size 3, 4, 7, 10, 13, 23, 36, etc, with the 36 note MOS being particularly even. | ||
The other no twos rank two temperament which | The other no twos rank two temperament which 13EDT "supports" is [[Arcturus]], which takes an ~5:3 as a generator. I speak advisedly of 13edt supporting this temperament because the lowest-error proper MOS of it is a 2L 11s triskaidecatonic scale. However, if you do not mind having a smeary 5, you will need only a 2L 7s (nonatonic) scale to make an understandable rendition of it. | ||
The named but not necessarily no twos rank two temperament which | The named but not necessarily no twos rank two temperament which 13EDT "supports" is [[Sirius]], which takes a generator between ~7:6 and ~6:5. Like Arcturus, I speak advisedly of 13EDT supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edx because it creates a strong sense of a second equal division of a y strictly less than x, and this sense of two different equal divisions trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways. If this "problem" has not been named yet, "cross-equivalence artifacting" would be a perfect name for it. | ||
The final interval which | The final interval which 13EDT can reasonably use to generate a rank two temperament is its false 3/2 of 5 degrees. By a weird coincidence, it will generate the [[5L 3s]] unfair father octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4/3 as it must in an EDO. This means, most importantly, that 16/15 cannot be assumed to be a "comma" tempered out by this false Father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen-Pierce temperament is its index-2 subtemperament. | ||
Due to the fact of its 9/7 generator, the temperament which is to BP what neutral temperaments are to syntonic temperaments does not become intelligibly a division of the tritave until extended to 17 tones whereas EDOs supporting various neutral temperaments have an "ordinary" heptatonic scale which is intelligibly a division of the octave. Additionally, 7 and 9 being consecutive odd numbers means that trying to force this temperament into a no-twos subgroup induces very poor "approximations" of less intelligible higher harmonics. To avoid this, this temperament should be assumed to be a temperment of the 3.5.7.8 subgroup tempering out 245/243 and 64/63, the familiar comma from EDOs supporting the Superpythagorean or Parapythagorean diatonic scale. | Due to the fact of its 9/7 generator, the temperament which is to BP what neutral temperaments are to syntonic temperaments does not become intelligibly a division of the tritave until extended to 17 tones whereas EDOs supporting various neutral temperaments have an "ordinary" heptatonic scale which is intelligibly a division of the octave. Additionally, 7 and 9 being consecutive odd numbers means that trying to force this temperament into a no-twos subgroup induces very poor "approximations" of less intelligible higher harmonics. To avoid this, this temperament should be assumed to be a temperment of the 3.5.7.8 subgroup tempering out 245/243 and 64/63, the familiar comma from EDOs supporting the Superpythagorean or Parapythagorean diatonic scale. | ||
| Line 28: | Line 28: | ||
Among the EDTs tempering out 245/243, 13EDT stands out. An apt analogy can be drawn with EDOs supporting meantone: 4EDT and 9EDT are to BP what 5EDO and 7EDO are to meantone. However, in contrast to meantone, the simplest EDT supporting the BP nonatonic scale - 13EDT, the traditional tempered BP scale - is the most accurate and lowest in tuning error until 56EDT. However, there are many EDTs supporting BP temperament which also support extensions to higher-limit temperaments; in particular 2, 3, and 4 times 13 in the form of 26EDT, 39EDT and 52EDT as well as 56EDT. For tempering out 16875/16807, 13EDT again stands out, though much better accuracy can be found in more complex divisions such as 114EDT or 127EDT. All of this explains the focus on 13EDT to the exclusion of other EDTs among practitioners of the art of nonoctave composition, but it must be noted that the analysis is only valid if consideration is confined to the 7-limit, which is exactly analogous to confining it to the 5-limit with EDOs. There's a whole other world out there which has not been much explored. | Among the EDTs tempering out 245/243, 13EDT stands out. An apt analogy can be drawn with EDOs supporting meantone: 4EDT and 9EDT are to BP what 5EDO and 7EDO are to meantone. However, in contrast to meantone, the simplest EDT supporting the BP nonatonic scale - 13EDT, the traditional tempered BP scale - is the most accurate and lowest in tuning error until 56EDT. However, there are many EDTs supporting BP temperament which also support extensions to higher-limit temperaments; in particular 2, 3, and 4 times 13 in the form of 26EDT, 39EDT and 52EDT as well as 56EDT. For tempering out 16875/16807, 13EDT again stands out, though much better accuracy can be found in more complex divisions such as 114EDT or 127EDT. All of this explains the focus on 13EDT to the exclusion of other EDTs among practitioners of the art of nonoctave composition, but it must be noted that the analysis is only valid if consideration is confined to the 7-limit, which is exactly analogous to confining it to the 5-limit with EDOs. There's a whole other world out there which has not been much explored. | ||
For example, 15EDT very well approximates the 5th and 13th harmonics, and 12EDT, the 13th and 17th. 39EDT makes for a fine 3.5.7.11.13 system, tempering out 245/243, 275/273, 847/845 and 1331/1343, and so supporting among other things the [ | For example, 15EDT very well approximates the 5th and 13th harmonics, and 12EDT, the 13th and 17th. 39EDT makes for a fine 3.5.7.11.13 system, tempering out 245/243, 275/273, 847/845 and 1331/1343, and so supporting among other things the [{{val|13 19 23 0 2}}, {{val|0 0 0 1 1}}] temperament supported by the whole suite of 13nEDTs: 13, 26, 39, 52, 65, 78 etc. | ||
'''One should bear in mind that, assuming tritave equivalence, when determining which harmonics are represented, the ratios of 3 in the denominator are fungible instead of those of 2.''' For example making the fifth harmonic 5:3 a "major sixth" by conventional (and arbitrarily silly for the purposes of xenharmony, even with octaves) pitch class terminology. | '''One should bear in mind that, assuming tritave equivalence, when determining which harmonics are represented, the ratios of 3 in the denominator are fungible instead of those of 2.''' For example making the fifth harmonic 5:3 a "major sixth" by conventional (and arbitrarily silly for the purposes of xenharmony, even with octaves) pitch class terminology. | ||
| Line 169: | Line 169: | ||
* [[241edt]] | * [[241edt]] | ||
* A list of [[Tritave_Reduced_Harmonics|tritave reduced harmonics]] for easy comparison of JI and temperaments in tritave based systems. | * A list of [[Tritave_Reduced_Harmonics|tritave reduced harmonics]] for easy comparison of JI and temperaments in tritave-based systems. | ||
* Also may be found convenient: http://www.nonoctave.com/tuning/twelfth.html | * Also may be found convenient: http://www.nonoctave.com/tuning/twelfth.html | ||