qin tunings

8/9/2019 Qin Tunings

1/15

Qin Tunings, Some

Theoretical Concepts 1 With Special Reference to Shen Qi Mi Pu

Introduction2

This article covers some theoretical concepts and mathematical calculations. For basic

qin tuning none of this is necessary. One just follows instructions such as those given

under Tuning a Qin. However, it is also interesting to consider the theoretical

possibilities of various systems of musical tuning (temperaments).

The three most commen tuning systems are

!. "ythagorean tuning ( sanfen sunyi: cycle of fifth tuning# compare meantone

tuning)#

$. %ust intonation tuning#

&. 'ual temperament tuning.

"ythagorean tuning was clearly the standard for tuning the qin, but it seems most liely

that notes actually played did not strictly follow this system if nothing else, the

common tendency to bend notes is evidence for this. *eanwhile, just intonation tuning

was also clearly used, most specifically in harmonics played at the &rd, +th, th or !!th

hui, something generally avoided in pieces developed during the -ing dynasty. s for

eual temperament, although it seems liely that it was first theori/ed by 0hu 0aiyu, a

prince who lived !1&+ 2 ca. !+!3, there is no evidence for its use in pre2modern 4hinese

music, let alone qin music. 5pecific mathematical differences between these three are

discussed below in 4omparing different tuning systems# the systems primarily

considered for qin tuning, "ythagorean and just intonation, are compared in a footnote.

Qin music today is largely pentatonic. nd although SQMP is flavored with many other

notes, its music still remains pentatonic at its core. 6estern and 4hinese pentatonic

scales are basically the same, as they are both the first five notes generated by going

through the cycle of fifths (do so re la mi)

Table 1: Pentatonic scale

!. 7umbers 5 ! $ & 1 + !8 $8 &8 18 +8 !9

$. 7ames 0hi :u gong shang jiao /hi yu gong8 shang8 jiao8 /hi8 yu8 gong9

&. 5olfeggio 5o ;a do re mi so la do8 re8 mi8 so8 la8 do9


2/15

varies, so my own transcriptions into staff notation maintain this sense of relativity.

Thus, although on my recording Music Beyond Sound the first string is consistently

tuned to about 1 vib>sec (appro?imately the 6estern @ two octaves below middle c),

this is variously considered as Ao, Be and 5o (depending on the mode), and my

transcriptions notate it as 4, A or =. 5ince the range of the qin is more than four

octaves, maing the lowest note two octaves below c this the only way to include all thenotes without the addition of a lot of octave2change marings. nd writing the music

with c meaning do (rather than absolute pitch) means there are fewer accidentals in the

transcription than there would be from any other method.

The qin (see illustration under Tuning a Qin) has seven strings, with the standard tuning

being 5 ! $ & 1 + , often played as 1 2 ! 5 ! $ . The first string, with the lowest pitch,

is the one furthest from the player# with sil strings it is often tuned to slightly above

C 11 vib>sec. Thus the note names in the charts and transcriptions, it must again be

emphasi/ed, indicate relative pitch only. The actual tuning depends on such variables as

the si/e of the instrument and the conseuent string length (varying from about


3/15

1$ Comparing different tuning s%stems

The basic tuning systems are "ythagorean, just intonation and eual temperament. s a

preface to the following comments, here is a comparison of these three standard note

relationships in terms of vibrations as well as fractions.

&$ P%thagorean note relationships

These relationships can now be used to generate notes according to what is

usually called in 'nglish the "ythagorean system, which derives notes based on

the cycle of fifths the fifths interval results from the ratio of three over two, and

the octave interval from two over one.

The cycle of fifths derives notes as follows (tae x vib>sec C ! , then multiply by

&>$)

! (do) 22 &>$ (so) 22 >< (re8) 22 $> (la8) 22 !>!+ (mi9)Jringing these within one octave by halving them whenever necessary changes

this to

! (do) 22 &>$ (so) 22 > (re) 22 $>!+ (la) 22 !>+< (mi)

Ordering these in ascending seuence, gives

! (do) 22 > (re) 22 !>+< (mi) 22 &>$ (so) 22 $>!+ (la)

To e?tend the system downward two notes, divide sol and la by !>$, thus

lowering them each an octave. This gives the following seven note seuence,

which is the standard qin tuning

&>< (5o) 22 $>&$ (;a) 22 ! (do) 22 > (re) 22 !>+< (mi) 22 &>$ (so) 22

$>!+ (la)

Gf do8 is &$3 vibrations per second (6estern d8@), the figures for this seuence

are

$


4/15

$& (;a) 22 &$3 (do) 22 &+3 (re) 22


5/15

2 note for line 1 !3! is actually !3!.$1# 13+ is 13+.$1# and &3< is &3&.1

5tring>pitch>fraction of do open !& !$ !! !3 / + 1 <

!. (5o D5E) C &>< (+


6/15

by the ninth position on the fifth string D&3&.1 vib>secE). lthough Table < has the first

string as 1 ( sol ), since Yi Lan is in shang mode the first string is considered as ! (do),

and this is the main note in this mode. Thus &33H/ is a just intonation third over the

main note, the latter is a "ythagorean third over the main note. How does one account

for thisK

6hen G first encountered such dissonances in my reconstructions G tried to see if they

could be avoided by re2tuning the qin. The problem was that removing a dissonance in

one place always brought a different one elsewhere. n account of such efforts is in a

section called "roblems with just intonation tuning, with especial reference to SQMP .

To sum up what is in the lined article, G have not yet found any qin pieces, e?cept

perhaps very short ones, in which G can use a form of just intonation tuning to avoid

dissonances caused by harmonics played at the just intonation studs. This is not to say

that such tuning was never used, only that G haven8t found that in any one piece it

succeeds in maing all the sorts of harmonic note pairs described above (plus open

strings) match. *y own feeling is that, after in some cases maing minor tuningadjustments so that clashes are minimi/ed, these dissonances are really an interesting

coloration of the sound

!$ Indicating the pitch of stopped sounds: comparing the old and ne3 s%stems

The present decimal system (see row 7ew below), in use since the !th century (see its

origins), can indicate finger positions with mathematical precision .+ and . mean

respectively +>!3 and >!3 of the distance between the th and th positions (hui). Gn

theory this system can be used to indicate very precise tonal differences, e.g., !3. for a

slightly sharpened '.

5ince there is no indication in the literature or the tablature itself that this sort of

precision was ever needed, the system used in SQMP (row Old) was in theory just as

precise as the contemporary one. Thus the decimal position .+ was in SQMP almost

always written , meaning 9the correct place between the th and th positions9# .

was rounded off to # and .& was written 2. 6hen SQMP deviated from this system,

using values found in row lt., it was usually on slides (see e?planation below the chart,

since G could not fit all the various lt figures within that one row).

Other early handboos may be less precise.

Gn particular, Zheyin Shizi Qinpu, althoughin pieces copied from SQMP it also follows that system, in pieces not in SQMP it

normally used the values in row lt., which can lead to much confusion. Thus, in

pieces, or versions of pieces, occurring for the first time in Zheyin Shizi Qinpu, .+ may

be written as , but it is more liely to be L (above ), !>$, or even 2 (below ),

with perhaps all three occurring on the same page.

Table 5: Standard positions on a qin string tuned to C (items mared M have

comments in the notes)

4 4@ A A@ ' F F@ = =@ @ J c c@ d d@ e f

7ew 3 (!&.M) (!&.1M) waiM !$.& !3. !3 .< .1 . .+ .& +. +.< +.$ 1. 1.+

http://www.silkqin.com/08anal/intn.htmhttp://www.silkqin.com/03qobj/hui.htm#changeshttp://www.silkqin.com/03qobj/hui.htm#decorfnhttp://www.silkqin.com/02qnpu/09zysz.htmhttp://www.silkqin.com/02qnpu/09zysz.htmhttp://www.silkqin.com/08anal/intn.htmhttp://www.silkqin.com/03qobj/hui.htm#changeshttp://www.silkqin.com/03qobj/hui.htm#decorfnhttp://www.silkqin.com/02qnpu/09zysz.htm


7/15

Old 3 (waiM) waiM waiM !$ !! !32!3

2 2 2 L +2 +2 + 12+

lt !&M 2M 2M 2M 2 +2 +2M + +L 12M

2N f@ g g@ a a@ b c8 c8@ d8 d8@ e8 f8 f8@ g8 g8@ a8 a8@ b8 c9

2N 1.& 1


8/15

ssuming this table refers to the &rd string, tuned to 4, and that the


9/15

5ome qin music originates with a qin player him2 or herself, some was adapted from

melodies already in e?istence. 5omeone adapting (or creating) a melody might find that

only a tuning which includes some dissonances would allow the melody to be set to the

qin. 'fforts might then be made at the sorts of re2tuning described above 22 with the

same problems 22 but the results published anyway. To understand this fully one should

see if a melody lie Yilan can 9improved9 by playing it in a way which avoidsharmonics at the just intonation studs.

$ Standard and nonstandard qin tuning

95tandard9 tuning is 5 ! $ & 1 + (also considered as 1 2 ! 5 ! $ ) regardless of

whether it is "ythagorean or just intonation. 97on2standard9 tunings are derived by

altering the tuning of one or more strings by a half tone, e?cept in manshang mode,

which lowers the second string a whole tone.

Today almost all qin pieces use standard tuning, the main e?ceptions being twomelodies that use ruiin tuning, in which the fifth string is raised a half tone. Gn

constrast, early qin music had a great variety of different tunings. For e?ample, Shen Qi

Mi Pu (!


10/15

*y relevant assumption here has been that tuning followed the "ythagorean system

with perhaps slight adjustments if an interval played at the just intonation studs was too

harsh. G appreciate most of these sounds for their special color, though G also thin they

sometimes may have resulted from music being transferred from other instruments

without regard to the resulting dissonances.

6ootnotes (5horthand references are e?plained on a separate page)

1$ This page concerns mathematical relationships between positions on the seven qin

strings. Jasically it e?plains why the tuning methods (定弦法子) described underTuning a qin result in the desired relative tuning (相對定音).

2$ 'ac7ground: -odern standard note fre*uencies in "ert8 9"8 ; #ib

7ote the following appro?imate modern concert pitch levels of a chromatic scale

beginning one octave below C

the same mathematical relationships $>! for the octave and &>$ for the fifth interval# and

both seem to give the same overall results. The mathematical relationships themselves

can be considered in terms either of freuency (vibrations per second) or of wave

length the results are reversed ($>! gives an octave higher, !>$ gives an octave lower),

giving in one case ascending pitch and the other descending, but the relations remain the

same. s will be seen below, there might also be differences in starting point, but in

practical terms once again the results are the same.

The earliest statement of sanfen sunyi is said to be the following from the boo of

!uanzi, 4hapter 1 (it follows a discussion of 9Nnote associations# translation from 6.llyn Bicett, !uanzi# "rinceton . "ress, p. $+&)

http://www.silkqin.com/12more/short.htmhttp://www.silkqin.com/08anal/tuning.htmhttp://www.silkqin.com/08anal/tuning.htmhttp://www.silkqin.com/08anal/tuning.htm#abspitchhttp://www.silkqin.com/08anal/tunings.htm#r2http://www.silkqin.com/02qnpu/16xltq/xl027gjx.htm#guanzifnhttp://www.silkqin.com/12more/short.htmhttp://www.silkqin.com/08anal/tuning.htmhttp://www.silkqin.com/08anal/tuning.htm#abspitchhttp://www.silkqin.com/08anal/tunings.htm#r2http://www.silkqin.com/02qnpu/16xltq/xl027gjx.htm#guanzifn


11/15

凡將起五音，凡首，先主一三!"#$%&''，%()*+,-!首%./，三分益!%一，0123，04，562三分789，:;，%()，%(.?，2三分789，:;，%(.@" To create the sounds of the five2note scale, first tae the primary unit and multiply it by

three. 4arried out four times, this will amount to a combination of nine times nine (or

eighty2one), thereby establishing the pitch of the huang zhong *A (+) tube in thelesser su ,- scale and its gong note. dding one2third ($) to mae !3 creates the

zhi note. 5ubtracting one2third (&+) results in the appropriate number ($) for producing

the shang note. dding one2third ($& 22 > 22 !+>$ 22 +!) by


12/15

There is a complete translation of =uan/i by 6. llyn Bicett, "rinceton niversity

"ress.

(Beturn)

!$ -eantone tuning (sometimes written 9mean tone9#FHI音JC法)

nother tuning system, often used in early 6estern music, follows what is calledmeantone temperament (FHI音C), in particular uarter2comma meantone. This andthe other tunings described here all try to resolve the tuning>intonation problems that

occur when playing harmonic or polyphonic music. s a soloist who in the traditional

repertoire would never play more than two notes simultaneously, the qin player did not

usually have to deal with such issues.

(Beturn)

5$ (ust intonation tuning (KC定音)For just intonation (sometimes also called natural tuning) the 4hinese term is KC chunl$ (9pure tones9). 5ection !.J. outlines its basics.

(Beturn)

$ )*ual temperament tuning (LMC定音)9'ual temperament9 is translated literally,LMC pinjun l$ orNOLMC shi%er

pingjun l$. 'ual temperament, though apparently discovered by the *ing prince 0hu

0aiyu, was never used on the qin. However, 5ection !.4. has some e?planation.

(Beturn)

/$ &ctual intonation used in earl% qin melodies

Gf one compares the old and new systems, it is easy to see that the lac of precision in

the old system would have prevented precise indications of mictrotones# on the other

hand, until modern times there seems to have been nothing written about the potential

for the decimal system to indicate microtones. Thus Tse 4hun :an8s analysis of variant

intonations, as with my own comments, must depend e?clusively on e?amining the

tablature itself, not on reading theoretical documents.

(Beturn)

$ ifferences bet3een P%thagorean and ust intonation tuning

The following chart shows the mathematical differences between notes organi/ed

according to the "ythagorean relations and according to just intonation# as mentioned

above, eual temperament was not used on the qin. &o (1) can be any note# the number

of vibrations per second (v>s) of any note on the chart is then the number resulting frommultiplying the number of v>s for do by the fraction indicated. The djustments row

shows how much mi, la and #i are lowered to give the simpler fractions of just

intonation.

7ote &o 'e Mi (a Sol La )i &o%

"ythagorean 1 @

Joth the "ythorean and just intonation systems reflect the apparent fact that the humanear finds consonance in sounds which are related to each other by simple mathematical

http://www.silkqin.com/08anal/tunings.htm#r3http://en.wikipedia.org/wiki/Meantone_temperamenthttp://en.wikipedia.org/wiki/Quarter-comma_meantonehttp://www.silkqin.com/08anal/tunings.htm#r4http://www.silkqin.com/08anal/tunings.htm#justinhttp://www.silkqin.com/08anal/tunings.htm#justinhttp://www.silkqin.com/08anal/tunings.htm#r5http://www.silkqin.com/09hist/qinshixu.htm#p64http://www.silkqin.com/09hist/qinshixu.htm#p64http://www.silkqin.com/08anal/tunings.htm#equalthttp://www.silkqin.com/08anal/tunings.htm#r6http://www.silkqin.com/08anal/tunings.htm#comparehttp://www.silkqin.com/03qobj/hui.htm#tsefnhttp://www.silkqin.com/08anal/tunings.htm#r7http://www.silkqin.com/08anal/tunings.htm#r3http://en.wikipedia.org/wiki/Meantone_temperamenthttp://en.wikipedia.org/wiki/Quarter-comma_meantonehttp://www.silkqin.com/08anal/tunings.htm#r4http://www.silkqin.com/08anal/tunings.htm#justinhttp://www.silkqin.com/08anal/tunings.htm#r5http://www.silkqin.com/09hist/qinshixu.htm#p64http://www.silkqin.com/09hist/qinshixu.htm#p64http://www.silkqin.com/08anal/tunings.htm#equalthttp://www.silkqin.com/08anal/tunings.htm#r6http://www.silkqin.com/08anal/tunings.htm#comparehttp://www.silkqin.com/03qobj/hui.htm#tsefnhttp://www.silkqin.com/08anal/tunings.htm#r7


13/15

relationships. The simplest of these are $>! and &>$. The ratio of vibrations per second

necessary to form what we call an octave is $>!, and for a fifth is &>$. Thus if (the

9main note9 fundamental note, here called tonic) is s and ' (a fifth higher) will be ++3 v>s. J (a fifth higher

than ') will then be 3 v>s# to bring this ' bac into the octave of to 8 you divide

the vibrations by half, giving

in the Harvard 4oncise Aictionary of *usic (!) the word 9harmonics9 has a similar

definition to the one given here, but the word 9harmonic9 is said to be the same as what

are here called 9overtones9. Here only two terms are used, as follows

!. 9"armonics9 in music practice a sound produced by lightly touching a string

while either bowing or plucing it is what is here called a 9harmonic9.

$. 9A#ertones9 in acoustic theory every musical sound has a fundamental pitch

for e?ample, the fundamental pitch of the note called 9modern concert 9 is said

to have a freuency of


14/15

actually describes a pure tone that can only be produced scientifically. Gn nature

each note is colored by a series of e?tra freuencies that are related in some

manner to the fundamental. These e?tra freuencies are what are here called the

9overtones9.

The main reasons for the confusion between these terms are probably as follows. First,the two concepts are related mathematically but are used differently in different

conte?ts# this is briefly discussed below. 5econd, precise definitions of overtones divide

them into two types, 9harmonic overtones9 and 9inharmonic overtones9. n additional

factor may perhaps be the fact that although on the qin e?tended passages in harmonics

are very common, in 6estern music such passages are uite rare.

"armonics and o#ertones: a brief discussion of their mathematical relationships.

!. "armonics

Gn music a harmonic is produced by lightly touching a string while either bowing

or plucing it. 6ith a stopped sound only the part of the string between where itis stopped and the side where it is pluced vibrates. 6ith a harmonic the whole

string vibrates. This means that to produce a harmonic the string can only be

touched in those places where doing so still allows the whole string to vibrate.

These places are the 9harmonic nodes9, located in places that divide the string

into simple fractions. The !& studs (hui) on a qin mar the places that divide the

string into the following fractions of the overall string length !>$, !>&, !>1,

!>+, and !>. The marers at $>&, &>1, &>1, 1, 1>+ and > duplicate the

above sounds. Thus, harmonics played at $>& produce the same pitch as those

played at !>& in both cases the string is vibrating in & eual segments (9loops9)#

liewise &>< is the same as !> the same as !>, and $>1, &>1 and 1 all the

same as !>1. 7ote that the basic (fundamental) sound is ! the sound of the open

string.

$. A#ertones

s mentioned above, in nature each note consists of a fundamental freuency

plus a series of e?tra freuencies that are related in some manner to the

fundamental. For practical reasons these e?tra freuencies are here simply called

9overtones9. s described more precisely in the 6iipedia article 9Overtone9,

n 8overtone8 is a partial (a 9partial wave9 or 9constituent freuency9)

that can be either a harmonic or an inharmonic. harmonic is an integermultiple of the fundamental freuency. n inharmonic overtone is a non2

integer multiple of a fundamental freuency.

Jy this definition the 9first harmonic9 is related to the fundamental by the

multiple !?!, so they are the same. The second harmonic is $?!, third harmonic

&?! and so forth. This mathematical relationship of the harmonic overtones to

the fundamental sound is same as the mathematical relationship of the string

divisions that produce the harmonics played on a qin. This can be seen by

inverting the si? fractions listed above as defining the divisions of the string that

produce harmonics $>!, &>!, !, 1>!, +>! and >! (note that the other seven

fractions duplicate these si?). Overtones whose freuencies have mathematicalrelationships other than these are called 9inharmonic9. The difference between

http://en.wikipedia.org/wiki/Overtonehttp://en.wikipedia.org/wiki/Overtone


15/15

the harmonics played on a qin and what are called 9harmonic overtones9 is that

the qin harmonics are discreet sounds, while the harmonic overtones all occur

together in one note.

10$ The main problem with the old system was that it was often used imprecisely andthere were inconsistencies. These problems seem to increase towards the end of the

*ing dynasty. Thus, for e?ample, the handboo Zhenchuan Zhengzong Qinpu usually

indicates all the intermediate finger positions with the word 9half9 (P an). Gf oneassumes the melodies are idiomatically similar to those described in earlier tablature,

then this is not a problem. However, it also maes it difficult to do research into such

issues as whether there were in fact modal developments taing place at that time, or if

there were regional (or personal) differences in intonation. (Gt should be noted that the

decimal system can only be useful for this if it used precisely according to theory# in

this regard see a comment above.

11$ Wai , 1+$1 and 1+

There is a clear e?ample near the end of 5ection $ of Shuixian Qu (my transcription

m.+).

.
http://www.silkqin.com/02qnpu/32zczz.htmhttp://www.silkqin.com/08anal/tunings.htm#imprecisehttp://www.silkqin.com/02qnpu/23wyqp/wy18sxq.htmhttp://www.silkqin.com/02qnpu/32zczz.htmhttp://www.silkqin.com/08anal/tunings.htm#imprecisehttp://www.silkqin.com/02qnpu/23wyqp/wy18sxq.htm

qin tunings

Documents