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Tuesday, 11 Jan 2000 08:21:22 0000 
[Paul Erlich, TD 479.23]Mandelbaum and/or Würschmidt call 128:125 the "major diesis," contradicting Keenan's recent definitions. Is there an accepted convention on this? Manuel?
In his appendix to Helmholtz [in the 'Table of Intervals not exceeding one Octave'], Ellis lists 125:128 as the...
[Ellis in Helmholtz 1885, p 453]Great Diesis, the defect of 3 major Thirds from an Octave, the interval between C# and Db in the meantone temperament, 3Td = C_{2}# : D^{1}b, ex.[= exact cents] 43.831
I happen to have missed Dave's definition of intervals, but apparently they don't conform exactly to those I'm familiar with.
Dave, Paul, and anyone else who needs to discuss intervals should have available Helmholtz 1885 and Rameau 1971. Ellis and Rameau give names to intervals that have pretty much been adopted by theorists as the standard ones, at least for 'commonpractice' harmony.
[Paul Erlich, TD 477.10]I'll try to give you a sense of why meantone (and thus Vicentino's JI scheme) works for anything in the Western repertoire without enharmonic respellings, and what happens when you do have enharmonic respellings. Basically, there are only two independent commas that come into play when trying to analyze or render music of the Western conservatory tradition in just intonation. Let's take the syntonic comma and the Pythagorean comma as the two basic ones (any two would do). The syntonic comma is four fifths minus a third, so we can write it as s=4vt. The Pythagorean comma is twelve fifths; write it as p=12v. Other important commas:
1. schisma = 8vt = sp
2. diaschisma = 4v2t = 2*sp
3. (minor or Great) diesis = 3t = 3*sp
4. major diesis = 4v4t = 4*spIf you think there might be others, check out Mathieu's _Harmonic Experience_, which is a thorough survey of Western harmony through JI glasses. Mathieu does not mention the "major diesis" but he does name one other, the "superdiesis", which is 8v2t = 2*s. But basically, we've got it all covered.
Structurally, the syntonic comma results in no notational change, while the Pythagorean comma changes a flat to the enharmonic sharp (e.g., Ab>G#). (You may want to verify this for yourself). From the equations above, we can see that commas 14 above all result in a notational change opposite that of the Pythagorean comma, i.e. they change a sharp to the enharmonic flat.
[Paul's equations are illustrated on the lattice below. Note that the first one should really be schisma = 8v+t = ps , so that the skhisma also acts like the Pythagorean comma, changing a flat to the enharmonic sharp.]
power of 5 2 1 0 1 2 3 4 5 15                 14                 13                 12   Pythagorean      comma      11                 10         p         o 9         w         e 8  skhisma  super   r     diesis    7         o         f 6                 3 5                 4    syntonic  major    comma   diesis  3                 2                 1                 0   [origin]  great       diesis   1                 2                 3                 4     diaschisma  
From the 'practical' point of view of "trying to analyze or render music of the Western conservatory tradition in just intonation", Paul is probably right that these 'commas' (the smallest and largest of which are actually forms of skhisma and diesis, respectively) are the only ones that need be considered.
But strictly speaking, depending on how far one takes the powers of 3 and 5 in any given 2dimensional lattice or periodicityblock, there are all sorts of 'commas' that may come into play.
Paul is using 'comma' here in a very generic sense to mean a small interval, which in many cases will be used as a unison vector to delimit the boundaries of a periodicityblock. Wilkinson 1988 uses the term ' anomaly' as a generic word designating several different of these: skhismas, commas, and dieses, and I use it too.
I've made a lattice diagram giving the centsvalues of all the 5limit intervals between 0 and 100 cents, within the arbitrary exponentlimits of +/15 for primebase 3 and +/7 for primebase 5. Hopefully some of you will find this useful.
In addition to the 'anomalies', this lattice also indicates the smaller of the 5limit 'semitones'.
[commentary from Paul Erlich:]It can be seen easily from the lattice that these intervals, as well as some lesserknown 'commas' like 243:250 and 3072:3125, cannot made up of various combinations of the ones described by Paul.
Western triadic music prior to Beethoven requires "bridging" solely through the syntonic comma, and hence is often performed in meantone temperament. Since Beethoven, "bridging" through syntonic comma and *any* (and therefore, all) of the other 'commas' paul mentions above (in connection with Mathieu) has been a feature of western triadic music, hence the use of 12tone equal (or well) temperament. The other 'commas' can be used for bridging in other, "invented" musical systems, motivating certain corresponding tuning systems as shown in the "Equal Temperament" entry. For example, you can see from the first chart and table on that page that "bridging" through 243:250 is characteristic of porcupine temperament, through 3072:3125 of magic temperament, and through both of them (and thus also any combination of the two) of 22tone equal temperament.
The lattice also makes it easy to see the periodicity inherent in the system, as the patterns of relationship between several small intervals repeats at several places in the lattice.
Lattice of centsvalues of 5limit intervals from 0 to 100 cents with arbitrary boundaries of 3^15...15 * 5^7...7 by Joe Monzo 2000.1.11  power of 5 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 15 34  75                          14                              13                              12       23  65                   11  53  94                        10                              9              13                8       2  43  84                 7 32  73                         6                              5                              4        22  63                  3 10   51  92        p                o 2               w                e 1               r                0        0  41  82  o                f 1  30  71                        3 2 100                             3                              4         20  61                 5  8  49  90                      6                              7                              8           39  80                9   28  69                       10 98                            11                              12          18  59                 13   6  47  88                     14 77                              15              
Below is a table giving the intervals listed by Erlich and by Ellis. (Edited, and incorporating additions I made in TD 484.1, giving Rameau's intervalnames.)
~cents ratio primefactor name 3^ 5^ Ellis Rameau 92 128:135  3 1  larger limma mean semitone 90 243:256 5 0  pythagorean limma 71 24:25 1 2  small semitone minor semitone (theory), aug. unison (practice) 63 625:648  4 4  (Erlich 'major diesis') least semitone 49 243:250 5 3  major diesis 43 6400:6561  8 2  (Mathieu 'superdiesis') 41 125:128  0 3  great deisis minor diesis 30 3072:3125 1 5  small diesis 23 524288:531441 12 0  pythagorean comma 22 80:81  4 1  syntonic comma comma 20 2025:2048 4 2  diaskhisma diminished comma 2 32768:32805  8 1  skhismaNote that 5^3 [= 125:128 = ~41 cents] was called 'minor diesis' by Rameau and 'Great Diesis' by Ellis. (These two different terms were noted by Paul Erlich in the posting I quoted.) Also note that 3^4*5^4 [= 625:648 = ~63 cents] was called 'least semitone' by Rameau and 'major diesis' by Erlich (and I presume Keenan).
My lattice shows many more, the names of which I don't know. Scala probably gives them. ...? If not, perhaps I can provide some new names. Manuel?
Here are some suggestions for a logical system encompassing more 'commas' than most other systems. It classifies intervals into seven broad groups (from smallest to largest): skhisma, kleisma, comma, small diesis, great diesis, small semitone, and limma.
Each interval is qualified by a pseudoGraecoLatin term indicating the exponent of 5 and its 'tivity', positive or negative. (Is there a real mathematical term for that?)
I don't particularly like these names, but the set of these intervals in this lattice fell roughly into 11 groups (based on the gaps in the graph of their centsvalues), which I condensed into these seven groups:
Suggested terminology for 5limit intervals from 0 to 100 cents with arbitrary boundaries of 3^15...15 * 5^7...7 by Joe Monzo 2000.1.11  ~cents ratio primefactor name proposed by Monzo 3^ 5^ 100 73728:78125  2 7  superseptapental limma 98 59049:62500 10 6  superhexapental limma 94 4194304:4428675  11 2  superbipental limma 92 128:135  3 1  superpental limma (Ellis larger limma, Rameau mean semitone) 90 243:256  5 0  Pythagorean limma 88 7971615: 8388608 13  1  subpental limma 84 6250:6561  8  5  subpentapental limma 82 15625: 16384  0  6  subhexapental limma 80 512578125:536870912  8  7  subseptapental limma 77 4782969: 5000000 14 7  superseptapental small semitone 73 262144:273375  7 3  supertripental small semitone 71 24:25  1 2  superbipental small semitone (Ellis small semitone, Rameau minor semitone) 69 19683: 20480  9 1  superpental small semitone 65 512000:531441  12  3  subtripental small semitone 63 625:648  4  4  subtetrapental small semitone (Erlich major diesis, Rameau least semitone) 61 253125:262144  4  5  subpentapental small semitone 59 8303765625:8589934592 12  6  subhexapental small semitone 53 536870912:553584375  11 5  superpentapental great diesis 51 16384:16875  3 4  supertetrapental great diesis 49 243:250  5 3  supertripental great diesis (Rameau major diesis) 47 1594323:1638400 13 2  superbipental great diesis 43 6400:6561  8  2  subbipental great diesis (Mathieu superdiesis) 41 125:128  0  3  subtripental diesis (Ellis great diesis, Rameau minor diesis) 39 4100625:4194304  8  4  subtetrapental great diesis 34 1099511627776:1121008359375  15 7  superseptapental small diesis 32 33554432:34171875  7 6  superhexapental small diesis 30 3072:3125  1 5  superpentapental small diesis (Ellis small diesis) 28 19683:20000  9 4  supertetrapental small diesis 23 524288:531441  12 0  Pythagorean comma 22 80:81  4  1  subpental comma (syntonic comma, Rameau comma) 20 2025:2048  4  2  subbipental comma (Ellis diaskhisma, Rameau diminished comma) 18 66430125:67108864 12  3  subtripental comma 13 78125: 78732  9  7  subseptapental kleisma 10 2097152:2109375  3 7  superseptapental kleisma 8 15552:15625  5 6  superhexapental kleisma (Tanaka kleisma) 6 1594323:1600000 13 5  superpentapental kleisma 2 32768:32805  8 1  superpental skhisma (Ellis skhisma) 0 1:1  0 0  reference pitch Lattice of proposed names of 5limit intervals from 0 to 100 cents with arbitrary boundaries of 3^15...15 * 5^7...7 by Joe Monzo 2000.1.11  power of 5 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 15 sm diesis  sm st.                          14                              13                              12       comma  sm st.                   11  lg diesis  limma                         10                              9              kleisma                8      schisma  lg diesis  limma                  7 sm diesis  sm st.                         6                              5                              4        comma  sm st.                  3 kleisma  lg diesis  limma         p                o 2               w                e 1               r                0        1/1   lg diesis  limma  o                f 1  sm diesis  sm st.                        3 2 100                             3                              4         comma  sm st.                 5 kleisma  lg diesis  limma                       6                              7                              8           lg diesis  limma                9   sm diesis  sm st.                       10 98                            11                              12          comma  sm st.                13  kleisma  lg diesis  limma                      14 sm semi                             15               REFERENCES  Helmholtz, Hermann. 1885. _On the Sensations of Tone as a Physiological Basis for the Theory of Music_. English translation by Alexander J. Ellis, of _Die Lehre von den Tonempfindungen..._, Braunschweig, 1863. 2nd edition, conforming to 4th German edition (1877). Reprint: 1954, Dover Publications, New York. ISBN# 0486607534 (back to text) Rameau, JeanPhilippe. 1971. _Treatise on Harmony_. English translation by Philip Gossett, of _Traite' de l'harmonie_, Paris, 1722. Dover Publications, New York. ISBN# 0486224619 (back to text) Wilkinson, Scott R. 1988. _Tuning In: Microtonality in Electronic Music_. Hal Leonard Books, Milwaukee. (back to text) monz Joseph L. Monzo Philadelphia monz@juno.com http://www.ixpres.com/interval/monzo/homepage.html "...I had broken thru the lattice barrier..."   Erv Wilson  
updated:
2003.02.17  added commentary from Paul Erlich
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