Intervals

From: Tom (tom-tom@gaea.ocn.ne.jp)
Date: Thu Nov 22 2001 - 04:44:30 PST


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Adrian Freedman are you out there or is there anyone who knows Adrian and
where he is out there? Adrian explained all about these intervals one day
while here and it made perfect sense then but the theory is largely
forgotten now from lack of use. So...onward...!

>From THE BOOK...with digressions

[When two notes have different pitch numbers (pitch number means the
frequency, such as 440 cycles per second), there is said to be an interval
between them. This gives rise to a sensation, very differently appreciated
by different individuals but in all cases the interval is measured by the
ratio of the pitch numbers, and, for some purposes, more conveniently by
other numbers called cents, derived from these ratios, as explained in App.
XX. sect C. The names of all the intervals usually distinguished are also
given in App. XX. sect. D., with the corresponding ratios and cents. These
names were in the first place derived from the ordinal number of the note in
the scales, or successions of continually sharper notes. The Octave is the
eighth note in the major scale. An octave is a set of notes lying within an
Octave. Observe that in this translation all names of intervals commence
with a capital letter, to prevent ambiguity, as almost all such words are
also used in other senses.--Translator.]

Appendix XX. section D. gives four pages of intervals in fine print,
probably 200 or more, most of which are not relevant here, for example, the
interval of 19:20 is that from the open string to the second fret on the
Tambour of Bagdad. Hmmm...interesting...? Here are some more familiar
ones...or are they?...in ascending order of width, hopefully for ease.

from the Table of Intervals not exceeding one Octave.

1:1 Fundamental note of the open string, assumed as C 66
1730:1731 Cent, hundredth of an equal Semitone, nearest approximate ratio
exactly 1:1.005755 (what does "nearest approximate ratio exactly" mean?)
5:6 Just minor Third
4:5 Just major Third
7:9 Septimal or supermajor Third
3:4 Just and Pythagorean Fourth
227:303 Equal Fourth, C : F, exact ratio 1:12th root of 2 to the 5th power
= 1:1.3348
289:443 Equal Fifth, C : G
2:3 Just and Pythagorean Fifth
63:100 Equal superfluous Fifth, C : G sharp, and also equal minor Sixth, C
: A flat, the same notes written differently
5:8 Just minor Sixth
22:37 Equal major Sixth, C : A
5:9 Acute minor seventh, used in descending minor scales
89:168 Equal major Seventh
1:2 The Octave

Someplace else it is written that the width of an interval is measured by
the number of cents it contains. Wow! This must mean that the interval Ro
: Re is 500 cents wide, sensible since there are five notes to finger up
from Ro to Tsu meri to Tsu chumeri to Tsu to Re meri to Re with each space
between notes containing 100 cents. Hmmm...OK.

Interval Widths

100 cents wide is exactly 100.0999, the nearest approximate in small numbers
to the ratio of the interval of an equal Semitone, exact ratio 1:1.059461,
approximate ratio 84:89
200 cents wide, an equal Tone, exact ratio 1:6th root of 2 = 1.22462
300 cents, the equal minor Third, A : C, ratio 37.44
400 cents, equal major Third, exact ratio 1:3rd root of 2 = 1:1.2599210,
approximate ratio 50:63
500 cents, equal Fourth, C : F or Ri to Tsu on a hassun, exact ratio 1:12th
root of 2 to the 5th = 1:1.3348, approximate ratio 227:303 (On the Just and
Pythagorean scales a Fourth is the ratio of 3:4 and is 498 cents wide.
600 cents, equal Tritone (Wha'tsa tritone???), Ri chumeri : Tsu kan,
(Careful in here, son, it's easy to get lost! Help! Someone please
check!), exact ratio 1:2nd root of 2 = 1.1412, approximate ratio 99:140
700 cents, equal Fifth, Ri otsu : Re kan, approximate ratio 289:433
800 cents, 63:100 equal superfluous Fifth, C : G sharp, and also equal minor
Sixth, C : A flat, the same notes written differently
900 cents, equal major Sixth, Ri : Chi kan, and also equal diminished
Seventh, C : B flat-flat = A : G flat, exact ratio 1: 4th root of 8 =
1.6818, approximate ratio 22:37
1000 cents, equal superfluous, or extreme sharp Sixth, Ri : Hi meri, or
minor Seventh, C : B flat, approximate ratio 55:98
1100 cents, equal major Seventh, ratio 89:168
1200 cents, the Octave, 1:2

Whew! Cant we adopt some sort of convention here like Re means otsu and re
means kan or whatever?

Now the trick is to get all this into shakuhachi lingo so we can talk at
ease about, say, the fingering possibilities for beginning and ending the
interval Chi meri kan Dokyoku : Hi, and so on for any length of flute. A
snap since chi meri (chimeri...?) is the same on all shakuhachi. However,
bear in mind that Hochiku have, in fact, heard of all this several times
before from various sources and don't believe a word of it.

Cheers!

Tom

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<HTML>
<HEAD>
<TITLE>Intervals</TITLE>
</HEAD>
<BODY>
Adrian Freedman are you out there or is there anyone who knows Adrian and w=
here he is out there? &nbsp;Adrian explained all about these intervals one d=
ay while here and it made perfect sense then but the theory is largely forgo=
tten now from lack of use. &nbsp;So...onward...!<BR>
<BR>
From THE BOOK...with digressions<BR>
<BR>
[When two notes have different pitch numbers (pitch number means the freque=
ncy, such as 440 cycles per second), there is said to be an interval between=
 them. &nbsp;This gives rise to a sensation, very differently appreciated by=
 different individuals but in all cases the <I>interval is measured by the r=
atio of the pitch numbers,</I> and, for some purposes, more conveniently by =
other numbers called <I>cents</I>, derived from these ratios, as explained i=
n App. XX. sect C. &nbsp;The names of all the intervals usually distinguishe=
d are also given in App. XX. sect. D., with the corresponding ratios and cen=
ts. &nbsp;These names were in the first place derived from the ordinal numbe=
r of the note in the scales, or successions of continually sharper notes. &n=
bsp;The Octave is the eighth note in the major scale. &nbsp;An octave is a s=
et of notes lying within an Octave. &nbsp;Observe that <I>in this translatio=
n</I> all names of intervals commence with a capital letter, to prevent ambi=
guity, as almost all such words are also used in other senses.--<I>Translato=
r.</I>]<BR>
<BR>
Appendix XX. section D. gives four pages of intervals in fine print, probab=
ly 200 or more, most of which are not relevant here, for example, the interv=
al of 19:20 is that from the open string to the second fret on the Tambour o=
f Bagdad. &nbsp;Hmmm...interesting...? &nbsp;Here are some more familiar one=
s...or are they?...in ascending order of width, hopefully for ease.<BR>
<BR>
<FONT SIZE=3D"5">from the Table of Intervals not exceeding one Octave</FONT>.=
<BR>
<BR>
1:1 &nbsp;Fundamental note of the open string, assumed as <I>C 66<BR>
</I>1730:1731 &nbsp;Cent, hundredth of an equal Semitone, nearest approxima=
te ratio exactly 1:1.005755 (what does &quot;nearest approximate ratio exact=
ly&quot; mean?)<BR>
5:6 &nbsp;Just minor Third<BR>
4:5 &nbsp;Just major Third<BR>
7:9 &nbsp;Septimal or supermajor Third<BR>
3:4 &nbsp;Just and Pythagorean Fourth<BR>
227:303 &nbsp;Equal Fourth, <I>C : F</I>, exact ratio 1:12th root of 2 to t=
he 5th power =3D 1:1.3348<BR>
289:443 &nbsp;Equal Fifth, <I>C : G<BR>
</I>2:3 &nbsp;Just and Pythagorean Fifth<BR>
63:100 &nbsp;Equal superfluous Fifth, <I>C : G sharp</I>, and also equal mi=
nor Sixth, <I>C : A flat</I>, the same notes written differently<BR>
5:8 &nbsp;Just minor Sixth<BR>
22:37 &nbsp;Equal major Sixth, <I>C : A<BR>
</I>5:9 &nbsp;Acute minor seventh, used in descending minor scales<BR>
89:168 &nbsp;Equal major Seventh<BR>
1:2 The Octave<BR>
<BR>
Someplace else it is written that the <I>width</I> of an interval is measur=
ed by the number of cents it contains. &nbsp;Wow! &nbsp;This must mean that =
the interval Ro : Re is 500 cents wide, sensible since there are five notes =
to finger up from Ro to Tsu meri to Tsu chumeri to Tsu to Re meri to Re with=
 each space between notes containing 100 cents. &nbsp;Hmmm...OK.<BR>
<BR>
Interval Widths <BR>
<BR>
100 cents wide is exactly 100.0999, the nearest approximate in small number=
s to the ratio of the interval of an equal Semitone, exact ratio 1:1.059461,=
 approximate ratio 84:89<BR>
200 cents wide, an equal Tone, exact ratio 1:6th root of 2 =3D 1.22462<BR>
300 cents, the equal minor Third, <I>A : C</I>, ratio 37.44<BR>
400 cents, equal major Third, exact ratio 1:3rd root of 2 =3D 1:1.2599210, ap=
proximate ratio 50:63<BR>
500 cents, equal Fourth, <I>C : F</I> or Ri to Tsu on a hassun, exact ratio=
 1:12th root of 2 to the 5th =3D 1:1.3348, approximate ratio 227:303 &nbsp;(On=
 the Just and Pythagorean scales a Fourth is the ratio of 3:4 and is 498 cen=
ts wide.<BR>
600 cents, equal Tritone (Wha'tsa tritone???), Ri chumeri : Tsu kan, (Caref=
ul in here, son, it's easy to get lost! &nbsp;Help! &nbsp;Someone please che=
ck!), exact ratio 1:2nd root of 2 =3D 1.1412, approximate ratio 99:140<BR>
700 cents, equal Fifth, Ri otsu : Re kan, &nbsp;approximate ratio 289:433<B=
R>
800 cents, 63:100 equal superfluous Fifth, <I>C : G sharp</I>, and also equ=
al minor Sixth, <I>C : A flat</I>, the same notes written differently<BR>
900 cents, equal major Sixth, Ri : Chi kan, and also equal diminished Seven=
th, <I>C : B flat-flat =3D A : G flat</I>, exact ratio 1: 4th root of 8 =3D 1.68=
18, approximate ratio 22:37<BR>
1000 cents, equal superfluous, or extreme sharp Sixth, Ri : Hi meri, or min=
or Seventh, <I>C : B flat</I>, approximate ratio 55:98<BR>
1100 cents, equal major Seventh, ratio 89:168<BR>
1200 cents, the Octave, 1:2<BR>
<BR>
Whew! &nbsp;Cant we adopt some sort of convention here like Re means otsu a=
nd re means kan or whatever?<BR>
<BR>
Now the trick is to get <U>all</U> this into shakuhachi lingo so we can tal=
k at ease about, say, &nbsp;the fingering possibilities for beginning and en=
ding the interval Chi meri kan Dokyoku : Hi, and so on for any length of flu=
te. &nbsp;A snap since chi meri (chimeri...?) is the same on all shakuhachi.=
 &nbsp;However, bear in mind that Hochiku have, in fact, heard of all this s=
everal times before from various sources and don't believe a word of it.<BR>
<BR>
Cheers! <BR>
<BR>
Tom=20
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