Major second: Difference between revisions
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A '''major second (M2)''' in the [[5L 2s|diatonic | A '''major second (M2)''' is an interval that spans one scale step in the [[5L 2s|diatonic]] scale with the major (wider) quality. It is generated by stacking 2 fifths [[octave reduction|octave reduced]], and depending on the specific tuning, it ranges from 171 to 240{{cent}} ([[7edo|1\7]] to [[5edo|1\5]]). It can be considered the large step of the diatonic scale. | ||
In [[just intonation]], an interval may be classified as a major second if it is reasonably mapped to 1\7 and [[24edo|4\24]] (precisely one step of the diatonic scale and two steps of the chromatic scale). The use of 24edo's 4\24 as the mapping criteria here rather than [[12edo]]'s 2\12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]]. | In [[just intonation]], an interval may be classified as a major second if it is reasonably mapped to 1\7 and [[24edo|4\24]] (precisely one step of the diatonic scale and two steps of the chromatic scale). The use of 24edo's 4\24 as the mapping criteria here rather than [[12edo]]'s 2\12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]]. | ||
As a concrete [[interval region]], it is typically near | As a concrete [[interval region]], it is typically near 200 ¢ in size, distinct from the [[Semitone (interval region)|semitone]] of roughly 100 ¢ and the [[neutral second]] of roughly 150 ¢. A rough tuning range for the major second is about 180 to 240 ¢ according to [[Margo Schulter]]'s theory of interval regions. | ||
This article covers intervals between 160 and | This article covers intervals between 160 and 260 ¢. The outer range of this might be too extreme to call "major seconds", but this is done so that one can find what they're looking for easily. | ||
== In just intonation == | == In just intonation == | ||
=== By prime limit === | === By prime limit === | ||
The | The Pythagorean ([[3-limit]]) major second is [[9/8]], which is 204 cents in size and corresponds to the mos-based interval category of the diatonic major second. It is generated by [[stacking]] two just perfect fifths of [[3/2]]. There is also a Pythagorean diminished third of [[65536/59049]], which is about 180 cents in size. While called a "third", it is within the range of major seconds. | ||
Other major seconds exist in higher [[ | Other major seconds exist in higher [[prime limit|limits]], however, for example: | ||
* The 5-limit '''ptolemaic major second''' is a ratio of [[10/9]], however in 5-limit harmony it is used alongside 9/8. It is about 182 cents. | * The 5-limit '''ptolemaic major second''' is a ratio of [[10/9]], however in 5-limit harmony it is used alongside 9/8. It is about 182 cents. | ||
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|} | |} | ||
== In | == In edos == | ||
The following table lists the best tuning of 10/9, 9/8, and 8/7, as well as other major seconds if present, in various significant [[ | The following table lists the best tuning of 10/9, 9/8, and 8/7, as well as other major seconds if present, in various significant [[edo]]s. | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | ! Edo | ||
! 10/9 | ! 10/9 | ||
! 9/8 | ! 9/8 | ||
| Line 109: | Line 109: | ||
| 15 | | 15 | ||
| 160{{c}} | | 160{{c}} | ||
| colspan="2" | 240{{c}} | | colspan="2" |240{{c}} | ||
| | | | ||
|- | |- | ||
| Line 124: | Line 124: | ||
| colspan="2" | 189{{c}} | | colspan="2" | 189{{c}} | ||
| 253{{c}} | | 253{{c}} | ||
| | | | ||
|- | |- | ||
| 22 | | 22 | ||
| Line 134: | Line 134: | ||
| colspan="2" | 200{{c}} | | colspan="2" | 200{{c}} | ||
| 250{{c}} | | 250{{c}} | ||
| | | | ||
|- | |- | ||
| 25 | | 25 | ||
| colspan="2" | 192{{c}} | | colspan="2" | 192{{c}} | ||
| 240{{c}} | | 240{{c}} | ||
| | | | ||
|- | |- | ||
| 26 | | 26 | ||
| colspan="2" | 185{{c}} | | colspan="2" | 185{{c}} | ||
| | | 231c | ||
| | | | ||
|- | |- | ||
| 27 | | 27 | ||
| Line 178: | Line 178: | ||
| 204{{c}} | | 204{{c}} | ||
| 226{{c}} | | 226{{c}} | ||
| 249{{c}} ≈ 15/13 | |||
|} | |} | ||
== In | == In mos scales == | ||
Being a small interval, major seconds generate a number of monosmall and monolarge | Being a small interval, major seconds generate a number of monosmall and monolarge [[mos]]. | ||
These tables start from the last monolarge | These tables start from the last monolarge mos generated by the interval range. | ||
Scales with more than 12 notes are not included. | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! Range | ! Range | ||
! colspan="3" | | ! colspan="3" | Mos | ||
|- | |- | ||
| 150–171{{c}} | | 150–171{{c}} | ||
| Line 218: | Line 218: | ||
The three simplest major second ratios are 10/9, 9/8, and 8/7, and these along with other more complex interpretations serve as [[generator]]s for a variety of [[regular temperament]]s. | The three simplest major second ratios are 10/9, 9/8, and 8/7, and these along with other more complex interpretations serve as [[generator]]s for a variety of [[regular temperament]]s. | ||
* The generator of the 7L | * The generator of the 7L 1s scale can be interpreted as a [[10/9]] major second, that is equated to [[11/10]] and [[12/11]] [[neutral second]]s by [[porcupine]], so that three generators reach [[4/3]]. Its tuning range is therefore somewhat ambiguous between major and neutral second. | ||
* The generator of the 6L | * The generator of the 6L 1s and 6L 7s scales can be interpreted in terms of [[2.5.7 subgroup|2.5.7]] [[didacus]], whose generator represents [[28/25]] and which splits the septimal tritone [[7/5]] in three, with one step making the generator 28/25 and two making [[5/4]]. This generator can also stand in for 10/9 and [[9/8]] in the 2.9.5.7 [[subgroup]], if it is treated as an index-2 restriction of [[septimal meantone]]. | ||
* The generator of the 5L | * The generator of the 5L 6s scale can be interpreted as [[8/7]] in [[2.3.7 subgroup|2.3.7]] [[slendric]], where three of them are equated to [[3/2]]. | ||
* The generator of the 5L | * The generator of the 5L 4s scale can be interpreted in terms of 2.3.7 [[semaphore]], where 8/7 is equated to the subminor third [[7/6]] so that two generators reach 4/3, or more accurately as 2.3.13/5 [[barbados]] if 8/7 is eschewed in favor of [[15/13]]. Either way, it is tuned as an [[interseptimal interval|interseptimal]] ambiguous between a major second and [[minor third]]. | ||
{{Todo|complete list|inline=1}} | {{Todo|complete list|inline=1}} | ||
{{Navbox intervals}} | {{Navbox intervals}} | ||