VI. Annäherung dieser Intervalle durch gleichstufige Temperaturen
Wenn man den „Fehler“ r, 2r, 3r usw. gleichmäßig aufteilt, kann man statt des Pythagoräischen Kommas das Holder-Komma oder Arabische Komma verwenden.
| Zahlenverhältnis | Centmaß | ||
| H = 21/53 | [H] = h = 22,6415094 |
Die Quint z.B. wird sehr gut durch 31 Holder-Kommata angenähert:
| Qui/H31 = (3/2)/231/53 = (353/284)1/53 = R1/53 | [Qui] – 31h = r/53 = 0,0682084 ¢ | ||
| Qui = H31· R1/53 | [Qui] = 31h + r/53 |
Wenn allgemein C = 3m/2n ≈ 1, also 3m ≈ 2n dann 3 ≈ 2n/m
| und die Quint 3/2 ≈ 2(n/m)-1 = 2(n-m)/m . | |
| Der Unterschied zwischen der reinen (3/2) und der auf diese Weise temperierten Quint beträgt daher | |
| Qui/2(n-m)/m = (3/2)/2(n-m)/m = 3/21+(n-m)/m = 3/2n/m = 3m/m/2n/m = C1/m | |
Das kann man für alle 3m/2n-„Kommata“ durchführen:
| „Komma“ | „Quint“ | Verhältnis | Differenz | |
| C = 3m/2n | 2(n-m)/m | (3/2)/(2(n-m)/m) | [C]/m = c/m |
| Qui = 31/21 | 20/1 | = | 1 | 1,5 | 701,9550009 |
| Qua-1 = 31/22 | 21/1 | = | 2 | 0,75 | – 498,0449991 |
| T = 32/23 | 21/2 | = | 1,4142135624 | 1,0606601718 | 101,9550009 |
| S-1 = 35/28 | 23/5 | = | 1,5157165665 | 0,9896309331 | – 18,0449991 |
| P = 312/219 | 27/12 | = | 1,4983070769 | 1,0011298906 | 1,9550009 |
| Q = 341/265 | 224/41 | = | 1,5004194331 | 0,9997204561 | – 0,4840235 |
| R = 353/284 | 231/84 | = | 1,4999409031 | 1,0000393995 | 0,06820841 |
| U = 3306/2485 | 2179/306 | = | 1,5000050111 | 0,9999966594 | – 0,005783448 |
| V = 3665/21054 | 2389/665 | = | 1,4999999015 | 1,0000000656 | 0,0001136473 |
| W = 315601/224727 | 29126/15601 | = | 1,5000000017 | 0,9999999988 | – 0,000002019000 |
| M = 331867/250508 | 218641/31867 | = | 1,4999999997 | 1,0000000002 | 0,0000003946815 |
Wenn man das im Quintenzirkel in beide Richtungen fortsetzt, erhält man für die restlichen Intervalle
mit c = [C] und o = 1200:
| O = | 2 | [O] = | o | ||
| Qui = | 2(n-m)/m · C1/m | [Qui] = | o · (n-m)/m | + c/m | |
| T = | 2(2n-3m)/m ·C2/m | [T] = | o · (2n-3m)/m | + 2c/m | |
| g62 = | 2(3n-4m)/m · C.3/m | [g62] = | o · (3n-4m)/m | + 3c/m | |
| g32 = | 2(4n-6m)/m · C4/m | [g32] = | o · (4n-6m)/m | + 4c/m | |
| g72 = | 2(5n-7m)/m · C5/m | [g72] = | o · (5n-7m)/m | + 5c/m | |
| Tr2 = | 2(6n-9m)/m · C6/m | [Tr2] = | o · (6n-9m)/m | + 6c/m | |
| A = | 2(7n-11m)/m · C7/m | [A] = | o · (7n-11m)/m | + 7c/m | |
| k62 = | 2(8n-12m)/m · C8/m | [k62] = | o · (8n-12m)/m | + 8c/m | |
| k32 = | 2(9m-14n)/m · C9/m | [k32] = | o · (9n-14m)/m | + 9c/m | |
| k72 = | 2(10n-15m)/m · C10/m | [k72] = | o · (10n-15m)/m | + 10c/m |
| O = | 2 | [O] = | o | ||
| Qua = | 2(2m-n)/m · C-1/m | [Qua] = | o · (2m-n)/m | – c/m | |
| k71 = | 2(4m-2n)/m ·C-2/m | [k71] = | o · (4m-2n)/m | – 2c/m | |
| k31 = | 2(5m-3n)/m · C-3/m | [k31] = | o · (5m-3n)/m | – 3c/m | |
| k61 = | 2(7m-4n)/m · C-4/m | [k61] = | o · (7m-4n)/m | – 4c/m | |
| S = | 2(8m-5n)/m · C-5/m | [S] = | o · (8m-5n)/m | – 5c/m | |
| Tr1 = | 2(10m-6n)/m · C-6/m | [Tr1] = | o · (10m-6n)/m | – 6c/m | |
| g71 = | 2(12m-7n)/m · C-7/m | [g71] = | o · (12m-7n)/m | – 7c/m | |
| g31 = | 2(13m-8n)/m · C-8/m | [g31] = | o · (13m-8n)/m | – 8c/m | |
| g61 = | 2(15m-9n)/m · C-9/m | [g61] = | o · (15m-9n)/m | – 9c/m | |
| S2 = | 2(16m-10n)/m · C-10/m | [S2] = | o · (16m-10n)/m | – 10c/m |
Für m = 12 und n = 19 ist c = p und o/m = 100 und es ergibt sich die gängige (12-)gleichstufige Temperatur:
| [S] = | o · (8m-5n)/m – 5c/m = | 100 – 5p/12 = | 100 | – | 9,775004 = | 90,224996 | |
| [A] = | o · (7n-11m)/m + 7c/m = | 100 + 7p/12 = | 100 | + | 13,685006 = | 113,685006 | |
| [S2] = | o · (16m-10n)/m – 10c/m = | 200 – 10p/12 = | 200 | – | 19,550009 = | 180,449991 | |
| [T] = | o · (2n-3m)/m + 2c/m = | 200 + 2p/12 = | 200 | + | 3,910002 = | 203,910002 | |
| [k31] = | o · (5m-3n)/m – 3c/m = | 300 – 3p/12 = | 300 | – | 5,865003 = | 294,134997 | |
| [k32] = | o · (9n-14m)/m + 9c/m = | 300 + 9p/12 = | 300 | + | 17,595008 = | 317,595008 | |
| [g31] = | o · (13m-8n)/m – 8c/m = | 400 – 8p/12 = | 400 | – | 15,640007 = | 384,359993 | |
| [g32] = | o · (4n-6m)/m + 4c/m = | 400 + 4p/12 = | 400 | + | 7,820003 = | 407,820003 | |
| [Qua] = | o · (2m-n)/m – c/m = | 500 – p/12 = | 500 | – | 1,955001 = | 498,044999 | |
| [Tr1] = | o · (10m-6n)/m – 6c/m = | 600 – 6p/12 = | 600 | – | 11,730005 = | 588,269995 | |
| [Tr2] = | o · (6n-9m)/m + 6c/m = | 600 + 6p/12 = | 600 | + | 11,730005 = | 611,730005 | |
| [Qui] = | o · (n-m)/m + c/m = | 700 + p/12 = | 700 | + | 1,955001 = | 701,955001 | |
| [k61] = | o · (7m-4n)/m – 4c/m = | 800 – 4p/12 = | 800 | – | 7,820003 = | 792,179997 | |
| [k62] = | o · (8n-12m)/m + 8c/m = | 800 + 8p/12 = | 800 | + | 15,640007 = | 815,640007 | |
| [g61] = | o · (15m-9n)/m – 9c/m = | 900 – 9p/12 = | 900 | – | 17,595008 = | 882,404992 | |
| [g62] = | o · (3n-4m)/m + 3c/m = | 900 + 3p/12 = | 900 | + | 5,865003 = | 905,865002 | |
| [k71] = | o · (4m-2n)/m – 2c/m = | 1000 – 2p/12 = | 1000 | – | 3,910002 = | 996,089998 | |
| [k72] = | o · (10n-15m)/m + 10c/m = | 1000 + 10p/12 = | 1000 | + | 19,550009 = | 1019,550009 | |
| [g71] = | o · (12m-7n)/m – 7c/m = | 1100 – 7p/12 = | 1100 | – | 13,685006 = | 1086,314994 | |
| [g72] = | o · (5n-7m)/m + 5c/m = | 1100 + 5p/12 = | 1100 | + | 9,775004 = | 1109,775004 |
Für m = 53 und n = 84 ist C = R, c = r und o/m = h und es ergibt sich:
| [S] = | o · (8m-5n)/m – 5c/m = | 4h – 5r/53 = | 4h | – 0,341042 = | 90,224996 | |
| [A] = | o · (7n-11m)/m + 7c/m = | 5h + 7r/53 = | 5h | + 0,477459 = | 113,685006 | |
| [S2] = | o · (16m-10n)/m – 10c/m = | 8h – 10r/53 = | 8h | – 0,682084 = | 180,449991 | |
| [T] = | o · (2n-3m)/m + 2c/m = | 9h + 2r/53 = | 9h | + 0,136417 = | 203,910002 | |
| [k31] = | o · (5m-3n)/m – 3c/m = | 13h – 3r/53 = | 13h | – 0,204625 = | 294,134997 | |
| [k32] = | o · (9n-14m)/m + 9c/m = | 14h + 9r/53 = | 14h | + 0,613876 = | 317,595008 | |
| [g31] = | o · (13m-8n)/m – 8c/m = | 17h – 8r/53 = | 17h | – 0,545667 = | 384,359993 | |
| [g32] = | o · (4n-6m)/m + 4c/m = | 18h + 4r/53 = | 18h | + 0,272834 = | 407,820003 | |
| [Qua] = | o · (2m-n)/m – c/m = | 22h – r/53 = | 22h | – 0,068208 = | 498,044999 | |
| [Tr1] = | o · (10m-6n)/m – 6c/m = | 26h – 6r/53 = | 26h | – 0,409250 = | 588,269995 | |
| [Tr2] = | o · (6n-9m)/m + 6c/m = | 27h + 6r/53 = | 27h | + 0,409250 = | 611,730005 | |
| [Qui] = | o · (n-m)/m + c/m = | 31h + r/53 = | 31h | + 0,068208 = | 701,955001 | |
| [k61] = | o · (7m-4n)/m – 4c/m = | 35h – 4r/53 = | 35h | – 0,272834 = | 792,179997 | |
| [k62] = | o · (8n-12m)/m + 8c/m = | 36h + 8r/53 = | 36h | + 0,545667 = | 815,640007 | |
| [g61] = | o · (15m-9n)/m – 9c/m = | 39h – 9r/53 = | 39h | – 0,613876 = | 882,404992 | |
| [g62] = | o · (3n-4m)/m + 3c/m = | 40h + 3r/53 = | 40h | + 0,204625 = | 905,865002 | |
| [k71] = | o · (4m-2n)/m – 2c/m = | 44h – 2r/53 = | 44h | – 0,136417 = | 996,089998 | |
| [k72] = | o · (10n-15m)/m + 10c/m = | 45h + 10r/53 = | 45h | + 0,682084 = | 1019,550009 | |
| [g71] = | o · (12m-7n)/m – 7c/m = | 48h – 7r/53 = | 48h | – 0,477459 = | 1086,314994 | |
| [g72] = | o · (5n-7m)/m + 5c/m = | 49h + 5r/53 = | 49h | + 0,341042 = | 1109,775004 |
So kann man durch Teilung der Oktav in gleiche Teile verschiedene Annäherungen der durch Potenzen der Zwei und der Drei beschriebenen Intervalle erhalten:
| Intervall | Centmaß | 5 Teile | 12 Teile | 41 Teile | 53 Teile | ||||
| S (28/35) | 90,224996 | 20/5 | 0,00 | 21/12 | 100,00 | 23/41 | 87,804878 | 24/53 | 90,566038 |
| A ( 37/211) | 113,685006 | 24/41 | 117,073171 | 25/53 | 113,207547 | ||||
| S2 (216/310) | 180,449991 | 22/12 | 200,00 | 26/41 | 175,609756 | 28/53 | 181,132076 | ||
| T (32/23) | 203,910002 | 21/5 | 240,00 | 27/41 | 204,878049 | 29/53 | 203,773585 | ||
| k31 (25/33) | 294,134997 | 23/12 | 300,00 | 210/41 | 292,682927 | 213/53 | 294,339623 | ||
| k32 (39/214) | 317,590078 | 211/41 | 321,951220 | 214/53 | 316,981132 | ||||
| g31 (213/38) | 384,359993 | 24/12 | 400,00 | 213/41 | 380,487805 | 217/53 | 384,905660 | ||
| g32 (34/26) | 407,820003 | 22/5 | 480,00 | 214/41 | 409,756098 | 218/53 | 407,547170 | ||
| Qua (22/3) | 498,044999 | 25/12 | 500,00 | 217/41 | 497,560976 | 222/53 | 498,113208 | ||
| Tr1 (210/36) | 588,269995 | 26/12 | 600,00 | 220/41 | 585,365854 | 226/53 | 588,679245 | ||
| Tr2 (36/29) | 611,730005 | 221/41 | 614,634146 | 227/53 | 611,320755 | ||||
| Qui (3/2) | 701,955001 | 23/5 | 720,00 | 27/12 | 700,00 | 224/41 | 702,439024 | 231/53 | 701,886793 |
| k61 (27/34) | 792,179997 | 28/12 | 800,00 | 227/41 | 790,243903 | 235/53 | 792,452830 | ||
| k62 (38/212) | 815,640007 | 228/41 | 819,512195 | 236/53 | 815,094340 | ||||
| g61 (215/39) | 882,404992 | 24/5 | 960,00 | 29/12 | 900,00 | 230/41 | 878,048781 | 239/53 | 883,018868 |
| g62 (33/24) | 905,865003 | 231/41 | 907,317073 | 240/53 | 905,660377 | ||||
| k71 (24/32) | 996,089998 | 210/12 | 1000,00 | 234/41 | 995,121951 | 244/53 | 996,226416 | ||
| k72 (310/215) | 1019,550009 | 25/5 | 1200,00 | 235/41 | 1024,39024 | 245/53 | 1018,867925 | ||
| g71 (212/37) | 1086,314994 | 211/12 | 1100,00 | 237/41 | 1082,92683 | 248/53 | 1086,792453 | ||
| g72 (35/27) | 1109,775004 | 238/41 | 1112,19512 | 249/53 | 1109,433962 | ||||
| O (2) | 1200,000000 | 212/12 | 1200,00 | 241/41 | 1200,00000 | 253/53 | 1200,000000 | ||