Yahoo Tuning Groups Ultimate Backup tuning small 5-limit intervals (anomalies)

> [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 smaller and larger 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 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.)

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. 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?)

~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

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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small 5-limit intervals (anomalies)

🔗Joe Monzo <monz@xxxx.xxxx>

🔗David C Keenan <d.keenan@xx.xxx.xxx>