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=== Helmholtz and Ellis ===
=== Helmholtz and Ellis ===
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 9:16 seemingly because of the 9 partial limit imposed on the table. It is also worth noting that 5:7 is labelled a subminor Fifth. 'Super', indicated in notation with a '+', raises an interval by 35:36, the septimal quarter tone, and 'sub', indicated with a '-' lowers by the same interval with the exception of the Supersecond 7:8, which lies 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 25:25, 80:81 above Smith's 64:45 minor Fifth. 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.
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 9:16 seemingly because of the 9 partial limit imposed on the table. It is also worth noting that 5:7 is labelled a subminor Fifth. 'Super', indicated in notation with a '+', raises an interval by 35:36, the septimal quarter tone, and 'sub', indicated with a '-' lowers by the same interval with the exception of the Supersecond 7:8, which lies 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 18:25, 80:81 above Smith's 45:64 minor Fifth, however in table 2 below, Ellis labels 18:25 a ''superfluous Fourth'', and it's inverse, 25:32, an ''acute diminished Fifth'', whilst 64:45 is labelled a ''diminished Fifth'' and its inverse 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.
{| class="wikitable"
{| class="wikitable"
|+Additional Intervals
|+Table 1. Additional Intervals
!Intervals
!Intervals
!Notation
!Notation
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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 182c. 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.
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 182c. 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.


The following table added by Ellis lists Pythagorean, 5-limit and tempered intervals, with Ellis' names also included.
The following table added by Ellis lists 5-limit perfect just, imperfect and tempered intervals, with Ellis' names also included.
{| class="wikitable"
{| class="wikitable"
|+
|+
Table 2.
!Intervals
!Intervals
!Ellis' Interval Names
!Ellis' Interval Names
Line 113: Line 114:
|Unison
|Unison
|Unison
|Unison
|c:c
|
|
|
|1:1
|
|0
|
|-
|-
|Minor Second
|Minor Second
|Tempered minor Second
|Tempered minor Second
|<nowiki>||c:c#</nowiki>
|
|
|
|
|
|100
|
|-
|-
|Minor Seocnd
|Minor Seocnd
|Just minor Second
|Just minor Second
|b<sub>1</sub>:c'
|
|
|
|15:16
|
|112
|
|-
|-
|Major Second
|Major Second
|Minor Tone
|Minor Tone
|d:e<sub>1</sub>
|
|
|
|9:10
|
|182
|
|-
|-
|Major Second
|Major Second
|Tempered major Second or whole Tone
|Tempered major Second or whole Tone
|<nowiki>||c:d</nowiki>
|
|
|
|
|
|200
|
|-
|-
|Major Second
|Major Second
|Major Tone
|Major Tone
|c:d
|
|
|
|8:9
|
|204
|
|-
|-
|Major Second
|Major Second
Line 157: Line 158:
|
|
|
|
|
|225:256
|
|224
|-
|-
|Minor Third
|Minor Third
Line 164: Line 165:
|
|
|
|
|
|64:75
|
|274
|-
|-
|Minor Third
|Minor Third
Line 171: Line 172:
|
|
|
|
|
|27:32
|
|294
|-
|-
|Minor Third
|Minor Third
Line 179: Line 180:
|
|
|
|
|
|300
|-
|-
|Minor Third
|Minor Third
Line 185: Line 186:
|
|
|
|
|
|5:6
|
|316
|-
|-
|Major Third
|Major Third
Line 192: Line 193:
|
|
|
|
|
|4:5
|
|386
|-
|-
|Major Third
|Major Third
Line 200: Line 201:
|
|
|
|
|
|400
|-
|-
|Major Third
|Major Third
Line 206: Line 207:
|
|
|
|
|
|25:32
|
|428
|-
|-
|Fourth
|Fourth
Line 213: Line 214:
|
|
|
|
|
|3:4
|
|498
|-
|-
|Fourth
|Fourth
Line 221: Line 222:
|
|
|
|
|
|500
|-
|-
|Fourth
|Fourth
Line 227: Line 228:
|
|
|
|
|
|20:27
|
|520
|-
|-
|Sharp Fourth or Flat Fifth
|Sharp Fourth or Flat Fifth
Line 234: Line 235:
|
|
|
|
|
|18:25
|
|568
|-
|-
|Sharp Fourth or Flat Fifth
|Sharp Fourth or Flat Fifth
Line 241: Line 242:
|
|
|
|
|
|32:45
|
|590
|-
|-
|Sharp Fourth or Flat Fifth
|Sharp Fourth or Flat Fifth
Line 249: Line 250:
|
|
|
|
|
|600
|-
|-
|Sharp Fourth or Flat Fifth
|Sharp Fourth or Flat Fifth
Line 255: Line 256:
|
|
|
|
|
|45:64
|
|610
|-
|-
|Sharp Fourth or Flat Fifth
|Sharp Fourth or Flat Fifth
Line 262: Line 263:
|
|
|
|
|
|25:36
|
|632
|-
|-
|Fifth
|Fifth
Line 269: Line 270:
|
|
|
|
|
|27:40
|
|680
|-
|-
|Fifth
|Fifth
Line 277: Line 278:
|
|
|
|
|
|700
|-
|-
|Fifth
|Fifth
Line 283: Line 284:
|
|
|
|
|
|2:3
|
|702
|-
|-
|Minor Sixth
|Minor Sixth
Line 290: Line 291:
|
|
|
|
|
|16:25
|
|772
|-
|-
|Minor Sixth
|Minor Sixth
Line 298: Line 299:
|
|
|
|
|
|800
|-
|-
|Minor Sixth
|Minor Sixth
Line 304: Line 305:
|
|
|
|
|
|5:8
|
|814
|-
|-
|Major Sixth
|Major Sixth
Line 311: Line 312:
|
|
|
|
|
|3:5
|
|884
|-
|-
|Major Sixth
|Major Sixth
Line 319: Line 320:
|
|
|
|
|
|900
|-
|-
|Major Sixth
|Major Sixth
Line 325: Line 326:
|
|
|
|
|
|16:27
|
|906
|-
|-
|Major Sixth
|Major Sixth
Line 332: Line 333:
|
|
|
|
|
|75:128
|
|926
|-
|-
|Major Sixth
|Major Sixth
Line 339: Line 340:
|
|
|
|
|
|128:225
|
|976
|-
|-
|Minor Seventh
|Minor Seventh
Line 346: Line 347:
|
|
|
|
|
|9:16
|
|996
|-
|-
|Minor Seventh
|Minor Seventh
Line 354: Line 355:
|
|
|
|
|
|1000
|-
|-
|Minor Seventh
|Minor Seventh
Line 360: Line 361:
|
|
|
|
|
|5:9
|
|1018
|-
|-
|Major Seventh
|Major Seventh
Line 367: Line 368:
|
|
|
|
|
|8:15
|
|1088
|-
|-
|Major Seventh
|Major Seventh
Line 375: Line 376:
|
|
|
|
|
|1100
|-
|-
|Octave
|Octave
Line 381: Line 382:
|
|
|
|
|
|1:2
|
|1200
|}
|}
The interval names in this table 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.
In ''Notes on the Observations of Musical Beats'', Proc. R. Soc. Lond. 1880, Ellis named many 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, Sub-minor Sixth 14:9, Sub-minor Third 7:6, Super-major Sixth 12:7, Sub-minor or Harmonic Seventh 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.  
In ''Notes on the Observations of Musical Beats'', Proc. R. Soc. Lond. 1880, Ellis named many 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, Sub-minor Sixth 14:9, Sub-minor Third 7:6, Super-major Sixth 12:7, Sub-minor or Harmonic Seventh 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.  


Line 390: Line 393:
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 instead of by 36:35.  
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 instead of by 36:35.  


Ellis, in a footnote to his translation also provides names for 11-limit intervals. The interval 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 11:18, and guess it's name to be a ''neutral Sixth''. 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:
Ellis, in a footnote to his translation of Helmholtz,'s treatise also provides names for a single 11-limit interval. The interval 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 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:
{| class="wikitable"
{| class="wikitable"
|+11-limit intervals
|+Table 3. 11-limit intervals
!Intervals
!Intervals
!Notation
!Notation
Line 447: Line 450:
* A perfect interval, when raised a semitone is labelled 'augmented', and when lowered a semitone, 'diminished'.
* A perfect interval, when raised a semitone is labelled 'augmented', and when lowered a semitone, 'diminished'.


=== Extended Meantone ===   
=== Dave Keenan's Interval Naming Systems ===   
 
From the 11-limit otonal chord 4:5:6:7:9:11, in addition to the familiar major, minor, subminor, supermajor and neutral thirds, Dave Kennan labelled 5:7 a sub-diminished fifth and 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.


Super and sub were further generalised by some theorists and musicians such that an interval a bit smaller than a major is referred to as a ''subminor third'', and an interval a bit larger than a minor third as a ''supraminor third''.  
Super and sub were further generalised by some theorists and musicians such that an interval a bit smaller than a major is referred to as a ''subminor third'', and an interval a bit larger than a minor third as a ''supraminor third''.  


Notice that 'supra' is used instead of 'super', but 'sub' is still used. Finally, in limited use are 'intermediates', where an interval in-between a major 3rd and a perfect fourth, for example, is referred to as a 'third-fourth'. There are undoubtedly other interval naming practice that exists that are not as well known to the author, which are likely to be less commonly used. Although all these microtonal interval naming concepts are in common use, there is not yet a complete system that defined them, only complete systems that depart from them.
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):
{| class="wikitable"
|+
!Index
!Prefix for unisons, fourths, fifths, octaves
!Prefix for seconds, thirds, sixths, sevenths, ninths
|-
| -4
|double diminished
|subdiminished
|-
| -3
|subdiminished
|diminished
|-
| -2
|diminished
|subminor
|-
| -1
|sub
|minor
|-
|0
|(perfect)
|neutral
|-
|1
|super
|(major)
|-
|2
|augmented
|supermajor
|-
|3
|superaugmented
|augmented
|-
|4
|double augmented
|superaugmented
|}


=== [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] ===
=== [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] ===
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