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:''See also [[Gravity family #Harry]] or [[Breedsmic temperaments #Harry]].''
{{Infobox regtemp
| Title = Harry
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Comma basis = [[2401/2400]], [[19683/19600]] (7-limit); <br>[[243/242]], [[441/440]], [[4000/3993]] (11-limit); <br>[[243/242]], [[351/350]], [[364/363]], [[441/440]]<br>(13-limit)
| Edo join 1 = 58 | Edo join 2 = 72
| Mapping = 2; -6 -17 -10 -15 -26
| Generators = 21/20 | Generators tuning = 83.1 | Optimization method = CWE
| MOS scales = [[2L 12s]], [[14L 2s]], [[14L 16s]], [[14L 30s]]
| Odd limit 1 = 9 | Mistuning 1 = 1.81 | Complexity 1 = 44
| Odd limit 2 = 13-limit 21 | Mistuning 2 = 3.30 | Complexity 2 = 58
}}
'''Harry''' is the rank-2 [[regular temperament|temperament]] with a [[period]] of half an [[octave]] and a [[generator]] somewhere between [[22/21]] and [[21/20]] (which are tempered together in harry), or around 83 [[cent]]s. Two generators are thus equal to [[11/10]] (which is [[4000/3993|made]] a third of [[4/3]]) and three of which [[1001/1000|made]] equal to [[15/13]] (which is [[676/675|made]] a half of 4/3). This means that harry splits 4/3 into 6 equal parts, a highly composite number, and splitting 2/1 into two equal parts (representing [[24/17]]~[[99/70]]) means it also splits 3/2 into two equal parts (representing [[11/9]]~[[49/40]]). Alternatively, it can be viewed as a [[cluster temperament]] with 14 clusters and a chroma that represents many important intervals including 81/80, 99/98, 100/99, and 121/120. In any case the first important [[mos]] of harry has the shape [[2L 12s]].


'''Harry''' temperament has a period of half an octave and a generator somewhere between 22/21 and 21/20 (which are tempered together in harry), or around 83 cents. Alternatively it can be viewed as a [[cluster temperament]] with 14 clusters and a chroma that represents many important intervals including 81/80, 99/98, 100/99, and 121/120. In any case the first important [[MOS]] of harry has the shape [[2L 12s]].
Harry was named after [[Harry Partch]], which is ironic given that Harry Partch was adamantly opposed to the very idea of tempering. This is perhaps not so insulting to Harry when you consider that these mathematical structures can also be used to arrange JI intervals into patterns ([[constant structure]]s) and create JI [[detempering]]s of the temperament.


Harry was named after Harry Partch, which is ironic given that Harry Partch was adamantly opposed to the very idea of tempering. This is perhaps not so insulting to Harry when you consider that these mathematical structures can also be used to arrange JI intervals into patterns ([[constant structure]]s) and create JI [[detempering]]s of the temperament.
This particular rank-2 temperament might be called "harry" because the lowest [[edo]] in which [[Harry Partch's 43-tone scale]] is represented distinctly is [[58edo]], and harry is one of the best temperaments supported by 58edo (it is 58 & 72). Alternatively, if you look at the tempered image of the 43-tone JI scale in this temperament, it is relatively compact and never "backtracks" from one of the 14 clusters to the previous one. In fact, the entire temperament can be derived from knowing that the fragment [12/11, 11/10, 10/9, 9/8] is supposed to be equidistant, and [14/11, 9/7] also has that same separation. The steps of those scale fragments are 121/120, 100/99, 81/80, and 99/98. Tempering these together means that 4000/3993, 243/242, and 9801/9800 are all tempered out, and harry is the unique 11-limit rank-2 temperament tempering those out.


This particular rank-2 temperament might be called "harry" because the lowest [[EDO]] in which [[Harry Partch's 43-tone scale]] is represented distinctly is [[58edo]], and harry is one of the best temperaments supported by 58edo (it is 58 & 72). Alternatively, if you look at the tempered image of the 43-tone JI scale in this temperament, it is relatively compact and never "backtracks" from one of the 14 clusters to the previous one. In fact, the entire temperament can be derived from knowing that the fragment [12/11, 11/10, 10/9, 9/8] is supposed to be equidistant, and [14/11, 9/7] also has that same separation.
See [[Gravity family #Harry]] for more technical data.
 
(The steps of those scale fragments are 121/120, 100/99, 81/80, and 99/98. Tempering these together means that 4000/3993, 243/242, and 9801/9800 are all tempered out, and the unique 11-limit rank-2 temperament tempering those out is harry.)


== Interval chain ==
== Interval chain ==
Line 76: Line 85:
| 9
| 9
| 748.04
| 748.04
| 54/35
| 54/35, 20/13
| 148.04
| 148.04
| 12/11
| 12/11
Line 117: Line 126:
|}
|}


== Chords ==
=== As a detemperament of 14et ===
{{main| Chords of harry }}
Harry is naturally considered a detemperament of [[14edo|14 equal temperament]], thus containing both diatonic and interordinal interval catgories. The small step at 1/2 octave minus seven ~21/20 generators serves as a spacer between intervals in the same category, representing 81/80~91/90~99/98~100/99~105/104~121/120.
 
{{Todo|complete section}}
 
== Chords and harmony ==
{{See also| Chords of harry }}


== Scales ==
== Scales ==
* [[Harry58]]
* [[Harry58]]


== Tuning spectrum ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~21/20 = 83.1249{{c}}
| CWE: ~21/20 = 83.1427{{c}}
| POTE: ~21/20 = 83.1560{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~21/20 = 83.1477{{c}}
| CWE: ~21/20 = 83.1589{{c}}
| POTE: ~21/20 = 83.1670{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~21/20 = 83.1175{{c}}
| CWE: ~21/20 = 83.1169{{c}}
| POTE: ~21/20 = 83.1164{{c}}
|}


{| class="wikitable center-all"
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]
! Generator<br>(¢)
! Comments
|-
|-
! [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
| 3\44
! generator<br>(¢)
|
! comments
| 81.818
| 44ceff val, lower bound of 7- to 11-odd-limit diamond monotone
|-
|-
|
| 9/7
| 9/7
| 82.458
| 82.458
|
|
|-
|-
|
| 11/10
| 11/10
| 82.502
| 82.502
|
|
|-
|-
|
| 15/13
| 15/13
| 82.580
| 82.580
|
|
|-
|-
| 4\58
|
| 82.759
| Lower bound of 13-odd-limit diamond monotone
|-
|
| 13/11
| 13/11
| 82.799
| 82.799
|
|
|-
|-
|
| 13/10
| 13/10
| 82.865
| 82.865
|
|
|-
|-
|
| 15/11
| 15/11
| 82.881
| 82.881
|
|
|-
|-
| 4/3
|
| 3/2
| 83.007
| 83.007
|
|
|-
|-
| 14/13
|
| 13/7
| 83.019
| 83.019
|
|
|-
|-
| 16/13
|
| 13/8
| 83.057
| 83.057
|
|
|-
|-
|
| 13/12
| 13/12
| 83.071
| 83.071
|
|
|-
|-
| 18/13
| 9\130
|
| 83.077
|
|-
|
| 13/9
| 83.099
| 83.099
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|-
| 8/7
|
| 7/4
| 83.117
| 83.117
|
|
|-
|-
| 16/15
|
| 15/8
| 83.119
| 83.119
|
|
|-
|-
|
| 15/14
| 15/14
| 83.120
| 83.120
|
|
|-
|-
|
| 5/4
| 5/4
| 83.158
| 83.158
| 5-, 7- and 9-odd-limit minimax
| 5-, 7- and 9-odd-limit minimax
|-
|-
|
| 7/5
| 7/5
| 83.216
| 83.216
|
|
|-
|-
| 6/5
|
| 5/3
| 83.240
| 83.240
|
|
|-
|-
|
| 11/8
| 11/8
| 83.245
| 83.245
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
|-
|
| 7/6
| 7/6
| 83.282
| 83.282
|
|
|-
|-
| 12/11
| 5\72
|
| 83.333
| Upper bound of 13-odd-limit diamond monotone
|-
|
| 11/6
| 83.404
| 83.404
|
|
|-
|-
| 14/11
|
| 11/7
| 83.502
| 83.502
|
|
|-
|-
| 10/9
|
| 9/5
| 83.519
| 83.519
|
|
|-
|-
| 6\86
|
| 83.721
| 86ceff val
|-
|
| 11/9
| 11/9
| 84.197
| 84.197
Line 227: Line 336:
* [[Lumatone mapping for harry]]
* [[Lumatone mapping for harry]]


[[Category:Temperaments]]
[[Category:Harry| ]] <!-- main article -->
[[Category:Harry| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Gravity family]]
[[Category:Gravity family]]
[[Category:Stearnsmic clan]]
[[Category:Breedsmic temperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Cataharry temperaments]]
[[Category:Cataharry temperaments]]

Latest revision as of 18:48, 8 April 2026

Harry
Subgroups 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
Comma basis 2401/2400, 19683/19600 (7-limit);
243/242, 441/440, 4000/3993 (11-limit);
243/242, 351/350, 364/363, 441/440
(13-limit)
Reduced mapping ⟨2; -6 -17 -10 -15 -26]
ET join 58 & 72
Generators (CWE) ~21/20 = 83.1 ¢
MOS scales 2L 12s, 14L 2s, 14L 16s, 14L 30s
Ploidacot diploid delta-hexacot
Minimax error 9-odd-limit: 1.81 ¢;
13-limit 21-odd-limit: 3.30 ¢
Target scale size 9-odd-limit: 44 notes;
13-limit 21-odd-limit: 58 notes

Harry is the rank-2 temperament with a period of half an octave and a generator somewhere between 22/21 and 21/20 (which are tempered together in harry), or around 83 cents. Two generators are thus equal to 11/10 (which is made a third of 4/3) and three of which made equal to 15/13 (which is made a half of 4/3). This means that harry splits 4/3 into 6 equal parts, a highly composite number, and splitting 2/1 into two equal parts (representing 24/17~99/70) means it also splits 3/2 into two equal parts (representing 11/9~49/40). Alternatively, it can be viewed as a cluster temperament with 14 clusters and a chroma that represents many important intervals including 81/80, 99/98, 100/99, and 121/120. In any case the first important mos of harry has the shape 2L 12s.

Harry was named after Harry Partch, which is ironic given that Harry Partch was adamantly opposed to the very idea of tempering. This is perhaps not so insulting to Harry when you consider that these mathematical structures can also be used to arrange JI intervals into patterns (constant structures) and create JI detemperings of the temperament.

This particular rank-2 temperament might be called "harry" because the lowest edo in which Harry Partch's 43-tone scale is represented distinctly is 58edo, and harry is one of the best temperaments supported by 58edo (it is 58 & 72). Alternatively, if you look at the tempered image of the 43-tone JI scale in this temperament, it is relatively compact and never "backtracks" from one of the 14 clusters to the previous one. In fact, the entire temperament can be derived from knowing that the fragment [12/11, 11/10, 10/9, 9/8] is supposed to be equidistant, and [14/11, 9/7] also has that same separation. The steps of those scale fragments are 121/120, 100/99, 81/80, and 99/98. Tempering these together means that 4000/3993, 243/242, and 9801/9800 are all tempered out, and harry is the unique 11-limit rank-2 temperament tempering those out.

See Gravity family #Harry for more technical data.

Interval chain

# Period 0 Period 1
Cents Approx. Ratios Cents Approx. Ratios
0 0.00 1/1 600.00 99/70, 140/99
1 83.12 21/20, 22/21 683.12 40/27
2 166.23 11/10 766.23 14/9
3 249.35 15/13 849.35 18/11, 44/27
4 332.46 40/33 932.46 12/7
5 415.58 14/11 1015.58 9/5
6 498.70 4/3 1098.70 66/35
7 581.81 7/5 1181.81 160/81
8 664.92 22/15 64.92 26/25, 27/26, 28/27
9 748.04 54/35, 20/13 148.04 12/11
10 831.16 21/13 231.16 8/7
11 914.28 22/13 314.28 6/5
12 997.39 16/9 397.39 44/35, 63/50
13 1080.51 28/15 480.51 33/25
14 1163.62 49/25, 88/45, 108/55 563.62 18/13
15 46.74 36/35 646.74 16/11

As a detemperament of 14et

Harry is naturally considered a detemperament of 14 equal temperament, thus containing both diatonic and interordinal interval catgories. The small step at 1/2 octave minus seven ~21/20 generators serves as a spacer between intervals in the same category, representing 81/80~91/90~99/98~100/99~105/104~121/120.

Chords and harmony

See also: Chords of harry

Scales

Tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~21/20 = 83.1249 ¢ CWE: ~21/20 = 83.1427 ¢ POTE: ~21/20 = 83.1560 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~21/20 = 83.1477 ¢ CWE: ~21/20 = 83.1589 ¢ POTE: ~21/20 = 83.1670 ¢
13-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~21/20 = 83.1175 ¢ CWE: ~21/20 = 83.1169 ¢ POTE: ~21/20 = 83.1164 ¢

Tuning spectrum

Edo
generator 		Eigenmonzo
(unchanged interval) 		Generator
(¢) 		Comments
3\44 81.818 44ceff val, lower bound of 7- to 11-odd-limit diamond monotone
9/7 82.458
11/10 82.502
15/13 82.580
4\58 82.759 Lower bound of 13-odd-limit diamond monotone
13/11 82.799
13/10 82.865
15/11 82.881
3/2 83.007
13/7 83.019
13/8 83.057
13/12 83.071
9\130 83.077
13/9 83.099 13- and 15-odd-limit minimax
7/4 83.117
15/8 83.119
15/14 83.120
5/4 83.158 5-, 7- and 9-odd-limit minimax
7/5 83.216
5/3 83.240
11/8 83.245 11-odd-limit minimax
7/6 83.282
5\72 83.333 Upper bound of 13-odd-limit diamond monotone
11/6 83.404
11/7 83.502
9/5 83.519
6\86 83.721 86ceff val
11/9 84.197

See also

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