Pythagoras system

Pythagorean system - a musical system, the theory of which is associated with the Pythagorean school of harmonics . From the time of late Antiquity, prominent music theorists ( Nicomachus , Yamvlich , Boethius and others) attributed it directly to Pythagoras .

The abstract mathematical idea of the Pythagorean system (like a fifth chain) developed in the era of Western European Baroque .

In some scientific articles, it is also referred to as the "Pythagorean system" .

It is usually represented as a sequence of a fifth (or quart), for example like this (a chain of 6 fifths from the sound of fa ):

F - C - G - D - A - E - H

or in the form of a diatonic scale:

C		D		E		F		G		A		H		C
one		9/8		81/64		4/3		3/2		27/16		243/128		2
	Whole tone		Whole tone		Limma		Whole tone		Whole tone		Whole tone		Limma
	8 : 9		8 : 9		243 : 256		8 : 9		8 : 9		8 : 9		243 : 256
	203.91 q		203.91 q		90.22 c		203.91 q		203.91 q		203.91 q		90.22 c

In Western music, the Pythagorean system is attributed the role of the foundation not only for the ancient monody , but also for the polyphonic music of the Middle Ages. Music theorists still continue to describe intervals, relying on the Pythagorean system , although singing, and then instrumental polyphonic tonal music no later than the 16th century began to master a clean system . Compared with the latter, Pythagoras is an octave-fifth structure generated by the natural intervals of a pure octave (1: 2) and a pure fifth (2: 3) ^[1] . For all those employed in the Pythagorean interval relations, the factorization numbers are based on prime numbers of no more than 3. For this reason, mainly in the English-speaking environment, the Pythagorean systems are also called tuning limit 3 ( 3-limit tuning ).

Content

1 Table of intervals of the Pythagorean system
2 See also
3 References
4 notes

Pythagorean interval table

The following table shows the intervals of the Pythagorean system, not exceeding an octave and obtained in no more than 18 fifth steps. Diatonic intervals (that is, occurring in the Pythagorean 7-step diatonic and obtained in no more than 6 fifth steps) are shown in bold. Chromatic intervals (occurring, along with diatonic intervals, in a 12-step Pythagorean octave scale, and obtained in 7-11 fifth steps) are indicated in regular type. The remaining “dichromatic” (or “enharmonic”) intervals obtained in 12-18 fifth steps are shown in italics. These latter ones (with the exception of the Pythagorean coma, corresponding to an increased septima without an octave, and a decreased nona) correspond to twice increased and reduced diatonic intervals.

Abbreviations: “m.” - small; "B." - large; “Mind.” - reduced; “Uv.” - enlarged.

The columns Q and O of the table show the numbers of quintes and octaves, respectively, by delaying which this interval is obtained (in this case, positive numbers correspond to a delay up, and negative numbers correspond to a down). For example, the reduced septima correspond to the values Q = −9 and O = 6, that is, the reduced septima is obtained by putting 9 quint down and 6 octaves up from a given sound (pitch); thus, it has a ratio of sound frequencies equal to

\left({\frac {3}{2}}\right)^{-9}\times \left({\frac {2}{1}}\right)^{6}=2^{15}\cdot 3^{-9}={\frac {32768}{19683}}.

{\ displaystyle \ left ({\ frac {3} {2}} \ right) ^ {- 9} \ times \ left ({\ frac {2} {1}} \ right) ^ {6} = 2 ^ { 15} \ cdot 3 ^ {- 9} = {\ frac {32768} {19683}}.}

Moreover, the number O (for intervals less than an octave) is uniquely determined by the number Q, being from it in a functional dependence determined by the formula:

\mathrm {O} =-\lfloor \mathrm {Q} \times (\log _{2}{3}-1)\rfloor ,

{\ displaystyle \ mathrm {O} = - \ lfloor \ mathrm {Q} \ times (\ log _ {2} {3} -1) \ rfloor,}

Where $\lfloor x\rfloor$ ${\ displaystyle \ lfloor x \ rfloor}$ Is the integer part of the number $x$ ${\ displaystyle x}$ ^[2] .

Further, each of the intervals indicated in the table is unambiguously represented as composed of T whole tones (indicated in column T ), L limm (column L ) and K Pythagorean comms (column K ), subject to restrictions

0\leqslant T\leqslant 6,\qquad 0\leqslant L\leqslant 1,\qquad -2\leqslant K\leqslant 1

{\ displaystyle 0 \ leqslant T \ leqslant 6, \ qquad 0 \ leqslant L \ leqslant 1, \ qquad -2 \ leqslant K \ leqslant 1}

.

As can be seen from the table, for diatonic intervals, one of three pairs of equalities holds: $L=0$ ${\ displaystyle L = 0}$ and $K=0$ ${\ displaystyle K = 0}$ either $L=1$ ${\ displaystyle L = 1}$ and $K=0$ ${\ displaystyle K = 0}$ either $L=0$ ${\ displaystyle L = 0}$ and $K={-1}$ ${\ displaystyle K = {- 1}}$ (that is, the diatonic interval is always equal to either an integer number of tones, or an integer number of tones with added limm, or less than an integer number of tones per Pythagorean comm). For chromatic intervals, moreover, the relations $L=1$ ${\ displaystyle L = 1}$ and $K=1$ ${\ displaystyle K = 1}$ either $L=1$ ${\ displaystyle L = 1}$ and $K={-1}$ ${\ displaystyle K = {- 1}}$ , and “dichromatic” (in italics) - also $L=0$ ${\ displaystyle L = 0}$ and $K=1$ ${\ displaystyle K = 1}$ either $L=0$ ${\ displaystyle L = 0}$ and $K={-2}$ ${\ displaystyle K = {- 2}}$ .

Title	Q	O	T	L	K	Attitude	Value in cents	Stage from c	Additional examples
unison prima	0	0	0	0	0	1: 1	0.00	c
Pythagorov Comm (SW. Septima without an octave) ^[3]	12	-7	0	0	one	531441: 524288	23.46	His	des — cis, fes — e, a — gisis
twice mind. third	-17	10	0	one	-one	134217728: 129140163	66.76	eseses ^[4]	cis — eses, eis — ges
Lime , m. second, smaller (diatonic) midtones	-5	3	0	one	0	256: 243	90.22	des	e — f, cis — d, des — eses
apotoma , uv. prima larger (chromatic) midtones	7	-four	0	one	one	2187: 2048	113.69	cis	cis — cisis, des — d, eses — es
mind. third	-10	6	one	0	-one	65536: 59049	180.45	eses	cis — es, e — ges
whole tone b. second	2	-one	one	0	0	9: 8	203.91	d	d — e, e — fis, B — c, des — es, cis — dis
twice uv. prima	fourteen	-8	one	0	one	4782969: 4194304	227.37	cisis	ces — cis, deses — d
twice mind. quart	-fifteen	9	one	one	-one	16777216: 14348907	270.67	feses	cis — fes, fis-b, cisis — f
half tone, m. third	-3	2	one	one	0	32:27	294.13	es	d — f, es — ges
SW. second	9	-5	one	one	one	19683: 16384	317.60	dis	des — e, es — fis
mind. quart	-8	5	2	0	-one	8192: 6561	384.36	fes	cis — f, fis — b, dis — ges
Deaton, b. third	four	-2	2	0	0	81:64	407.82	e	d — fis, eis-gisis
twice uv. second	16	-9	2	0	one	43046721: 33554432	431.28	disis	ces — dis, es — fisis
twice mind. quint	-13	8	2	one	-one	2097152: 1594323	474.58	geses	cis — ges, disis — a
quart	-one	one	2	one	0	4: 3	498.04	f	d — g, ces — fes
SW. third	eleven	-6	2	one	one	177147: 131072	521.51	eis	des — fis, deses — f
twice mind. sixth	-eighteen	eleven	3	0	-2	536870912: 387420489	564.81	aseses ^[4]	cisis — as, cis — ases
mind. quint (commutative newt ^[5] )	-6	four	3	0	-one	1024: 729	588.27	ges	cis — g, H — f, e — b
newt, uv quart	6	-3	3	0	0	729: 512	611.73	fis	f — b, des — g
twice uv. third	eighteen	-10	3	0	one	387420489: 268435456	635.19	eisis	des — fisis, eses — gis
mind. sixth (the wolf quint of the Pythagoras system)	-eleven	7	3	one	-one	262144: 177147	678.49	ases	cis — as, Gis — es
quint	one	0	3	one	0	3: 2	701.96	g	d — a, dis — ais
twice uv. quart	13	-7	3	one	one	1594323: 1048576	725.42	fisis	des — gis, deses — a
twice mind. seventh	-16	10	four	0	-2	67108864: 43046721	768.72	heseses ^[4]	cis — heses, cisis — b
m. sexta	-four	3	four	0	-one	128: 81	792.18	as	d — b, dis-h
SW. quinta (tetraton)	8	-four	four	0	0	6561: 4096	815.64	gis	des — a, eses — b
mind. seventh	-9	6	four	one	-one	32768: 19683	882.40	heses	cis — b, Gis — f
b. sixth	3	-one	four	one	0	27:16	905.87	a	d — h, Es — c
twice uv. quint	fifteen	-8	four	one	one	14348907: 8388608	929.33	gisis	des — ais, deses — a
twice mind. octave	-fourteen	9	5	0	-2	8388608: 4782969	972.63	ceses ¹	Dis — des, Disis — d
m. septima	-2	2	5	0	-one	16: 9	996.09	b	G — f, Des — ces
SW. sexta (pentaton)	10	-5	5	0	0	59049: 32768	1019.55	ais	des — h, deses — b
mind. octave	-7	5	5	one	-one	4096: 2187	1086.31	ces ¹	Cis — c, des — deses
b. seventh	5	-2	5	one	0	243: 128	1109.78	h	cis — his
twice uv. sixth	17	-9	5	one	one	129140163: 67108864	1133.24	aisis	ces — ais, esses — cis
mind. nona	-12	8	6	0	-2	1048576: 531441	1176.54	deses ¹	Dis — es, Eis — f
octave	0	one	6	0	-one	2: 1	1,200.00	c ¹

Links

Margo Schulter . Pythagorean Tuning and Medieval Polyphony
Hugo Riemann Interval Tables

Notes

↑ Natural intervals, or intervals of a natural scale , between the 1st and 2nd, 2nd and 3rd overtones are indicated by the ratios 1: 2 and 2: 3, respectively.
↑ The above formula is obtained by logarithm of the inequality $1\leqslant \left({\frac {3}{2}}\right)^{\mathrm {Q} }\cdot 2^{\mathrm {O} }<2$ ${\ displaystyle 1 \ leqslant \ left ({\ frac {3} {2}} \ right) ^ {\ mathrm {Q}} \ cdot 2 ^ {\ mathrm {O}} <2}$ uniquely determining the dependence of O on Q.
↑ The increased Septima of the Pythagorean system (for example, c - his ) is wider than an octave ( c - c ¹ ) per Pythagorean commune.
↑ ¹ ² ^{3 The} spelling of the letter designation of a step separated from s by a given interval (twice a reduced third, sixth or septima) requires the indication of a “triple flat” ( -eseses ) , indicating a decrease in the corresponding diatonic step (in this case, respectively, e , a and h ) on three chromatic tones; for examples of the same intervals between other steps that do not require “triple signs of alteration”, see the “Additional Examples” column.
↑ That is, a newt reduced by (Pythagorean) comm.

[1] Natural intervals, or intervals of a natural scale , between the 1st and 2nd, 2nd and 3rd overtones are indicated by the ratios 1: 2 and 2: 3, respectively.

[2] The above formula is obtained by logarithm of the inequality $1\leqslant \left({\frac {3}{2}}\right)^{\mathrm {Q} }\cdot 2^{\mathrm {O} }<2$ ${\ displaystyle 1 \ leqslant \ left ({\ frac {3} {2}} \ right) ^ {\ mathrm {Q}} \ cdot 2 ^ {\ mathrm {O}} <2}$ uniquely determining the dependence of O on Q.

[3] The increased Septima of the Pythagorean system (for example, c - his ) is wider than an octave ( c - c ¹ ) per Pythagorean commune.

[3apo-4] ¹ ² ^{3 The} spelling of the letter designation of a step separated from s by a given interval (twice a reduced third, sixth or septima) requires the indication of a “triple flat” ( -eseses ) , indicating a decrease in the corresponding diatonic step (in this case, respectively, e , a and h ) on three chromatic tones; for examples of the same intervals between other steps that do not require “triple signs of alteration”, see the “Additional Examples” column.

[5] That is, a newt reduced by (Pythagorean) comm.

Pythagoras system

Content

Pythagorean interval table

See also

Links

Notes

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