5-limit intervals, 100 cents and under

> [Paul Erlich, TD 479.23]
> Mandelbaum and/or W�rschmidt call 128:125 the "major diesis,"
> contradicting Keenan's recent definitions. Is there an accepted
> convention on this? Manuel?

In his appendix to Helmholtz, Ellis lists [in the 'Table
of Intervals not exceeding one Octave'] 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(sub 2)# : D(super 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
'common-practice' 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=4v-t. The Pythagorean comma is twelve fifths;
> write it as p=12v. Other important commas:
>
> 1. schisma = -8v-t = s-p
> 2. diaschisma = -4v-2t = 2*s-p
> 3. (minor or Great) diesis = -3t = 3*s-p
> 4. major diesis = 4v-4t = 4*s-p
>
> If 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 8v-2t = 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
> 1-4 above all result in a notational change opposite that of
> the Pythagorean comma, i.e. they change a sharp to the
> enharmonic flat.

From a 'practical' point of view, Paul is probably right that
these 'commas' (the smallest and largerst 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 2-dimensional lattice or
periodicity-block, 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 periodicity-block.
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 cents-values of all
the 5-limit intervals between 0 and 100 cents, within the
arbitrary exponent-limits of +/-15 for prime-base 3 and
+/-7 for prime-base 5. Hopefully some of you will find this
useful.

In addition to the 'anomalies', this lattice also indicates
the smaller of the 5-limit 'semitones'. It can be seen easily
from the lattice that all the intervals are made up of various
combinations of the ones described by Paul, and indeed, all
of them can be described in terms of combinations of the
syntonic and Pythagorean commas, or the syntonic comma and
skhisma. 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.

(Use Courier New font, or another fixed-width font. If it
doesn't form a perfect grid in your browser, then that's
because my or your email software added unintended line-feeds.
Simply copy and paste the lattice to a text file and remove
the superfluous line-feeds to correct the view. Use a font-size
that is small enough to make it a perfect grid, or turn off word
wrap.)

Lattice of cents-values of 5-limit 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.

~cents ratio prime-factor name

2 32768:32805 3^ 8 * 5^ 1 skhisma
20 2025:2048 3^-4 * 5^-2 diaskhisma
22 80:81 3^ 4 * 5^-1 Syntonic Comma
23 524288:531441 3^12 Pythagorean Comma
30 3072:3125 3^-1 * 5^ 5 Ellis's 'Small Diesis'
41 125:128 5^-3 Ellis's 'Great Deisis'
43 6400:6561 3^ 8 * 5^-2 Mathieu's 'superdiesis'
63 625:648 3^ 4 * 5^-4 'major diesis' (Erlich's #4)
71 24:25 3^-1 * 5^ 2 small semitone
90 243:256 3^-5 Pythagorean Limma
92 128:135 3^ 3 * 5^ 1 Larger Limma

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: skhisma, kleisma, comma,
small diesis, great diesis, small semitone, and limma (from
smallest to largest).

Each interval is qualified by a pseudo-Graeco-Latin 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 cents-values),
which I condensed into these seven groups, to try to retain
familiar names without introducing new ones. I then devised
a logical system of qualification to accomodate the great
variety. The stem '-pental' immediately betrays the
5-limit dimensionality of these intervals, 'Pythagorean'
designating the ones that don't include 5 as a factor.

Suggested terminology for 5-limit intervals from 0 to 100 cents
with arbitrary boundaries of 3^-15...15 * 5^-7...7
by Joe Monzo 2000.1.11
----------------------------------------------------------------

~cents prime-factor name

100 3^- 2 * 5^ 7 super-septapental limma
98 3^-10 * 5^ 6 super-hexapental limma
94 3^ 11 * 5^ 2 super-bipental limma
92 3^ 3 * 5^ 1 super-pental limma (Ellis larger limma)
90 3^- 5 Pythagorean limma
88 3^-13 * 5^- 1 sub-pental limma
84 3^ 8 * 5^- 5 sub-pentapental limma
82 5^- 6 sub-hexapental limma
80 3^- 8 * 5^- 7 sub-septapental limma
77 3^-14 * 5^ 7 super-septapental small semitone
73 3^ 7 * 5^ 3 super-tripental small semitone
71 3^- 1 * 5^ 2 super-bipental small semitone (Ellis small semitone)

69 3^- 9 * 5^ 1 super-pental small semitone
65 3^ 12 * 5^- 3 sub-tripental small semitone
63 3^ 4 * 5^- 4 sub-tetrapental small semitone (Erlich major diesis)
61 3^- 4 * 5^- 5 sub-pentapental small semitone
59 3^-12 * 5^- 6 sub-hexapental small semitone
53 3^ 11 * 5^ 5 super-pentapental great diesis
51 3^ 3 * 5^ 4 super-tetrapental great diesis
49 3^- 5 * 5^ 3 super-tripental great diesis
47 3^-13 * 5^ 2 super-bipental great diesis
43 3^ 8 * 5^- 2 sub-bipental great diesis (Mathieu superdiesis)
41 5^- 3 sub-tripental diesis (Ellis great diesis)
39 3^- 8 * 5^- 4 sub-tetrapental great diesis
34 3^ 15 * 5^ 7 super-septapental small diesis
32 3^ 7 * 5^ 6 super-hexapental small diesis
30 3^- 1 * 5^ 5 super-pentapental small diesis (Ellis small diesis)
28 3^- 9 * 5^ 4 super-tetrapental small diesis
23 3^ 12 Pythagorean comma
22 3^ 4 * 5^- 1 sub-pental comma (syntonic comma)
20 3^- 4 * 5^- 2 sub-bipental comma (Ellis diaskhisma)
18 3^-12 * 5^- 3 sub-tripental comma
13 3^ 9 * 5^- 7 sub-septapental kleisma
10 3^ 3 * 5^ 7 super-septapental kleisma
8 3^- 5 * 5^ 6 super-hexapental kleisma (Tanaka kleisma)
6 3^-13 * 5^ 5 super-pentapental kleisma
2 3^ 8 * 5^ 1 super-pental skhisma (Ellis skhisma)
0 n^0 reference pitch

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# 0-486-60753-4

Rameau, Jean-Philippe. 1971.
_Treatise on Harmony_.
English translation by Philip Gossett, of
_Traite' de l'harmonie_, Paris, 1722.
Dover Publications, New York.
ISBN# 0-486-22461-9

Wilkinson, Scott R. 1988.
_Tuning In: Microtonality in Electronic Music_.
Hal Leonard Books, Milwaukee.

-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 |
--------------------------------------------------

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