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| Today a small number of competing interval naming schemes exist for the description of microtonal music. After a review of the historical development of interval names, and of current proposed schemes, a scheme is developed, taking the best and leaving alone the worst aspects of the existing standards. In addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees and the additional qualifiers 'c' and 'C'. Using these ''SHEFKHED interval names'' or ''Smith/Helmholtz/Ellis/Fokker/Keenan/Hearne Extended-diatonic interval names'', almost all small to medium sized [[Equal Temperaments|equal temperaments]] (ETs) can be named such that 'S' and 's' and/or 'C' and 'c' correspond to a displacement of an interval up or down a single degree of the ET, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are expressed consistently in in any tuning that supports them. The scheme, which can also be easily mapped to many of the current interval naming standards, facilitating translation between them, should improve pedagogy and communication in microtonal music.
| | #redirect [[FKH Extended-diatonic Interval Names]] |
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| ==Background: Interval names from antiquity to today==
| | [[Category:Interval naming]] |
| ===The origin of diatonic interval names===
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| [[File:Mesopotamian interval names table.jpg|thumb|500x500px|Mesopotamian interval names, from http://www.historyofmusictheory.com/?page_id=130, accessed October 7, 2018.|link=https://en.xen.wiki/w/File:Mesopotamian_interval_names_table.jpg]]Music theory describing the use of heptatonic-diatonic scales, including interval names, has been traced back as far as 2000BC, deciphered from a Sumerian cuneiform tablet from Nippur by Kilmer (1986). From Kummel (1970) we know that 'the names given to the seven tunings/scales were derived from the specific intervals on which the tuning procedure started' (Kilmer, 1986). This formed the basis of their musical notation ([http://www.jstor.org/stable/985853. Kilmer, 2016]). The table to the right following table displays the Ancient Mesopotamian interval names accompanied by their modern names.
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| Kilmer also writes that 'the ancient Mesopotamian musicians/“musicologists” knew what we call today the Pythagorean series of fifths, and that the series could be accomplished within a single octave by means of “inversion.” '. The Mesopotamian's music and theory was passed down through the Babylonians and the Assyrians to the Ancient Greeks, as well as their mathematics, particularly concerning musical and acoustical sound [[ratios]] (Ibid, [http://math-cs.aut.ac.ir/~shamsi/HoM/Hodgkin%20-%20A%20History%20of%20Mathematics%20From%20Mesopotamia%20to%20Modernity.pdf Hodgekin, 2005]).
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| Such mathematical and musical ideas are attributed to Pythagoras, who undoubtedly made them popular., although many scholars suggest he may have learned these ideas from his Babylonian and Egyptian mentors. None the less, Pythagoras' idea that that by dividing the length of a string into ratios of halves, thirds, quarters and fifths created the musical intervals of an octave, a perfect fifth, an octave again, and a major third form the basis of Ancient Greek music theory (http://www.historyofmusictheory.com/?page_id=20). His tuning of the diatonic scale by only octaves and perfect fifths ([[Pythagorean tuning]]) is influential through to today.
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| ===Ancient Greek interval names===
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| Intervals in Ancient Greek music were written either as string length ratios, after Pythagoras, or as positions in a [[tetrachord]].
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| [[2/1]], the [[octave]], was named ''diapason'' meaning ''<nowiki/>'''through all [strings]'
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| [[3/2]], the [[perfect fifth]] was labelled ''diapente,'' meaning 'through 5 [strings]'
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| [[4/3]], the [[perfect fourth]], was labelled ''diatessaron'', meaning 'through 4 [strings]'
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| ''dieses,'' 'sending through', refers to any interval smaller than about 1/3 of a perfect fourth
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| ''tonos'' referred both to the interval of a whole tone, and something more akin to [[mode]] or key in the modern sense ([http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf Chalmers, 1993])
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| ''ditone'' referred to the interval made by stacking two [[9/8]] whole tones, resulting in [[81/64]], the Pythagorean major third. ([[Joe Monzo|Monzo]], [http://www.tonalsoft.com/ http://www.tonalsoft.com])
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| [[256/243]] - the ''limma'', which is the ratio between left over after subtracting two 9/8 tones (together making a ditone) a perfect fourth, the ''diatonic semitone'' of the Pythagorean diatonic scale
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| [[2187/2048]] - the ''apotome'', which is the ratio between the tone and the limma, the ''chromatic semitone'' of the Pythagorean diatonic scale
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| The Ancient greek term ''diatonon'', meaning 'through tones', refers to the genus with two whole tones and a semitone, or any genus in which no interval is greater than one half of the fourth (Chalmers, 1993). The Pythaogrean diatonic scale is the scale that may be built from one two Pythagorean tetrachords, and the left over interval of 9/8.
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| ===Zarlino and [[Meantone]]===
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| [[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb|566x573px|''Le institutioni harmoniche,'' Zarlino, 1558, Cap. 15: Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1, pg. 25.|link=https://en.xen.wiki/w/File:Della_propriet%C3%A0_del_numero_Senario_&_della_sue_parti;_&_come_in_esse_si_ritroua_ogni_consonanze_musicale,_figura_1.png]]Intervals were referred to by the Ancient Greek names through the the 18th century, as Latin names. By the Renaissance it had been discovered that a Pythagorean diminished fourth sounded sweet, and approximated the string length ratio [[5/4]]. This just tuning for the major third was sought after, along with the complementary [[6/5]] tuning for the minor third, and octave complements to both - [[8/5]] for the minor sixth and [[5/3]] for the major sixth. Influential Italian music theorist and composer Gioseffo Zarlino put forth that choirs tuned the diatonic scale to the tuning built from this tetrachord, the ''[[Zarlino|intense diatonic scale]]'', also known as the ''syntonic or syntonus diatonic scale'' or the ''Ptolemaic sequence'':
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| 1/1 9/8 5/4 4/3 3/2 5/3 [[15/8]] 2/1
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| Included in this scale, however, were ''wolf intervals:'' imperfect consonances that occurred as tunings of the same interval as perfect consonances. For example, between 1/1 and 3/2, 4/3 and 1/1, 5/3 and 5/4; and 5/4 and 15/8 occurs the perfect fifth, 3/2, whereas between 9/8 and 5/3 occurs the wolf fifth, [[40/27]], flat of 3/2 by [[81/80]]. This was also the interval by which four 3/2 fifths missed [[5/1]] (the interval two octaves above 5/4). It was named the ''syntonic comma'' after Ptolemy's ''syntonus'' or ''intense diatonic tetrachord'' which consists of the intervals 9/8, [[10/9]] and [[16/15]], where 9/8 and 10/9 differ by this interval. By making the syntonic comma a unison the wolf fifth could be made a perfect fifth. It was discovered that this could be achieved by flattening (tempering) the perfect fifth by some fraction of this comma such that four of these fifths less two octaves gave an approximation of 5/4. Where two fifths less an octave give 9/8, the next two add another 10/9 to result in the 5/4. 9/8 and 10/9 were referred to as the ''major tone'' (''tunono maggiore'') and ''minor tone'' (''tunono minore''), respectively, and where this tuning led to them being equated, it was referred to as Meantone temperament, which is said to 'temper out' the syntonic comma. Zarlino advocated the flattening of the fifth by 2/7 of a comma, leading to [[2-7 Syntonic Comma Meantone|2/7-comma Meantone]], but also described [[1-3 Syntonic Comma Meantone|1/3-comma]] and 1/4-comma Meantone as usable (Zarlino, 1558).
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| The diagram on the right, from Zarlino's 1558 treatise ''Le istitutioni harmoniche'' associates many intervals with their tuning as perfect consonances. The perfect tuning for the ditone was considered then to be 5/4, rather than 81/64. The interval for which 6/5 is considered a perfect tuning was referred to as a ''semiditone'' (labelled also in ''Le istitutioni harmoniche'' by as ''Trihemituono)''. This may seem odd to us now, but in Latin 'semi' referred not to 'half', but to 'smaller', so 'semiditone' translated to something like 'smaller ditone'. Additionally 'semitone' referred to the interval smaller than the 'tone'. Like the tone, this interval possessed two alternative perfect tunings: 16/15, the difference between 15/8 and 2/1, or 5/4 and 4/3, and [[25/24]], the difference between 6/5 and 5/4. 16/15 was referred to as the ''major semitone'' (''semituono maggiore'') and 25/24 as the ''minor semitone (semituono maggiore'').
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| In addition to the Latin interval names, derived from the Ancient Greek interval names, we see on the diagram a single interval name in Italian: ''Essachordo maggiore'', referring to the ratio 5/3, which we are tempted to translate to 'major sixth'. Chapter 16, ''Quel che sia Consonanze semplice, e Composta; & che nel Senario si ritouano le sorme di tutte le somplici consonanze; & onde habbia origine l'Essachordo minore'', puts forward that the ''Essachordo minore,'' or perhaps 'minor sixth' be tuned to 8/5.
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| In the 1691 ''Lettre de Monsieur Huygens à l'Auteur [Henri Basnage de Beauval] touchant le Cycle Harmonique,'' theorist Christiaan Huygens gave names and ratios to common intervals and mapped them to [[31edo|31-tET]], which very closely approximates 1/4-comma Meantone. Translated from French, 3/2 was labelled a Fifth, 4/3 a Fourth, 5/4 a major Third, 6/5 and minor Third, 5/3 a major Sixth and 8/5 a minor Sixth. Here we really begin to see today's interval names.
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| ===English interval names in the Baroque===
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| [[File:Harmonics, or The Philosophy of Musical Sounds, Section 2 figure 3.png|thumb|517x548px|''Harmonics, or The Philosophy of Musical Sounds'', Edition 2, Smith, 1759, Section 2: On the Names and Notation of consonance and their intervals, Fig. 2 & 3 , pg. 10|link=https://en.xen.wiki/w/File:Harmonics,_or_The_Philosophy_of_Musical_Sounds,_Section_2_figure_3.png]]After English superseded Latin as the the main language of scholarship, the Latin interval names were rejected and the convention we saw in Zarlino's Italian for naming the smaller of a pair of sizes of an interval 'minor' and the larger 'major' was further applied.
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| Music theorist and mathematician Robert Smith provides the diagram and table on the right in his 1749 ''Harmonics, or, The Philosophy of Musical Sounds,'' with the description:<blockquote>'Fig. 2. If a musical string ''CO'' and it's parts ''DO'', ''EO'', ''FO'', ''GO'', ''AO'', ''BO'', ''cO'', be in proportion to one another as the numbers 1, 8/9, 4/5, 3/4, 2/3, 3/5, 8/15, 1/2, their vibrations will exhibit the system of 8 sounds which musicians donate by the letters ''C'', ''D'', ''E'', ''F'', ''G'', ''A'', ''B'', ''c''.</blockquote><blockquote>Fig. 3. And supposing those strings to be ranged like ordinates to a right line ''Cc'', and their distances ''CD'', ''DE'', ''EF'', ''FG'', ''GA'', ''AB'', ''BC'', not to be the differences of their lengths, as in fig. 2. but to be the magnitudes proportional to the intervals of their sounds, the received Names of these intervals are shewn in the following Table; and are taken from the numbers of the strings or sounds in each interval inclusively; as a Second, Third, Fourth, Fifth, &c, with the epithet of ''major'' or ''minor'', according as the name or number belongs to a greater of smaller total interval; the difference of which results chiefly from the different magnitudes of the major and minor second, called the Tone and Hemitone.'</blockquote>We note that Smith uses the tuning of the diatonic scale that Zarlino put forward: the Ptolemaic Sequence.
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| We may be surprised to see 4:3 here labelled as a minor Fourth, and 3/2 as a major Fifth, but it is obvious that this naming is more consistent than today's. Smith adds that<blockquote>'Any one of the ratios in the third column of the foregoing Table, except 80 to 81, or any one of them compounded once of oftener with the ratio 2 to 1 or 1 to 2, is called a Perfect ratio when reduced to it's least terms. And when the times of the single vibrations of any two sounds have a perfect ratio, the consonance and it's interval too after called Perfect; and is called Imperfect or Tempered when that perfect ratio and interval is a little increased or decreased.'</blockquote><blockquote>'Any small increment of decrement of a perfect interval is called respectively the Sharp or Flat Temperament of the imperfect consonance, and is measured most conveniently by the proportion it bears to the comma'</blockquote>Therefore in this system 3/2 is the ''Perfect major Fifth'' and 5/4 the ''Perfect major Third''. 81/64 might be labelled a ''comma sharp major Third'', [[32/27]] a ''comma flat minor Third'', and the 1/4-comma Meantone fifth a ''1/4-comma flat major Fifth''. The interval naming scheme Smith describes may be immediately applied to 5-limit microtonal systems. There is an inconsistency, however, where it seems that 9/8 should be called a ''Perfect major Second,'' but that, while [[9/5]] be named a ''comma sharp minor Seventh'', it's inverse, 10/9, is a ''Perfect minor Tone.''
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| In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and [[27/16]] the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described:
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| Diatonic Semitone = Pythagorean Semitone + Comma
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| = Apatomè Pythagoria - Skhisma
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| Chromatic Semitone = Apatomè Pythagoria - Comma
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| = Pythagorean Semitone + Skhisma
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| Pythagorean Semitone = Comma + Skhisma
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| = Apatomè Pythagoria - Pythagorean Semitone
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| ===Helmholtz and Ellis===
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| [[File:Helmholtz consonances table.png|thumb|617x469px|link=https://en.xen.wiki/w/File:Helmholtz_consonances_table.png|Table detailing the influence of the different consonances on each other, from ''On the Sensations of Tone as a Psychological Basis for the Theory of Music'', Helmholtz, 1863, Translated by Ellis, 1875, Chap. X, pg. 187.]]Through the investigations of Galileo (1638), Newton, Euler (1729), and Bernouilli (1771), theorist Hermann von Helmholtz was aware that ratios governing the lengths of strings existed also for the vibrations of the tones they produced. His investigation of the harmonic series associated with these ratios of vibration led him to the consideration of ratios above the [[5-limit]]. In his seminal ''On the Sensations of Tone as a Physiological Basis for the Theory of Music'', published in German in 1863 and translated into English in 1875 by Alexander Ellis, he listed just intervals as show in the table to the right. It is interesting to note that 8:9 is labelled a 'Second' rather than a 'major Second'. The minor Seventh is shown as 5:9 rather than as [[16/9|9:16]] seemingly because of the 9 partial limit imposed on the table. It is also worth noting that [[7/5|5:7]] is labelled a subminor Fifth. 'Super', indicated in notation with a '+', raises an interval by [[36/35|35:36]], the septimal quarter tone, and 'sub', indicated with a '-' lowers by the same interval with the exception of the Supersecond [[8/7|7:8]], which lies [[64/63|63:64]], the septimal comma above the Second. The subminor fifth is not included in this as no minor Fifth is shown. If we assume that 'sub' lowers an interval 35:36, then the minor Fifth would be [[25/18|18:25]], 80:81 above Smith's [[64/45|45:64]] minor Fifth, however in table 2 below, Ellis labels 18:25 a ''superfluous Fourth'', and it's inverse, [[32/25|25:32]], an ''acute diminished Fifth'', whilst 64:45 is labelled a ''diminished Fifth'' and its inverse [[45/32|32:45]] a ''false Fourth or Tritone.'' If we label 9:10 as a 'major Second', and 7:8 as a 'supermajor Second' then they differ by 35:36, the major Second is the inverse of the minor Seventh, and the supermajor Second is the octave inverse of the subminor Seventh. Perhaps 'major' has been left off name of the major Fifth, and minor off the name of the minor Fourth since the time of Smith. We can add to this table the remaining octave inversions as well as the super Fourth and sub Fifth.
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| {| class="wikitable"
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| |+Table 1. Additional Intervals
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| !Intervals
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| !Notation
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| !Ratio
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| !Cents in the interval
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| |-
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| |Major Second
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| |D
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| |9:10
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| |182
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| |-
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| |Sub Fourth
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| |F-
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| |[[21/16|16:21]]
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| |471
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| |-
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| |Supermajor Fourth
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| |F#+
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| |[[10/7|7:10]]
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| |617
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| |-
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| |Super Fifth
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| |G+
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| |[[32/21|21:32]]
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| |729
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| |-
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| |Subminor Sixth
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| |A♭-
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| |[[14/9|9:14]]
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| |765
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| |-
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| |Supermajor Sixth
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| |A+
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| |[[12/7|7:12]]
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| |933
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| |}Helmholtz defined the ''perfect consonances'' as the Octave, Twelfth and Double Octave as well as Fourth and Fifth. The major Sixth and major Third are next called ''medial consonances'', considered to in the era of Pythagorean tuning to be ''imperfect consonances'', which Helmholtz defined instead to be the minor Third and the minor Sixth.
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| Regarding the tuning of the intervals however, those corresponding to simple ratios of vibration are, as in Smith, referred to as perfect, however hey are also described as 'justly-intoned', or by Ellis as '[[just]]'. The perfect tuning for the semitone is listed as 16/15, or 112c. The perfect tunings are compared to the Pythagorean tunings, where the Pythagorean tuning of the major Third and sixth are described as 81/80 above the perfect tunings, and of the minor Third, minor Sixth and semitone to be 81/80 below the perfect tunings. Helmholtz notes that the Pythagorean tunings are closer to the equal tempered tunings than the perfect tunings. Helmholtz also describes the Pythagorean Tritone as of 612c.
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| Ellis includes an [[Ellis' interval table|additional table]] providing names for many different just and tempered intervals, perfect and imperfect. The interval names do not appear to follow any sort of consistent naming system, rather intervals seem to be named case-by-case. They also largely do not correspond to the interval names used by Helmholtz.
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| In ''Notes on the Observations of Musical Beats'', Proc. R. Soc. Lond. 1880, Ellis named many [[7-limit interval names|just intervals]] of the [[7-limit]] (including 3 and 5-limit intervals):<blockquote>‘Fifth 3:2, Fourth 4:3, Major Third 5:4, Minor Third 6:5, Major Sixth 5:3, Sub-Fifth 7:5, Super-Fourth 10:7, Super-major Third [[9/7|9:7]], Sub-minor Sixth 14:9, Sub-minor Third [[7/6|7:6]], Super-major Sixth 12:7, Sub-minor or Harmonic Seventh [[7/4|7:4]], Super-major Second 8:7, Major Tone 9:8, Minor Tone 10:9, Small Major Seventh 9:5 and Diatomic (sic.) Semitone 16:15’</blockquote>He later lists ‘The Major Sevenths 16:9 and 15:8’. The labeling of 16:9 as a Major Seventh and 9:5 as a Small Major Seventh is interesting and at odds with Smith's interval names. Given that 9:5 is larger than 16:9, and no Minor Seventh is mentioned, we can assume 16:9 was mislabeled as a Major Seventh and was understood to be a Minor Seventh, as referred to by Smith.
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| There is an inconsistency associated with the labeling of 9:5 as a Small Major Seventh also, as it lies a 3:2 Fifth above the 6:5 Minor Third, and we know a fifth and a minor third when added together to give a minor, rather than major seventh.
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| Ellis here uses the name for 8:7 we suggested above, Super-major Second, and includes our suggested Sub-minor Sixth and Super-major Sixth, however rather than Subminor Fifth and Supermajor Fourth, 7:5 and 10:7 are labelled Sub-Fifth and Super-Fourth, where in this instance sub and super are seen to raise and lower by [[21/20|21:20]] instead of by 36:35.
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| Ellis, in a footnote to his translation of Helmholtz,'s treatise also provides names for a single [[11-limit]] interval. The interval [[27/22|22:27]], of 355c, introduced by Zalzal, says Ellis was termed a ''neutral Third'' by Herr J. P. N. Land originally in ''Over de Toonladders der Arabische Musiek'' (On the Scales of Arabic Music) in 1880. An interval a fourth higher than this is mentioned, but a ratio is not given, and it is not named. We can ourselves however find it's ratio as [[18/11|11:18]], and guess it's name to be a ''neutral Sixth'', given that it lies a perfect Fourth above the neutral Third. Following a similar process as in our completion of Helmholtz table above, and assuming that the octave inverse of a neutral Third should be a neutral Sixth we may introduce the following 11-limit intervals that see common use among music theorists and microtonal musicians through to today:
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| {| class="wikitable"
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| |+Table 2. 11-limit intervals
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| !Intervals
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| !Notation
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| !Ratio
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| !Cents in the interval
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| |-
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| |Neutral Second
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| |D♭^
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| |[[88/81|81:88]]
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| |143
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| |-
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| |Neutral Second
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| |Dv
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| |[[12/11|11:12]]
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| |151
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| |-
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| |Neutral Third
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| |E♭^
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| |[[11/9|9:11]]
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| |347
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| |-
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| |Neutral Third
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| |Ev
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| |22:27
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| |355
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| |-
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| |Neutral Sixth
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| |A♭^
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| |27:44
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| |845
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| |-
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| |Neutral Sixth
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| |Av
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| |11:18
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| |853
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| |-
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| |Neutral Seventh
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| |B♭^
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| |[[11/6|6:11]]
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| |1049
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| |-
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| |Neutral Seventh
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| |Bv
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| |44:81
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| |1057
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| |}Each interval name has two sizes that differ by the comma 243/242. The notation included in the table is from HEWM notation, developed as an extension to the Helmholtz-Ellis use of '+' and '-' by Joe Monzo (http://www.tonalsoft.com/enc/h/hewm.aspx<nowiki/>).'^' indicates raising 'v' a lowered of [[33/32]]. In HEWM notation '+' and '-' are refined to mean raising and lowering of 81/80 respectively and '>' and '<' are added instead to indicate raising and lowering of 64/63. Letter names correspond instead of the the Ptolemaic sequence, as in Smith's and Helmholtz' descriptions, but to a Pythagorean tuning of the diatonic scale, where '#' and '♭' and respectively raise and lower the apotome, 2187/2048. HEWM notation is not accompanied by an interval naming system.
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| ===Common interval names today===
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| These interval names are used by theorists and microtonal musicians today, though 7/5 and 10/7 are given many different names, today also considered to be an augmented fourth and diminished fifth, lesser septimal tritone and greater septimal tritone, or simply as tritones. The fourth and fifth are today called perfect fourth and perfect fifth, and Smith's major Fourth and minor Fifth referred to as augmented fourth and diminished fifth respectively. As can be seen in Tchaikovsky's ''A Guide to the Practical Study of Harmony,'' by the beginning of end of the 19th century the familiar conventions for the naming of intervals were set, wherein
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| *Seconds, thirds, sixths and sevenths appear in the diatonic in two sizes, the larger labelled 'major' and the smaller, 'minor'.
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| *Major, when raised by a semitone, becomes 'augmented', and minor, lowered by a semitone, 'diminished'.
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| *The smaller of the two sizes of fourth and the larger of the two sizes of fifth are labelled 'perfect', along with the unison and octave.
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| *A perfect interval, when raised a semitone is labelled 'augmented', and when lowered a semitone, 'diminished'.
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| ==Current proposed schemes for the naming of microtonal intervals==
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| ===Fokker/[[Dave Keenan|Keenan]] [http://www.dkeenan.com/Music/IntervalNaming.htm Extended-diatonic interval-names]===
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| Considering the 11-limit [[Otonality and utonality|otonal]] chord 4:5:6:7:9:11 a chain of thirds, in addition to the familiar major, minor, subminor, supermajor and neutral thirds, Dave Kennan labelled 5:7 a sub-diminished fifth and [[11/7|7:11]] an augmented fifth. 7:10, the inversion of 5:7, is labelled a diminished. 5:7, therefore, is also an augmented fourth. In terms of sevenths, 4:7 is subminor, 5:9 is minor and 11:6 is neutral.
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| From this, Keenan defines a consistent interval naming system, meaning one which obeys diatonic interval arithmetic (In each column, the parenthesised prefix is the one that is implied when there is no prefix). When adding intervals the indexes are added together to give the index of the resulting interval. Keenan also adds corrections for each interval class to the indexes in order to account for inconsistencies that occur within diatonic interval arithmetic when concerning intervals greater than an octave, so that his system, unlike regular diatonic interval names, may be completely consistent.
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| As can be seen above, sub, super, augmented and diminished have also carried inconsistent meaning historically, where in Keenan's system they always alter intervals by the same amount.
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| {| class="wikitable"
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| |+Table 3. Fokker/Keenan Extended-diatonic interval-names indexes
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| !Index
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| !Prefix for unisons, fourths, fifths, octaves
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| !Prefix for seconds, thirds, sixths, sevenths, ninths
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| |-
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| | -4
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| |double diminished
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| |subdiminished
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| |-
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| | -3
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| |subdiminished
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| |diminished
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| |-
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| | -2
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| |diminished
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| |subminor
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| |-
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| | -1
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| |sub
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| |minor
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| |-
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| |0
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| |(perfect)
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| |neutral
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| |-
| |
| |1
| |
| |super
| |
| |(major)
| |
| |-
| |
| |2
| |
| |augmented
| |
| |supermajor
| |
| |-
| |
| |3
| |
| |superaugmented
| |
| |augmented
| |
| |-
| |
| |4
| |
| |double augmented
| |
| |superaugmented
| |
| |}The index values correspond most directly to degrees of 31-tET, whose interval names by this method are given in the following table:
| |
| {| class="wikitable"
| |
| |+Table 4. Fokker/Keenan Extended-diatonic interval-names in 31-tET
| |
| !31-tET degree
| |
| !Ratios
| |
| !Names
| |
| !
| |
| |-
| |
| |0
| |
| |1:1
| |
| |
| |
| |unison
| |
| |-
| |
| |1
| |
| |48:49 44:45 36:35 33:32
| |
| |(diminished second)
| |
| |super unison
| |
| |-
| |
| |2
| |
| |27:28 24:25 20:21
| |
| |subminor second
| |
| |(augmented unison)
| |
| |-
| |
| |3
| |
| |15:16 14:15
| |
| |minor second
| |
| |
| |
| |-
| |
| |4
| |
| |11:12 10:11
| |
| |neutral second
| |
| |
| |
| |-
| |
| |5
| |
| |9:10 8:9
| |
| |major second
| |
| |
| |
| |-
| |
| |6
| |
| |7:8
| |
| |supermajor second
| |
| |(diminished third)
| |
| |-
| |
| |7
| |
| |6:7
| |
| |(augmented second)
| |
| |subminor third
| |
| |-
| |
| |8
| |
| |5:6
| |
| |
| |
| |minor third
| |
| |-
| |
| |9
| |
| |9:11
| |
| |
| |
| |neutral third
| |
| |-
| |
| |10
| |
| |4:5
| |
| |(subdiminished fourth)
| |
| |major third
| |
| |-
| |
| |11
| |
| |11:14 7:9
| |
| |(diminished fourth)
| |
| |supermajor third
| |
| |-
| |
| |12
| |
| |16:21
| |
| |subfourth
| |
| |(augmented third)
| |
| |-
| |
| |13
| |
| |3:4
| |
| |perfect fourth
| |
| |
| |
| |-
| |
| |14
| |
| |8:11
| |
| |super fourth
| |
| |
| |
| |-
| |
| |15
| |
| |5:7
| |
| |augmented fourth
| |
| |(subdiminished fifth)
| |
| |-
| |
| |16
| |
| |7:10
| |
| |(superaugmented fourth)
| |
| |diminished fifth
| |
| |-
| |
| |17
| |
| |11:16
| |
| |
| |
| |sub fifth
| |
| |-
| |
| |18
| |
| |2:3
| |
| |
| |
| |perfect fifth
| |
| |-
| |
| |19
| |
| |21:32
| |
| |(diminished sixth)
| |
| |super fifth
| |
| |-
| |
| |20
| |
| |9:14 7:11
| |
| |subminor sixth
| |
| |(augmented fifth)
| |
| |-
| |
| |21
| |
| |5:8
| |
| |minor sixth
| |
| |(superaugmented fifth)
| |
| |-
| |
| |22
| |
| |11:18
| |
| |neutral sixth
| |
| |
| |
| |-
| |
| |23
| |
| |3:5
| |
| |major sixth
| |
| |
| |
| |-
| |
| |24
| |
| |7:12
| |
| |supermajor sixth
| |
| |(diminished seventh)
| |
| |-
| |
| |25
| |
| |4:7
| |
| |(augmented sixth)
| |
| |subminor seventh
| |
| |-
| |
| |26
| |
| |9:16 5:9
| |
| |
| |
| |minor seventh
| |
| |-
| |
| |27
| |
| |11:20 6:11
| |
| |
| |
| |neutral seventh
| |
| |-
| |
| |28
| |
| |8:15
| |
| |
| |
| |major seventh
| |
| |-
| |
| |29
| |
| |14:27
| |
| |(diminished octave)
| |
| |supermajor seventh
| |
| |-
| |
| |30
| |
| |18:35
| |
| |sub octave
| |
| |(augmented seventh)
| |
| |-
| |
| |31
| |
| |1:2
| |
| |octave
| |
| |
| |
| |}The interval names shown in brackets could be said to be 'secondary', the others, 'primary'.
| |
| | |
| After releasing his system Keenan was informed that is was identical to the extended-diatonic interval-naming scheme of Adriaan Fokker but for the acknowledgment of more 11-limit ratios.This system depends on the tempering out of 81/80, where the diatonic major third, from four stacked fifths, approximates the just major third, 5/4. It also depends on the existence of neutral intervals, i.e., that the perfect fifth or equivalently, the chromatic semitone, subtends an even number of degrees of the ET. To simply to our familiar naming scheme for 12-tET, we observe that it applies to [[24edo|24-tET]] equally as directly as in 31-tET, where the prefixes correspond to degrees of the ET. Exactly the same is also true for [[38edo|38-tET]], twice [[19edo|19-tET]], a meantone which very closely approximates 1/3-comma Meantone. Meantone temperament wherein the fifth is divided into two equally sized neutral thirds is referred to as neutral temperament. Whereas meantone temperament is generated by the fifth, in neutral temperament the generator is half this interval, the neutral third. Where it was seen above that there are two neutral thirds, 9:11 and 22:27 that differ by 243/242, neutral temperament is at its most simple the temperament defined by this equivalence: the tempering out of 243/242, as meantone is defined by the tempering out of 81/80. The temperament that tempers out both 81/80 and 243/242 is called [[Mohajira|''Mohajira'']], upon which Keenan's scheme can be said to be based. As well as 24-tET, 31-tET and 38-tET, Mohajira is supported by [[7edo|7-tET]] and [[17edo|17-tET]].
| |
| | |
| The primary interval names resulting in this system's application to these ETs is now show for easy comparison, where 'M', 'm', 'P', 'N', 'A', 'd', 'S' and 's' are shorthand for major, minor, perfect, neutral, augmented, diminished, super and sub, respectively:
| |
| | |
| 7-tET: P1 N2 N3 P4 P5 N6 N7 P8
| |
| | |
| 17-tET: P1 m2 N2 M2 m3 N3 M3 P4 S4 s5 P5 m6 N6 M6 m7 N7 M7 P8
| |
| | |
| 24-tET: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 S4 A4/d5 s5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8
| |
| | |
| 31-tET: P1 S1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 S4 A4 d5 s5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8 P8
| |
| | |
| 38-tET: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 S4 A4 SA4/sd5 d5 s5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8
| |
| | |
| to Meantone tunings that are not Mohajira tunings, the regular diatonic interval names can be applied, but with the addition of double augmented and double diminished from Fokker/Keenan's system.
| |
| | |
| 19-tET: P1 A1 m2 M2 A2/d3 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 A6/d7 m7 d8 P8 (every second step of 38edo)
| |
| | |
| [[26edo|26-tET]]: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 AA4/AA5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8
| |
| | |
| In the above primary interval names for equal tunings, it should be noted that no interval of interval-class ''n-1'' is subtended by a larger number of degrees than an interval of class ''n''. I define that an interval-name set for which this is true is said to be ''well-ordered''. The possibility for Well-ordered interval-name sets is a desirable property for interval naming schemes to possess and is possess by all proposals discussed in this paper.
| |
| | |
| Keenan adds further that if it is desired to distinguish between ratios that are in 31-tET approximated by the same number of steps, an addition prefix be added to describe the prime limit of the approximated interval. For 3-limit intervals, the obvious choice is 'Pythagorean', for 5-limit Keenan chooses 'classic', for 7, 'septimal, 11, 'undecimal' and 13, 'tridecimal'. When the highest prime is the same, Keenan suggests adding 'small' and 'large' as final prefixes for this purpose.
| |
| | |
| In non-Meantone tunings, the two definitions of major third - 4:5 and 64:81, the just (or classic) and Pythagorean major thirds no-longer correspond. If intervals are to receive unique names then to one or both of these major thirds must be added a prefix. Keenan has been involved with the development of both types of systems. Only when the major is defined by it's mapping as fourth fifths, i.e. 81/64, can conserve interval arithmetic.
| |
| ===[[Miracle]] [http://dkeenan.com/Music/Miracle/MiracleIntervalNaming.txt interval naming]===
| |
| Keenan later describes how the scheme can be extended to also cover 72-tET and 41-tET. In 31-tET the fifth may divided into six minor seconds. This temperament is called ''Miracle'', and is also supported by [[41edo|41-tET]] and [[72edo|72-tET]]. The first six generators of Miracle give the following intervals: P1 m2 SM2 m3 S4 P5, as can be seen in the table above. 31-tET may be covered by 15 generators downwards and 15 generators upwards from 1:1. In 72-tET, either side of the intervals that that result from these 31 notes, called Miracle[31] 15|15, lie unnamed intervals that may be found first at either 31 or 41 generators further upwards or downwards and in 41-tET, either at 10 or 31 generators. If, one degree of 41 or 72-tET above an interval or Miracle[31] 15|15 lies an unnamed interval that can be first found by an additional 31 generators upwards, it is given the same name as the interval directly below it, with the addition of the prefix 'n', for 'narrow'. Similarly, 'W' for 'wide' prefixes an unnamed interval one degree, 31 generators below.
| |
| | |
| 41-tET: P1 S1 nsm2 sm2 m2 N2 nM2 M2 SM2 sm3 nm3 m3 N3 M3 nSM3 SM3 s4 P4 nS4 S4 A4 d5 s5 Ws5 P5 S5 sm6 Wsm6 m6 N6 M6 Wm6 SM6 sm7 m7 Wm7 N7 M7 SM7 WSM7 s8 P8
| |
| | |
| In 41-tET, fourth fifths make a wide major third, rather than a major third, and interval arithmetic is no longer conserved. The same is true for 72-tET, so we have still yet to a system able to conserve interval arithmetic in non-meantone ETs. Though many edos can be covered, many still cannot, including the [[Superpyth|''Superpythagorean'']] edos, where the fifth is sharper than just, and four fifths give an approximation to 7:9, the super major third, tempering out the septimal comma, 63:64.
| |
| ===[[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak]===
| |
| One system which in it's naming of meantone and non-meantone edos is able to conserve interval arithmetic, sagispeak, was developed largely by [[George Secor]], with input from Dave Keenan and others as an interval naming system that maps 1-1 with the Sagittal microtonal music notation system. Sagittal notation was developed as a generalised diatonic-based notation system applicable equally to just intonation, equal tunings and rank-''n'' [[temperaments]]. Dozens of different accidentals can be used on a regular diatonic [[Staff notation|staff]] to notate up to extremely fine divisions, however in most cases only a handful are needed. In sagispeak, each accidental is presented by a prefix, made up of a single letter, in most cases, followed by either 'ai' if the accidental raises a note, or 'ao' if it lowers a note. As in HEWM notation, Pythagorean intonation is assumed as a basis. Then the prefixes depart from Pythaogrean intonation, altering by commas and introducing other primes. In place of the prefixes 'sub' and 'super', generally signifying an alteration of 36/35 from 5-limit intervals or 64/63 for 3-limit, Sagittal features an accidental of 64/63, which may be used to take a Pythagorean major interval to a supermajor, minor to subminor, or perfect to super or sub. The prefix 'tao' indicates a decrease of 64/63 and and the prefix 'tai' an increase. Whereas in previous interval naming schemes 'major' and 'minor' were synonymous with the 5-limit tunings, in sagispeak they map instead to Pythagorean. A prefix is needed then to take a Pythagorean intoned interval to a 5-limit tuning. Where 5/4 is 81/80 below the the Pythagorean third, the prefixes 'pai' and 'pao' (where 'p' is for 'pental', as in, involving prime 5), which raise or lower a note by 81/80 respectively. Similarly, 'vai' and 'vao', which raise or lower a note by 33/32 respectively, leading to ratios of 11.
| |
| | |
| Because it is built off of the diatonic scale, sagispeak conserves diatonic interval arithmetic, i.e. familiar relations in the diatonic scale, i.e. M2 + m3 = P4. As in Fokker/Keenan Extended-diatonic Interval-names, diatonic interval arithmetic is also extended, where, for example, tai-major 2 + tao-minor 3 = P4 (8/7 + 7/6 = 4/3), where opposite alterations cancel each other out, and diatonic interval arithmetic is conserved, a very useful property for a microtonal interval naming system to possess. Another helpful property of sagispeak is its generalised applicability to edos, just intonation and other tunings, where the same intervals maintain their spelling across different tunings. Despite these benefits however, many see Sagittal and Sagispeak as overly complex (even though the entire extended system need hardly ever be applied), and requiring too many new terms to be learnt.
| |
| | |
| We have seen that there are two competing definitions of a major third, the ratio '5/4' or the interval built from fourth stacked fifths, that may or may not correspond to 5/4. In meantone systems, those we are used to, they correspond, but in most edos they do not. Interval naming systems wherein the major third is defined as an approximation to 5/4 rather than as four fifths minus two octaves may benefit from a familiar name for 5/4, but they are unable to conserve diatonic interval arithmetic.
| |
| | |
| For comparison, 31-tET, is shown below in sagispeak:
| |
| | |
| 31-tET: P1 tai-1/vai-1 tao-m2 m2 vai-m2/vao-M2 M2 tai-M2 tao-m3 m3 vai-m3/vao-M3 M3 tai-M3 tao-4 P4 vai-4 A4 d5 vao-5 P5 tai-5 tao-m6 m6 vai-m6/vao-M6 M6 tai-M6 tao-m7 m7 vai-m7.vao-M7 M7 tai-M7 vao-8/tao-8 P8
| |
| ===Dave Keenan's most recent system===
| |
| [[File:Dave Keenan edo interval names prefix diagram.png|thumb|580.99x580.99px|Prefix diagram from ''One way to name the interval of any EDO from 5 to 72'', Keenan, 2016, pg. 4.]]
| |
| In 2016 Dave Keenan proposed an alternative generalised [http://dkeenan.com/Music/EdoIntervalNames.pdf microtonal interval naming system for edos]. In what might be understood as a generalisation of his extended-diatonic interval-naming system described above onto any equal tuning. Employing as prefixes the familiar 'sub', 'super', and 'neutral'. His scheme is based on the diatonic scale, however the diatonic interval names are not defined by their position in a cycle of fifths like is Sagispeak. In Keenan's system the ET's best 3/2 is first labelled P5, and the fourth P4. The interval half-way between the tonic and fifth is labelled the neutral third, or 'N3', and halfway between the fourth and the octave N6. Then the interval a perfect fifth larger than N3 is labelled N7, and the interval a fifth smaller than N6 labelled N2. The neutral intervals then lie either at a step of the ET, or between two steps. After this the remaining interval names are decided based on the distance they lie in pitch from the 7 labelled intervals, which make up the ''Neutral scale'', P1 N2 N3 P4 P5 N6 N7, which, like the diatonic, is an MOS scale, which may be labelled [[Neutral7|Neutral[7]]] 3|3 using [[Modal UDP Notation|Modal UPD notation]]. To name an interval in an ET, the number of steps of 72-tET that most closely approximate the size of the interval difference from a note of the neutral scale is first found. Then the prefix corresponding to that number of steps of 72-tET is applied to the interval name. The diagram to the right details this process. An interval just smaller than a major third in Keenan's system is labelled a ''narrow major third'', and an interval just wider than a 6/5 minor third a ''wide minor third'', however he notes that 'narrow' and 'wide' are only necessary in edos greater than 31. This system is equivalent to the Fokker/Keenan Extended-diatonic interval-naming system and Miracle interval naming when applied to any of the ETs they were able to cover.
| |
| | |
| Keenan's system is an elegant way to keep the 'major 3rd' label for 5/4, where labels depend on the size of the best fifth, however it suffers from it's applicability only to ETs, and that it does not conserve interval arithmetic. Another potentially undesirable result of the system is that the major second approximates 10/9, and a ''wide major second'' 9/8, where as 9/8 is almost always considered a major second, and 10/9 often a narrow or small major second. One such system that considers 10/9 a narrow major second is that of Aaron Hunt.
| |
| ===Size-based systems===
| |
| Microtonal theorist [[Aaron Andrew Hunt]] devised [http://musictheory.zentral.zone/huntsystem4.html the Hunt system], which includes interval name assignments for JI (just intonation) and edos based on [[41edo]]. Compared to Keenan's 72 interval names, Aaron's system includes 41. His system is based directly on 41edo, and unlike Keenan's system, interval are given the name of the closest step of 41edo, and no account is taken of the size of the edos fifth. In 41edo, Major, minor, augmented and diminished intervals are those obtained through the approximately Pythagorean cycle of fifths. Intervals one step of 41edo above these are given the prefix 'small', one step larger are given the prefix 'large', two steps smaller the prefix 'narrow' and two larger the prefix 'wide'. As a result, 5/4 is labelled a 'small major 3rd', or SM3 (not to be confused with a super major third, a label that does not exist in this system).
| |
| | |
| Neo-medieval musicians and early music historian and theorist [[Margo Schulter]] described her own [http://www.bestii.com/~mschulter/IntervalSpectrumRegions.txt interval naming scheme] built on approximations to JI intervals. Each interval names corresponds to an approximate size, and no particular edo is referenced. In her scheme middle major thirds range in size from 400-423 cents, and small major thirds from 372-400c. 5/4 is labelled a small major third, 81/64 a middle major third and 9/7 a large major third. Margo's scheme includes small, middle and large varieties of major, minor and neutral 2nds, 3rds, 6ths, 7ths; perfect fourth and fifths; and tritones, as well as a sub fifth and super fourth a dieses and comma and an octave less dieses and comma and ''interseptimals'', which correspond to intermediates, her name referencing the fact that they may each approximate two ratios of 7.
| |
| | |
| In Hunt's system when used in 41edo or JI diatonic interval arithmetic is conserved, but in other tunings it may not be, and Margo's system may not conserve diatonic interval arithmetic either. Both systems may be applied to arbitrary tunings, but the same intervals (defined, perhaps by a MOS scale) may not be given the same interval names across different tunings. [[User:PiotrGrochowski/Extra-Diatonic Intervals|Other]] size-based systems also exist, but are less thoroughly described and less well known. In all these systems, interval arithmetic is not conserved across all tunings.
| |
| ===Ups and Downs===
| |
| One final interval naming system, associated with the [[Ups and Downs Notation]] system, belonging to microtonal theorist and musician [[KiteGiedraitis|Kite Giedraitis]], like Sagittal is based on deviations from the diatonic scale. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings ([[12edo]], 19edo or 31edo for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo]], [[22edo]], 41edo, 72edo), or even an up-major 3rd (e.g. [[21edo]]). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). 'Up' and 'down' prefixes may be used before mid also, i.e. 'v~ 3). P1, P4, P5 and P8 are simply labelled '1', '4', '5' and '8'. This system benefits from it's simplicity as well as it's conservation of interval arithmetic. Rank-2 temperaments may also be described, with the addition of of an additional pair of qualifiers - '/' and '\'. A rank-2 scale, such as a MOS scale may appear different than this rank-2 notation when approximated in an equal (rank-1) tuning. Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be be given different names.
| |
| | |
| [[Igliashon Jones]] is a supporter of this system, but for the relabeling of 'down' as 'sub' and 'up' as 'super' and 'mid' as 'neutral', so that more common names are used, wherein 'super' infers a raise of 1 step of the edo, and 'sub' a lowering of one step. In this 'Extra-diatonic' system 'super' and 'sub' may be doubly applied, as in Ups and Downs, but they may not be applied before 'neutral' where in Ups and Downs they may be applied before 'mid'.
| |
| | |
| 31edo in Ups and Downs:
| |
| | |
| 1 ^1 vm2 m2 ~2 M2 ^M2 vm3 m3 ~3 M3 ^M3 v4 4 ^4 A4 d5 v5 5 ^5 vm6 m6 ~6 M6 ^M6 vm7 m7 ~7 M7 ^M7 v8 8
| |
| | |
| In Jones' relabeling 31edo appears as in Fokker/Keenan Extended-diatonic Interval-names.
| |
| ==SHEFKHED interval names==
| |
| ===Introduction===
| |
| It is possible for the best aspects of all interval naming systems to be employed in a single system, namely:
| |
| *Backwards compatibility with familiar diatonic interval-names (Fokker/Keenan, Miracle interval naming, Keenan's most recent, size-based systems)
| |
| *Conservation of interval arithmetic (Fokker/Keenan, Sagispeak, Ups and Downs)
| |
| *Generalisation across all small to medium ETs (Sagispeak, Keenan's most recent, Size-based systems, Ups and Downs)
| |
| *Consistency through translation across tunings (Sagispeak)
| |
| *Prefixes that imply augmentation and diminution by a single step of an ET (Fokker/Keenan, Ups and Downs)
| |
| *Possibility for well-ordered interval name sets (all proposals)
| |
| *More than one possible name for intervals (Fokker/Keenan, Miracle interval naming, Sagittal, Keenan's most recent, Ups and Downs)
| |
| Such a system is developed through the extension of Fokker/Keenan Extended-diatonic Interval-names to define prefixes by alterations by specific commas as in Sagispeak, with the addition of prefix names of Smith and Keenan to enable application to Pythagorean and Superpythagorean systems. Where the prefixes of the Fokker/Keenan system were introduced by Helmholtz/Ellis, attribution to them is added, leading to Smith/Helmholtz/Ellis/Fokker/Keenan/Hearne Extended-diatonic interval names, or SHEFKHED interval names (Hearne is me).
| |
| ===Prefixes===
| |
| *The unison is labelled P1 for perfect unison, and the octave P8 for perfect octave.
| |
| *A tuning's best approximation to 3/2 is labelled P5, for perfect fifth, and it's octave-complement labelled P4, for perfect fourth.
| |
| *From the Pythagorean diatonic scale using a tuning's best 3/2 fifth, the two sizes of second, third, sixth and seventh are labelled major, or 'M', for the larger, and minor, or 'm' for the smaller.
| |
| *Any perfect or major interval raised by the apotome, the interval between the major and minor intervals of a single interval-class is labelled 'A' for augmented, and any perfect or minor interval lowered by the same is labelled 'd' for diminished.
| |
| *Any augmented interval may be made doubly augmented, with short-hand 'AA' by the further raising of an apotome and any diminished interval made doubly diminished, with short-hand 'dd' by the further lowering of an apotome. This process may be iterated ad nauseum. At this stage we have simply rigorously defined diatonic interval names. Thankfully what remains of the definition leads to more desirable alternatives for most occasions in which one might find these iteratively diminished and augmented intervals.
| |
| *Perfect, major and augmented intervals may be given the prefix 'super', with shorthand 'S' which infers an augmentation by the septimal comma, 64/63, whereas perfect, minor and diminished intervals are lowered by the same interval when given the prefix 'sub', with short-form 's'.
| |
| *Major and augmented intervals may be given the prefix 'classic', with short-form 'c', inferring a diminution by the syntonic comma, 81/80, whereas minor and diminished may also be given the prefix 'classic' but with short-hand 'C', inferring an augmentation by 81/80. This results in the labeling of 10/9, 6/5, 5/4, 8/5, 5/3 and 9/5 as classic major second, classic major third, classic minor third, classic minor sixth, classic major sixth and classic minor seventh, as per Keenan's suggestion when a comparison to Pythagorean is needed.
| |
| *Perfect intervals may also be given the prefixes 'C' and 'c' to imply augmentation and diminution by the syntonic comma, however, where this interval is given the name 'comma' by Smith and Bosanquet, and where Smith's interval-naming scheme involves prefixes of ''m''/''n''-comma sharp and ''m''/''n''-comma flat, the long-form for these prefixes when applied to perfect intervals is 'comma-sharp' and 'comma-flat' respectively.
| |
| *For seconds, thirds, sixths and sevenths, any interval exactly half-way major and minor is labelled 'neutral', with short-form 'N'.
| |
| *This is extended such that for the intervals exactly half-way between the perfect unison and the augmented unison, the perfect fourth and the augmented fourth, the perfect fifth and the diminished fifth; and the perfect octave and the diminished octave are also given the label 'neutral' with short-form 'N'. In all cases 'N' marks a splitting-in-half or the apotome, and it's presence implies neutral temperament and the tempering out of 243/242. Accordingly it implies a diminution from perfect, major, augmented of 33/32, as well as an augmentation from perfect, minor or diminished of 33/32, but may not be used to imply those alterations in any other cases.
| |
| *To extend to the 13-limit, we add that to cP, cM and cA intervals may be added the 'sub' or 's' prefix, in this instance indicating a diminution of [[65/64]], and that to CP, Cm and Cd intervals may be added the 'super' or 'S' prefix, indication an augmentation of the same interval. Accordingly the difference between 65/64 and 64/63, 4096/4095, the ''tridecimal schisma'', is tempered out. Accordingly 16/13 is labelled a 'sub classic major third', or scM3. In tunings where the syntonic comma is tempered out, such that (cP, cM, cA, CP, Cm, Cd) = (P, M, A, P, m, d), the 'c' or 'C' prefixes are dropped in the short-form.
| |
| *Where N indicates a splitting of the apotome and of the perfect fifth, interval names indicating the splitting of the limma and of the perfect fourth are included for remaining unnamed intervals, reflecting limited, but existing practice. The interval half-way between P1 and m2 is given the short-form '1-2', and long-form 'unison-second' or 'unicond' for short. Similarly the interval half-way between M7 and P8 is given the short-form '7-8', and long-form 'seventh-octave' or 'sevtave' for short. The interval splitting the fourth, lying half-way between M2 and m3 is given the short-form '2-3', and long-form 'second-third', or 'serd', and it's octave complement, lying half-way between M6 and m7 is given the short-form '6-7', and long-form 'sixth-seventh', or 'sinth'. The interval half-way between M3 and P4 is given the short-form '3-4' and long-form 'third-fourth' or 'thourth', and it's octave-complement, the interval half-way between P5 and m6 is given the short-form '5-6', and long-form 'fifth-sixth', or 'fixth'. These interval names can be associated with [[The Archipelago|Barbados]] temperament, indicating the tempering out of 676/675, generated by 2-3, half of the fourth, associated with the ratio 15/13. These ''intermediates'' lie 40/39 above major intervals and the perfect unison and fifth, and below minor intervals and the perfect fourth and octave. 3-4, for example, is associated with the ratio 13/10.
| |
| *For completeness, '4-5', with long-form 'fourth-fifth' or 'firth' is added, though it is separate to the other intermediates, splitting not the limma, but the dieses (between A4 and d5), or the octave. It does not map to any particular ratios and is not needed as a primary interval name, but is included to be used as an optional secondary interval name when there are no others.
| |
| *In any prefix is used before 'P' then 'P' is removed in both the short-form and long-form names.
| |
| ===Application===
| |
| As in Keenan/Fokker and Ups and Downs, intervals may be given multiple names. The following details the order to which certain names are privileged above others.
| |
| | |
| Interval names are ranked in eight tiers.
| |
| #Perfect
| |
| #Neutral
| |
| #Major, minor, A4 and d5.
| |
| #'S', 's', 'C', 'c' 'SC' and 'sc' prefixes to major, minor, perfect intervals and to A4 and d5
| |
| #Intermediates
| |
| #Remaining augmented and diminished intervals (for when the chroma is subtended by more than a single (positive) step of the edo)
| |
| #'S', 's', 'C', 'c' 'SC' and 'sc' prefixes to augmented and diminished intervals
| |
| #Intervals augmented and diminished more than singularly
| |
| When more than one interval name corresponds to a specific interval, the names are privileged in order of the tiers. By this ordering, the first available name is the ‘primary’ for that interval, the second available ‘secondary’ and third 'tertiary'.
| |
| | |
| On top of this, well-ordered interval-name sets are desired, leading to interval names in lower tires being used in preference to higher-tier names in some cases.
| |
| | |
| All ''regular diatonic'' edos (edos whose best fifth is greater than 4 degrees of 7edo and less than 3 degrees of 5edo, such that the diatonic scale has 5 large and 2 small steps) up to 46 can be simply given primary well-ordered interval names. All of those that I've seen used have their primary well-ordered interval-names below, with the addition of 53edo, which is as far as I want to go and can go with this system without extending it further.
| |
| | |
| 12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8
| |
| | |
| 17edo: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8
| |
| | |
| 19edo: P1 S1/sm2 m2 M2 SM2/sm3 m3 M3 SM3/s4 P4 A4 d5 P5 S5/sm6 m6 M6 SM6/sm7 m7 M7/s8 P8
| |
| | |
| 22edo: P1 m2 Cm2 cM2 M2 m3 Cm3 cM3 M3 P4 C4 cA4/Cd5 c5 P5 m6 Cm6 cM6 M6 m7 Cm7 cM7 M7 P8
| |
| | |
| 24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 N4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8
| |
| | |
| 26edo: P1 S1 sm2 m2 M2 SM2 sm3 m3 M3 SM3 s4 P4 A4 SA4/sd5 d5 P5 S5 sm6 m6 M6 SM6 sm7 m7 M7 SM7 s8 P8
| |
| | |
| 27edo: P1 m2 N1 N2 cM2 M2 m3 Cm3 N3 cM3 M3 P4 N4 scA4/Cd5 cA4/SCd5 N5 P5 m6 Cm6 N6 cM6 M6 m7 Cm7 N7 N8 M7 P8
| |
| | |
| 29edo: P1 C1/S1/sm2 m2 Cm2 cM2 M2 SM2/sm3 m3 Cm3 cM3 M3 SM3/s4 P4 C4 cA4/d5 A4/Cd5 c5 P5 S5/sm6 m6 Cm6 cM6 M6 SM6/sm7 m7 Cm7 cM7 M7 SM7/S8/c8 P8
| |
| | |
| 31edo: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8
| |
| | |
| 34edo: P1 C1 m2 Cm2 N2 cM2 M2 2-3 m3 Cm3 N3 cM3 M3 3-4 P4 C4 N4/d5 cA4/Cd5 A4/N5 c5 P5 5-6 m6 Cm6 N6 cM6 M6 6-7 m7 Cm7 N7 cM7 M7 c8 P8
| |
| | |
| 36edo: P1 S1 sm2 m2 Sm2 sM2 M2 SM2 sm3 m3 Sm3 sM3 M3 SM3 s4 P4 S4 sA4 A4/d5 SA4 s5 P5 S5 sm6 m6 Sm6 sM6 M6 SM6 sm7 m7 Sm7 sM7 M7 SM7 s8 P8
| |
| | |
| 38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8
| |
| | |
| 41edo: P1 C1/S1 N1 m2 Cm2 N2 cM2 M2 SM2 sm3 m3 Cm3 N3 cM3 M3 SM3 s4 P4 C4 N4 cA4 Cd5 N5 c5 P5 S5 sm6 m6 Cm6 N6 cM6 M6 SM6 sm7 m7 Cm7 N7 cM7 M7 N8 c8/s8 P8
| |
| | |
| 43edo: P1 S1 1-2 sm2 m2 Sm2 sM2 M2 SM2 2-3 sm3 m3 Sm3 sM3 M3 SM3 3-4 s4 P4 S4 sA4 A4 d5 Sd5 s5 P5 S5 5-6 sm6 m6 Sm6 sM6 M6 SM6 6-7 sm7 m7 Sm7 sM7 M7 SM7 7-8 s8 P8
| |
| | |
| 46edo: P1 C1 sm2 m2 Cm2 SCm2 scM2 sM2 M2 SM2 sm3 m3 Cm3 SCm3 scM3 cM3 M3 SM3 s4 P4 C4 SC4 scA4/d5 cA4/Cd5 A4/SCd5 SA4/sc5 c5 P5 S5 sm6 m6 Cm6 SCm6 scM6 sM6 M6 SM6 sm7 m7 Cm7 SCm7 scM7 cM7 M7 SM7 c8 P8
| |
| | |
| 53edo: P1 C1/S1 1-2 sm2 m2 Cm2 SCm2 scM2 sM2 M2 SM2 2-3 sm3 m3 Cm3 SCm3 scM3 cM3 M3 SM3 3-4 s4 P4 C4 SC4 scA4 cA4/d5 A4/Cd5 SCd5 SA4/sc5 c5 P5 S5 5-6 sm6 m6 Cm6 SCm6 scM6 sM6 M6 SM6 6-7 sm7 m7 Cm7 SCm7 scM7 cM7 M7 SM7 7-8 c8/s8 P8
| |
| | |
| We can see that
| |
| *17edo, 24edo, 27edo, 31edo, 34edo (through 17edo), and 38edo are neutral tunings from the use of 'N'. We can find the scale Neutral[17] 8|8: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8 in all of these edos.
| |
| *19edo, 24edo (through 12edo), 26edo, 31edo, 36edo (through 12edo), 38edo (through 19edo) and 43edo are meantone tunings through the use of 'S' and 's'.
| |
| *22edo, 27edo and 34edo (through 17edo) are superpythagorean tunings from the use of 'C' and 'c'.
| |
| *29edo, 41edo, 46edo and 53edo are Pythagorean tunings through the use of both 'S' and 's'; and 'C' and 'c'.
| |
| *34edo, 43edo and 53edo are barbados tunings through the use of intermediates. We can find the scale Barbados[9] 4|4, P1 1-2 M2 2-3 3-4 P4 P5 5-6 6-7 m7 7-8 P8 in all of those edos, but not necessarily in the primary interval names.
| |
| Every edo in which we see SM2/sm3 also supports barbados, where this interval is the generator, at half a fourth, however rather than 15/13 the generator is more simple represented as 8/7~7/6. The temperament generated by the semi-fourth wherein it represent both SM2 and sm3 (tempering out [[49/48]]) is called Semaphore.
| |
| | |
| Semaphore[9] 4|4 has primary interval names P1 S1/sm2 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 m7 SM7/s8 P8, which can be seen in 19edo, 24edo and 29edo, so we know they are Semaphore tunings.
| |
| | |
| Well-ordered primary and their secondary interval names for 22edo, 41edo and 53edo are shown below in more detail.
| |
| {| class="wikitable"
| |
| |+22edo
| |
| !Degree
| |
| !Primary interval name
| |
| !Short-form
| |
| !Secondary interval name
| |
| !Short-form
| |
| !Cents
| |
| !Approximate 2.3.7.11.17
| |
| Ratios
| |
| |-
| |
| |0
| |
| |perfect unison
| |
| |P1
| |
| |super octave
| |
| |S1
| |
| |0
| |
| |1/1, 64/63
| |
| |-
| |
| |1
| |
| |minor second
| |
| |m2
| |
| |comma-sharp unison/super minor second
| |
| |C1/sm2
| |
| |54.55
| |
| |33/32, 34/33, 25/24, 81/80
| |
| |-
| |
| |2
| |
| |classic minor second
| |
| |Cm2
| |
| |diminished third
| |
| |d3
| |
| |109.09
| |
| |18/17, 17/16, 16/15, 15/14
| |
| |-
| |
| |3
| |
| |classic major second
| |
| |cM2
| |
| |augmented unison
| |
| |A1
| |
| |163.64
| |
| |11/10, 10/9
| |
| |-
| |
| |4
| |
| |major second
| |
| |M2
| |
| |super major second
| |
| |SM2
| |
| |218.18
| |
| |9/8, 8/7, 17/15
| |
| |-
| |
| |5
| |
| |minor third
| |
| |m3
| |
| |super minor third
| |
| |sm3
| |
| |272.73
| |
| |7/6, 20/17
| |
| |-
| |
| |6
| |
| |classic minor third
| |
| |Cm3
| |
| |diminished fourth
| |
| |d4
| |
| |327.27
| |
| |6/5, 17/14, 11/9
| |
| |-
| |
| |7
| |
| |classic major third
| |
| |cM3
| |
| |augmented second
| |
| |A2
| |
| |381.82
| |
| |5/4
| |
| |-
| |
| |8
| |
| |major third
| |
| |M3
| |
| |super major third
| |
| |SM3
| |
| |436.36
| |
| |9/7, 14/11, 22/17
| |
| |-
| |
| |9
| |
| |perfect fourth
| |
| |P4
| |
| |sub fourth
| |
| |s4
| |
| |490.91
| |
| |4/3, 21/16
| |
| |-
| |
| |10
| |
| |comma-sharp fourth
| |
| |C4
| |
| |diminished fifth
| |
| |d5
| |
| |545.45
| |
| |11/8, 15/11, 27/20
| |
| |-
| |
| |11
| |
| |classic augmented fourth
| |
| classic diminished fifth
| |
| |cA4
| |
| Cd5
| |
| |super classic augmented fourth
| |
| sub classic diminished fifth
| |
| |ScA4
| |
| sCd5
| |
| |600
| |
| |7/5, 17/12, 45/32
| |
| 10/7, 24/17, 64/45
| |
| |-
| |
| |12
| |
| |comma-flat fifth
| |
| |c5
| |
| |augmented fourth
| |
| |A4
| |
| |654.55
| |
| |16/11, 22/15, 40/27
| |
| |-
| |
| |13
| |
| |perfect fifth
| |
| |P5
| |
| |super fifth
| |
| |S5
| |
| |709.09
| |
| |3/2, 32/21
| |
| |-
| |
| |14
| |
| |minor sixth
| |
| |m6
| |
| |sub minor sixth
| |
| |sm6
| |
| |763.64
| |
| |11/7, 14/9, 17/11
| |
| |-
| |
| |15
| |
| |classic minor sixth
| |
| |Cm6
| |
| |diminished seventh
| |
| |d7
| |
| |818.18
| |
| |8/5
| |
| |-
| |
| |16
| |
| |classic major sixth
| |
| |cM6
| |
| |augmented fifth
| |
| |A5
| |
| |872.73
| |
| |5/3, 18/11, 28/17
| |
| |-
| |
| |17
| |
| |major sixth
| |
| |M6
| |
| |super major sixth
| |
| |SM6
| |
| |927.27
| |
| |12/7, 17/10
| |
| |-
| |
| |18
| |
| |minor seventh
| |
| |m7
| |
| |sub minor seventh
| |
| |sm7
| |
| |981.82
| |
| |7/4, 16/9, 30/17
| |
| |-
| |
| |19
| |
| |classic minor seventh
| |
| |Cm7
| |
| |diminished octave
| |
| |d8
| |
| |1036.36
| |
| |20/11, 9/5
| |
| |-
| |
| |20
| |
| |classic major seventh
| |
| |cM7
| |
| |augmented sixth
| |
| |A6
| |
| |1090.91
| |
| |15/8, 32/17, 17/9, 28/15
| |
| |-
| |
| |21
| |
| |major seventh
| |
| |M7
| |
| |super major seventh / comma-flat octave
| |
| |SM7/d8
| |
| |1145.45
| |
| |33/17, 64/33, 48/25, 160/81
| |
| |-
| |
| |22
| |
| |perfect octave
| |
| |P8
| |
| |sub octave
| |
| |s8
| |
| |1200
| |
| |2/1, 63/32
| |
| |}This interval names in this table tell us what the 7-limit ratios do, that 64/63 in tempered out, meaning it is a superpythagorean tuning and that 81/80 and 25/24 are represented by a single degree. They also show us that the chromatic semitone or apotome is 3 degrees wide.
| |
| {| class="wikitable"
| |
| |+41edo
| |
| !Degrees
| |
| !Interval names
| |
| !Short-form
| |
| !Cents
| |
| !Approximated Ratios
| |
| |-
| |
| |0
| |
| |perfect unison
| |
| |P1
| |
| |0.00
| |
| |[[1/1]]
| |
| |-
| |
| |1
| |
| |comma-sharp unison/super unison
| |
| |C1/S1
| |
| |29.27
| |
| |[[81/80]], 64/63
| |
| |-
| |
| |2
| |
| |neutral unison, subminor second
| |
| |N1, sm2
| |
| |58.54
| |
| |[[25/24]], [[28/27]], [[33/32]]
| |
| |-
| |
| |3
| |
| |minor second
| |
| |m2
| |
| |87.80
| |
| |[[21/20]], [[22/21]]
| |
| |-
| |
| |4
| |
| |classic minor second, augmented unison
| |
| |Cm2, A1
| |
| |117.07
| |
| |[[16/15]], [[15/14]]
| |
| |-
| |
| |5
| |
| |neutral second
| |
| |N2
| |
| |146.34
| |
| |[[12/11]]
| |
| |-
| |
| |6
| |
| |classic major second, diminished third
| |
| |cM2, d3
| |
| |175.61
| |
| |[[10/9]], [[11/10]]
| |
| |-
| |
| |7
| |
| |major second
| |
| |M2
| |
| |204.88
| |
| |[[9/8]]
| |
| |-
| |
| |8
| |
| |super major second
| |
| |SM2
| |
| |234.15
| |
| |[[8/7]]
| |
| |-
| |
| |9
| |
| |sub minor third
| |
| |sm3
| |
| |263.41
| |
| |[[7/6]], [[32/25]]
| |
| |-
| |
| |10
| |
| |minor third
| |
| |m3
| |
| |292.68
| |
| |[[32/27]]
| |
| |-
| |
| |11
| |
| |classic minor third, augmented second
| |
| |Cm3, A2
| |
| |321.95
| |
| |[[6/5]]
| |
| |-
| |
| |12
| |
| |neutral third
| |
| |N3
| |
| |351.22
| |
| |[[11/9]], [[27/22]]
| |
| |-
| |
| |13
| |
| |classic major third, diminished fourth
| |
| |cM3, d4
| |
| |380.49
| |
| |[[5/4]]
| |
| |-
| |
| |14
| |
| |major third
| |
| |M3
| |
| |409.76
| |
| |[[14/11]], [[81/64]]
| |
| |-
| |
| |15
| |
| |super major third
| |
| |SM3
| |
| |439.02
| |
| |[[9/7]]
| |
| |-
| |
| |16
| |
| |sub fourth
| |
| |s4
| |
| |468.29
| |
| |[[21/16]]
| |
| |-
| |
| |17
| |
| |perfect fourth
| |
| |P4
| |
| |497.56
| |
| |[[4/3]]
| |
| |-
| |
| |18
| |
| |comma-sharp fourth, augmented third
| |
| |C4, A3
| |
| |526.83
| |
| |[[15/11]], [[27/20]]
| |
| |-
| |
| |19
| |
| |neutral fourth
| |
| |N4
| |
| |556.10
| |
| |[[11/8]]
| |
| |-
| |
| |20
| |
| |classic augmented fourth, diminished fifth
| |
| |cA4, d5
| |
| |585.37
| |
| |[[7/5]], 45/32
| |
| |-
| |
| |21
| |
| |classic diminished fifth, augmented fourth
| |
| |Cd5, A4
| |
| |614.63
| |
| |[[10/7]], 64/45
| |
| |-
| |
| |22
| |
| |neutral fifth
| |
| |N5
| |
| |643.90
| |
| |[[16/11]]
| |
| |-
| |
| |23
| |
| |comma-flat fifth, diminished sixth
| |
| |c5, d6
| |
| |673.17
| |
| |[[22/15]], [[40/27]]
| |
| |-
| |
| |24
| |
| |perfect fifth
| |
| |P5
| |
| |702.44
| |
| |[[3/2]]
| |
| |-
| |
| |25
| |
| |super fifth
| |
| |S5
| |
| |731.71
| |
| |[[32/21]]
| |
| |-
| |
| |26
| |
| |sub minor sixth
| |
| |sm6
| |
| |760.98
| |
| |[[14/9]], [[25/16]]
| |
| |-
| |
| |27
| |
| |minor sixth
| |
| |m6
| |
| |790.24
| |
| |[[11/7]], [[128/81]]
| |
| |-
| |
| |28
| |
| |classic minor sixth, augmented fifth
| |
| |Cm6, A5
| |
| |819.51
| |
| |[[8/5]]
| |
| |-
| |
| |29
| |
| |neutral sixth
| |
| |N6
| |
| |848.78
| |
| |[[18/11]], 44/27
| |
| |-
| |
| |30
| |
| |classic major sixth, diminished seventh
| |
| |cM6, d7
| |
| |878.05
| |
| |[[5/3]]
| |
| |-
| |
| |31
| |
| |major sixth
| |
| |M6
| |
| |907.32
| |
| |[[27/16]]
| |
| |-
| |
| |32
| |
| |super major sixth
| |
| |SM6
| |
| |936.59
| |
| |[[12/7]]
| |
| |-
| |
| |33
| |
| |sub minor seventh
| |
| |sm7
| |
| |965.85
| |
| |[[7/4]]
| |
| |-
| |
| |34
| |
| |minor seventh
| |
| |m7
| |
| |995.12
| |
| |[[16/9]]
| |
| |-
| |
| |35
| |
| |classic minor seventh, augmented sixth
| |
| |Cm7, A6
| |
| |1024.39
| |
| |[[9/5]], [[20/11]]
| |
| |-
| |
| |36
| |
| |neutral seventh
| |
| |N7
| |
| |1053.66
| |
| |[[11/6]]
| |
| |-
| |
| |37
| |
| |classic major seventh, diminished octave
| |
| |cM7, d8
| |
| |1082.93
| |
| |[[15/8]]
| |
| |-
| |
| |38
| |
| |major seventh
| |
| |M7
| |
| |1112.20
| |
| |[[40/21]], [[21/11]]
| |
| |-
| |
| |39
| |
| |neutral octave, super major seventh
| |
| |N8, SM7
| |
| |1141.46
| |
| |[[48/25]], [[27/14]], 64/33
| |
| |-
| |
| |40
| |
| |comma-flat octave/sub octave
| |
| |c8/s8
| |
| |1170.73
| |
| |[[160/81]], 63/32
| |
| |-
| |
| |41
| |
| |perfect octave
| |
| |P8
| |
| |1200
| |
| |2/1
| |
| |}
| |
| | |
| Secondary interval names are not available for every note without going into double and triple augmented and diminished intervals. I include only up to singly augmented and diminished and leave most secondary interval names out. Accordingly I do not write them in a separate column. We can see from the interval names that 64/63 and 81/80 are represented both by a single degree and the augmented unison by three, that it is a neutral tuning, and that it is a [[Schismatic]] tuning, where the diminished fourth approximates 5/4.
| |