130.75.229.87: /* Cauchy’s stress theorem—stress tensor */

2014-01-31T18:30:20Z

Cauchy’s stress theorem—stress tensor

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[[File:Harry Partch Institute-3.jpg|thumb|right|The [[Quadrangularis Reversum]], an instrument constructed by [[Harry Partch]] based on the 11-limit tonality diamond]]
In [[music theory]] and [[musical tuning|tuning]], a '''tonality diamond''' is a two-dimensional diagram of [[ratio]]s in which one dimension is the [[Otonality]] and one the Utonality.<ref>Rasch, Rudolph (2000). "A Word or Two on the Tunings of Harry Partch", ''Harry Partch: An Anthology of Critical Perspectives'', p.28. Dunn, David, ed. ISBN 90-5755-065-2.</ref> Thus the [[limit (music)|n-limit]] tonality diamond is an arrangement in diamond-shape of the set of [[rational number]]s ''r'', <math>1 \le r < 2</math>, such that the odd part of both the [[numerator]] and the [[denominator]] of ''r'', when reduced to lowest terms, is less than or equal to the fixed [[odd number]] ''n''. Equivalently, the diamond may be considered as a set of [[pitch class]]es, where a pitch class is an [[equivalence class]] of pitches under [[octave]] equivalence. The tonality diamond is often regarded as comprising the set of [[consonance and dissonance|consonances]] of the n-limit. Although originally invented by [[Max Friedrich Meyer]],<ref>[http://www.chrysalis-foundation.org/Meyer-s_Diamond.htm "Musical Mathematics: Meyer's Diamond"], ''Chrysalis-Foundation.org''.</ref> the tonality diamond is now most associated with [[Harry Partch]].

==The diamond arrangement==
Partch arranged the elements of the tonality diamond in the shape of a [[rhombus]], and subdivided into (n+1)<sup>2</sup>/4 smaller rhombuses. Along the upper left side of the rhombus are placed the odd numbers from 1 to n, each reduced to the octave (divided by the minimum power of 2 such that <math>1 \le r < 2</math>). These intervals are then arranged in ascending order. Along the lower left side are placed the corresponding reciprocals, 1 to 1/n, also reduced to the octave (here, ''multiplied'' by the minimum power of 2 such that <math>1 \le r < 2</math>). These are placed in descending order. At all other locations are placed the product of the diagonally upper- and lower-left intervals, reduced to the octave. This gives all the elements of the tonality diamond, with some repetition. Diagonals sloping in one direction form [[Otonality and Utonality|Otonalities]] and the diagonals in the other direction form Utonalities. One of Partch's instruments, the [[diamond marimba]], is arranged according to the tonality diamond.

===5-limit===
{| border="0" cellspacing="0" cols="7" frame="void" rules="none"
|-
| || || [[perfect fifth|3/2]] ||
|-
| || [[major third|5/4]] || || [[just minor third|6/5]] ||
|-
| [[unison|1/1]] || || 1/1 || || 1/1 ||
|-
| || [[just minor sixth|8/5]] || || [[major sixth|5/3]] ||
|-
| || || [[perfect fourth|4/3]] ||
|}

This diamond contains three [[Identity_(tuning)#Identity|identities]] (1, 3, 5).

===7-limit===
{| border="0" cellspacing="0" cols="7" frame="void" rules="none"
|-
| || || || [[harmonic seventh|7/4]] || || ||
|-
| || || [[perfect fifth|3/2]] || || [[septimal tritone|7/5]] || ||
|-
| || [[major third|5/4]] || || [[just minor third|6/5]] || || [[septimal minor third|7/6]] ||
|-
| [[unison|1/1]] || || 1/1 || || 1/1 || || 1/1
|-
| || [[just minor sixth|8/5]] || || [[major sixth|5/3]] || || [[septimal major sixth|12/7]] ||
|-
| || || [[perfect fourth|4/3]] || || [[septimal tritone|10/7]] || ||
|-
| || || || [[septimal major second|8/7]] || || ||
|}

This diamond contains four identities (1, 3, 5, 7).

===11-limit===
[[File:Partchdiamond.svg|thumb|center|350px|Tonal basis of [[Harry Partch]]'s tuning system: 11-limit tonality diamond]]

This diamond contains six identities (1, 3, 5, 7, 9, 11). Harry Partch used the 11-limit tonality diamond, but flipped it 90 degrees.

===15-limit===
15/8
7/4 5/3
13/8 14/9 3/2
3/2 13/9 7/5 15/11
11/8 4/3 13/10 14/11 5/4
5/4 11/9 6/5 13/11 7/6 15/13
9/8 10/9 11/10 12/11 13/12 14/13 15/14
1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1
16/9 9/5 20/11 11/6 24/13 13/7 28/15
8/5 18/11 5/3 22/13 12/7 26/15
16/11 3/2 20/13 11/7 8/5
4/3 18/13 10/7 22/15
16/13 9/7 4/3
8/7 6/5
16/15

This diamond contains eight identities (1, 3, 5, 7, 9, 11, 13, 15).

==Geometry of the tonality diamond==
The five- and seven-limit tonality diamonds exhibit a highly regular geometry within the [[modulatory space]], meaning all non-unison elements of the diamond are only one unit from the unison. The five-limit diamond then becomes a regular [[hexagon]] surrounding the unison, and the seven-limit diamond a [[cuboctahedron]] surrounding the unison.{{citation needed|date=November 2012}}

==Properties of the tonality diamond==
{{see|Farey sequence}}

Three properties of the tonality diamond and the ratios contained:
#All ratios between neighboring ratios are [[superparticular number|superparticular ratio]]s, those with a difference of 1 between [[numerator]] and [[denominator]].<ref name="Rasch">Rasch (2000), p.30.</ref>
#Ratios with relatively lower numbers have more space between them than ratios with higher numbers.<ref name="Rasch"/>
#The system, including the ratios between ratios, is symmetrical within the octave when measured in cents ''not'' in ratios.<ref name="Rasch"/>

For example:
{| class="wikitable" style="text-align:center"
!colspan=17| 5-limit tonality diamond, ordered least to greatest
|-
! Ratio
|colspan=2| 1/1
|colspan=2| 6/5
|colspan=2| 5/4
|colspan=2| 4/3
|colspan=2| 3/2
|colspan=2| 8/5
|colspan=2| 5/3
|colspan=2| 2/1
|-
! Cents
|colspan=2| 0
|colspan=2| 315.64
|colspan=2| 386.31
|colspan=2| 498.04
|colspan=2| 701.96
|colspan=2| 813.69
|colspan=2| 884.36
|colspan=2| 1200
|-
! Width
|
|colspan=2| 315.64
|colspan=2| 70.67
|colspan=2| 111.73
|colspan=2| 203.91
|colspan=2| 111.73
|colspan=2| 70.67
|colspan=2| 315.64
|
|}
#The ratio between 6/5 and 5/4 (and 8/5 and 5/3) is 25/24.
#The ratios with relatively low numbers 4/3 and 3/2 are 203.91 cents apart, while the ratios with relatively high numbers 6/5 and 5/4 are 70.67 cents apart.
#The ratio between the lowest and 2nd lowest and the highest and 2nd highest ratios are the same, and so on.

==Size of the tonality diamond==
If φ(''n'') is [[Euler's totient function]], which gives the number of positive integers less than n and [[coprime|relatively prime]] to n, that is, it counts the integers less than n which share no common factor with n, and if d(n) denotes the size of the n-limit tonality diamond, we have the formula
:<math> d(n) = \sum_{m<n \ odd} \phi(m).</math>
From this we can conclude that the rate of growth of the tonality diamond is asymptotically equal to <math> \frac{2}{\pi^2} n^2 </math>. The first few values are the important ones, and the fact that the size of the diamond [[Quadratic growth|grows as the square]] of the size of the odd limit tells us that it becomes large fairly quickly. There are seven members to the 5-limit diamond, 13 to the 7-limit diamond, 19 to the 9-limit diamond, 29 to the 11-limit diamond, 41 to the 13-limit diamond, and 49 to the 15-limit diamond; these suffice for most purposes.

==Translation to string length ratios==
[[Yuri Landman]] rewrites Partch's diamond to clarify its theoretical relationship to string lengths (as Partch used in his Kitharas) and his [[Moodswinger]] instrument. Landman flips the ratios (5/4 becomes 4/5) and takes the [[complement (music)|complement]] string part (1/5 instead of 4/5) to make them easier to understand.<ref>[http://www.hypercustom.com/11limitdiamondandcolors.html "Partch's Tonality Diamond Related to my Moodswinger Color Schedule"], ''Hypercustom.com''. {{deadlink|date=November 2012}}</ref>{{Citation needed|date=June 2011}}

==See also==
*[[Numerary nexus]]
*[[Lattice (music)]]

==References==
{{reflist}}

{{Harry Partch}}
{{Musical tuning}}
{{Pitch space}}

{{DEFAULTSORT:Tonality Diamond}}
[[Category:Harry Partch]]
[[Category:Pitch space]]

Cauchy stress tensor - Revision history

130.75.229.87: /* Cauchy’s stress theorem—stress tensor */